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Lessons for gravity from entanglement
A thesis submitted for the degree of
Doctor of Philosophy
in the Faculty of Sciences
Arpan Bhattacharyya
Centre for High Energy Physics
Indian Institute of Science
Bangalore - 560012. India.
June 2015
ii
Declaration
I hereby declare that the work presented in this thesis “Lessons for gravity from entangle-
ment” is based on the research work by me under the supervision of Prof. Aninda Sinha and
with my collaborators at the Centre for High Energy Physics, Indian Institute of Science,
Bangalore, India. It has not been submitted elsewhere as a requirement for any degree or
diploma of any other Institute or University. Proper acknowledgements and citations have
been made in appropriate places while borrowing research materials from other investiga-
tions.
Date : Arpan Bhattacharyya
Certified by :
Prof. Aninda. Sinha
Centre for High Energy Physics
Indian Institute of Science
Bangalore - 560012
India
iii
iv
List of publications
This thesis is based on the following publications :
1. “ Entanglement entropy in higher derivative holography ”
A. Bhattacharyya, A. Kaviraj and A. Sinha.
arXiv:1305.6694 [hep-th] JHEP 1308, 012 (2013)
2. “ On generalized gravitational entropy, squashed cones and holography ”
A. Bhattacharyya, M. Sharma and A. Sinha.
arXiv:1308.5748 [hep-th] JHEP 1401, 021 (2014)
3. “ Constraining gravity using entanglement in AdS/CFT ”
S. Banerjee, A. Bhattacharyya, A Kaviraj, K. Sen and A. Sinha.
arXiv:1401.5089 [hep-th] JHEP 1405, 029 (2014)
4. “ On entanglement entropy functionals in higher derivative gravity theories”
A. Bhattacharyya and M. Sharma.
arXiv:1405.3511 [hep-th] JHEP 1410, 130 (2014)
5. “ Renormalized Entanglement Entropy for BPS Black Branes ”
A. Bhattacharyya, S. S. Haque and A. Veliz-Osorio.
arXiv:1412.2568 [hep-th] Phy. Rev. D 91, 045026 (2015)
The following works were done during my PhD but is not included in this thesis :
1. “ On c-theorems in arbitrary dimensions ”
A. Bhattacharyya, L. Y. Hung, K. Sen and A. Sinha.
arXiv:1207.2333 [hep-th] Phy. Rev. D 86, 106006 (2012)
2. “ Entanglement entropy from the holographic stress tensor ”
A. Bhattacharyya and A. Sinha.
arXiv:1303.1884 [hep-th] Class. Quantum Grav 30, 235032 (2013)
3. “ Entanglement entropy from surface terms in general relativity ”
A. Bhattacharyya and A. Sinha.
arXiv:1305.3448 [hep-th] IJMPD 22 12, 1342020 (2013)
4. “ Attractive holographic c-functions ”
A. Bhattacharyya, S. S. Haque, V. Jejjala, S. Nampuri and A. Veliz-Osoio.
arXiv:1407.0469 [hep-th] JHEP 1411, 138 (2014)
v
vi
5. “ Viscosity bound for anisotropic superfluids in higher derivative gravity ”
A. Bhattacharyya and D. Roychowdhury.
arXiv:1410.3222 [hep-th] JHEP 1503, 063 (2015)
6. “ Lifshitz Hydrodynamics And New Massive Gravity ”
A. Bhattacharyya and D. Roychowdhury.
arXiv:1503.03254
Synopsis
One of the recent fundamental developments in theoretical high energy physics is the AdS/CFT
correspondence [1, 2, 3, 4] which posits a relationship between Quantum Field Theories (QFT)
in a given dimension and String Theory on a higher dimensional anti- de Sitter (AdS) space-
time. This has revolutionised our understanding of QFTs (more specifically conformal field
theories (CFTs)) and string theory/gravity, and has far reaching consequences for explo-
rations into a vast array of physical phenomena. Using the elegant formalism provided by
this powerful duality, often called “holography”, one can now use fundamental physical ob-
servables in QFT to better understand the nature of quantum gravity. The theoretical tools
provide a translation of calculable field theoretic observables into the language of gravity
thereby leading to the construction of holographic models for several interesting QFTs.
Entanglement is a fundamental physical property of all quantum systems. From models
of various condensed matter systems to its application as a tool for secure and fast communi-
cation in quantum information theory [5], it serves as an intersection point between different
subfields of physics [6]. From the AdS/CFT point of view quantum entanglement connects
geometry with quantum information, providing a window to understand how the bulk gravity
physics emerges from the holographic field theoretic viewpoint. Probing various aspects of
this connection in detail will be the broad theme of this thesis.
For extended, many-body systems, the most well known measure of quantum entangle-
ment is the “Entanglement Entropy” (EE) which is also the best understood measure within
the holographic framework. In early 2006, Ryu and Takayanagi (RT) gave a simple and
elegant prescription for computing this quantity using AdS/CFT duality within Einstein
gravity [7, 8]. They proposed that EE for a subsystem within an extended system (QFT),
is computed by the (proper) area of a static, codimension- 2, “extremal” surface inside the
dual AdS spacetime. The RT proposal has passed several non-trivial consistency checks, for
example strong sub-additivity, area law to name a few [9]. A remarkable aspect of the pro-
posal is the ease with which EE can now be calculated, while it is well known that obtaining
EE from first principles in QFT presents several technical challenges which have so far been
surmounted only in some 2d field theories using the “replica method” [10, 11, 12].
The most intriguing aspect of the RT proposal is its striking similarity to Bekenstein-
Hawking (BH) entropy which is proportional to the area of a black hole horizon, further
confirming an intimate relationship between entropy and geometry [13, 14, 15, 16]. This
leads to the natural question: what is the connection between EE and BH entropy? This
question has been sharpened recently by Lewkowycz and Maldacena (LM) via the concept of
Generalized Gravitational Entropy which extends the QFT replica trick to a replica symmetry
vii
for the dual space-time [17]. This was used to prove the RT conjecture successfully by deriving
the correct extremal surface equation for two derivative gravity theories. In this thesis I have
studied the generalization of LM method for higher derivative gravity theories [18, 19, 20,
21, 22, 23] describing holographic duals (of QFT’s with finite number of colours) and finite
’t Hooft coupling which takes the AdS/CFT correspondence beyond the usual supergravity
limit. If one wants to use AdS/CFT to study real life systems then it is absolutely necessary to
incorporate the finite coupling effect into the theory and hence the study of higher derivative
effects becomes very important. In these two papers [21, 22] I have formulated a proof for
the existence of the entropy functionals for certain higher derivative theories extending LM
method. We have shown that the for a certain special class of higher derivative theories
there exist well defined entropy functionals. To extend this proof for more general theories
of gravity opens up a possibility of breaking replica symmetry in the bulk space-time [24].
For higher derivative gravity, black hole entropy for a large class of stationary black
holes with bifurcate killing horizon is given by the well known Wald prescription [25, 26, 27]
which relates the concept of the Noether charge with the black hole entropy. Iyer and Wald
proposed a generalization for dynamical horizons. This throws up the question whether
there is a relation between these EE functionals and the Noether charge, and whether we can
derive them using the approach of Iyer and Wald. For a certain class of theories I have shown
that there exists a relation between these two [28] but a more rigorous proof is needed. This
somewhat firms up the area-entropy relation for arbitrary surfaces and proves the existence of
holographic EE functionals for higher curvature theories thereby extending the applicability
of Iyer-Wald formalism beyond the bifurcation surface.
Apart from this, it is well known that there exist several measures of quantum entangle-
ment, each satisfying a variety of mathematical inequalities and conditions [5]. Translating
these into the language of holography constrains the dual gravity theory and will lead to
general statements about the consistency of the theory. In this thesis I have discussed one
such measure namely Relative entropy [29], the positivity of which has led to constraints on
the underlying gravity theory [30]. Also entanglement entropy is a very useful tools for prob-
ing renormalization group (RG) flow from the holographic point of view [34, 31, 32, 35, 33].
We end with exploring the concept of renormalized entanglement entropy [36, 37] and its
application in probing RG flow in the context of N = 2 gauged supergravity [38].
References
[1] J. M. Maldacena, “The large N limit of superconformal field theories and supergravity,”
Adv. Theor. Math. Phys. 2, 231 (1998), [arXiv:hep-th/9711200]
[2] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “Gauge theory correlators from
noncritical string theory,” Phys. Lett. B428 (1998) 105, [arXiv:hep-th/9802109]
[3] E. Witten, “ Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. 2 (1998)
253, [arXiv:hep-th/97802150]
[4] O. Aharony, S. S. Gubser, J. Maldacena H. Ooguri and Y. Oz, ” Large N field
theories, String theory and gravity,” Phys. Rept, 323 (2000) 183-386, [arXiv:hep-th/
9905111].
[5] Michael A. Nielsen and Isaac L. Chuang, “ Quantum Computation and Quantum
Information”, Cambridge University Press, 23-Oct-2000
[6] J. Eisert, M. Cramer and M. B. Plenio, “Area laws for the entanglement entropy - a
review,” Rev. Mod. Phys. 82 (2010) 277 [arXiv:0808.3773 [quant-ph]] and the references
there in.
[7] S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from
AdS/CFT,” Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001].
[8] T. Nishioka, S. Ryu and T. Takayanagi, “Holographic Entanglement Entropy: An
Overview,” J. Phys. A 42 (2009) 504008 [arXiv:0905.0932 [hep-th]].
[9] S. Ryu and T. Takayanagi, “Aspects of Holographic Entanglement Entropy,” JHEP
0608 (2006) 045 [hep-th/0605073] and the references there in.
[10] C. Holzhey, F. Larsen and F. Wilczek, Nucl. Phys. B 424 (1994) 443 [hep-th/9403108].
[11] P. Calabrese and J. L. Cardy, “ Entanglement entropy and conformal field theory”,
Journal of Physics A: Mathematical and Theoretical, Volume 42, Issue 50, article id.
504005, 36 pp. (2009).
[12] P. Calabrese and J. L. Cardy, “Entanglement entropy and quantum field theory,” J.
Stat. Mech. 0406 (2004) P06002 [hep-th/0405152].
[13] Luca Bombelli, Rabinder K. Koul, Joohan Lee, and Rafael D. Sorkin, “Quantum
source of entropy for black holes”, Phys. Rev. D 34, 373
[14] M. Srednicki, “Entropy and area,” Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048].
[15] S. N. Solodukhin, “Entanglement entropy of black holes,” Living Rev. Rel. 14 (2011) 8
[arXiv:1104.3712 [hep-th]] and the references there in.
[16] E. Bianchi and R. C. Myers, “On the Architecture of Spacetime Geometry,” Class.
Quant. Grav. 31 (2014) 21, 214002 [arXiv:1212.5183 [hep-th]].
[17] A. Lewkowycz and J. Maldacena, “Generalized gravitational entropy,” JHEP 1308
(2013) 090 [arXiv:1304.4926 [hep-th]].
[18] L. Y. Hung, R. C. Myers and M. Smolkin, “On Holographic Entanglement Entropy and
Higher Curvature Gravity,” JHEP 1104 (2011) 025 [arXiv:1101.5813 [hep-th]].
[19] X. Dong, “Holographic Entanglement Entropy for General Higher Derivative Gravity,”
JHEP 1401 (2014) 044 [arXiv:1310.5713 [hep-th], arXiv:1310.5713].
[20] J. Camps, “Generalized entropy and higher derivative Gravity,” JHEP 1403 (2014) 070
[arXiv:1310.6659 [hep-th]].
[21] A. Bhattacharyya, A. Kaviraj and A. Sinha, “Entanglement entropy in higher derivative
holography,” JHEP 1308 (2013) 012 [arXiv:1305.6694 [hep-th]].
[22] A. Bhattacharyya and M. Sharma, “On entanglement entropy functionals in higher
derivative gravity theories,” JHEP 1410 (2014) 130 [arXiv:1405.3511 [hep-th]].
[23] R. X. Miao and W. z. Guo, “Holographic Entanglement Entropy for the Most General
Higher Derivative Gravity,” arXiv:1411.5579 [hep-th].
[24] J. Camps and W. R. Kelly, “Generalized gravitational entropy without replica symme-
try,” JHEP 1503 (2015) 061 [arXiv:1412.4093 [hep-th]].
[25] R. M. Wald, “Black hole entropy is the Noether charge,” Phys. Rev. D 48 (1993) 3427
[gr-qc/9307038].
[26] V. Iyer and R. M. Wald, “Some properties of Noether charge and a proposal for dynam-
ical black hole entropy,” Phys. Rev. D 50 (1994) 846 [gr-qc/9403028].
[27] V. Iyer and R. M. Wald, “A Comparison of Noether charge and Euclidean methods
for computing the entropy of stationary black holes,” Phys. Rev. D 52 (1995) 4430
[gr-qc/9503052].
[28] A. Bhattacharyya, M. Sharma and A. Sinha, “On generalized gravitational entropy,
squashed cones and holography,” JHEP 1401 (2014) 021 [arXiv:1308.5748 [hep-th]].
[29] D. D. Blanco, H. Casini, L. Y. Hung and R. C. Myers, “Relative Entropy and Hologra-
phy,” JHEP 1308 (2013) 060 [arXiv:1305.3182 [hep-th]].
T. Faulkner, M. Guica, T. Hartman, R. C. Myers and M. Van Raamsdonk, “Gravitation
from Entanglement in Holographic CFTs,” JHEP 1403 (2014) 051 [arXiv:1312.7856
[hep-th]].
[30] S. Banerjee, A. Bhattacharyya, A. Kaviraj, K. Sen and A. Sinha, “Constraining gravity
using entanglement in AdS/CFT,” JHEP 1405 (2014) 029 [arXiv:1401.5089 [hep-th]].
[31] H. Casini and M. Huerta, “A Finite entanglement entropy and the c-theorem,” Phys.
Lett. B 600 (2004) 142 [hep-th/0405111].
[32] H. Casini and M. Huerta, “A c-theorem for the entanglement entropy,” J. Phys. A 40
(2007) 7031 [cond-mat/0610375].
[33] R. C. Myers and A. Sinha, “Holographic c-theorems in arbitrary dimensions,” JHEP
1101 (2011) 125 [arXiv:1011.5819 [hep-th]].
[34] H. Casini and M. Huerta, “On the RG running of the entanglement entropy of a circle,”
Phys. Rev. D 85 (2012) 125016 [arXiv:1202.5650 [hep-th]].
[35] H. Casini and M. Huerta, “Positivity, entanglement entropy, and minimal surfaces,”
JHEP 1211 (2012) 087 [arXiv:1203.4007 [hep-th]].
[36] H. Liu and M. Mezei, “A Refinement of entanglement entropy and the number of degrees
of freedom,” JHEP 1304 (2013) 162 [arXiv:1202.2070 [hep-th]].
[37] H. Liu and M. Mezei, “Probing renormalization group flows using entanglement entropy,”
JHEP 1401 (2014) 098 [arXiv:1309.6935 [hep-th], arXiv:1309.6935].
[38] A. Bhattacharyya, S. Shajidul Haque and A. Veliz-Osorio, “Renormalized Entanglement
Entropy for BPS Black Branes,” Phys. Rev. D 91 (2015) 4, 045026 [arXiv:1412.2568
[hep-th]].
Acknowledgements
First and foremost, I would like to convey my sincere thanks to my advisor, Aninda Sinha for
his generous support as well as outstanding guidance during the entire tenure of my doctoral
research. Being his first PhD student was always a stimulating experience. I thank him
for guiding me throughout my doctoral work and helping me to complete my PhD in just
3 years. Apart from learning a great deal of physics from him, he has helped me a lot to
improve my soft skills.
I am also thankful to Menika Sharma, Ling-Yan (Janet) Hung, Shajid Haque, Alvaro Veliz
Osorio, Vishnu Jejjala, Suresh Nampuri and Dibakar Roychowdhury for useful collaborations
and numerous productive discussions which has added a lot towards my understanding of
the subject itself.
I would also like to thank Rajesh Gopakumar, Jose Edelstein, Axel Kleinschmidt, Johanna
Erdmenger, Tadashi Takayanagi, Heng-Yu Chen, Janet Hung, Shamik Banerjee, Vishnu
Jejjala for inviting me to give seminars. I am also thankful to them for numerous stimulating
discussions in various occasions. I am also thankful to Ashoke Sen, Rob Myers and Joan
Camps for valuable discussions.
I would also like to thank all the professors of the Department of Physics and the Centre
for High Energy Physics for providing beautiful courses. In particular, I would like to thank
Prof. Justin David for providing a beautiful course on QFT. I thank the Chairman of CHEP,
B Ananthanarayan for striving to maintain a vibrant and simulating atmosphere in the
department. I am indeed grateful to my Integrated PhD batchmates and colleagues at the
Centre for High Energy Physics for creating friendly and competitive atmosphere. During my
stay at IISc, I found various departmental activities like the weekly math-phys meets, journal
club sessions, seminars and colloquia etc. as quite stimulating and in particular playing a
very crucial role in developing the scientific mind. Finally, I would specially like to thank
Apratim and Shouvik for helping me enormously with all the diagrams and the Latex.
I thank the Indian Institute of Science for their generous financial support for attending
numerous conferences and visiting other research institutes in India and abroad during my
tenure.
Finally, I’m very thankful to my parents for giving me constant support and never giving
up hope on me.
Arpan Bhattacharyya
Bangalore, June 2015.
xiii
xiv
xv
To my parents.
xvi
xvii
“Somewhere, something incredible is waiting to be known”.
– Carl Sagan
xviii
Contents
1 Introduction 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Quantum Entanglement . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Entanglement entropy and Holography . . . . . . . . . . . . . . . . 9
2 Holographic entanglement entropy functionals: A derivation 29
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Entropy functional for general theories of gravity . . . . . . . . . . . . . . . 32
2.3 Test of the entropy functional for R2 theory . . . . . . . . . . . . . . . . . . 34
2.3.1 Minimal surface condition from the entropy functional . . . . . . . . 36
2.3.2 Minimal surface condition from the Lewkowycz-Maldacena method . 41
2.3.3 The stress-energy tensor from the brane interpretation . . . . . . . . 49
2.4 Quasi-topological gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.4.1 The entropy functional . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.4.2 Universal terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.4.3 Minimal surface condition . . . . . . . . . . . . . . . . . . . . . . . . 54
2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3 Entanglement entropy from generalized entropy 69
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2 Generalized entropy and Fefferman-Graham expansion . . . . . . . . . . . . 69
3.2.1 Four derivative theory . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.2.2 New Massive Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.2.3 Quasi-Topological Gravity . . . . . . . . . . . . . . . . . . . . . . . . 75
3.2.4 α′3 IIB supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.3 Comment about singularities in the metric . . . . . . . . . . . . . . . . . . . 76
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4 Connection between entanglement entropy and Wald entropy 81
xix
xx CONTENTS
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.2 Wald Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.3 Four derivative theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.3.1 Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.3.2 Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.4 Quasi-Topological gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.5 α′3 IIB supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.6 Connection with Ryu-Takayanagi . . . . . . . . . . . . . . . . . . . . . . . . 87
4.7 Comments on the connection with the Iyer-Wald prescription . . . . . . . . . 89
4.8 Universality in Renyi entropy . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5 Constraining gravity using entanglement entropy 95
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.2 Smoothness of entangling surface . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6 Relative entropy 101
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.2 Relative entropy considerations . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.3 Relative entropy in Gauss-Bonnet holography . . . . . . . . . . . . . . . . . 109
6.3.1 Linear order calculations . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.3.2 Quadratic corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.3.3 Constant Tµν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.3.4 Shockwave background . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.3.5 Correction from additional operators . . . . . . . . . . . . . . . . . . 117
6.4 Relative entropy for an anisotropic plasma . . . . . . . . . . . . . . . . . . . 118
6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7 Coding holographic RG flow using entanglement entropy 131
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.2 Renormalized Entanglement Entropy . . . . . . . . . . . . . . . . . . . . . . 132
7.3 BPS black objects in AdS4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
CONTENTS xxi
7.4 REE for BPS black branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8 Conclusions 145
xxii
1 Introduction
1.1 Introduction
The concept of entanglement is a very old one and dates back to early 1930’s when quantum
mechanics was born. One of the measures of quantum entanglement is the entanglement
entropy. Although it has played a crucial role in understanding some aspects of quantum
mechanics, its application remains rather limited mainly due to its non local behaviour until
late 90’s.
In early 1970’s Bekenstein proposed that the black hole entropy (SBH) follows an area
law [1] .
SBH =A
4GN
(1.1)
where A is the area of the horizon and GN is the Newton constant. This formula is quite
counter-intuitive as entropy is usually an extrinsic quantity, depends on the volume of the
system. Later Stephen Hawking showed that a black hole emits radiation with a well defined
temperature [2], thereby establishing the concept of black hole entropy. It was observed that
the calculation of the entropy associated with the radiation emitted from the black hole is
plagued by the presence of ultraviolet divergences. These divergences can be associated with
the particles close to the horizon [3] and one has to regulate them to get a finite answer
for the entropy. Then in early 1980’s Bombelli, Koul, Lee and Sorkin in their seminal work
[4] showed that one can possibly understand that black hole entropy using the concept of
‘entanglement’. To the observers outside the black hole horizon there is no information about
the spacetime inside the horizon. They considered scalar fields in the black hole background
and traced out the spacetime inside the horizon, thereby defining a “reduced density” matrix
for the system. Using this, they computed the von-Neumann entropy and it was shown that
the entropy follows an area law. Later it was generalized by Srednicki [5] for massless scalar
fields in flat spacetime. He also showed that if one divides the spacetime in two parts, then
the entropy associate with the reduce density matrix for one of these two parts is proportional
to the area of the boundary between these two halves. Later this concept of “ entanglement
entropy”(EE) was made concrete by Callan, Holzhey, Larsen and Wilczek [6] and separately
by Susskind and Uglum [7]. From their work it is evident that the EE exhibits a universal
1
2 1.1. INTRODUCTION
behaviour, it goes as a logarithm of correlation length in 1 + 1 dimensions after suitably
regulating the ultraviolet divergences. But still its application remain rather limited as in
general it is very hard to compute this quantity for generic field theories.
But in early 2000, Calabrese and Cardy used the “ Replica Trick” formulated by Callan,
Holzey, Larson and Wilczek successfully to compute the EE for many cases in the context of
1+1 dimensional conformal field theory (CFT) [8], thereby increasing its physical importance.
After that EE has been calculated extensively not only for 1+1 dimensional CFT but also for
various other simple quantum field theories both analytically and numerically. Also in recent
times it has been successfully computed numerically using the lattice technique for quantum
many body systems [9]. Although the techniques employed for computing EE is very hard
and yet to be developed fully, still in recent times, we have lots of data regarding quantum
entanglement and EE coming from both analytical and numerical approaches [10, 11].
Figure 1.1: Entanglement and its diverese applications
in physics. (Picture courtesy- From the talk given by
Prof. Robert. C. Myers in the Conference “Entangle-
ment from gravity”, ICTS, 2014, Bangalore, India)
Recently EE has found many
applications in various branches of
physics like, quantum information,
condensed matter system, statis-
tical physics and in AdS/ CFT
[9, 10, 11, 12, 13, 14]. It serves as an
intersecting point between various
subfields of physics. In recent times
it has played a crucial role in un-
derstanding the nature of hologra-
phy (AdS/ CFT correspondence).
AdS/ CFT correspondence, com-
monly known as “ Holography” is
one of the most important dualities
in physics. It postulates that grav-
ity emerges from a certain class of
field theories. It is still not known
rigorously how holography works
from first principles. EE has the
merit to shed light on this problem
as it connects quantum information
of the system with geometry. There
exists an interesting connection be-
tween geometry and EE and our
goal in this thesis will be to explore some aspects of this connection. Recently apart from EE,
CHAPTER 1. INTRODUCTION 3
many different tools of quantum entanglement like, entanglement negativity, differential en-
tropy, quantum error coding, relative entropy, cMERA (continuous multiscale entanglement
renormalization ansatz) etc, have been used in an attempt to build geometry from the field
theory data, is some sense trying to prove the holographic principle [15, 16, 17, 18, 19, 20].
But still EE plays the central character in this program and we will use this to learn many
lessons of gravity in the holographic set up.
1.1.1 Quantum Entanglement
Let us first consider a quantum mechanical system consists of two spin 12
particles. We
denote the total Hilbert space by H . Now HA and HB denote the Hilbert spaces of the two
individual particles . Also,
H = HA ⊗HB.
Now consider the following state belonging to H,
|ψ1 >=1√2
(| ↑ + ↓>)A ⊗ (| ↓ + ↑>)B . (1.2)
This state is not an entangled state as there is no correlation between the two particles in
this state. On the other hand, if we consider the following state
|ψ2 >=1√2
(| ↑A↓B> −| ↑B↓A>), (1.3)
then one cannot factorize this state in terms of the individual particle states. So this state is
an example of “ entangled state”. In other words, entanglement is a property of a quantum
mechanical system that tells us, that one cannot describe the underlying pair of particles
belonging to this particular state independently. All the physical properties of the two
individual particles are correlated with each other. This argument can be extended for any
number of particles and interesting things shows up when one consider many body systems
due to the non local nature of the entanglement.
One of the measures of entanglement is the entanglement entropy (EE). Let us first see
how we can define this. First step is to define a “ density matrix” for the full system. If |ψ >is the wavefunction characterizing the total system then the density matrix can be defined
in the following way,
ρ = |ψ >< ψ|. (1.4)
Next step is to define the “ reduced density matrix”. Suppose we want to compute the EE for
the sub-system A. Then we first trace out the degrees of freedom corresponding the system
B.
ρA = TrBρ =< b|ρ|b >, (1.5)
4 1.1. INTRODUCTION
where ρA is the reduced density matrix for the subsystem A. Now we define EE by defining
the “von-Neumann entropy” which is,
SEE(A) = −TrAρA ln ρA . (1.6)
Here we have completed the trace over the subsystem A. Now this is a good time to cook
up some examples and elucidate the process of computing EE. First consider the state as
mentioned in Eq. (1.2). We trace out B and the corresponding reduced density matrix is ,
ρA =1
2
(1 1
1 1
). (1.7)
Now to compute the EE as defined in Eq. (1.6) one first diagonalize this and find the eigen-
values. In terms of the eigenvalues Eq. (1.6) becomes
SEE = −2∑i=1
λi lnλi. (1.8)
This in turn gives,
SEE = 0. (1.9)
So the entropy is zero and hence the state is not entangled.
Now consider the state as shown in Eq. (1.3). Corresponding reduced density matrix is,
ρA =1
2
(1 0
0 1
). (1.10)
The entropy is,
SEE = ln 2. (1.11)
It is an entangled state, in fact it is a maximally entangled state. So whenever SEE is nonzero
the state is entangled.
Now let us consider a more complicated system, a system of two coupled oscillators. This
type of systems are considered in [5] and we will review their calculation here to demonstrate
the increasing difficulty of computing this quantity when one consider quantum many body
systems.
Let us start by writing the hamiltonian that describes the two coupled oscillators.
H =1
2
[p2
1 + p22 + k1(x2
1 + x22) + k2(x1 − x2)2
], (1.12)
1 and 2 respectively denote the two oscillators. Then one defines the canonical coordinate
xA =x1 + x2√
2, xB =
x1 + x2√2
. (1.13)
CHAPTER 1. INTRODUCTION 5
In terms of these coordinates the ground state wave function can be written as
ψ0(x1, x2) = (ωAωBπ2
)1/4e−(ωAx2A+ωBx
2B)/2, (1.14)
where ωA = k1/21 and ωB = (k1 + 2k2)1/2 are the two frequencies corresponding to the two
normal modes. Now suppose we integrate out the oscillator 2. The reduced density matrix
is defined as,
ρ(x1, x′1) =
∫ ∞−∞
dx2ψ0(x1, x2)ψ0(x′1, x2)∗. (1.15)
This gives
ρ(x1, x′1) = (
δ
π)1/2e−α(x21+x′21 )/2+βx1x′1 , (1.16)
where β = 14
(ωA−ωB)2
(ωA+ωB)and δ = α − β = 2ωAωB
(ωA+ωB). Then we have to just compute the SEE
as defined in Eq. (1.6). To do that we have to find the eigenvalues of this reduced density
matrix. In this case we are fortunate, as one can easily solve this problem and eigenvalues
are given as,
λn =(
1− β
α + (α2 − β2)1/2
)( β
α + (α2 − β2)1/2
)n. (1.17)
Then
SEE = −∑n
λn lnλn. (1.18)
After performing this sum, which is somewhat tedious, we get
SEE(k1
k2
) = − ln(
1− β
α + (α2 − β2)1/2
)− β
α− β + (α2 − β2)1/2ln( β
α + (α2 − β2)1/2
).
(1.19)
Ultimately SEE is just a function of the ratio of k1 and k2.
Now the stage is prepared for us to generalize this concept for field theory. In the field
theory the problem becomes much more difficult and subtle. One key issue is to factorize the
Hilbert space. One way is to discretize the system over a lattice.1 However one can still use
the von-Neumann formula as defined in Eq. (1.6), but one has to deal with the ultraviolet
divergences that are present in the field theory.
As shown in the Fig. (1.2), we can consider a particular region in the field theory denoted
by A. To compute the EE for this region we trace out the remaining portion. The system is
discretized over the full space. The SEE(A) for the subsystem A is roughly proportional to
the number of links cut by the boundary of the region A. So it is telling us that, indeed EE
1 Several ambiguities might enter in the calculation because of the discretization, specially for the gauge
theory. But still one can extract a meaningful answer for EE in the field theory.
6 1.1. INTRODUCTION
Figure 1.2: System is discretized on a lattice and H =∑
iHi
is proportional to the area of the boundary dividing the two regions. If the total system is
in the pure state, then one can show
SEE(A) = SEE(B), (1.20)
where B denotes the remaining portion of the spacetime. From this one can intuitively guess
that SEE is proportional to the number of degrees of freedom that live at the boundary
between the two regions.
Although there is no formal proof of area law, it has been checked for many instances.
For almost all the cases when one considers a ground state of a local hamiltonian, one indeed
gets the area law. It is more or less robust, although the violation of it has been observed
for the excited states and also for non-local hamiltonians [21]. Let us close this section by
briefly sketching an argument for the area law. We will follow [5] and consider scalar field
theory. To show this let us go back to the oscillator case as almost all the field theory can
be described effectively using the coupled oscillators model. To start with let us write down
the Hamiltonian,
H =1
2
∫d3x[π2 + |∇φ(x)|2]. (1.21)
Here φ(x) denotes the scalar field and π is the canonical momentum. After this we express
this Hamiltonian in terms of the partial wave expansion of the scalar field,
φlm = x
∫dΩZlmφ(x). (1.22)
Zlm are the spherical harmonics. The above relation stems from the fact that we can expand
the scalar field in the basis of spherical harmonics. A similar relation can be written for the
CHAPTER 1. INTRODUCTION 7
conjugate momentum. We impose the canonical quantization relation,
[φlm(x), πl′m′(x′)] = iδll′δmm′δ(x− x′). (1.23)
In terms of this partial wave,
H =∑l,m
Hlm. (1.24)
Now,
Hlm =1
2
∫ ∞0
dx(π2lm(x) + x2[∂x(
φlmx
)]2 +l(l + 1)
x2φlm(x)
). (1.25)
Then we discretize the system. We put it on a lattice with a lattice spacing 1M
. The M plays
the role of the uv cutoff. The boundary condition imposed on φlm(x) is such that it vanishes
when x ≥ L where L is the length of the box in which the system is placed. Also
L = (N + 1)1
M, (1.26)
whereN is a integer and this relation shows that the system is discretized. So the Hamiltonian
becomes
Hlm =M
2
N∑j=1
[π2lm + (j +
1
2)2(φlm,j
j− φlm,j+1
j + 1
)2
+l(l + 1)
j2φ2lm,j
]. (1.27)
Now this looks exactly the same as the N coupled oscillators hamiltonian. We can proceed
as before extending the result of 2 coupled oscillators. We trace over the first n number of
sites to obtain the EE. Finally we get
SEE(n,N) =∑l
al(n)[− ln al(n) + 1], (1.28)
where al(n) = n(n+1)(2n+1)2
64l2(l+1)2+O( 1
l6). At this level we are only interested in the leading result
in l, as that will give after summing all values of l’s, the area like term. We perform the sum
over l numerically. We define a radius R midway between the outermost point which was
traced over and the innermost point which was not as,
R = (n+1
2)
1
M. (1.29)
Then it can be shown that the leading term of the EE is,
SEE = 0.30M2R2. (1.30)
From this it is clearly evident, that the EE corresponding to the ground state wavefunction
of a local hamiltonian of the scalar field satisfies the area law.
A more intuitive way to understand the area law [22] is to consider a particular entangling
region A as shown in Fig. (1.3). For simplicity let us stick to the massless scalar field model
8 1.1. INTRODUCTION
Figure 1.3: Modes straddling the boundary ∂A is responsible for SEE
in 3 + 1 dimensions. We expand the scalar field in terms of its modes. These modes are
quasi-localized and each has an momentum ~k. Now we know,
|k| = 2π
λ, (1.31)
where λ is the usual wavelength. The total number of modes inside the region A is given by,
N =
∫ kmax
kmin
dN =
∫ kmax
kmin
V d3k
(2π)3, (1.32)
where V is the volume of A. Now kmin = 2π2R
and kmin = 2πε
. As all the wavelengths are
localized inside A, maximum wavelength can atmost be equivalent to 2R and the minimum
wavelength is 0. But then kmin will be divergent, hence we have to put a uv cut-off ε. From
this the necessity of a cut-off becomes quite clear. Now we count the fraction of the modes
which resides at the boundary of A, responsible for the EE.
Ns ≈∫ kmax
kmin
αAdN
V, (1.33)
where Ns denotes the number of modes straddling the boundary and it is a fraction of the
total number of modes living inside A.
α denotes the thickness of the boundary (α << R) and A denotes its area. Now only the
mode localized near the boundary is responsible for SEE, so we can approximate α by 2πk
.
Also d3k = 4πk2dk and αd3k = 8π2kdk. We next perform the integration and the entropy is
proportional to Ns upto some phase-space factors. We get,
SEE ≈2πA
ε2. (1.34)
So EE is proportional to the area, hence proportional to number of degrees of freedom residing
at the boundary between the two region. It can be also shown in the same way that for 2 + 1
dimensions it is proportional to the circumference and in 1 + 1 it goes as logarithm.
CHAPTER 1. INTRODUCTION 9
Lastly, SEE satisfies one more important property, namely strong subadditivity. For a
bipartite system it tells us that,
SEE(A) + SEE(B) ≥ SEE(A ∪B) + SEE(A ∩B). (1.35)
This result can be extended for any arbitrary number of subsystems. This inequality provides
a non trivial constraints on SEE and any consistent holographic proposal should pass this
test.
1.1.2 Entanglement entropy and Holography
In this section we will review various facts about holographic entanglement entropy. The
main goal of this thesis is to understand the connection between EE and geometry, thereby
learning important lessons about underlying gravity theory in the holographic set up. By
holography we will mean AdS/ CFT correspondence. Two main character of this play is
Anti-de Sitter space (AdS) and conformal field theory (CFT). So before proceeding further
let us briefly comment on the structure of the conformal group and AdS spacetime [23].
Conformal group and structure of AdS
Conformal isometries keep the metric invariant upto a scale transformation. The conformal
transformations form a group by themselves. Poincare group comes as a subgroup under
the broad structure of the conformal group. The conformal transformation preserves the
angle between the two curves. These transformations consist of the following four kinds of
transformations.
Translation→ x′µ = xµ + aµ.
Lorentz → x′µ = Rµ
νxν , where infinitesimal matrix Rµν is antisymmetric.
Dilatation → x′µ = cxµ.
Special Conformal Transformation(SCT) → x′µ = xµ−cµx2
1−2 c.x+c2x2.
For SCT the conformal factor is (1−2 c.x+c2x2)2. SCT is nothing but a translation preceded
and followed by an inversion. The corresponding generators for the infinitesimal transforma-
tions are listed below. For a generic field
Translation(P µ) → −i∂µ.
Rotation (Jµν)→i(xµ∂ν − xν∂µ) + Sµν .
10 1.1. INTRODUCTION
Dilatation(D)→−i(d+ (x.∂)).
SCT(Kµ)→−i((2xµxν − 2gµνx2)∂ν + 2d.xµ) + 2xνSµν ,
where Sµν is an anti-symmetric spin matrix for a given field and satisfies the Lorentz
algebra. d is a real number that depends on the nature of the fields that are present in
the underlying theory.2 These generators satisfy the following commutation relations among
themselves.
[D,D] = 0 ,
[P a, P b] = 0 ,
[D,P a] = iP a,
[Jab, P c] = −i(gacP b − gbcP a) ,
[Jab, J cd] = −i(gadJ bc + gbcJad − gacJ bd − gbdJac) ,[Jab, D] = 0 ,
[D,Ka] = −iKa ,
[iKa, Kb] = 0 ,
[Ka, P b] = 2i(gabD − Jab) .
(1.36)
For example, in 2 + 1 dimensional flat spacetime we have the following 10 conformal genera-
tors,
J1 = ∂a = iPa ,
J2 = xb∂a − xa∂b = −iJab ,J3 = −(xa∂a) = −iD ,
J4 = (2xa(xd∂d)− (xdxd)∂a) = iKa.
(1.37)
a, b runs from 1 to 3. These generators satisfy the usual conformal commutation rules. Now
we will see what are the corresponding isometry generators of AdS4. We first write the AdS4
metric in poincare coordinates. This is the coordinate system we will often use throughout
this thesis.
ds2 =L2(dz2 + dx2 + dy2 + dt2)
z2. (1.38)
t denotes the Euclidean time. Then we do the following substitution
r =L
z. (1.39)
It gives
ds2 =L2
r2dr2 +
r2
L2(dt2 + dx2 + dy2). (1.40)
2 e.g for Fermion d = 32 and for Boson d = 1 .
CHAPTER 1. INTRODUCTION 11
Next we list all the 10 generators.
J1 = ∂t ,
J2 = ∂x ,
J3 = ∂y ,
J4 = x∂t − t∂x ,J5 = y∂x − x∂y ,J6 = t∂y − y∂t ,J7 = r∂r − t∂t − x∂x − y∂y ,
J8 = rt∂r −1
2t2∂t − tx∂x − ty∂y ,
J9 = rx∂r − tx∂t −1
2x2∂x − xy∂y ,
J10 = ry∂r − ty∂t − xy∂x −1
2y2∂y.
(1.41)
We make suitable identifications and t → it, such that the generators satisfy the usual
SO(3, 2) algebra [Jab, Jcd] = i[gadJbc+gbcJad−gacJbd−gbdJac] , where a, b, c, d ∈ 0, 1, 2, 3, 4,So basically they satisfy the same algebra as the CFT generators in one lower dimensions.
AdS/CFT
Now we describe what exactly this correspondence is. There are many dualities that exist
in the physics [24]. Among them AdS/CFT connects a strongly coupled field theory with a
weakly coupled gravity in Anti-de Sitter (AdS) space time [25]. It is a strong weak duality.
It has been observed that there exists an equivalence between a strongly coupled N = 4
supersymmetric SU(N) Yang-Mills (SYM) theory and Type IIB string theory on AdS5×S5
in the large N limit. Now consider a stack of N D3-branes. Open strings describe the
excitations of the D3-branes and the low energy dynamics is governed by N = 4 SYM gauge
theory. For this theory one can define a ’t-Hooft coupling λ = g2YMN = gsN. We can do a
perturbation theory when λ << 1 (also gs << 1). On the other hand we have closed string
excitations in the vacuum. This gives rise to the gravity multiplate in 10 dimensions, low
energy description of which is effectively given by Type IIB supergravity. One can construct
a metric solution for this theory for which the near horizon geometry looks like,
ds2 = α′[r2
√4πgsN
(−dt2 + dx21 + dx2
2 + dx23) +
√4πgsN
dr
r2+√
4πgsNdΩ25] (1.42)
We have assumed that α′ → 0 so that we can neglect stringy effects and work in the su-
pergravity regime. Identifying L2 = α′√
4πgsN, where L is the AdS radius we can see that
12 1.1. INTRODUCTION
metric defined in Eq. (1.42) describes AdS5 × S5 geometry. Also string length ls =√α′ and
this description is valid when, (Lls
)4
= 4πgsN >> 1. (1.43)
This means that classical gravity description is valid when the AdS length scale is much
bigger than the string length and one can use this supergravity language when the ’t-Hooft
coupling becomes very large. Also we know g2YMN = 4πgsN. All these things point to the
fact that we have a classical gravity description when L >> ls in the bulk and it is equivalent
to a strongly coupled gauge theory at the boundary. Although this conjecture has not been
proved yet, it passes several important checks, for eg, matching of the spectrum of chiral
operators, correlation functions, supersymmetric indices etc. We obtain a precise dictionary
between field theory correlators and correlators of fields living inside the AdS space time.
One example is that, currents in the conformal field theory (CFT) side correspond to a gauge
field living inside the bulk spacetime.
One can easily see that the isometry group of AdS5 is SO(4, 2) and isometry group for S5
is SO(6). On the other hand the gauge theory remains invariant under the action of SO(4, 2)
conformal group and also possess an SO(6) R-symmetry. So we have obtained a geometric
realization of the field theory degrees of freedom.
Based on this, one can study systems described by strongly coupled field theories by using
equivalent classical gravity description. Holography is being used to study hydrodynamic
transports of quark gluon plasma, phase transitions in condensed matter systems etc [26]. 3
Holographic entanglement entropy
Although there exist several evidences supporting holographic principle, but it is still not
clear how gravity emerges from field theory. To understand this several tools have been
employed, EE is one of them. In AdS/CFT set up we will investigate EE and will see that it
will provide us with a nice geometrical problem. We will see that we can extract important
information about the underlying geometry, hence the gravity theory using this quantity.
Now as the AdS/CFT is a two way street, let us start by reviewing some basics about EE in
the CFT side of the story.
3 An analogous duality has been observed between the near horizon geometry of AdS3 × S3 ×M and
that of the low energy description on the branes in D1-D5 system in terms of 1+1 dimensional CFT. Also in
recent times holographic principles are being applied for other spacetimes, for eg, Lifshitz, de-Sitter etc [27].
CHAPTER 1. INTRODUCTION 13
Entanglement entropy in CFT
Most general structure of EE for a CFT in even spacetime dimensions is ,
SEE = c1Rd−2
εd−2+ · · ·+ c2 ln
(Rε
)+ · · · (1.44)
First term is the “ area term”. R denotes the radius of the entangling surface i.e the surface
for which the EE is computed. ε is the uv-cutoff. In even dimensions one gets a “logarithmic
term” known as universal term in EE. The coefficient proportional to this term is cut-off
independent and carries the information of the central charges of the underlying CFT. So
this term is also known as “ universal term” and we will be interested in computing this
term throughout this thesis for various theories of gravity. Also we will be considering time
independent scenarios and choose a particular time slice t = 0. Let us take an example. In
d = 4 dimensions, c2 takes the following form [28],
c2 = A
∫d2xR+ C
∫d2x(W abcdhachbd −K2
s +1
2KsabKsab). (1.45)
We have chosen t = 0 slice and rest of the 3 dimensional space is divided into two halves. So
the boundary between the entangling region and the rest of the space is a two dimensional
space. We will call it a “codimension-2” surface. The integration defined in Eq. (1.45) is
essentially over this boundary. c2 depends on the geometrical property of this codimension-2
surface. R is the Ricci scalar, Wabcd is the Weyl tensor and Ksab is the extrinsic curvature of
this surface. hac is the projection operator to the surface from the 4 dimensional spacetime.
hab = ηab − nsanbs. (1.46)
s denotes the two transverse directions and a, b are the surface indices. The extrinsic curva-
ture can be defined as,
Ksab = eαaeβb h
µαh
νβ∇µnsν . (1.47)
It has also an index (s) corresponding to the two transverse directions and two normals are
defined for that. Ks is the trace of Ksab. hµα is the bulk to surface projection operator and eαais the tangent vector 4. A and C are related to the two anomaly coefficient that are present
for the 4 dimensional CFT.
A =A
16π2, C =
C16π2
. (1.48)
A is known as Euler anomaly and C is know as Weyl anomaly. They show up in the non
vanishing part of trace of the stress energy tensor.
< T ii >=C
16π2W 2 − A
16π2E4 (1.49)
4eαa = ∂Xα
∂Xa , where µ and a denote respectively bulk and surface indices.
14 1.1. INTRODUCTION
where, E4 = RijklRijkl−4RijR
ij +R2 is the 4 dimensional Euler tensor and W 2 is the square
of 4 dimensional Weyl tensor. i, j, k, l denote the 4 dimensional indices.
In general to compute EE form field theory one uses “ Replica Trick” [6, 8]. For that,
first step is to define Renyi-entropy (Sn) ,
Sn = − 1
n− 1ln trAρ
nA. (1.50)
Figure 1.4: Path integral formulation of
density matrix (“Aspects of holographic
entanglement entropy”, S. Ryu and
T. Takayanagi, arxiv:-:hep-th/0605073.)
Then one has to evaluate this quantity on a “
Replica space ”. First we have to write down the
reduced density matrix in path integral formalism
[6, 8]. For simplicity consider a 1 + 1 dimensional
space. We will closely follow the notations and con-
ventions of [13]. We choose a tE = 0 slice and
consider an interval (A) as our entangling region
as shown in the figure. We denote all the dynam-
ical fields that are present collectively as φ(tE, x)
where tE is the Euclidean time. We also impose
the following boundary condition on the fields liv-
ing inside A.
φ0(tE = 0+, x) = φ+(x). (1.51)
and for other fields living outside A we demand
that,
φ0(x) = φ′(x). (1.52)
The ground state wave function can be written as,
Ψ(φ0(x)) =
∫ φ0(tE=0+,x)=φ+(x)
tE=−∞Dφe−S(φ(x)). (1.53)
Complex conjugate of this is defined as,
Ψ(φ0(x))∗ =
∫ tE=∞
φ0(tE=0−,x)=φ−(x)
Dφe−S(φ(x)). (1.54)
From this one can construct the density matrix easily,
ρAφ+φ− =1
Z1
∫ tE=∞
tE=−∞Dφe−S(φ(x))Πx∈Aδ(φ(tE+ = 0, x− φ+(x))δ(φ(tE− = 0, x− φ−(x)).
(1.55)
CHAPTER 1. INTRODUCTION 15
We have taken care of the fact, that the boundary conditions imposed are obeyed by inserting
the delta functions. Then to evaluate the n’th Renyi-entropy we have to find the product of
the n copies of this density matrix glued with each other by making suitable identifications
which we do in the next step.
TrAρnA = ρAφ1+φ1− · · · ρAφn+φn− . (1.56)
Also we have imposed φi− = φi+1,+ for all i = 1, · · ·n. So this quantity gives rise to the
“Replica space ” which is similar to an n sheeted Riemann surface and has a discrete Z(n)
symmetry coming from permuting the n copies of the replica. So finally we have to compute
the path integral on this replicated manifold which is denoted by Zn.
T rAρn = (Z1)−n
∫replicaspace
Dφe−S(φ) =Zn
(Z1)n. (1.57)
This is an important formula for computing EE from field theory using the replica trick. We
will see in the later sections, what is its implication in the context of holography. Replica
method has been employed successfully for computing EE for 1 + 1 dimensional CFT ’s [8],
but for higher dimensions it is hard to apply this method [29].
Entanglement entropy in Holography
Now let us turn our attention to the holography, main hero of our story. Importance of EE
in this context is profound. We will see that it geometrize the problem in the bulk space
time. Before going into the details let us take an example, which will show the connection
between EE and geometry. We start by drawing an analogy with the quantum mechanics.
We consider CFTs on two spatially disconnected regions [30]. Next we consider a wave-
function for this system of the form,
|Ψ >= |ΨA > ×|ΨB > (1.58)
where A and B denote the individual wavefunctions of the two CFTs. So it is evident that
the state is not an entangled state as it is written as a direct product of two states. This
kind of state in holography corresponds to a disconnected geometry. Now let us consider two
disconnected CFTs placed on Sd. Ei is the energy corresponding to the ith eigenstate. Now
let us consider the following state,
|ψ >=∑i
e−βEi2 |Ei × |Ei > . (1.59)
This state is not a direct product state. So it is an “ entangled state”. From holography we
know that this corresponds to a thermofield double state and the dual geometry is an eternal
16 1.1. INTRODUCTION
Figure 1.5: On the left hand side, we have shown a thermofield double state and its holo-
graphic dual eternal black hole is shown on the right hand side. (Picture courtesy- “ Building
up spacetime with quantum entanglement ” by Mark Van Raamsdonk, arXiv:-1005.3035)
black hole [31]. So this shows that quantum superposition of two states of two classically
disconnected CFTs corresponds to a classically connected geometry.
This analogy makes things more interesting as one can possibly hope to understand
geometry using EE. In the context of AdS/CFT one can ask, whether it is possible to associate
a concept of entropy for any arbitrary region of field theory sitting at the boundary of AdS?
If so, then the next obvious thing is to ask which portion of the bulk spacetime capture that
information? The answer comes in the form of Ryu-Takayanagi (RT) proposal for Einstein
gravity [32]. The proposal is very simple, it says that one can attach a notion of entanglement
entropy for any arbitrary region at the boundary of AdS at a constant time slice and the
corresponding entropy is given by the area of some special codimension 2 surface (γA) inside
the bulk spacetime. So consider a d dimensional bulk space time. Then the EE associated
with a region A at the boundary is given by,
SEE(A) =2π
`d−2p
Area(γA). (1.60)
`p is the planck length. To remind ourselves , this proposal is made for Einstein gravity.
S = − 1
2`d−1p
∫dd+1x[R− 2Λ]. (1.61)
The AdS metric is a solution of the action mentioned in Eq. (1.61) with a negative cosmo-
logical constant
Λ = −d(d− 1)
2L2.
CHAPTER 1. INTRODUCTION 17
Figure 1.6: holographic prescription for EE:
γA is the extremal surface
Now let us demystify this proposal and
see how it works. Let us for simplicity
consider a AdS5 metric and a spherical re-
gion (A) at the boundary as shown in the
Fig. (1.6).
ds2 =L2
z2(dz2+dt2+dr2+r2dθ2+r2 sin(θ)2dφ2).
(1.62)
z is the radial coordinate of the AdS space.
We consider a constant time slice and set
the Euclidean time t = 0. Now to evalu-
ate EE associated with the region A we will
employ RT method which tells us to find a
particular surface (γA) extending inside the
bulk spacetime which minimizes the area en-
closed. Area of that particular surface will give us the EE and γA is known as the “ minimal
surface”. To elucidate this further, we put t = 0 and r = f(z) in the (1.62) to obtain an
induce metric for the minimal surface. Then we evaluate the area for this.
SEE =8π2L3
`3p
∫dzf(z)2
√1 + f ′(z)2
z3. (1.63)
We minimize SEE defined in Eq. (1.63), thereby obtaining an equation for f(z). Solving that
equation with the following boundary condition ,
f(z = 0) = f ′(z = 0) = 0, (1.64)
where the prime denotes the derivative with respect to z, we get 5
f(z) =√R2 − z2. (1.65)
R is the radius of the spherical region (A) at the boundary. We plug this into Eq. (1.63) and
expand the resulting expression around z=0 which is divergent. We introduce a uv-cutoff ε
and finally we get,
SEE =4π2L3
`3p
[R2
ε2− ln(
R
ε)]. (1.66)
The leading term follows the area law and also we get an universal term.
One can consider more general surfaces and still get the expected results for the universal
term (1.45). RT proposal passes several consistency checks, it produces correct universal
5For any generic entangling surface, the extremization condition in the context of Einstein gravity can be
written as Ks = 0 where, Ks is the trace of the extrinsic curvature of this codimension-2 surface.
18 1.1. INTRODUCTION
term for EE, which in the context of AdS3/CFT2 reproduces many know results coming
from CFT computations [13]. In higher dimensions all these holographic results give us some
insights about the field theory. Also RT method has been successfully applied for computing
EE in excited state [33, 34], for time dependent cases (covariant RT proposal [35]) and for
many other interesting cases 6. It also satisfies the strong subadditivity relation [36]. More
importantly it has provide us with a geometrical interpretation of boundary data which is
the first step towards understanding how gravity (geometry) emerges form the field theory.
We will also consider various higher derivative corrections to Einstein gravity. As we are
considering effective theories of gravity, these corrections naturally arise as we integrate out
higher momentum modes in Wilsonian RG flow. These higher derivative corrections describe
holographic duals of field theory with finite ’t-Hooft coupling ( they correspond to QFTs with
finite number of colour) and can arise either as 1N
corrections or when one considers string
loop corrections [38]. But these terms take the AdS/CFT description beyond supergravity
limit, thereby providing us with a good platform to understand the effect of finite coupling
on the underlying field theory.
We will explore EE in this context of higher derivative gravity, hoping to understand
some important lessons about these effective theories. For eg, one can add an Gauss-Bonnet
term to the Einstein Lagrangian [39, 40].
SEGB = − 1
2`3p
∫d5x[R− 2Λ + λL2(RµναβR
µναβ − 4RαβRαβ +R2)
](1.67)
where λ is the Gauss -Bonnet coupling. AdS is still a solution for this theory provided the
following relations hold,L
L= f 2
∞ (1.68)
and
1− f∞ + λf 2∞ = 0.7 (1.69)
To compute EE for this theory one has to start with a suitbale entropy functional. For this
case we have famous Jacobson-Myers functional (JM) [41].
SJM =2π
`3p
∫d3x√h[1 + λL2R
]. (1.70)
√h is the induce metric for the extremal surface and R is Ricci scalar of the extremal
surface. Now we will consider two different type of entangling region - sphere and cylinder.
We know that there are two types of anomaly coefficients in 3 + 1 dimensional CFT. For
generic higher curvature theories, unlike Einstein gravity, these two anomaly coefficients are
6Interested readers are referred to this thesis [37]7We choose the particular root which smoothly goes to one when λ→ 0.
CHAPTER 1. INTRODUCTION 19
numerically different from each other. From Eq. (1.45) it can be shown that, for a cylindrical
entangling surface the first term of Eq. (1.45) vanishes and one picks out the contribution of
Weyl anomaly in the universal part of the EE. On the other hand if one chooses a spherical
entangling surface the second term in Eq. (1.45) vanishes and one picks out the contribution of
Euler anomaly in the universal part. So we will consider these two different types of entangling
surface to explore the nature of the universal terms in the context of the higher derivative
theories. Extremizing the JM functional we get the following results for the universal parts
[41].
ScylinderEE = −CH2R
ln(R
ε) , SsphereEE = −4A ln(
R
ε), (1.71)
where, C = π2L3
f3/2∞ `3p
(1 − 2f∞λ) and A = π2L3
f3/2∞ `3p
(1 − 6f∞λ) are respectively Weyl and Euler
anomaly. So JM functional produces the expected universal terms. For any arbitrary higher
curvature theories one can construct an entropy functional which produces the correct uni-
versal part. But the question remains whether one can derive them or not. We will try to
formulate such a derivation and that will teach us some important lessons about underlying
gravity theories.
So we have now introduced all the characters of the story. We are now ready to explore the
interplay between gravity and entanglement and hoping to uncover some interesting physics.
Summary of the thesis
Before ending this section let us summarize the key points of the thesis at this stage.
1. In Chapter 2 we will try to derive the proposed entropy functionals for general theories
of gravity using the generalized gravitational entropy method proposed in [42]. We
will show that it is possible to derive the entropy functional only for a certain class of
gravity theories and discuss the implications of that.
2. In Chapter 3 we will demonstrate how to compute the universal terms of EE using the
generalized entropy method.
3. Identifying EE with the generalized entropy opens up the possibility of connecting
EE with the black hole entropy. For general theories of gravity black hole entropy is
given by the Wald formula [43]. We will explore the possibility of connecting EE with
the Wald entropy in Chapter 4, thereby opening up the possibility of deriving EE by
Noether charge method.
4. In Chapter 5 we will see how put constraints on the couplings of the higher derivative
terms using extremal surfaces.
20 REFERENCES
5. In Chapter 6 we will took a small step towards understanding how bulk equation of
motion arises from EE, which will give us some intuition about the emergence of gravity
from field theory. We will use the concept of relative entropy to understand this, which
is nothing but the change of entanglement entropy between vacuum and excited states.
Using relative entropy we will demonstrate how one can constrain underlying gravity
theory.
6. After that we will discuss briefly how to code holographic RG flow using EE in the
Chapter 7 and end with summarizing the main results and some open questions.
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2 Holographic entanglement entropy
functionals: A derivation
2.1 Introduction
It is has been observed that there exists a striking similarity between black hole entropy
and entanglement entropy [1]. In the context of AdS/CFT, the entanglement entropy1 for
a boundary field theory which is dual to Einstein gravity can be calculated using the well-
known Ryu-Takayanagi proposal (RT) [18, 19]. This proposal states that the entanglement
entropy SEE of any region on the boundary of AdS can be calculated by evaluating the area
of a minimal surface in the bulk which is homologous to this boundary region:
SEE =Area
4GN
. (2.1)
Building upon earlier attempts [20, 21, 22, 23], this proposal was recently proved in [24], for
a general entangling surface. The entanglement entropy formula in Eq. (2.1) is of the same
form as the formula for calculating the entropy of a black hole. In the black hole case, there
exists a simple generalization of this area law for calculating the entropy of a black hole in
any general higher-derivative gravity theory, known as the Wald entropy [25, 26, 27]. It is
natural to ask then if one can generalize the Ryu-Takayanagi prescription to higher-derivative
gravity theories by simply replacing the RHS of Eq. (2.1) with the Wald entropy. However,
this is known not to be the case [28, 29].
Recently, a general formula for calculating the holographic entanglement entropy (HEE)
in higher-derivative gravity theories was proposed in [30, 31]. It was also conjectured that
the minimal entangling surface can be determined by interpreting this formula as the entropy
functional for the higher derivative gravity theory and extremizing it. At present there exists
no general proof of this proposal. Main objective of this chapter is to carry out various tests
to determine the validity of this conjecture and formulate a general proof.
Lewkowycz and Maldacena (LM) [24] have proposed a derivation of the Ryu-Takayangi
(RT) prescription [18] for computing entanglement entropy (EE) [2] in holography [19]. A
1There exists a huge literature on entanglement entropy. For background and interesting applications see
[2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17].
29
30 2.1. INTRODUCTION
generalization of black hole entropy is proposed in the context where there is no U(1) sym-
metry in the bulk. In the Euclidean theory, although there is no U(1) symmetry, one imposes
a periodicity condition of 2πn with n being an integer on the Euclidean time direction at
the boundary. This time direction shrinks to zero at the boundary. By suitably choosing
boundary conditions on the fields, LM propose to identify the on-shell Euclidean action with
a generalized gravitational entropy.
In calculations of entanglement entropy in quantum field theories, one frequently uses
the replica trick which entails introducing a conical singularity in the theory 2. An earlier
attempt to prove the RT formula was made by Fursaev [20]. In recent times, in the context
of AdS3/CFT2 there have been further developments in [8] towards a proof. In the context
of holography, this corresponds to taking the n→ 1 limit. In this case, LM suggest that the
time direction shrinks to zero on a special surface. The equation for this surface is derived
in Einstein gravity by showing that there is no singularity in the bulk equations of motion.
This surface has vanishing trace of the extrinsic curvature and corresponds to a minimal
surface–which is precisely what comes from minimizing the RT area functional. Next we will
try to generalize this for various higher derivative gravity theories.
We will first work with general four-derivative theory. It is also sufficient for our purpose
to consider only four-derivative theory as that will capture all the essential issues that we
like to bring up. The conjectured form of the holographic entropy functional for general
R2 theory first appeared in [33]. The formula of [30, 31] also reduces to this functional for
general R2 theory. For the purpose of this thesis, we will refer to this functional as the FPS
(Fursaev-Patrushev-Solodukhin) functional after the authors of the paper where it was first
proposed. In [34] it was shown that this entropy functional leads to the expected universal
terms in the entanglement entropy for cylindrical and spherical entangling surfaces, so the
FPS functional passes this basic first test. The obvious next step is to determine whether
the surface equation of motion derived from extremizing this functional is the same as that
derived using the generalized gravitational entropy method (which we will refer to as the LM
method) of [24].
General R2 theory depends on three parameters: λ1, λ2 and λ3. Gauss-Bonnet gravity is
a special point in this parameter space [35] and the FPS functional reduces to the Jacobson-
Myers functional at this point. For Gauss-Bonnet gravity, the question whether the surface
equation of motion one gets from the Jacobson-Myers functional matches with the surface
equation of motion derived using the LM method was addressed in [34, 36, 37]. We will look
2The only example where a derivation of EE exists without using the replica trick is for the spherical
entangling surface [4, 22] although in [32] it has been explained how this procedure is connected with the
replica trick. A proposal has been made in [7] for the equation for the entangling surface which does not
depend on the replica trick.
CHAPTER 2. HOLOGRAPHIC ENTANGLEMENT ENTROPY FUNCTIONALS: ADERIVATION 31
at the Gauss-Bonnet case first to emphasize several interesting points for this theory. For
this theory, the surface equation of motion that one gets from the Jacobson-Myers functional
matches with what one gets from the LM method, provided that terms cubic in the extrinsic
curvature are suppressed. We will find that for general R2 theory using a procedure similar to
the Gauss-Bonnet case leads to a match in the leading-order terms on both sides, where we
designate terms cubic in the extrinsic curvature as sub-leading. However, as we will show, in
the case of R2 theory, the LM method also yields an extra condition that cannot be satisfied
at arbitrary points of the parameter space. The conclusion is, therefore, that for a general
R2 theory the conditions that follow from the LM method do not correspond exactly to the
surface equation of motion derived from the FPS functional.
An alternative method to demonstrate that the FPS functional is the correct entropy
functional for R2 theory is to show that it can be interpreted as the action of a cosmic brane.
This method was employed in [30], where it was referred to as the cosmic brane method. We
will re-examine this procedure for R2 theory and show that the result we get is consistent
with what we get using the LM method.
What happens when we go to a six-derivative gravity theory? In this case, we consider
quasi-topological gravity [38] which is again a special point in the parameter space of R3
theories. We first construct the entropy functional for quasi-topological gravity using the
formula proposed in [30, 31]. We then show that this functional reproduces the expected
universal terms for this theory for the cylindrical and spherical entangling surfaces. This is
in agreement with the result of [39] that the entropy functional proposed in [30, 31] leads to
the correct universal terms for a general higher-derivative theory. We also find the minimal
surface condition for this theory using the LM method and show that it deviates from what
is expected from the HEE functional.
This chapter is organized as follows. In Sec. (2.2) we review the general entropy functional
proposed in [30, 31]. Our main focus in this chapter is general four-derivative gravity theory,
for which the entropy functional is the FPS functional. In Sec. (2.3) we find the surface
equation of motion for R2 theory by extremizing the FPS functional on the codimension-2
surface. We then compare it with what we obtain using LM prescription. We also make
some remarks on the Gauss-Bonnet case. We then investigate the cosmic-brane method of
[30]. In Sec. (2.4), we repeat our analysis for quasi-topological gravity. Lastly, in Sec. (2.5)
we summarize our findings and discuss their implications.
32 2.2. ENTROPY FUNCTIONAL FOR GENERAL THEORIES OF GRAVITY
2.2 Entropy functional for general theories of gravity
In this section we will review the general entropy formula proposed in [30, 31]. First we
summarize the argument leading up to this proposal, following [30]. For details the reader is
referred to [24, 30, 31, 33]. Some applications of this entropy formula are in [40].
In field theory, the entanglement entropy SEE = −Tr[ρ log ρ] can be calculated as the
n→ 1 limit of the Renyi entropy. The Renyi entropy in turn can be computed as
Sn = − 1
n− 1(logZn − n logZ1) . (2.2)
Here Zn is the partition function of the field theory on the manifold Mn which is the n-
fold cover of the original spacetime manifold M1. In the holographic dual theory one can
construct a suitable bulk solution Bn with boundary Mn. The manifold Mn at integer n has
a Zn symmetry that cyclically permutes the n replicas. In [24] it was proposed that this
replica symmetry extends to the bulk Bn. Orbifolding the bulk by this symmetry results in
a space Bn = Bn/Zn , that is regular except at the fixed points of the Zn action. The fixed
points form a codimension 2 surface with a conical defect in the bulk. This is the surface
that is ultimately identified with the minimal entangling surface in the n→ 1 limit. Further,
one can use gauge-gravity duality [41] to identify the field theory partition function on Mn
with the on-shell bulk action on Bn in the large-N limit
Zn ≡ Z[Mn] = e−S[Bn] . (2.3)
It is now straightforward to calculate the entanglement entropy. By construction, one can
identify
S[Bn] = nS[Bn] (2.4)
at integer n, where S[Bn] is the classical action for the bulk configuration Bn not including
any contribution from the conical defect. By analytically continuing Bn to non-integer n,
Eq. (2.4) can be used to define S[Bn]. Using Eqs. (2.2) and (2.3) and expanding around
n = 1, one gets
SEE = limn→1
n
n− 1
(S[Bn]− S[B1]
)= ∂εS[Bn]
∣∣∣ε=0
(2.5)
where ε ≡ n−1. The quantity S[Bn] can be calculated for the bulk theory by writing the bulk
metric locally around the surface in gaussian normal coordinates and introducing a conical
defect. It can be shown [24, 30] that ∂εS[Bn] gets a contribution entirely from the tip of the
cone. To compute it, therefore, one employs a metric regularized at the tip of the cone.
This calculation is similar to that employed in [42] for calculating the Wald entropy
from a regularized cone metric. Indeed, evaluating S[Bn] for a bulk theory with the cone
metric to linear order in ε and using Eq. (2.5) will result in two terms. The first is SWald:
CHAPTER 2. HOLOGRAPHIC ENTANGLEMENT ENTROPY FUNCTIONALS: ADERIVATION 33
the Wald entropy for the theory. However, as was noted in [33], there is a second way for
a linear contribution to arise. A term in the bulk lagrangian that is of order ε2 can get
enhanced to order ε after integrating over the transverse directions. Following [30], we label
the contribution of such terms as SAnomaly. At this point, the calculation of the form of
SEE is basically finished. Eq. (2.5) can be used to find the entanglement entropy for any
higher-derivative theory, including ones whose lagrangians involve derivatives of the Riemann
tensor. However, for a general higher-derivative theory it can be computationally difficult to
compute SAnomaly directly using Eq. (2.5).
In [30, 31] a simpler prescription for calculating the holographic entanglement entropy
for higher-derivative theories of gravity in d + 1 dimensions, for which the lagrangian (L)
contains only contractions of the Riemann tensor, was given. The formula is:
SEE =2π
`dp
∫ddy√h
∂L
∂Rzzzz
+∑α
(∂2L
∂Rzizj∂Rzkzl
)α
8KzijKzklqα + 1
. (2.6)
The notation used in the above equation and also in the rest of the chapter is as follows:
We use Greek Letters µ, ν, ρ, σ, · · · to label the bulk indices. We use the Latin letters
a, b, ......m, n to label the indices of the codimension 2 surface, while reserving the letters
p, q, r, s to denote the indices of the transverse directions. In these directions, we use the com-
plex coordinates z and z. The bulk metric is denoted by gµν .The metric on the codimension-2
entangling surface is denoted by hij while the surface itself is denoted by Σ. The bulk Rie-
mann tensor is denoted by Rµνρσ while the intrinsic Riemann tensor of the surface is denoted
by Rikjl. The extrinsic curvatures of the surface are denoted by Krij, where the first index la-
bels the extrinsic curvature in the transverse directions. We follow the curvature conventions
in [43].
The first term in Eq. (2.6) is the Wald entropy. The second term is the correction to the
Wald entropy and is evaluated in the following way: The second derivative of the lagrangian Lis a polynomial in components of the Riemann tensor. We expand the components Rpqij, Rpiqj
and Rikjl using
Rpqij = Rpqij +KpjkKkqi −KpikKkqj ,Rpiqj = Rpiqj +KpjkKkqi −Qpqij ,Rikjl = Rikjl +KpilKpjk −KpijKpkl . (2.7)
Here, Qpqij ≡ 12∂p∂qgij|Σ. Rpqij and Rpiqj can also be defined in terms of metric variables,
but the exact definition is not needed here. The variable α is used to label the terms in the
expansion. For each term labelled by α, qα is defined as the number of Qzzij and Qzzij, plus
one half of the number of Kpij, Rpqri, and Rpijk. The final step is to sum over α with weights
34 2.3. TEST OF THE ENTROPY FUNCTIONAL FOR R2 THEORY
1/(1 + qα). The quantities Rpqij, Rpiqj, and Rikjl can then be eliminated using Eq. (2.7),
resulting in an expression involving only components of Rµνρσ, Kpij and Qpqij.To yield the entanglement entropy, the formula in Eq. (2.6) should be evaluated on the
minimal entangling surface. This surface is supposed to be determined following the LM
method. [30, 31] also contain the proposal that the minimal surface can be determined
by extremizing SEE as given in Eq. (2.6) — SEE therefore being the entanglement entropy
functional for a general theory of gravity. Rest of this chapter is mainly based on the work
done with Prof. Aninda Sinha and Dr. Menika Sharma [34, 44].
2.3 Test of the entropy functional for R2 theory
In this section we consider general R2 theory in five dimensions. The lagrangian for this
theory is
L = L1 + L2 , (2.8)
where
L1 = R +12
L2(2.9)
is the usual Einstein-Hilbert lagrangian with the cosmological constant appropriate for five-
dimensional AdS space and
L2 =L2
2
(λ1RαβγδR
αβγδ + λ2RαβRαβ + λ3R
2)
(2.10)
is the R2 lagrangian.
The proposed entropy functional for this theory is
SEE,R2 = SWald,R2 + SAnomaly,R2 , (2.11)
where
SWald,R2 =2π
`3P
∫d3x√h
1 + L2
2(2λ3R + λ2Rµνn
νrn
rµ + 2λ1Rµνρσnµrn
νsn
rρnsσ),
(2.12)
and SAnomaly,R2 =2π
`3P
∫d3x√hL2
2
(− 1
2λ2KrKr − 2λ1KsijKsij
). (2.13)
As mentioned earlier, this entropy functional leads to the correct universal terms. To demon-
strate this, we will write down first the bulk AdS metric as,
ds2 =L2
z2(dz2 + dτ 2 + hijdx
idxj) (2.14)
CHAPTER 2. HOLOGRAPHIC ENTANGLEMENT ENTROPY FUNCTIONALS: ADERIVATION 35
where, L is the AdS radius and hij is a three dimensional metric given below. We will use
Greek letters for the bulk indices and Latin letters for the three dimensional indices. For the
calculation of EE for a spherical entangling surface we write the boundary hij in spherical
polar coordinates as,spherehijdx
idxj = dρ2 + ρ2dΩ22 , (2.15)
where dΩ22 = dθ2 + sin2 θdφ2 is the metric of a unit two-sphere and θ ∈ [0, π] and φ ∈ [0, 2π].
For a cylindrical entangling surface,
cylinderhijdxidxj = du2 + dρ2 + ρ2dφ2 . (2.16)
u is the coordinate along the direction of the length of the cylinder. For a cylinder of length
H, u ∈ [0, H]. Here L = L√f∞.
We put ρ = f(z), τ = 0 in the metric and minimize (2.11) on this codimension 2 surface
and find the Euler-Lagrange equation for f(z). Using the solution for f(z) we evaluate (2.11)
to get the EE.
For the sphere, we get f(z) =√f 2
0 − z2 which gives the universal log term,
SEE = −4a ln(f0
δ) . (2.17)
For the cylinder, f(z) = f0 − z2
4f0+ ... which gives,
SEE = −cH2R
ln(f0
δ) . (2.18)
a =π2L3
f3/2∞ `3
P
(1− 2f∞(λ1 + 2λ2 + 10λ3)) and c =π2L3
f3/2∞ `3
P
(1 + 2f∞(λ1 − 2λ2 − 10λ3)) . (2.19)
and δ is the UV cut-off comes from the lower limit of the z integral. f0 is the radius of the
entangling surface. These are the expected results [28, 35].
In this section, we will further test this entropy functional by determining whether the
surface equation of motion one gets from extremizing this functional is the same as the
surface equation of motion one gets following the LM method. In Sec. (2.3.1), we extremize
the functional for R2 theory. In this particular section, we will first find the surface equation of
motion for this functional in a general spacetime background. However, the Ryu-Takayanagi
proposal and its generalizations are most precisely formulated in the AdS/CFT context, so
eventually we will specialize to the AdS background. In Sec. (2.3.2) we find the surface
equation of motion using the LM method. In this case, we will always assume that the bulk
is AdS space. Since a variation of the LM method – called the cosmic-brane method – was
used in [30] to formulate a proof that the FPS functional is the correct entropy functional
for R2 theory, we also investigate this method in Sec. (2.3.3).
36 2.3. TEST OF THE ENTROPY FUNCTIONAL FOR R2 THEORY
2.3.1 Minimal surface condition from the entropy functional
To extremize the functional in Eq. (2.11), we follow the methods of [45, 46, 47]. We denote
the surface we are going to extremize w.r.t the action in Eq. (2.11) by Σ. The induced metric
on Σ is
hij = eµi eνj gµν , (2.20)
where gµν is the bulk metric and eµi ≡ ∂iXµ are the basis vectors tangent to the surface
Σ, Xµ being the bulk coordinates. On the surface Σ, the gir component of the bulk metric
vanishes. The two normals to the surface are denoted by nµr where r = 1, 2 are the transverse
directions. The metric tensor in the tangent space spanned by the normal vectors (the metric
tensor of the normal bundle of the sub-manifold Σ) is the Kronecker delta:
δrs = ε nµrnνsgµν (2.21)
We work in Euclidean signature and set ε = +1. We use the inverse metric δrs, to raise
indices in the normal directions: nrµ = δrsnµs . Note that, repeated s indices always imply
summation over the transverse directions: nµsnνs ≡ nµ1n
ν1 + nµ2n
ν2. In this notation, the
completeness relation relating gµν , the inverse of the bulk metric, to hij, the inverse of the
induced metric, is
gµν = hijeµi eνj + nµsn
νs. (2.22)
The Gauss and Weingarten equations are
∇ieµj = ∂ie
µj + Γµνρe
ρi eνj − Γkije
µk = −Krijnµr
∇inµs = ∂in
µs + Γµρνe
ρin
νs − Γrsin
µr = Kjsie
µj . (2.23)
Here, ∇ is the Van der Waerden-Bortolotti covariant derivative [45] which acts on a general
tensor T s···ri···j as
∇kTs···ri···j = ∇kT
s···ri···j + ΓspkT
p···ri···j + · · ·+ ΓrpkT
s···pi···j , (2.24)
where ∇ is the usual covariant derivative associated with the surface Christoffel. This
Christoffel is related to the bulk Christoffel Γµσν as
Γijk = (∂jeµk + Γµσνe
σj eνk)e
iµ . (2.25)
The Chrisoffel Γris is the Christoffel induced in the normal bundle. It is related to the bulk
Christoffel as
Γris = (∂inµs + Γµσνe
σi n
νs)n
rµ . (2.26)
This Christoffel can be interpreted geometrically as the freedom to perform rotations of the
normal frame. It is, therefore, equivalent to a gauge field Ak, commonly referred to as a twist
potential. This field is defined as:
Ak ≡1
2εrs∂rgks , so that Γsjr = δpsεrpAj , (2.27)
CHAPTER 2. HOLOGRAPHIC ENTANGLEMENT ENTROPY FUNCTIONALS: ADERIVATION 37
where εrs is the Levi-Civita symbol.
The Gauss identity relating the bulk Riemann with all indices projected in the tangential
directions with the surface Riemann is
Rµνρσeµkeνi eρl eσj = Rkilj −KrklKrij +KrikKrjl . (2.28)
The Codazzi-Mainardi relation is
∇kKrij −∇iKrkj = Rµνρσeµkeνi eσj n
ρr . (2.29)
From Eq. (2.28) we get the Gauss-Codazzi identity
R = R− 2Rµνnνrnµr +Rµνρσn
µrnνsnρrnσs +KrKr −KsijKijs . (2.30)
We now consider an infinitesimal variation of the surface Σ given by Xµ −→ Xµ + δXµ.
The change δXµ is
δXµ = ξrnµr + ξieµi . (2.31)
where ξr and ξi are small parameters. For deriving the equation describing the minimal
surface we are only concerned with the variation in the normal direction, since the tangential
variation leads to a constraint equation. The variation then reduces to
δXµ = ξrnµr , (2.32)
The variation δXµ in the surface will induce a variation in the basis vectors eµi . This can
be computed by finding the basis vectors at Xµ + δXµ and parallel transporting them back
to Xµ. Taking the difference between the parallel-transported quantity and the original basis
vector at the coordinate Xµ, using the identities in Eq. (2.23) and then restricting to normal
variation results in
δeµi = nµs∇iξs + eµjK
jsiξ
s . (2.33)
The details of this calculation are in [45]. As stated in Eq. (2.24), the covariant derivative ∇acts on ξs as
∇iξs = ∂iξ
s + Γsirξr . (2.34)
The variation in any other tensor quantity can be calculated in a similar way, by parallel
transporting the quantity at the new coordinate back to the old coordinate and taking the
difference. This gives the variation in the bulk metric as zero. We write down the result
for other variations. For details the reader is referred to [45]. The variation of the induced
metric is
δhij = 2ξrKrij ,δ√h = ξr
√hKr . (2.35)
38 2.3. TEST OF THE ENTROPY FUNCTIONAL FOR R2 THEORY
The variation of the extrinsic curvature is
δKsij = (−∇i∇jδsr +KsikKkrj +Rµνρσn
sµnσr eρi eνj )ξ
r ,
δKs = (−∇i∇iδsr −KsikKkir + hijRµνρσn
sµnσr eρi eνj )ξ
r . (2.36)
The covariant derivatives ∇ act all the way to the right.
Using these variations we can now compute the change in the action. For this we first
rewrite the Rµνnνrn
rµ and Rµνρσnµrn
νsn
rρnsσ terms in the action given in Eq. (2.12) as
Rµνnνrnµr = R− hijRµνe
νi eµj
Rµνρσnµrnνsnρrn
σs = R− 2hijRµνe
νi eµj + hikhjlRµνρσe
µi eνj eρkeσl (2.37)
using the completeness relation in Eq. (2.22). The variation of a term such as hijRµνeνi eµj is
given by
δ(hijRµνeνi eµj ) = (δhij)Rµνe
νi eµj + 2hijRµνδ(e
νi )e
µj + hijδ(Rµν)e
νi eµj (2.38)
The first two variations can be computed using Eqs. (2.35) and (2.33) respectively. For
evaluating the last term we need the variation of the bulk Ricci Tensor which is given by
δ(Rµν) = nσr ∇σRµνξr . (2.39)
The variation in the bulk Ricci scalar and Riemann tensor take a similar form. All these
variations are under the integral sign in Eq. (2.12) and we perform a integration by parts
where needed, discarding the term that is a total variation. Then using the variations given
above we obtain 3:
δ(√hR) =
√h KsRξs + nµs
√h∇µRξ
s ,
δ(√hRµνn
νrnµr ) =√h KsRµνn
νrnµr ξs + 2
√h∇i(Rµνn
νseµi )ξs −
√hnσsh
ijeµi eνj ∇σRµνξ
s + nµs√h ∇µRξ
s ,
δ(√hRµνρσn
µrnνsnρrnσs ) =√h KsRµνρσn
µrnνqnρrnσq ξ
s − 4√h∇i(Rµνρσn
µs eνj eρi eσkh
jk)ξs+
4√h∇i(Rµνn
νseµi )ξs +
√hhikhjleµi e
νj eρkeσl n
αs ∇αRµνρσξ
s −2nνsh
ijeµi eρj∇νRµρξ
s +√hnµs ∇µRξ
s . (2.40)
Similarly the variations for the terms present in the action in Eq. (2.13) are
δ(√hKsKs) =−2
√h∇i∇iKrξr +
√hKrKsKsξr − 2
√hKsKsijKijr ξr−
2√hKsRµνρσh
ijnµr eνi n
ρseσj ξ
r ,
δ(√hKsijKsij) =−2
√h∇i∇jKijr ξr +
√hKrKsijKsijξr − 2
√hKsijKsik Kkjr ξr−
2√hKsijRµνρσn
µr eνi n
ρseσj ξ
r . (2.41)
3We thank Joan Camps for valuable discussions
CHAPTER 2. HOLOGRAPHIC ENTANGLEMENT ENTROPY FUNCTIONALS: ADERIVATION 39
Adding these variations up with the appropriate factors will give us the equation for the
minimal surface for the action in Eq. (2.11) in a general spacetime background.
As a check of these equations we now demonstrate that the above results lead to the
correct surface equation of motion in the Gauss-Bonnet case. For Gauss-Bonnet the entropy
functional for general R2 theory reduces to the Jacobson-Myers functional
SJM =2π
`3P
∫d3x√h(1+λL2(R−2Rµνn
νrnµr +Rµνρσnµrnνsnρrn
σs +KsKs−KsijKsij) . (2.42)
This functional is valid in a general space-time background. Note that the integrand is equal
to√h(1 + λL2R) on using the Gauss-Codazzi identity Eq. (2.30). The surface equation of
motion for this theory using this form of the functional is [34],
K + λL2(RK− 2RijKij) = 0 . (2.43)
We now find the surface equation of motion by directly varying Eq. (2.42). Using the variation
equations Eqs. (2.40–2.41) and simplifying using the identities in Eqs. (2.28–2.29) we get
√h Ks ξs + λL2
[√h KsR ξs − 2
√hRjkKjks ξs
+√hhikhjleµi e
νj eρkeσl n
αs ∇αRµνρσξ
s − 2√h∇i(Rµνρσn
µs eνj eρi eσkh
kj)ξs
− 2√hKrRµνρσh
ijnµs eνi n
ρreσj ξ
s + 2√hKrijRµνρσn
µs eνi n
ρreσj ξ
s
+ 2√hRµνρσe
µj eνi eρkeσl h
ilKjks ξs]. (2.44)
The first three terms give precisely the equation of motion for Gauss-Bonnet theory. The
rest of the terms add up to zero, as we show in the following. We use the Bianchi identity
on the ∇αRµνρσ factor of the fourth term giving
∇αRµνρσ = −∇σRµναρ − ∇ρRµνσα (2.45)
and then rewrite each of these terms as
hikhjleµi eνj eρkeσl n
αs ∇σRµναρ = eσl ∇σ(hikhjleµi e
νj eρkn
αsRµναρ)− eσl ∇σ(hikhjleµi e
νj eρkn
αs )Rµναρ .
(2.46)
The expression in brackets in the first term of the R.H.S is a bulk scalar and therefore this
term can be written as
∂l(hikhjleµi e
νj eρkn
αsRµναρ) = −∇i(Rµνρσn
µs eνj eρi eσkh
jk) + Γrslhikhjleµi e
νj eρkn
αrRµναρ+
Γmjlhikhjleµi e
νme
ρkn
αsRµναρ , (2.47)
Inserting these expressions in Eq. (2.44) after expanding the second term on the R.H.S of
Eq. (2.46) and using the identities in Eq. (2.23) will lead to cancellation of all terms except
for the terms in the first line of Eq. (2.44).
40 2.3. TEST OF THE ENTROPY FUNCTIONAL FOR R2 THEORY
AdS background
We now specialize to the case of AdS background which is the background we will use while
finding the equation of motion using the LM method. In AdS space the Riemann tensor is
Rµνρσ = −C(gµρgνσ − gµσgνρ) , (2.48)
where we have defined C = f∞/L2. Here, L is the length scale associated with the cosmo-
logical constant and is related to the AdS radius L as L = L√f∞. The variable f∞ satisfies
the following equation for R2 theory
1− f∞ +1
3f 2∞(λ1 + 2λ2 + 10λ3) = 0 . (2.49)
For ease of comparison with later results, we also rewrite the variation in√hR given in
Eq. (2.40) using the Gauss-Codazzi relation Eq. (2.30). The minimal surface equation is then
Kr + L2λ3(RKr − 2RijKrij + 2∇2Kr − 2∇i∇jKrij−KrK2 + 2KrijK
ij2 +KrK2 − 2Kr3 − 18CKr)+
λ2(12∇2Kr − 1
4KrK2 + 1
2KrijK
ij2 − 11
2CKr)+
λ1(2∇i∇jKrij −KrK2 + 2Kr3 − 4CKr) = 0 , (2.50)
where we have defined K2 = KsijKsij,Kij2 = KsKsij, K2 = KsKs and Kr3 = KsijKsik Krkj. Note
that these are a set of two equations one for each of the extrinsic curvatures K1,K2.
In AdS space we can make a further simplification using Eq. (2.29). The R.H.S of this
equation disappears on using Eq. (2.48). We then get the identity ∇k∇kKr = ∇i∇jKrij on
taking a further covariant derivative of the L.H.S. As explained in Appendix 2.5, in the LM
method for a time-independent metric, we can set K1 = K2 = K. We, therefore, also drop
the r index and Eq. (2.50) simplifies to
K + L2λ3(RK− 2RijKij −K3 + 3KK2 − 2K3 − 18CK)+
λ2(12∇2K − 1
4K3 + 1
2KK2 − 11
2CK)+
λ1(2∇2K −KK2 + 2K3 − 4CK) = 0 . (2.51)
We have also verified this equation by determining the bulk extremal surfaces for different
types of boundary entangling regions (sphere, cylinder and slab).
For the Gauss-Bonnet case: λ1 = λ, λ2 = −4λ and λ3 = λ, this equation reduces to the
known result in Eq. (2.43). Note that terms cubic in the extrinsic curvature as well as the
CK terms are not present in that equation. The Gauss-Bonnet case is special in this sense.
CHAPTER 2. HOLOGRAPHIC ENTANGLEMENT ENTROPY FUNCTIONALS: ADERIVATION 41
No such simplification occurs if we set the value for Weyl2 theory, λ1 = λ, λ2 = −4λ/3 and
λ3 = λ/6:
K +λL2
6(RK− 2RijKij + 8∇2K +K3 − 7KK2 + 10K3 + 2CK) = 0 . (2.52)
The CK term, in particular, stands out. If we trace the provenance of this term, it comes
from terms of the form KsRµνρσnµrn
νqn
ρrnσq and Kijs Rµνρσnµr eνi n
ρseσj in Eqs. (2.40) and (2.41)
— terms that have components normal to the surface. Nevertheless, for AdS background
these reduce to a term intrinsic to the surface. In fact, using the Gauss-Codazzi identity,
Eq. (2.30), we can rewrite this CK term as ∼ K3 +RK.
So far, we have only considered normal variations of the surface. Considering tangential
variations leads to a constraint equation. For R2 theory this constraint equation is indistin-
guishable from the condition in Eq. (2.29) which is the Codazzi-Mainardi relation.
2.3.2 Minimal surface condition from the Lewkowycz-Maldacena
method
We will now derive the surface equation for R2 using the LM method. As already mentioned,
the main idea of Ref. [24] is that one can obtain the minimal surface condition by extending
the replica trick to the bulk. The bulk will then have a Zn symmetry. Orbifolding by this
symmetry will lead to a spacetime in which the fixed points form a codimension-2 surface
with a conical deficit. In the n → 1 limit this surface can be identified with the entangling
surface. The metric of this surface can be parametrized in gaussian normal coordinates as
follows:
ds2 = e2ρ(z,z)dzdz + e2ρ(z,z)Ω(zdz − zdz)2+ (gij +Krijxr +Qrsijxrxs)dyidyj +
2e2ρ(z,z)(Ai + Brixr)(zdz − zdz)dyi + · · · . (2.53)
Here ρ(z, z) = − ε2
ln(zz) and ε = n−1, while x1 = z and x2 = z. This is the most general form
of the metric upto terms second order in z(z) [30, 31, 33, 48]. The · · · denote higher-order
terms. As we will see later, for R2 theory we also need to include third-order terms in the
metric expansion. The quantity Kij in this metric is identified with the extrinsic curvature,
while Ai is identified with the twist potential. Both of these are standard quantities that
characterize the embedding of the surface in the bulk. The quantities Ω,Bri and Q in the
second-order terms in the metric are not arbitrary, but can be written in terms of Krij,Aiand the components of the curvature tensors.
The bulk equation of motion will now contain divergences arising out of the conical
singularity of the form εz, εz2
. However, the matter stress-energy tensor is expected to be
42 2.3. TEST OF THE ENTROPY FUNCTIONAL FOR R2 THEORY
finite. Therefore, we must set all divergences to zero. This condition fixes the location of the
entangling surface.
The bulk equation of motion for general four-derivative theory is [49]:
Rαβ −1
2gαβR−
6
L2gαβ −
L2
2Hαβ = 0 , (2.54)
where
Hαβ = λ1
(1
2gαβRδσµνR
δσµν − 2RασδµRβσδµ − 4∇2Rαβ + 2∇α∇βR +
4RδαRβδ + 4RδσRδαβσ
)+
λ2
(∇α∇βR + 2RδσRδαβσ − ∇2Rαβ +
1
2gαβRδσR
δσ − 1
2gαβ∇2R
)+
λ3
(− 2RRαβ + 2∇α∇βR +
1
2gαβR
2 − 2gαβ∇2R). (2.55)
Gauss-Bonnet theory
Our eventual goal is to find the surface equation of motion for general R2 theory, but it is
illuminating to look at the Gauss-Bonnet case first. The Gauss-Bonnet case was addressed
in [34, 36, 37] using a metric linear in z(z). In this section, we will find the surface equation
of motion for this theory using the metric in Eq. (2.53).
We first show that the second-order metric in Eq. (2.53) suffices for Gauss-Bonnet theory
and inclusion of higher-order terms in this conical metric will not affect the surface equation
of motion that we find for this theory from the LM method. The bulk equation of motion
for Gauss-Bonnet theory can be obtained from Eq. (2.54) by setting λ1 = λ, λ2 = −4λ and
λ3 = λ giving:
Hαβ = 4RδαRβδ − 4RδσRδαβσ − 2RRαβ − 2RασδµRβ
σδµ+
12gαβ(RδσµνR
δσµν − 4RδσRδσ +R2) . (2.56)
The surface equation of motion is derived by finding the divergences in this equation that
arise on using the conical metric in the limit z = z → 0. Terms higher than second-
order in the metric will not contribute to the curvature tensors to zeroeth-order in z(z),
although they might contribute at higher order. This is because the curvature tensors are of
dimension 1/Length2 while third-order terms in the metric will be of order 1/Length3. The
explicit values of the curvature tensors are listed in Appendix (2.5). These are calculated
using a conical metric which is third-order in z(z). Note also, that the curvature tensors
contain at most divergences of order 1/z. In the above equation of motion all terms are the
product of two curvature tensors. Since each curvature tensor can only contribute at most a
CHAPTER 2. HOLOGRAPHIC ENTANGLEMENT ENTROPY FUNCTIONALS: ADERIVATION 43
1/z divergence and no third(or higher)-order term occurs at zeroeth order in any curvature
tensor, third(and higher)-order terms will be absent in the divergence equations.
By the same logic one can see that second-order terms will contribute to the divergence
equations. However, in this case, cancellations between terms remove most of the second-
order quantities, leaving only the quantities Qzzij and Qzzij in the divergence equations.
In the z = z → 0 limit K1 = K2 as explained in Appendix 2.5, so we drop the index r on
Kr. The divergence in the zz component from Hαβ term in the bulk eom is
Hzz =ε
z
[λ(RK− 2KijRij)
]+ε
z
[e−2ρ(z,z)λ
−K3 + 3KK2 − 2K3
]. (2.57)
Setting this divergence to zero should yield the condition for the extremal surface. There is
no divergence in the zz component. The divergence in the zi component is
Hzi =ε
z
[e−2ρ(z,z)λ
2K∇iK − 2K∇jKji − 2Kji∇jK + 2Kij∇kKkj −
2Kkj∇iKkj + 2Kjk∇jKki]
. (2.58)
This divergence is equivalent to the constraint equation one gets for the entropy functional
(which doesn’t have to be necessarily the Jacobson-Myers functional) using tangential vari-
ations of the surface and vanishes similarly by Eq. (2.29). Finally, the divergence in the ij
component is
Hij =4ε
z
[e−4ρ(z,z)λ
2KikKklKlj + hijKK2 −KijK2 − hijK3 −KKikKkj − 4hijKQzz
+ 4hijKklQklzz − 8KkiQkzzj + 4KijQzz + 4KQzzij]
+
2ε2
z2
[e−4ρ(z,z)λ
2KijK − 2KikKkj − hijK2 + hijK2
]. (2.59)
In the above equation we have set Qzzij = Qzzij. Using the value of the Rzizj component
of the Riemann tensor from Appendix 2.5 and setting the background to AdS space, using
Eq. (2.48), we can show thatQzzij = 14KikKkj and as a result the ε
zdivergence exactly vanishes.
However the ε2
z2divergence will remain.
The final step is to take the ε, z → 0 limit. Depending on the ordering one chooses, there
are two ways to do this. One way is to take z → 0 limit first. Physically, this corresponds to
looking for a divergence in the bulk equation of motion while there is still a small but non-
zero conical deficit parameter ε. The second way is take ε→ 0 first. The limit is, therefore,
an iterated limit – the final result depends on the order in which the limit is taken, so there
is an inherent ambiguity in this procedure. In fact, this ambiguity can be made even larger
in scope if we take the limit simultaneously in ε and z. Mathematically, the divergence is a
function of the two variables: ε and z. In this ε-z plane there are an infinite number of paths
44 2.3. TEST OF THE ENTROPY FUNCTIONAL FOR R2 THEORY
along which we can take the limit. At least on a mathematical level, there exists no reason
why the limit should only be taken along the z = 0 or ε = 0 path.
The path z = 0 is, however, the simplest way to take the limit so as to obtain εz→∞. In
this case, all terms containing εz
are leading divergences while terms containing e−2ρ(z,z)ε/z =
(zz)εε/z contribute to sub-leading divergences. Therefore, in this way of taking limits, setting
the Hzz divergence to zero will yield two different conditions for the minimal surface. The
first condition which, after adding the Einstein term, corresponds to the surface equation of
motion is
K + L2λ(RK− 2KijRij) = 0 . (2.60)
This agrees with the surface equation that comes from the Jacobson-Myers functional. How-
ever, there will also be an extra constraint [6] of the form
−K3 + 3KK2 − 2K3 = 0 , (2.61)
coming from the sub-leading divergence. The Hij divergence will also lead to a similar
constraint. The above condition can only be true for very special surfaces and therefore is an
over-constraint on the surface. In fact, if these two conditions were to be true simultaneously,
the surface equation of motion we would end up getting is:
cK + αλL2(K3 − 3KK2 + 2K3) = 0. (2.62)
To get this form of the equation, we have used the Gauss identities on AdS space. Here,
c = (1 − 2f∞λ) is proportional to the Weyl anomaly and α is a variable that can take any
arbitrary numerical value. The surface equation of motion corresponding to the Jacobson-
Myers functional can be recovered if α = 1. However, at present nothing within the LM
method sets the value of this parameter to one. Note that if α was zero, the minimal surface
that we would get is the same as in the Einstein case. It also the minimal surface that would
follow if one were minimizing just the Wald part of the entropy functional.
In the above paragraph we outlined one way in which the LM method could potentially
give rise to the correct surface equation of motion. Let us now explore if we can change
the limit-taking procedure itself to get the correct equation. This can be accomplished by
choosing a different path in the ε-z plane to take the limit. Taking the limit along the
path ε = 0 will simply kill off all divergences; this is not surprising since physically this
corresponds to turning off the conical deficit in the metric. However, we can pick a path in
the ε, z plane that will kill off the sub-leading divergence but preserve the leading divergence.
For example, as was shown in [36], taking any path of the form (z)2ε = ( zε)1+v, where v is a
number greater than one, will keep only the leading divergence. At this point, we can offer
no justification of why one should choose this particular way of taking limits. We are merely
CHAPTER 2. HOLOGRAPHIC ENTANGLEMENT ENTROPY FUNCTIONALS: ADERIVATION 45
demonstrating that there does exist a way to take limits in the ε, z plane that leads to the
correct surface equation of motion in the Gauss-Bonnet case. This way of taking limits is
equivalent to discarding terms suppressed by e−2ρ(z,z) and was also used in [30] to show that
the LM method leads to the same surface equation of motion in the Lovelock case as can
be derived from the entropy functional for Lovelock gravity [27, 30, 50]. This is the way of
taking limits that we will use. However, unless one can specify a mechanism or a physical
interpretation which reproduces this way of taking limits (which is possible if the metric itself
is re-defined), the argument that the LM method reproduces the correct surface equation of
motion for Gauss-Bonnet theory remains incomplete.
The same ambiguity in taking limits exists for general R2 theory. To remain consistent
with the Gauss-Bonnet point, for R2 theory we will continue to take limits as stated in the
paragraph above. However, in the general case this is not an ideal solution. As we will see,
∼ K3 terms always occur with the e−2ρ(z,z) factor in the divergence equations for R2 theory.
This means that if we use the above way of taking limits we will never get such terms at any
point in the parameter space. As we saw in Eq. (2.51), the surface equation of motion for R2
theory does contain such terms. However, our goal for general R2 theory is to see to what
extent we are able to reproduce the surface equation of motion in Eq. (2.51), while taking
the limit in such a manner that the result at the Gauss-Bonnet point agrees with what comes
from the Jacobson-Myers functional. It is clear, though, that the question of taking limits in
the LM method deserves more study.
The general case
We now work out the divergence equations for the R2 case. For general R2 theory all second-
order quantities will enter into the divergences. We can anticipate the effect that terms
containing Ω and B will have on the surface equation of motion coming from the LM method.
Consider the following components of the bulk Riemann tensor around z = z = 0:
Rpqrs
∣∣∣(z=0,z=0)
= −3e4ρ(z,z)εpqεrsΩ ,
Rpqri
∣∣∣(z=0,z=0)
= 3e2ρ(z,z)εpqBri , (2.63)
where, εab is defined as εzz = −εzz = e−2ρ(z,z)gzz. The quantity Ω is therefore equivalent
to −1/3Rµνρσnµrn
νsn
ρrn
σs evaluated at z, z = 0. We can determine a numerical value for the
quantities Bri and Ω in the metric by demanding that the bulk Riemann tensor be the AdS
solution at zeroeth order. Since for AdS space the Riemann tensor is given by Eq. (2.48), we
can write the components of the bulk Riemann tensor on the L.H.S of Eq. (2.63) in terms of
the components of the bulk metric. Expanding the metric using Eq. (2.53) and keeping only
46 2.3. TEST OF THE ENTROPY FUNCTIONAL FOR R2 THEORY
the zeroeth-order terms in z, z we get
Ω = − 1
12C and Bri = 0 (2.64)
Therefore, Bri can be set to zero. In writing the divergences, we also ignore4 Qzzij and Qzz
ij ,
the remaining component being Qij = Qzzij = Qzz
ij .
For R2 theory the derivative order of the equation of motion is four. That means we
should include order z3 terms in the metric, since they can contribute to the divergences.
These terms can be parametrized as
ds2 = e4ρ(z,z)∆pqrstxpxqxrdxsdxt +Wrspijx
rxsxpdyidyj + 2e2ρ(z,z)Crsixrxs(zdz − zdz)dyi .
(2.65)
This is the most general form of the third-order terms in the metric. Here, we have written
the e2ρ(z,z) dependence of each term explicitly. As for the second-order quantities, the third-
order quantities ∆pqrst,Wrspij and Crsi can be found by calculating the curvature tensors, but
to linear order in z(z). Then, for example, e4ρ(z,z)∆pqrst ≡ −1/6∂p(Rµνρσnµqn
νrn
ρsn
σt ) evaluated
at z = z = 0. Note that the factor of e4ρ(z,z) will cancel from both sides on using the AdS
background. In fact, this particular term vanishes altogether in this background. On using
the metric with the third-order terms listed above to find the divergences in the equation
of motion we find that the Crsi, Wzzzij and Wzzzij do not contribute. The terms that are
relevant are Wzzzij and Wzzzij, because as will show below they will lead to unsuppressed
CK terms. Without loss of generality, we can set them to be equal and denote this term as
Wij.
4As we saw for the Gauss-Bonnet case, these terms will be present in the divergences, but they will not
change our conclusions for R2 theory.
CHAPTER 2. HOLOGRAPHIC ENTANGLEMENT ENTROPY FUNCTIONALS: ADERIVATION 47
For general R2 theory, the divergence in the zz component from the Hαβ term in the bulk
eom is
Hzz =ε
z
[− 1
2(λ2 + 4λ3)∇2K + (2λ1 + λ2 + 2λ3)∇i∇jKij + λ3(RK− 2KijRij) +
4(−2λ1 + 3λ2 + 14λ3)KijAiAj − 6(λ2 + 4λ3)KAiAi +
8(3λ1 + 2λ2 + 5λ3)KΩ]−
ε
z2
[e−2ρ(z,z)
(2λ1 − λ2 − 6λ3)K2 + 1
2(λ2 + 4λ3)K2 + 2(λ2 + 4λ3)Q
]+
ε
z
[e−2ρ(z,z)
− λ3K3 + (λ2 + 7λ3)KK2 − 2(3λ1 + 2λ2 + 6λ3)K3 +
(6λ1 + 5λ2 + 14λ3)KijQij − 32(λ2 + 4λ3)KQ −
4(λ2 + 4λ3)W]
. (2.66)
The divergences in the other components are
Hzz =2ε
z
[e−2ρ(z,z)
(λ3 + 1
4λ2)K3 + (λ1 − 3
2λ2 − 7λ3)KK2 +
2(−2λ1 + λ2 + 6λ3)K3 + (2λ1 − 3λ2 − 14λ3)KklQkl +
32(λ2 + 4λ3)KQ+ 8(λ2 + 4λ3)W
], (2.67)
Hzi =2ε
z
[e−2ρ(z,z)
− 1
2(2λ1 + λ2)Kki∇kK − (3λ1 − λ2 − 6λ3)Kkl∇iKkl −
14(3λ2 + 8λ3)K∇iK + (5λ1 + λ2)Kki∇lKlk − λ1K∇kKki +
(9λ1 + 2λ2)Kkj∇kKji − (λ2 + 4λ3)∇iQ− (4λ1 + λ2)∇kQki −(10λ1 − 2λ2 − 18λ3)AiK2 − 1
2(3λ2 + 12λ3)AiK2 +
8(4λ1 + λ2)KijKjkAk − 2(λ2 + 4λ3)AiQ]
, (2.68)
Hij =4ε
z
[e−4ρ(z,z)
(1
3λ1 + 1
4λ2 + 2
3λ3)hijK3 − (7λ1 + 2λ2 + 2λ3)KKikKkj +
2(16λ1 + 4λ2 + λ3)KikKklKlj − (λ1 + 3λ2 + 10λ3)hijKK2 −(3λ1 − 2λ3)KijK2 − 1
3(λ1 − 18λ2 − 70λ3)hijK3 +
2(4λ1 + λ2)QijK + 2(λ2 + 4λ3)hijKQ− 8(4λ1 + λ2)KikQkj −(λ2 + 4λ3)KijQ− 7(λ2 + 4λ3)hijKklQkl + 32(4λ1 + λ2)Wij+
32(λ2 + 4λ3)hijW]
. (2.69)
Whether or not the divergences in the ij, zi and zz components vanish before taking the
ε→ 0 limit will depend upon the exact values of the second-order terms. The zi divergence,
48 2.3. TEST OF THE ENTROPY FUNCTIONAL FOR R2 THEORY
in particular, should be equivalent to the constraint equation coming from the tangential
variations and should vanish by the Codazzi relation in Eq. (2.29). As in the Gauss-Bonnet
case, the divergence in the ij component is not expected to fully vanish by itself. We therefore
take the limit as prescribed in the last section. This reduces the divergences in the zi and ij
components to zero. However, because of the presence of the W term there still remains an
unsuppressed divergence in the zz component. This divergence can only go to zero if K = 0
or the theory is at the Gauss-Bonnet point.
We now examine the divergences in the zz component, to be able to compare it with
the surface equation of motion derived using the FPS functional. First looking at the 1/z
divergence in that component, one can see that it contains the unsuppressed terms KijAiAj
and KAiAi which are not present in Eq. (2.51). However, these terms can be eliminated
in favor of other variables. Consider the Rzizj component of the Riemann tensor expanded
around z = 0, z = 0:
Rzizj
∣∣∣(z=0,z=0)
= 12e2ρ(z,z)Fij − 2e2ρ(z,z)AiAj + 1
4KzikKkzj − 1
2Qzzij . (2.70)
Using Eq. (2.48) again and multiplying both sides by Kij, we find that the AiAjKij term
can be written as ∼ CK + e−2ρ(z,z)K3 + e−2ρ(z,z)QK. The AiAiK terms can be written in a
similar fashion. Since only the CK term is unsuppressed we find
AiAjKij =CK4
+ · · · and
AiAiK =3CK
4+ · · · , (2.71)
where the dots denote the suppressed terms.
Next looking at the e−2ρ(z,z)/z divergence we find that the W term will contribute to
the surface equation of motion, since this term contains a e2ρ(z,z) factor that enhances the
divergence to 1/z. This term can be determined by using the following equation
∂zRzz
∣∣∣(z=0,z=0)
= −W + 2e2ρ(z,z)KijAiAj − 2e2ρ(z,z)ΩK + · · · . (2.72)
The R.H.S of this equation disappears in the AdS background. Using Eqs. (2.64) and (2.71)
we find
W =2e2ρ(z,z)CK
3+ · · · . (2.73)
The Q terms that are also present in this divergence do not contribute since as we show
below they are expected to contain only ∼ K2 terms and therefore remain suppressed.
Substituting these values in Eq. (2.66), and adding the Einstein term we find that the
1/z divergence of the zz component gives rise to the following surface equation of motion:
K + L2(2λ1 + 12λ2)∇2K + λ3(RK− 2KijRij) + λ1C1K + λ2C2K + λ3C3K = 0 (2.74)
CHAPTER 2. HOLOGRAPHIC ENTANGLEMENT ENTROPY FUNCTIONALS: ADERIVATION 49
where C1 = −4C, C2 = −11C/2 and C3 = −18C. The coefficients of the ∇2K,RK, KijRij
and CK terms in the above equation all match with those in Eq. (2.51). Because of the way
we are taking limits, the K3 terms that are present in Eq. (2.51) are not present here.
Finally we look at the ε/z2 divergence in the zz component. For this divergence to vanish,
we get the condition
(2λ1 − λ2 − 6λ3)K2 + 12(λ2 + 4λ3)K2 + 2(λ2 + 4λ3)Q = 0 . (2.75)
To satisfy this condition at arbitrary points of the parameter space, one has to demand thatQbe a function of ∼ K2 terms, and also λ1, λ2 and λ3. Demanding that Q be independent of λ1,
λ2 and λ3, will pick out a special point in the parameter space (apart from the Gauss-Bonnet
point where this condition is trivially satisfied).
To summarize the results for R2 theory:
1. Apart from the absence of ∼ K3 terms, Eq. (2.74) that we found using the LM method
is exactly the surface equation of motion that results from the FPS functional.
2. There are some problematic extra divergences. The zz component of the bulk equation
of motion has a divergence that can only disappear at the Gauss-Bonnet point. There
is also a second-order 1/z2 divergence in the zz component. This can be taken to fix the
value of the term Q; however, it is not possible to do this in a way that is independent
of the parameters of R2 theory.
2.3.3 The stress-energy tensor from the brane interpretation
In [24], it was noted that a equation of motion of a cosmic string is the same as the equation
for the minimal entangling surface. This is because a cosmic string produces a spacetime
with a conical defect with a metric of the form in Eq. (2.53). The equation of motion is
given by minimizing its action. For Einstein gravity this is just the Nambu-Goto action and
equation of motion of a cosmic string is
K = 0. (2.76)
This condition minimizes the surface area of the string as it sweeps through spacetime. The
same thing holds for a cosmic brane.
As was done in [30], where it was referred to as the cosmic brane method, this fact
can be exploited to construct the entropy functional from the bulk equation of motion. In
this section, we will check this construction of [30]. The idea is that the bulk equation of
motion in Eq. (2.54) should lead to the cosmic brane as a solution, to linear order in ε. In
50 2.3. TEST OF THE ENTROPY FUNCTIONAL FOR R2 THEORY
particular, this means that L.H.S of Eq. (2.54) should be equal to the stress-energy tensor of
the brane. Since the brane is a localized source, the stress-energy tensor will contain delta
functions. Once we have found the stress-energy tensor we can identify the associated action
via Tαβ = δSδgαβ
.
Let us see how this works in the Gauss-Bonnet case. In the bulk equation of motion,
terms such as ∂z∂zρ(z, z) correspond to delta functions. We set δ(z, z) = e−2ρ(z,z)∂z∂zρ(z, z).
Note that δ(z, z) defined this way contains a factor of ε.
The delta divergences in the ij component of the bulk equation of motion to linear order
in ε are then:
Tij = δ(z, z)− 4λ (hijR− 2Rij)+
− 2λ e−2ρ(z,z)(hijK2 − hijK2 + 2KijK − 2KikKkj ). (2.77)
To identify this as the stress-energy tensor coming from the Jacobson-Myers functional (in-
terpreted as a cosmic brane action), the second term should go to zero. This term carries
a factor of e−2ρ(z,z) as compared to the first term and according to our way of taking limits
is suppressed. Our result is then in agreement with the claim in [30] that the cosmic-brane
method can be used to show that the Jacobson-Myers functional is the right entropy func-
tional for Gauss-Bonnet theory. However, as we will see there are problems for the general
four-derivative theory.
For R2 theory, the delta divergences in the ij component are
CHAPTER 2. HOLOGRAPHIC ENTANGLEMENT ENTROPY FUNCTIONALS: ADERIVATION 51
Tij = δ(z, z)[− 4λ3 (hijR− 2Rij)− 16(6λ1 + 11λ2 + 38λ3)hijΩ +
e−2ρ(z,z)− (λ2 + 2λ3)hijK2 + 2(λ2 + 3λ3)hijK2 −
2(12λ1 + 4λ2 + 4λ3)Qij − 2(λ2 + 4λ3)Qhij −2(4λ1 + λ2 + 2λ3)KijK + 2(14λ1 + 4λ2 + 4λ3)KkjKkj ) +
16(20λ1 + 11λ2 + 24λ3)AiAj ]
+
e−2ρ(z,z)∂zδ(z, z) + ∂zδ(z, z)
− 2(2λ1 + λ2 + 2λ3)Kij + (4λ3 + λ2)hijK
−
4e−2ρ(z,z) ∂z∂zδ(z, z)(λ2 + 4λ3) . (2.78)
Again, barring the term suppressed by e−2ρ(z,z), we have checked that the result for this
component is of the same form as that produced on calculating the stress-energy tensor from
an action equivalent to the FPS functional. The derivative of delta terms like ∂zδ(z, z) are
typical in the stress-energy tensor of actions containing terms that depend on the extrinsic
curvature [47]. However, the zz and zz components of the bulk equation of motion also
contain delta divergences that are not suppressed:
Tzz = − 4∂2zδ(z, z)(2λ1 + λ2 + 2λ3)− 2∂zδ(z, z)(4λ1 + λ2)K (2.79)
and
Tzz = − 2∂zδ(z, z) + ∂zδ(z, z)
(2λ1 + λ2 + 2λ3
)K +
4 ∂z∂zδ(z, z)(2λ1 + λ2 + 2λ3) . (2.80)
Taking the delta divergences in all components into account, the Tµν we have found does not
look like the stress-energy tensor for a cosmic brane corresponding to a three-dimensional
surface in the five-dimensional bulk. Note that the extra divergences all vanish for the Gauss-
Bonnet theory. The Gauss-Bonnet result therefore stands. However, any attempt to use this
method to show that the FPS functional is the correct entropy functional for R2 theory
should be able to account for these extra delta divergences.
2.4 Quasi-topological gravity
The lagrangian for quasi-topological gravity [38] contains terms cubic in the Riemann tensor.
It can be used to study a class of CFT’s involving three parameters in four dimensions. It
has many interesting features including the fact that its linearized equation of motion is
two-derivative order. Unitarity for this theory was studied in [51].
52 2.4. QUASI-TOPOLOGICAL GRAVITY
In Sec. (2.4.1), we find the HEE functional for quasi-topological theory using Eq. (2.6)
and compute the universal terms is Sec. (2.4.2). In Sec. (2.4.3), we find the surface equation
of motion for this theory using the LM method.
2.4.1 The entropy functional
The action for quasi-topological theory in five dimensions is
SQT = − 1
2`3P
∫d5x(L1 + L2 + ν Z5
), (2.81)
where L1 is the Einstein-Hilbert action given in Eq. (3.9) and L2 is the Gauss-Bonnet la-
grangian as in Eq. (3.10) with λ1 = λ3 = λ , λ2 = −4λ. The last term is the R3 lagrangian:
Z5 =µ0RαβγδRγδ
µνRµναβ + µ1Rα
βγδRβ
ηδζRα
ηγζ + µ2RαβγδR
αβγδR +
µ3RαβγδRαβγ
ηRδη + µ4RαβγδR
αγRβδ + µ5RαβRβ
γRγα + µ6R
βα R
αβ R + µ7R
3 . (2.82)
There are two different consistent R3 theories. For the first theory
µ0 = 0 , µ1 = 1 , µ2 = 38, µ3 = −9
7, µ4 = 15
7, µ5 = 18
7, µ6 = −33
14, µ7 = 15
56(2.83)
and the coupling constant is ν = 7µL4
4, while for the second theory
µ0 = 1 , µ1 = 0 , µ2 = 32, µ3 = −60
7, µ4 = 72
7, µ5 = 64
7, µ6 = −54
14, µ7 = 11
14(2.84)
and the coupling constant ν = 7µL4
8.
The R3 part for the HEE functional is
SEE,R3 =2πν
`3P
∫d3x√h (LWald,R3 + LAnomaly,R3) , (2.85)
where
LWald,R3 = 6µ0RzzαβRzzαβ + 3µ1
(Rzαz
βRzαzβ −Rzαz
βRzαzβ)
+ µ2
(RαβρσR
αβρσ −4RRzz
zz
)+ 2µ3
(Rα
zzzRα
z −RαzzzRα
z + 12Rαβρ
zRαβρz
)+ µ4(2Rz
αzβRαβ +
(Rzz)
2 −RzzRzz) + 3µ5RzαRzα + µ6
(RαβR
αβ + 2RRzz
)+ 3µ7R
2 . (2.86)
The symbols z and z in the above expression label the two orthogonal directions while the
indices α, β, ... are the usual bulk indices. The expression for the anomaly part is
LAnomaly,R3 =µ0(12K2ijQij − 6K4)− µ1(3
2K4 − 3
2K2
2 + 3KijKklRikjl)−µ2(6K2
2 − 2K2K2 − 8K2Q+ 4K2R)−µ3(2K4 + 1
2K2
2 −K2Q− 2K2ijQij − 2KijQij K + 2K2
ijRij)−µ4(2K3K −K2K2 − 2KijQij K + 2KijRij K)−µ5(3
4K2K2 − 3
2K2Q)− µ6(3
2K2K2 − 1
2K4 − 2K2Q+K2R) . (2.87)
CHAPTER 2. HOLOGRAPHIC ENTANGLEMENT ENTROPY FUNCTIONALS: ADERIVATION 53
where K4 = KijKjlKlkKki . In calculating the anomaly part from Eq. (2.6), we have used the
value of Bi = 0 that we found in Sec. (2.3.2). This is the reason that while terms involving
Bi are supposed to contribute to Eq. (2.6), the above equation does not contain any terms
containing Bi. The full HEE functional has contributions from the Einstein and R2 part also
which are given in Eqs.(2.12) and (2.13).
2.4.2 Universal terms
In this section, we will demonstrate that our HEE functional for the quasi-topological gravity
produces the correct universal terms. For the general structure and calculation of the uni-
versal term of the entanglement entropy in four dimensions, see [52]. These central charges
can be easily calculated using the technique of [53].
We follow the procedure given in [34] for R2 theory. Here we sketch the main steps of this
calculation. We will minimize Eq. (2.85) for a bulk surface with a spherical and cylindrical
boundary. We will carry out this procedure for the five-dimensional bulk AdS metric as
shown in the section (2.3).
Following the analysis of section (2.3) we set ρ = f(z), τ = 0 in the metric in Eq. (2.14)
and minimize the entanglement entropy functional (whose R3 part is given in Eq. (2.85)) on
this codimension 2 surface to find the Euler-Lagrange equation for f(z). Using the solution
for f(z) we evaluate the entropy functional to get the EE.
For the spherical boundary, we get f(z) =√f 2
0 − z2 which gives the EE as
SEE = −4a ln(f0
δ) . (2.88)
Here, δ is the UV cut-off that comes from the lower limit of the z integral and f0 is the radius
of the entangling surface. The value of a is
a =π2L3
f3/2∞ `3
P
(1− 6f∞λ+ 9f 2∞µ) . (2.89)
For this case, the entire contribution comes from the Wald entropy as the extrinsic curvatures
are identically zero.
For the cylindrical boundary, we find f(z) = f0 − z2
4f0+ ... leading to
SEE = −cH2R
ln(f0
δ) . (2.90)
The value of c corresponding to the theory in Eq. (2.83) is
c =π2L3
f3/2∞ `3
P
1− 2f∞λ+ 9f 2
∞µ+ f 2∞µ(42µ1 − 336µ2 − 56µ3)
, (2.91)
54 2.4. QUASI-TOPOLOGICAL GRAVITY
while that corresponding to the theory in Eq. (2.84) is
c =π2L3
f3/2∞ `3
P
1− 2f∞λ+ 9f 2
∞µ− f 2∞µ(168µ2 + 28µ3)
. (2.92)
The 1 + 9f 2∞µ part is the usual Wald entropy contribution, while the remaining part comes
from the anomaly part. After putting in the values of µ’s given in Eqs.(2.83) and (2.84) we
obtain
c =π2L3
f3/2∞ `3
P
(1− 2f∞λ− 3f 2∞µ) (2.93)
for both theories.
These results for the universal terms agree with those calculated in [34] for the two quasi-
topological theories. Note from Eqs. (2.91) and (2.92) that only a few terms from LAnomaly,R3
have contributed to the universal term. Terms of the form ∼ K4 do not contribute to this
calculation at all. Since Q ∼ K2, terms of the form ∼ K2Q also do not contribute.
2.4.3 Minimal surface condition
We now find the surface equation of motion for quasi-topological gravity using the LM
method. For ease of calculation, we set all second-order quantities and cross-components
in the metric in Eq. (2.53) to zero. The bulk equation of motion for this theory is [49]:
Rαβ −1
2gαβR−
6
L2gαβ −
L2
2Hαβ − νFαβ = 0 , (2.94)
where Fαβ is defined in [38, 49]. The εz
divergence in the zz component of the equation of
motion coming from the Fαβ term is
F 1zz =
ε
z
[(3
2µ1 − µ2 − µ3 − 3
2µ5 − 4µ6 − 12µ7)Rij∇2Kij − (1
2µ2 + µ6 + 6µ7)RijKijR +
(µ2 + 16µ6 + 3µ7)RijRijK + 1
2(µ6 + 1
2µ4)K∇2R+ (µ4 + µ3 + 4µ2)∇i∇iK −
(3µ1 − 8µ− 2− 3µ3 − µ4 + 32µ5)∇lRlijk∇kKij − 1
2(µ4 + 3µ1)Kkl∇i∇jRkijl −
(34µ1 − 5
2µ2 − µ3 − 1
2µ4 − 3
4µ5 − 2µ6 − 6µ7)R∇i∇jKij +
14(µ4 + 2µ3 + 8µ2)Kij∇i∇jR
]. (2.95)
While we haven’t computed the surface equation of motion that one gets on minimizing the
functional in Eq. (2.85), this is not very hard to do using the methods of Sec. (2.3.1) and
Mathematica5. The main point is, however, that the surface equation of motion that one
5We have used the Xact package for a number of calculations in this chapter
CHAPTER 2. HOLOGRAPHIC ENTANGLEMENT ENTROPY FUNCTIONALS: ADERIVATION 55
will get from the entropy functional will contain K4 terms that are absent in the above
divergence.
Other divergences are also present in the zz component:
F 2zz =
ε
z2
[e−2ρ(z,z)
12(3µ1 + 19µ2 + 2µ3 + 14
3µ6)RijklKikKjl − (7
2µ6 + 18µ7)RK2 +
(2µ2 + 32µ6 + 6µ7)RK2 + (µ4 − 3
2µ5)K∇2K −
(43µ2 − µ4 + 3
2µ5 + 4
3µ6)Kij∇i∇jK − (µ4 − 3
2µ5)K∇i∇jKij +
(3µ1 − 23µ2 − µ3 + µ4 − 3
2µ5 − 2
3µ6)∇kKij∇kKij −
(3µ1 + 23µ2 + µ3 + 3µ4 − 3
2µ5 + 2
3µ6)∇iKij∇kKjk −
(8µ2 − µ4 + 32µ5)RijKijK + (3µ1 + 8µ2 − µ4 − 2µ6)RijKjkKik
]+
ε
z3
[e−4ρ(z,z)
(3µ1 − 2µ2 − 2µ3)K3 + (µ4 − 3µ5 − 2µ6)KK2
]. (2.96)
As for R2 theory, these divergences can be used to determine second and higher-order terms
in the metric. At linear-order in the metric, divergences in all other components of the
equation of motion go to zero if we take the limit as mentioned in Sec. (2.3.2).
2.5 Discussion
We found the surface equation of motion for general R2 theory and quasi-topological gravity
using the generalized gravitational entropy method of [24]. We found that these do not match
exactly with what can be derived by extremizing the HEE functional for these theories – the
HEE functional being calculated using the formula proposed in [30, 31].
Let us summarize our findings regarding R2 theory. First, the leading-order terms on both
sides do match. In fact, barring ∼ K3 terms, the surface equation of motion that follows
from the LM method is precisely the surface equation of motion that follows from the FPS
functional.
The main problem with the LM method is that there are divergences in components other
than the zz component, for a general higher-derivative theory. In the Gauss-Bonnet case,
there are ways we can take the limit to set these divergences to zero. However, the effect of
taking the limit in this way is to remove all ∼ K3 divergences from all components of the
equation of motion. This means that we do not get any ∼ K3 terms in the surface equation
of motion using the LM method. No matter what the HEE functional for R2 theory is, it
is unlikely that no ∼ K3 terms will occur in its surface equation of motion at any point
in its parameter space. Even after taking the limit as prescribed, for general R2 theory,
there remain extra divergences in the bulk equation of motion. It is impossible to set these
56 2.5. DISCUSSION
divergences to zero at all points of the parameter space, although this can be done for specific
points like the Gauss-Bonnet point.
As we discussed, the absence of ∼ K3 terms is the R2 equation of motion is an artifact
of the way limits have to be taken in the LM method for the Gauss-Bonnet case. The limit
can also be taken in such a way so as to preserve ∼ K3 terms. It is worth recapitulating the
results this way of taking the limit gives for Gauss-Bonnet theory. As we showed, using the
second-order conical metric, the bulk equation of motion for Gauss-Bonnet theory, before
taking the limit, has divergences only in the zz and ij components. There is no divergence
in the zz component, while the divergence in the zi component turns out to be a constraint
equation that vanishes by itself on using the Codazzi-Mainardi relation on AdS space. This
same constraint equation results from the Jacobson-Myers functional, as well, on taking
tangential variations of the surface. It is not clear what the relevance of the divergence in the
ij component is in the LM method. Were we to ignore this divergence, the surface equation
of motion that would result from the zz component for Gauss-Bonnet theory, after taking
the limit, is cK = 0, where c is proportional to the Weyl anomaly. This equation is clearly
not what comes from the Jacobson-Myers functional. However, the resulting minimal surface
is what one obtains on extremizing just the Wald entropy part of the functional. It would
be interesting to check whether the zz component of R2 theory also leads to the same result.
One of the pending issues with the LM method is to fix the ambiguity present in the limit-
taking procedure. However, fixing this by itself does not seem enough to simultaneously cure
the two problems present for R2 theory: the absence of ∼ K3 terms and the presence of extra
constraints; although, it can remove one of these problems from the list. The ambiguity in
the limit-taking procedure is not unique to the LM method. Similar, though not exactly the
same, issues occur in studies of co-dimension two branes in the context of brane-world gravity
[54]. It is possible that a further modification to the LM method will fix these problems;
on the contrary, it may be that one cannot get rid of it in any way. The problem of extra
divergences is related to the derivative order of the bulk equation of motion and seems to
spring from the pathology of the general R2 theory itself. In this sense, it is not surprising
that we encounter it for general higher-derivative theories. Higher-derivative theories are
known to suffer from problems regarding unitarity [55, 56, 57]. These problems seem to be
manifested in the LM method in the inability to remove all divergences, that occur on using
the conical metric, from the bulk equation of motion.
What does our analysis say about the validity of the formula proposed in [31, 30] as the
entanglement entropy functional? For general R2 theory as we demonstrated the leading-
order terms match on both sides, which stops short of being a validation of the proposal
for this theory. This test, at present, is similar in scope in refining conjectured entropy
functionals for higher-derivative theories as the test whether the entropy functional leads to
CHAPTER 2. HOLOGRAPHIC ENTANGLEMENT ENTROPY FUNCTIONALS: ADERIVATION 57
the correct universal terms. Also we showed, for quasi-topological theory the universal terms
are not sensitive to terms of the form ∼ K4 in the entropy functional (similar statement
applies for other higher-derivative theories) and one can change these terms and still have
the universal terms come out to be correct.
The LM method, therefore, in its current form has limitations that make it ineffective in
testing proposed entropy functionals for generic higher-derivative gravity theories. The fact
that the LM method only works for specific theories may indicate one of two things. One
possibility is that entropy functionals only exist for specific theories such as Lovelock theories,
for which the result of the surface equation of motion from the existing entropy functional
and the LM method coincide. The other possibility, as mentioned before, is that the LM
method needs some modification. In this context, it is also desirable that alternate methods
to test entropy functionals be developed. Our argument can be extended for other Lovelock
theories in general dimensions and can be shown that extremization condition can be derived
consistently. But for for general theories it fails. In short LM method in its current form
only picks out some particular theories of gravity.
After our work [44], it was shown in the [58], that one can possibly modify the LM
method by breaking replica symmetry inside the bulk spacetime, but keeping the symmetry
preserved at the boundary. This generates new terms in the conical metric and enhances the
Zn symmetry group inside the bulk. Using this proposal one can show that, it is possible
to cancel the offending K3 terms for the Gauss-Bonnet theory without taking the limits as
shown in this chapter. This methods works only when one treats the Gauss-Bonnet coupling
perturbatively. But still it remains to be investigated, whether it is possible to cure the
problem for general four-derivative theories.
Appendix
A: Conical Metric
Near the conical singularity the bulk metric can also be written as
ds2 = ρ(x, y)−2ε(dx2 + dy2) + ρ(x, y)−2εapj dxpdxj + gij dx
idxj . (2.97)
The two-dimensional part is written in cartesian coordinates x and y and ρ(x, y) =√x2 + y2.
We have written the metric upto terms first order in x(y). The co-dimension two surface (Σ)
is located at x = 0 and y = 0. The metric gij can be written down order by order in x(y)
after expanding around the surface Σ as
gij = hij + x ∂xgij∣∣Σ
+ y ∂ygij∣∣Σ
+ · · · . (2.98)
58 2.5. DISCUSSION
The surface tensor hij is independent of x and y. The variable apj ∼ O(x).
The extrinsic curvatures for the co-dimension two surface (Σ) are defined as
Ksij = eβj∇insβ∣∣Σ
= eβj (∂insβ − Γδαβeαi nsδ)
∣∣Σ. (2.99)
Expanding the Christoffel in terms of the metric and using the fact that the first term eβj ∂insβ
vanishes it follows that
Kxij =1
2∂x gij
∣∣Σ, Kyij =
1
2∂ygij
∣∣Σ. (2.100)
We now make the simplifying assumption that the metric gij is independent of the co-ordinate
y. Under this assumption, the extrinsic curvature Kyij vanishes as ∂ygij vanishes.
The complex coordinates z and z used in the metric in Eq. (2.14) are related to x and y
as
z = x+ iy, z = x− iy . (2.101)
In these coordinates the metric is
ds2 = e2ρ(z,z)(dzdz) + gijdxidxj + 2e2ρ(z,z)Ai(zdz − zdz)dyi , (2.102)
where
gij = hij + zKzij + zKzij + · · · . (2.103)
The extrinsic curvatures in this coordinate system are related to Kxij and Kyij as
Kzij =Kxij + iKyij
2, Kzij =
Kxij − iKyij2
. (2.104)
Since Kyij = 0 we have
Kzij = Kzij . (2.105)
Similar considerations apply to the second-order quantities Q.
B. Curvature Tensors
In this appendix, we list components of the curvature tensors for the metric in Eq. (2.53),
that do not appear in the main text. We retain only terms uptil zeroeth-order in z, z.
The components of the Christoffels are
Γzzz = − εz, Γz zz = − ε
z, Γzij = −e−2ρ(z,z) Kzij , Γz ij = −e−2ρ(z,z) Kzij ,
Γizj =1
2Kizj , Γizj =
1
2Kizj , Γijk =
1
2gil(∂jglk + ∂kglj − ∂lgjk) ,
Γzzi = −2Ai , Γzzi = 2Ai . (2.106)
REFERENCES 59
The components of the Riemann tensor are
Rpqij = 2e2ρ(z,z)εpqFij + (KpjkKkqi −KpikKkqj) ,Rzizj = 1
4KzjkKkzi −Qzzij − ε
2zKzij ,
Rzizj = 12e2ρ(z,z)Fij − 2e2ρ(z,z)AiAj + 1
4KzjkKkzi − 1
2Qzzij ,
Rpijk = 12(∇kKpij −∇jKpik) ,
Rikjl = Rikjl + 12e−2ρ(z,z)(KzilKzjk +KzilKzjk −KzijKzkl −KzijKzkl) , (2.107)
where Fij ≡ ∂iAj − ∂jAi.The components of the Ricci tensor are
Rzi = 12(∇jKzji −∇iKz) ,
Rzz = 14KzijKijz − 1
2Qzz − ε
2zKz ,
Rzz = 14Kzij Kijz − 1
2Qzz − 2e2ρ(z,z)(AiAi − 3Ω) ,
Rij = e−2ρ(z,z)(KkzjKzik +KkzjKzik − 1
2KzijKz − 1
2KzijKz − 2Qzzij
)+Rij − 8AiAj . (2.108)
As in the main text, ∇ used in the above equations is the Van der Waerden-Bortolotti
covariant derivative [45] defined in Eq. (2.24).
The Ricci scalar is
R = R+ 24Ω− 16AiAi − e−2ρ(z,z)(KzKz − 3KzijKijz + 4Qzz
). (2.109)
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[51] T. C. Sisman, I. Gullu and B. Tekin, “All unitary cubic curvature gravities in D di-
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S. N. Solodukhin, “Entanglement entropy, conformal invariance and extrinsic geome-
try,” Phys. Lett. B 665, 305 (2008) [arXiv:0802.3117 [hep-th]].
[53] S. ’i. Nojiri and S. D. Odintsov, “On the conformal anomaly from higher derivative
gravity in AdS / CFT correspondence,” Int. J. Mod. Phys. A 15, 413 (2000) [hep-
th/9903033].
[54] For representative papers see: P. Bostock, R. Gregory, I. Navarro and J. Santiago,
“Einstein gravity on the codimension 2-brane?,” Phys. Rev. Lett. 92, 221601 (2004)
[hep-th/0311074].
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sion,” JHEP 0508, 075 (2005) [hep-th/0502170].
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Phys. Rev. D 41 (1990) 3720.
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3 Entanglement entropy from
generalized entropy
3.1 Introduction
In the Chapter.(2) we have shown that the identification of the entanglement entropy with the
generalized gravitational entropy has opened up the avenue for systematically generalizing
holographic entanglement entropy (EE) for more general bulk theories of gravity other than
Einstein gravity. This understanding is crucial in order to understand systematics of how
finite coupling effects in the field theory modify entanglement entropy. In this chapter we will
only focus on those particular theories which arise from “classical” and local higher derivative
corrections to the bulk theory1. Now our main objective will be to compute the universal
term for EE for various higher derivative gravity theories using generalized entropy method.
Rest of this chapter is mainly based on the work done with Prof. Aninda Sinha and Dr.
Menika Sharma [2].
3.2 Generalized entropy and Fefferman-Graham expan-
sion
Let us start by recapitulating generalized entropy method one more time. We will see fol-
lowing [3], the generalized gravitational entropy is defined as,
S = −n∂n(ln[Z(n)]− n ln[Z(1)])n=1 , (3.1)
where ln[Z(1)] is identified with the Euclidean gravitational action for which the period of
the Euclidean time is 2π and the boundary condition for other fields collectively denoted as φ
present in the action is φ(0) = φ(2π) . ln[Z(n)] is identified with the Euclidean gravitational
action In for which the period of the Euclidean time is 2πn and the boundary condition for φ
is still φ(0) = φ(2π) . This is the usual replica trick. Translating this fact for the holographic
1there are other type of corrections which arise from “quantum” or loop corrections to the effective action
which would include non-local effects. Interested readers are referred to [1].
69
70 3.2. GENERALIZED ENTROPY AND FEFFERMAN-GRAHAM EXPANSION
case we can define In for a regularized geometry on a cone whose opening angle is 2π/n. We
can analytically continue this for non integer n and then can compute the entropy. Also while
evaluating ln[Z(n)] we can perform the time τ integral from 0 to 2π and multiply it by n so
that ln[Z(n)] = n ln[Z]2π . The entropy calculated using this method is equal to the area of
some codimension 2 surface where the time circle shrinks to zero which can be shown to be
the minimal surface in Einstein gravity [3]. In this section we will show that this procedure
also gives the correct entanglement entropy for higher curvature gravity theories. To compute
the EE we have to start with some specific boundary geometry for the nth solution. Then we
can construct our bulk spacetime using the Fefferman-Graham expansion. We will consider
the following two 4-dimensional metrics following [4],
ds2cylinder = f(r, b)dr2 + r2dτ 2 + (f0 + rnd1−n cos(τ))2dφ2 + dz2
ds2sphere = f(r, b)dr2 + r2dτ 2 + (f0 + rnd1−n cos(τ))2(dθ2 + sin2 θdφ2)
(3.2)
where, f(r, b) = r2+b2n2
r2+b2. For b→ 0 and n→ 1 limit these two metrics reduce to the cylinder
and the sphere. This metric is known as “squashed cone” metric. The key point in Eq. (3.2)
as compared to earlier regularizations e.g., [3] is the introduction of a regulator in the extrinsic
curvature terms. This is needed since otherwise the Ricci scalar would go like (n− 1)/r and
would be singular. Another important point is that b is a regulator which at this stage does
not have an restriction except that f(0, b) = n2. In AdS/CFT we do not expect an arbitrary
parameter to appear in the metric. b is here a dimensionful quantity having the dimension
of r . So b must be proportional to f0(n− 1)α>0 such that it goes to zero as n→ 1. We can
take the metrics in Eq. (3.2) as boundary metrics and construct the bulk spacetime using the
Fefferman-Graham expansion. Notice that our starting point is a smooth metric. At the end
of the calculation, when we remove the regulators and compute EE, we will separately check
what the contribution from the singularities is going to be. In the best case scenario, although
the boundary metric will be singular once the regulator is removed, the bulk metric will at
most be mildly singular, namely the on-shell bulk action will not be singular, following the
terminology used in [3]. As in [3] we could have done a conformal transformation to pull out
a factor of r2 such that the r, τ part of the metric looks like dτ 2 + dr2
r2which would make the
time-circle non-shrinking. We can use this form of the metric with a suitable regularization
and do the calculation after verifying that there are no singularities in the bulk. Since this is
a conformal transformation of a smooth metric, the results for the universal part of the EE
will remain unchanged. One can write the bulk metric as,
ds2 = L2dρ2
4ρ2+
(g(0)ij + ρg
(2)ij + .....)
ρdxidxj . (3.3)
CHAPTER 3. ENTANGLEMENT ENTROPY FROM GENERALIZED ENTROPY 71
To evaluate the log term we will need the g(2)ij coefficient and here we will use Eq. (3.2) as
g(0)ij . We will consider here a 5 dimensional bulk lagrangian. In this case,
g(2)ij = − L
2
2(R
(0)ij −
1
6g
(0)ij R
(0)) ,
where R(0)ij and R(0) are constructed using g
(0)ij . Note that in all subsequent calculations g
(2)ij
will play an important role. The structure of g(2)ij is independent of the form of the higher
derivative terms present in the action. Only terms proportional to n − 1 in the on-shell
bulk action contributes to the SEE. The calculation is similar in spirit to the way that Weyl
anomaly is extracted in AdS/CFT, e.g., [5] except that the n−1 dependence comes from the
neighbourhood of r = 0 in the bulk action. In the next section we proceed to give details of
this.
Regularization procedure
To illustrate the regularization procedure in some detail, we start with some simple examples
involving curvature polynomials 2. We calculate g(2)ij and evaluate the following integral ,
I1 =
∫d5x√g RµνR
µν . (3.4)
Following3 [4], in the integrand, we put r = bx then expand around b = 0 and pick out the
O(b0) term. The r integral is between 0 < r < r0. This makes the upper limit of the x
integral to be r0/b which goes to infinity. We will be interested in the log term so we extract
first the coefficient of 1ρ
term which has the following form,
I1 = b
∫dρ
ρdτ d2y
∫ ∞0
dx (n− 1)2ζ(x, n)(bx)2n−3 +O((n− 1)3) . (3.5)
We have here shown only the leading term. Note that at this stage the integrand is propor-
tional to (n− 1)2 whereas we need get something proportional to (n− 1). The integral over
x will give a factor of 1/(n − 1). We will now expand ζ(x, n) around n = 1 and then carry
out the integral over x. After expanding around n = 1 this leads to
I1 = (n− 1)ζ1 +O(n− 1)2 + · · · . (3.6)
Note that the rn factor in the cylindrical and the spherical parts in Eq. (3.2) were crucial in
reaching this point. ζ1 is just a quantity independent of the regularization parameters b, d,
2We thank Sasha Patrushev for discussions on this topic.3Alternatively we could have done the expansion around x = 0 first, since it was assumed in [4] that the
metric is valid between 0 < r < b f0. Then we could have integrated x in the neighbourhood of x = 0.
The results are identical.
72 3.2. GENERALIZED ENTROPY AND FEFFERMAN-GRAHAM EXPANSION
ε, ε′. The same procedure is applied for other curvature polynomial integrals. For example,
I2 =
∫√g d5xRµνρσR
µνρσ = (n− 1)ζ2 +O(n− 1)2 + · · · ,
I3 =
∫√g d5xR2 = O(n− 1)2 + · · · .
(3.7)
3.2.1 Four derivative theory
Let us now consider the general R2 theory lagrangian as shown below.
L = L1 + L2 , (3.8)
where
L1 = R +12
L2(3.9)
is the usual Einstein-Hilbert lagrangian with the cosmological constant appropriate for five-
dimensional AdS space and
L2 =L2
2
(λ1RαβγδR
αβγδ + λ2RαβRαβ + λ3R
2)
(3.10)
is the R2 lagrangian.
Also we will henceforth consider only a 5 dimensional bulk spacetime unless mentioned
otherwise. The boundary of this spacetime is at ρ = 0 . We then evaluate the total action
and extract the 1ρ
term and carry out the τ integral. We put r = b x and expand Eq. (3.8)
around b = 0 . Then we pick out the O(b0) term.
S = − 1
2`3P
∫dρ
ρdx d2y (n− 1)2a1
(bx)2n
x3+O((n− 1)3) , (3.11)
where
a1 =A(x)
18 b2f5/2∞ f0 (1 + x2)4
. (3.12)
A(x) is a function of x . For the cylinder we get,
A(x) = πL3(f 2∞(λ1
(4x8 + 16x6 + 43x4 + 36x2 + 9
)− 2
(20x8 + 80x6 + 161x4 + 108x2 + 27
)(λ2 + 5λ3)
)+ 6f∞
(5x8 + 20x6 + 38x4 + 24x2 + 6
)− 3
(8x8 + 32x6 + 59x4 + 36x2 + 9
) ).
(3.13)
We then carry out the x integral.
S = − 1
2`3P
∫dρ
ρd2y
A1(x, n)
36 b2 f5/2∞ (n2 − 1) f0 x2
∣∣∣∣∞0
, (3.14)
CHAPTER 3. ENTANGLEMENT ENTROPY FROM GENERALIZED ENTROPY 73
where
A1(x, n) = πL3(n− 1)2(bx)2n
[(n− 1)x4
2F1
(2, n+ 1;n+ 2;−x2
) (f 2∞(5λ1 − 14(λ2 + 5λ3))
+ 6f∞ − 3)
+ 2(n− 1)x42F1
(3, n+ 1;n+ 2;−x2
) (f 2∞(5λ1 − 14(λ2 + 5λ3))
+ 6f∞ − 3)
+ 2F1
(4, n+ 1;n+ 2;−x2
) (4f 2∞λ1x
4(1− n)− 40f 2∞λ2x
4(1− n)
− 200f 2∞λ3x
4(1− n) + 30f∞x4(1− n)− 24x4(1− n)
)− 9f 2
∞λ1(1 + n)
+ 54f 2∞λ2(1 + n) + 270f 2
∞λ3(1 + n)− 36f∞(1 + n) + 27(1 + n)
].
(3.15)
For the cylinder after doing the expansion around n = 1 and the remaining integrals (note
that ρ = z2 in the coordinates used in [6] and so ln δρ = 2 ln δ),
SEE = −cH2R
ln(f0
δ) . (3.16)
Here we have used 1 = f∞− 13f 2∞(λ1 +2λ2 +10λ3) and c is given in Eq. (2.19). For the sphere
we proceed similarly. In this case, expanding Eq. (3.11) around b = 0 we get ,
S = · · · − 1
2`3P
∫dρ
ρdx d2y (n− 1)2a1
(bx)2n
x3+O((n− 1)3) , (3.17)
where
a1 =A(x)
72 b2f5/2∞ f 4
0 (1 + x2)4. (3.18)
A(x) is a function of x . For the sphere we get,
A(x) = −πL3 sin(θ)
[300 b4λ3x
10f 2∞ − 45 b4x10f∞ + 600 b4λ3x
8f 2∞ − 90 b4x8f∞
+ 300 b4λ3x6f 2∞ − 45 b4x6f∞ + 36 b4x10 + 72 b4x8 + 36 b4x6 − 680 b2λ3R
2x10f 2∞
+ 84 b2R2x10f∞ − 1920 b2λ3R2x8f 2
∞ + 216 b2R2x8f∞ − 120 b2λ3R2x6f 2
∞ + 36 b2R2x6f∞
+ 1120 b2λ3R2x4f 2
∞ − 96 b2R2x4f∞ − 60 b2R2x10 − 144 b2R2x8 − 36 b2R2x6 + 48 b2R2x4
+ 2λ1f2∞(− 3b4
(x2 + 1
)2x6 + 2b2R2
(7x6 + 24x4 − 3x2 − 20
)x4 + 4R4
(x8 − 73x6 + 242x4
+ 361x2 + 54))
+ 4λ2f2∞(15b4
(x2 + 1
)2x6 − 2b2R2
(17x6 + 48x4 + 3x2 − 28
)x4
+ 4R4(13x8 − 13x6 + 230x4 + 301x2 + 54
))+ 4320λ3R
4f 2∞ + 1040λ3R
4x8f 2∞ − 192R4x8f∞
− 1040λ3R4x6f 2
∞ − 168R4x6f∞ + 18400λ3R4x4f 2
∞ − 2760R4x4f∞ + 24080λ3R4x2f 2
∞
− 3144R4x2f∞ − 576R4f∞ + 168R4x8 + 264R4x6 + 2208R4x4 + 2328R4x2 + 432R4)
].
(3.19)
74 3.2. GENERALIZED ENTROPY AND FEFFERMAN-GRAHAM EXPANSION
After doing the x integral,
S = − 1
2`3P
∫dρ
ρd2y
A1(x, n)
144 b2 f5/2∞ n (n+ 1) f 4
0 x2
∣∣∣∣∞0
, (3.20)
where A1(x, n) is a function of x and n .
A1(x, n) = πL3(n− 1) sin(θ)(bx)2n
[− 8(n+ 1)R4
(f 2∞(λ1(145x2(n− 1) + 54n) + 2(λ2 + 5λ3)
(85x2(n− 1) + 54n))− 3f∞(n(35x2 + 24)− 35x2) + n(75x2 + 54
)− 75x2
)+ 2F1
(4, n+ 1;n+ 2;−x2
)(−72(n− 1)nR4x4
((λ1 + 2 (λ2 + 5λ3)) f 2
∞ − 3f∞ + 3))
+ 2F1
(3, n+ 1;n+ 2;−x2
)(8(n− 1)nR2x4(f 2
∞(λ1(15b2 + 328R2)− 2 (λ2 + 5λ3)(21b2 − 232R2
)) + 6
(3b2 − 46R2
)f∞ − 9b2 + 192R2)) + 2F1
(1, n+ 1;n+ 2;−x2
)((n− 1)nx4(−36b4 + 60b2R2 + f∞(45b4 − 84b2R2 + 2f∞(λ1(3b4 − 14b2R2 + 580R4)
− 2(λ2 + 5λ3)(15b4 − 34b2R2 − 340R4))− 840R4) + 600R4)) + 2F1
(2, n+ 1;n+ 2;−x2
)(−3(n− 1)nx4(2f 2
∞(λ1(b4 + 2b2R2 − 264R4)− 2(λ2 + 5λ3)(5b4 − 2b2R2 + 168R4))
+ 3(5b4 − 4b2R2 + 136R4)f∞ − 12(b4 − b2R2 + 24R4)))
].
(3.21)
For the sphere after doing the expansion around n = 1 and the remaining integrals ,
SEE = −4 a ln(f0
δ) , (3.22)
where we have used 1 = f∞ − 13f 2∞(λ1 + 2λ2 + 10λ3) and a is given in Eq. (2.19). Thus we
get the expected universal terms using the regularization proposed in [4].
3.2.2 New Massive Gravity
As an example for a calculation of generalized gravitational entropy in other dimensions, we
consider the New Massive Gravity action in three dimensions [7] and use the notation in [8]
S = − 1
2`P
∫d3x√g[R +
2
L2+ 4λL2(RµνR
µν − 3
8R2)].
Here 1− f∞ + f 2∞λ = 0. The entropy functional for this is not intrinsic as compared to the
three dimensional Einstein gravity and is given by
SEE =2π
`P
∫dx√gxx[1 + 4λL2([Rµνn
µsn
νs −
1
2KsKs]−
3
4R)]. (3.23)
CHAPTER 3. ENTANGLEMENT ENTROPY FROM GENERALIZED ENTROPY 75
The integral is over the one dimensional entangling region. s denotes the two transverse
direction. We calculate the generalized gravitational entropy following the same procedure
as used above. The two dimensional squashed cone metric is given by
ds2 = f(r, b)dr2 + r2dτ 2 .
f0 in this case also corresponds to the radius of the entangling surface.
In 3 dimensions [9, 10]
g(2)ij = − L
2
2R(0)g
(0)ij + tij (3.24)
Only divergence and trace of tij are known.
g(0)ij t
ij = R(0) , ∇itij = 0 .
R(0) = − 2b2 (n2 − 1)
(b2n2 + r2)2 . (3.25)
Using 1− f∞ + f 2∞λ = 0 and we get,
S = · · ·+ 1
2`P
∫dρ
ρ
∫ 2π
0
dτ
∫ r=f0
r=0
drL(rb2(n2 − 1)(1 + 2f∞λ)
f1/2∞√b2 + r2(b2n2 + r2)3/2
+ · · · . (3.26)
Note that tij does not enter in the calculation of the universal term. After doing the integrals
we get
S = · · ·+∫dρ
ρ
[πL (1 + 2f∞λ)
`P√f∞
1
n−
√b2 + f0
2
b2n2 + f02
]+ · · · . (3.27)
Then expanding around b = 0 and n = 1 we get the correct universal term
SEE =c
3ln(
f0
δ) , (3.28)
where, c3
= 2πL(1+2f∞λ)
f1/2∞ `P
.
3.2.3 Quasi-Topological Gravity
The six-derivative action for quasi-topological gravity is given below [11],
S = − 1
2`3P
∫d5x√g[R +
12
L2+L2λ
2GB +
L47µ
4Z5
](3.29)
where,
GB = RµνρσRµνρσ − 4RµνR
µν +R2 and
Z5 = RµνρσRν
ασβRα
µβρ +
3
8RµνρσR
µνρσR− 9
7RµνρσR
µνραR
σα +15
7RµνρσR
µρRνσ
+18
7RµσR
σαRµα −
33
14RαβR
αβR +15
56R3 .
(3.30)
76 3.3. COMMENT ABOUT SINGULARITIES IN THE METRIC
Following exactly the same procedure we can derive the holographic entanglement entropy
for this six derivative gravity theory and obtain the correct universal terms..
For the sphere we get,
SEE = − 4π2L3
f3/2∞ `3
P
(1− 6f∞λ+ 9f 2∞µ) ln(
f0
δ) . (3.31)
For the cylinder
SEE = − π2L3H
2f3/2∞ `3
PR(1− 2f∞λ− 3f 2
∞µ) ln(f0
δ) . (3.32)
3.2.4 α′3 IIB supergravity
The action for this follows from [12]
S = − 1
2`3P
∫d5x√g[R +
12
L2+ L6γκ5
](3.33)
where,
κ5 = CαβµνCρβµσCαδγ
ρCνδγσ −
1
4CαβµνC
αβρσC
µρδγC
νσδγ .
Cαβµν is the Weyl tensor in 5 dimensions. In the context of IIB string theory, γ = 18ζ(3)α′3/L6.
For this theory we find that the universal parts of EE do not get corrected compared to the
Einstein case. This is expected since from the perspective of the AdS/CFT correspondence,
the C4 correction correspond to 1/λ corrections and the anomalies are not expected to receive
such corrections. Recently the effect of the C4 correction on Renyi entropy was analysed in
[13].
3.3 Comment about singularities in the metric
There are singularities in the five dimensional metric coming entirely from g(2)ij . We expand
the metric around r = 0 . Upto the leading order the metric is shown below.
For the sphere (diagonal components are gρρ, grr, gττ , gθθ, gφφ ),
L2
4f∞ρ20 0 0 0
0 (n−1) cos(τ)L2
f0 rf∞+ 1
ρ0 0 0
0 0 r2
ρ− L2(n−1) r cos(τ)
f0f∞0 0
0 0 0 f02
ρ0
0 0 0 0 f02 sin2(θ)ρ
. (3.34)
REFERENCES 77
For the cylinder,
L2
4f∞ρ20 0 0 0
0 (n−1) cos(τ)L2
2f0 rf∞+ 1
ρ0 0 0
0 0 r2
ρ− L2(n−1) r cos(τ)
2f0f∞0 0
0 0 0 f02
ρ0
0 0 0 0 1ρ
. (3.35)
The grr component is singular in r. The other components are non singular. However it is
easy to see that the determinant does not have a singularity at r = 0. The singularity in the
metric gives rise to singularities in the components of the Riemann tensor. We have explicitly
checked that these singularities do not enter in the higher derivative actions considered in
this paper. Hence these are mild singularities in the sense used in [3]. Note that in order to
calculate the universal part of EE in four dimensions only g(2)ij is important.
3.4 Discussion
The newly proposed regularization in [4] yields the expected universal terms in the EE
in higher derivative gravity theories dual to four dimensional CFTs. We considered the
Fefferman-Graham metric with the regularized metrics in [4] as the boundary metric. Then
we computed the generalized gravitational entropy as proposed in [3]. The universal log
terms worked out to be as expected. We showed that upto the order we are interested in, the
singularities in the metric are mild. As pointed out in [3] we could also have done a conformal
transformation of the boundary metric with conical singularity such that it is non-singular
and then done the calculation. We expect the results to be identical. One can possibly
use other regularization scheme and compute EE using generalized entropy to obtain correct
results, but we will demonstrate the importance of this particular regularization in the next
chapter when we will try to connect EE with the Wald entropy formula.
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80 REFERENCES
4 Connection between entanglement
entropy and Wald entropy
4.1 Introduction
As discussed in the Chapter.(2), there is a striking similarity between black hole entropy
and entanglement entropy (EE) in Einstein gravity. In the black hole case, there exists a
simple generalization of the area law for calculating the entropy of a black hole in any general
higher-derivative gravity theory, known as the Wald entropy [1, 2, 3]. It is by now well known,
in an arbitrary theory of gravity, taking the Wald entropy functional in AdS space will give
rise to the wrong universal terms in EE [4, 5]. A prescription is given for evaluating EE
in [6, 7], suitably modifying the Wald entropy functional for general surfaces based on the
generalized entropy principle. In the Chapter.(2) of this thesis we have shown that, for any
arbitrary theory of gravity it is not possible to derive those conjectured entropy functionals
[8]. On the other hand, based on the Noether charge prescription one can derive the Wald
entropy functionals, there by increasing its geometrical significance. In this chapter our main
objective is to explore the possibility of connecting EE with the Wald entropy as that we will
give us a concrete holographic derivation of EE based on Noether charge method, enabling
us to avoid the conflicts as pointed out in Chapter.(2).
In this chapter we will consider the background constructed out of the squashed cone
metric in Eq. (3.3) and evaluate the Wald entropy for various theories of gravity in this
background on a particular surface. This will produce the correct universal terms in EE.
After that we will comment on the connection with Iyer-Wald prescription for computing
entropy for black holes with dynamical horizon, opening up a possibility of connecting EE
rigorously with the Noether charge.
In the Chapter.(3) we have shown that if we evaluate generalized entropy [9] in the
squashed cone background we will get the correct universal terms for EE. In this chapter it
will be clear why that particular background is important, as that will play a crucial role in
our quest to connect EE with the Wald entropy.
81
82 4.2. WALD ENTROPY
4.2 Wald Entropy
Before proceeding further, let us mention the key steps in Wald’s derivation of the entropy
functionals for black holes for general theories of gravity [2]. Let us consider a diffeomorphism
invariant lagrangian L. We denote all the dynamical fields collectively by φ. In general φ
carries indices depending on the nature of the fields present. Varying L with respect to φ we
get, ∫ddxδL =
∫ddxE. δφ+
∫ddx∇µΦµ (4.1)
First piece gives the equation of motion and the second piece is a surface term. Now one can
construct a Noether current out of this surface term as follows
Jµ = Φµ(φ,Lζφ)− ζµL, (4.2)
where ζµ denotes the killing vectors associated with the diffeomorphism invariance. Using
the equation of of motion one can write down,
J = dQ (4.3)
where Q is the Noether charge. At this stage several ambiguities enter in the calculation.
For example, one can add a closed form to Q to define a new charge Q→ Q+ dΨ such that,
J = dQ (4.4)
still holds. Also one can add a total derivative term to L which will not affect the equation
of motion but it will change the expression for Q such that Eq. (4.3) still holds. For more
detailed discussion of these ambiguities interested readers are referred to the original reference
[2].
But it was shown by Iyer and Wald in the [2] that all these ambiguities vanish on a
bifurcation surface and one can write down a first law like relation for the entropy.
δ
∫Q =
κ
2πδSwald (4.5)
where κ is the surface gravity related to the temperature of the black hole and the charge is
integrated over the horizon of the black hole. From this we one can read of the Wald entropy
solely in terms of geometrical quantities.
Swald =
∫dd−2x
√h
∂L
∂Rαβγδ
εαβ εγδ . (4.6)
This expression is evaluated on the black hole horizon, which is a codimension-2 surface.
Here εαβ = n1αn
2β − n2
αn1β is the binormal corresponding to the two transverse directions 1, 2 .
CHAPTER 4. CONNECTION BETWEEN ENTANGLEMENT ENTROPY AND WALDENTROPY 83
One crucial point is that, Wald formula is applicable only on a bifurcation surface. For
a static black hole, the horizon is a bifurcation surface for which the extrinsic curvatures
vanish. For surfaces with non zero extrinsic curvature one cannot apply this formula. Now
let us see what it gives for our case. We will evaluate Eq. (4.6) on the background constructed
in Eq. (3.3) on a codimension-2 surface r = 0 , τ = 0. Rest of this chapter is mainly based
on the work done with Prof. Aninda Sinha and Dr. Menika Sharma [10].
4.3 Four derivative theory
The Wald entropy is calculated from Eq. (3.8). We have to first evaluate the following,
∂L
∂Rαβγδ
=1
2(gαγgβδ − gαδgβγ) + L2
[λ1R
αβγδ +1
4λ2
(gβδRαγ − gβγRαδ − gαδRβγ + gαγRβδ
)+
1
2λ3R
(gαγgβδ − gαδgβγ
) ](4.7)
and after some simplifications we get,
Swald =2π
`3P
∫d3x√h(1 +
L2
2(2λ3R + λ2Rµνn
νrn
rµ + 2λ1Rµνρσnµrn
νsn
rρnsσ)). (4.8)
In this section we will show that starting with the boundary metrics in Eq. (3.2) we can
construct a bulk spacetime on which Swald will produce the expected universal parts for the
entanglement entropy for both cylindrical and spherical region. Note that (4.8) differs from
(2.11) by the O(K2) terms.
4.3.1 Cylinder
As we will show, a particular form of the regularization b = α(n − 1)1/2 appearing in the
Eq. (3.2), where α is some number which we will determine later (it will turn out to be surface
dependent but theory independent), is needed to get the correct universal term. Recall that
the only restriction on b was that f(r, b) has to be n2 in the r = 0 limit. However, in
holographic calculations we expect that the bulk metrics will only depend on the AdS radius,
the radius of the entangling region and n. As such we can expect that the only way that
b→ 0 would arise in holographic calculations is such that b is some positive power of (n−1).
Now we will evaluate Eq. (4.8) using Eq. (3.3) using the cylinder metric to be its boundary.
Then we extract the coefficient of the 1ρ
term. We set τ = 0 . There is no integral over r
84 4.3. FOUR DERIVATIVE THEORY
in the Wald entropy as the entangling surface is located at r = 0, τ = 0 . We put r = b x .
After that we expand around x = 0 and then expand around n = 1 . We retain only the n
independent part as other terms vanish in n → 1 limit. Below we quote some intermediate
steps after expanding in ρ, r and n respectively. It is important to take the limits in r, n in
that particular in order to get the correct result [11]. After doing the ρ expansion we pick
out the 1ρ
term of (4.8) which is shown below.
Swald = · · ·+ 2π
`3P
∫dρdφdz
A(x, n)
ρ+O(ρ) + · · · , (4.9)
where
A(x, n) =L3 (n2 − 1) d−n ((4λ2 + 20λ3 − 2λ1) f∞ − 1) (2f0d
n − d (n2 + n+ x2 − 2) (bx)n)
24b2f3/2∞ (n2 + x2)2
.
Then expanding A(x, n) around x = 0 we get,
A(x, n) =L3 (n2 − 1) f0 ((4λ2 + 20λ3 − 2λ1) f∞ − 1)
12 b2n4f3/2∞
+ · · · . (4.10)
If
b =2f0√
3
√n2 − 1β(n) ,
where β(1) = 1 we get upon further expanding A(x, n) around n = 1
A(x, n) = −L3 (1 + 2 (λ1 − 2 (λ2 + 5λ3)) f∞)
16f0f3/2∞
+O(n− 1) + · · · . (4.11)
Notice that the choice for b was independent of the theory, i.e., in this case of λi’s. Finally
we get,
Swald = −π2L3H(1 + 2f∞(λ1 − 2λ2 − 10λ3))
2f3/2∞ `3
Pf0
ln(f0
δ) . (4.12)
This is precisely what is expected.
4.3.2 Sphere
We proceed similarly for the sphere case. First we expand in ρ and pick out the 1ρ
term.
Swald = · · ·+ 2π
`3P
∫dρdθdφ
A(x, n)
ρ+O(ρ) + · · · . (4.13)
CHAPTER 4. CONNECTION BETWEEN ENTANGLEMENT ENTROPY AND WALDENTROPY 85
Here
A(x, n) =L3d−2n sin(θ)
12b2f 3/2x2(n2 + x2)2
[4λ1f∞(b2x2d2n(n2 + x2)2 − d2 (n4 − n3x2 + 3n2x2 + nx2
+ x4 − x2)(bx)2n + d (n2 − 1)Rx2(n2 + n+ x2 − 2)(b dx)n − (n2 − 1)R2x2d2n)
− (2(λ1 + 2(λ2 + 5λ3))f∞ − 1)(−2b2x2d2n (n2 + x2)2 + d2(n4(3x2 + 2) + n3x2
+ 3n2x4 − nx2 − x4 + x2)(bx)2n + d (n2 − 1)Rx2(n2 + n+ x2 − 2)(b d x)n
− (n2 − 1)R2x2 d2n)]
(4.14)
Then expanding A(x, n) around x = 0 we get 1,
A(x, n) =L3 sin(θ)
(2b2n4 (4 (λ1 + λ2 + 5λ3) f∞ − 1) + (n2 − 1) f0
2 ((−2λ1 + 4λ2 + 20λ3) f∞ − 1))
12b2f3/2∞ n4
.
(4.15)
Only the x independent term is shown. If (for consistency checks see below)
b = f0
√n2 − 1β(n) (4.16)
where β(1) = 1, expanding around n = 1 we get,
A(x, n) = −L3 sin(θ) (1− 2 (λ1 + 2 (λ2 + 5λ3)) f∞)
4f3/2∞
+O(n− 1) + · · · . (4.17)
As in the cylinder case, notice that the choice for b is theory independent. Finally we get,
Swald = −4π2L3(1− 2f∞(λ1 + 2λ2 + 10λ3))
f3/2∞ `3
P
ln(f0
δ) (4.18)
We have fixed b for both the cylinder and the sphere case. In all the subsequent calculations
of Wald entropy we will use these same values for b.
1Remember that at this stage n = 1 + ε. Thus we will drop x2n compared to x2.
86 4.4. QUASI-TOPOLOGICAL GRAVITY
4.4 Quasi-Topological gravity
The Wald entropy is calculated for (3.29) using (4.6) . For this case,
∂L
∂Rαβγδ
=1
2(gαγgβδ − gαδgβγ) + L2
[λ1R
αβγδ +1
4λ2
(gβδRαγ − gβγRαδ − gαδRβγ + gαγRβδ
)+
1
2λ3R
(gαγgβδ − gαδgβγ
) ]+
7µL4
4
[(3µ1(RαργσRβ δ
ρ σ −RαρδσRβ γρ σ)) +
µ2
2[(gαγgβδ − gαδgβγ)
RµνρσRµνρσ + 4RRαβγδ] +
µ3
4[gβδRαρσµRγ
ρσµ − gβγRαρσµRδρσµ − gαδRβρσµRγ
ρσµ + gαγRβρσµRδρσµ
− 2RγρRαβδρ + 2RδρRαβγ
ρ + 2RβρRαργδ − 2RαρRβ
ργδ] +
µ4
2(Rρσ[gβδRα
ργσ − gβγRα
ρδσ
− gαδRβργσ + gαγRβ
ρδσ] + [RαγRβδ −RαδRβγ]) +
3µ5
4[gβδRασRγ
σ − gβγRασRδσ
− gαδRβσRγσ + gαγRβσRδ
σ] +µ6
2
[R(gβδRαγ − gβγRαδ + gαγRβδ − gαδRβγ
)+ (gαγgβδ − gαδgβγ)RµνR
µν]
+3
2µ7(R2[gαγgβδ − gαδgβγ])
].
(4.19)
Now the coefficients are,
µ1 = 1 , µ2 =3
8, µ3 = −9
7, µ4 =
15
7, µ5 =
18
7, µ6 = −33
14, µ7 =
15
56,
and λ2 = −4λ1, λ3 = λ1 = λ. Proceeding similarly as mentioned for the R2 theory we get
the expected universal terms.
For the cylinder 2,
Swald = − π2L3H
2f3/2∞ `3
PR(1− 2f∞λ− 3f 2
∞µ) ln(f0
δ) . (4.20)
For the sphere,
Swald = − 4π2L3
f3/2∞ `3
P
(1− 6f∞λ+ 9f 2∞µ) ln(
f0
δ) . (4.21)
Again note that the choice for α did not depend on the theory.
2The c and a coefficients for an arbitrary higher derivative theory can be easily calculated using the
short-cut mentioned in the appendix of [12].
CHAPTER 4. CONNECTION BETWEEN ENTANGLEMENT ENTROPY AND WALDENTROPY 87
4.5 α′3 IIB supergravity
The Wald entropy is calculated for (3.33) using (4.6) . For this case,
∂L
∂Rαβγδ
=1
2(gαγgβδ − gαδgβγ) + L6γ
[13
(gβγCαµδνCνρσηCµρση − gβδCαµγνCνρσηCµ
ρση
+ gαδCβµγνCνρσηCµρση − gαγCβµδνCνρσηCµ
ρση) +1
6(gαγgβδ − gαδgβγ)(CσµνρCσηνζCηρζµ
− 1
2Cµν
ρσCµνηζCηρζσ) +1
6(gβδCαρζσCρσµνC
γζµν − gαδCβρζσCρσµνC
γζµν
− gβγCαρζσCρσµνCδζµν + gαγCβρζσCρσµνC
δζµν) +
1
6(gβδCαρζσCγµ
ρνCζσµν
− gαδCβρζσCγµρνCζσµν − gβγCαρζσCδµ
ρνCζσµν + gαγCβρζσCδµ
ρνCζσµν)
+ (CαρµσCβµδηCγ
ρησ − CβρµσCαµδηCγ
ρησ − CαρµσCβµγηCδ
ρησ + CβρµσCαµγηCδ
ρησ)
− 1
2(CγδσζCβ
ζµρCασµρ + CαβσζCδ
ζµρCγσµρ) +
2
3(gαδCβρζνCρσνµC
γµζσ
− gβδCαρζνCρσνµCγµζσ + gβγCαρζνCρσνµC
δµζσ − gαγCβρζνCρσνµC
δµζσ)].
(4.22)
Proceeding similarly as mentioned for the R2 theory we get the expected universal terms.
For the cylinder,
Swald = −π2L3H
2`3PR
ln(f0
δ) . (4.23)
For the sphere,
Swald = −4π2L3
`3P
ln(f0
δ) . (4.24)
As expected, for this case the universal terms are independent of the higher derivative cor-
rection.
4.6 Connection with Ryu-Takayanagi
The Ryu-Takayanagi calculation involves the minimization of an entropy functional3. For
both the sphere and the cylinder, one can check that minimizing the Wald area functional in
the Fefferman-Graham background for squashed cones leads to the correct universal terms
provided we choose b as mentioned above. Recall that the Wald entropy functional in AdS
spacetime was not the correct one [4, 5]. However, our background is not AdS and it turns
out that the Wald entropy functional leads to the correct universal terms. We show this for
3We thank Rob Myers for discussions on this section.
88 4.6. CONNECTION WITH RYU-TAKAYANAGI
the cylinder, the sphere case working similarly. Putting r = R(ρ) = r0 + r1ρα around ρ = 0
leads to r0 = 0 and the equation
cnrn1ραn+1 − 4r2
1Rcnα(α− 2)ρ2α = 0 ,
where we have shown the leading terms which would contribute around n = 1. If we set
n = 1 we recover the result α = 1, r1 = −1/(4f0) for a cylinder–this is expected. The n = 1
boundary geometry is just flat space with the dual bulk being AdS. Hence we expect to
recover the RT result. However if n = 1 + ε, then it is easy to see that either r1 = 0 or α = 2
or r1 = −1/(4f0) and α = 1 + ε. As in the RT case, only the linear term in R(ρ) would have
affected the universal term–since α 6= 1 if n = 1 + ε we find that there is no linear term.
For n 6= 1 the minimal surface is at r = 0 = τ. This is the reason why the Wald entropy on
the r = 0 = τ surface and the RT entropy functional approach give the same result for the
universal terms in the squashed cone background. We now point out a direct comparison
between the calculation done in AdS spacetime and that in the squashed cone background
for the sphere in what follows.
The Ryu-Takayanagi prescription was implemented in the following way for a spheri-
cal entangling surface. Consider the AdS5 metric with the boundary written in spherical
coordinates
ds2 =L2
z2(dz2 + dt2 + dr2 + r2dθ2 + r2 sin2 θdφ2) . (4.25)
Now put r = f(z) = f0 + f2z2 + · · · and t = 0 and minimize the relevant entropy functional.
Implicitly our analysis says that this surface and the r = 0 = τ surface in the coordinate
system we have been using are related. Since in both cases the extrinsic curvatures vanish
we can attempt to make a direct comparison. In order to do this we make a coordinate
transformation:dz
z
√1 + f ′(z)2 =
dρ
2ρ. (4.26)
Around ρ = 0 we will find z2 = ρ− 2f 22ρ
2 + · · · and f(z)2/z2 = f02/ρ+ 2f0f2(1 + f0f2) + · · · .
Now around ρ = 0, the metric on the r = 0 = τ surface takes the form
ds2 = L2[dρ2
4ρ2+K(ρ)(dθ2 + sin2 θdφ2)] , (4.27)
where
K(ρ) =f0
2
ρ− L2
6b2n4(2b2n4 + (n2 − 1)f0
2) .
This also shows that for n 6= 1 minimal surface is at r = 0 = τ . Now choosing b as in
Eq. (4.16), expanding upto O((n − 1)0) and comparing with the RT calculation we find
f2 = −1/(2f0). This is exactly what we would have got if we minimized the RT area
functional (or the relevant higher derivative entropy functional) in AdS space. This also
serves as a consistency check for the choice of b.
CHAPTER 4. CONNECTION BETWEEN ENTANGLEMENT ENTROPY AND WALDENTROPY 89
4.7 Comments on the connection with the Iyer-Wald
prescription
Why does the Wald entropy functional lead to the correct result in our case? Wald’s formula
in Eq. (4.6) is valid for a surface which is a local bifurcation surface on which the Killing field
vanishes. For a bifurcation surface, the extrinsic curvatures vanish. SEE mentioned in (2.11)
differs from Swald only by the extrinsic curvature terms. The Noether charge method of [1]
needs a bifurcation surface to remove various ambiguities [2, 3]. According to the prescription
of Iyer and Wald [2], in order to compute the entropy for horizons which are not bifurcate,
e.g., dynamical horizons, the curvature terms in ∂L∂Rabcd
are replaced by their boost invariant
counterparts [2]. To do this we have to construct a boost invariant metric from our original
metric. Let gab be our starting d dimensional metric with the two normals n1a, n
2b . The boost
invariant part of gab will only have terms with the same number of n1, n2. We then consider
a d − 2 dimensional surface and find a neighbourhood of it O such that for any points x
belonging to this neighbourhood, we can find a point P which lies on a unit affine distance
on a geodesic with a tangent vector va on the d− 2 dimensional plane perpendicular to this
surface under consideration. Now we assign a coordinate system U, V, x1, ...xd−2 for the point
x where U, V are the components of va along n1a and n2
a. A change of normals under the
boosts na1 → αna1, nb2 → α−1na2 will change the coordinates as follows U → αU, V → α−1V .
Now we Taylor expand gab around Uand V ,
gab = g(0)ab + U∂g + V ∂g + UV ∂∂g + ........... . (4.28)
We have shown the expansion schematically. Under boosts, the terms linear in U, V do not
remain invariant. The prescription in [2] is to drop these terms. The UV term is invariant
under the boost. One important point to note is that , ψa = U( ∂∂U
)a−V ( ∂∂V
)a is a Killing field
of the metric. This means that Lie derivative of gab with respect to ψ is zero. Effectively,
we have constructed a new spacetime in which the original dynamical horizon becomes a
bifurcate Killing horizon.
The evidence for the existence of this bifurcation surface would be that extrinsic curva-
tures for this surface in the bulk background vanishes. Our entangling surface is a codimension-
2 surface. Now we calculate the extrinsic curvatures for this surface in the bulk Fefferman-
Graham metric. There will be two of them—one along the direction of the normal (τ)n for
τ = 0 and the other one along the normal (r)n for r = 0. We start with the 5 dimensional
metrics given in Eq. (3.3). The non-zero components of the normals are
(τ)nτ =1√gττ
, (r)nr =1√grr
.
With these we calculate the two extrinsic curvatures (τ)Kµν and (r)Kµν . Then we put r = b x
90 4.8. UNIVERSALITY IN RENYI ENTROPY
and τ = 0 as before. As the entangling surface is located at r = 0, τ = 0 we further do an
expansion around x followed by an expansion in n. Now (τ)Kab = 0 whereas (r)Kab = A(x, n, ρ)
is some function of x , n and ρ . First we expand it around x = 0 and then we do an expansion
around n = 1 . We find that (r)Kab = 0 .
Thus effectively the Fefferman-Graham construction is the same as the Iyer-Wald pre-
scription, provided we take the limits in the manner prescribed in [11]. The replacement of
rKijdxidxj by rnKijdx
idxj plays a key role in this construction. Recall that this was needed
to keep the boundary Ricci scalar finite. Also another important point to notice that for
the squashed cone metric there is no time like killing vector as the metric components are
dependent on τ . The Wald-Iyer prescription calls for calculating the Wald functional in the
context of black hole entropy where there exists a time like killing vector. But in the metric
(3.2) the cos(τ) factor which breaks the time translational symmetry is accompanied by a
factor of rn . In our calculation we have taken the r → 0 limit first and then the n→ 1 limit.
Thus the cos(τ) multiplied by rn is suppressed in this way of taking limits. For this reason
we have an approximate time-translational symmetry in our new space time.
Upto this point the discussion is independent of the choice of b. Now when one wants to
evaluate the Wald entropy functional with this squashed cone metric one needs to specify b
as mentioned in the previous sections for the sphere and the cylinder to obtain the correct
universal terms. As there is no integral over r in the Wald entropy functional, the final result
obtained will be b dependent as we have found and hence we have to choose b accordingly.
4.8 Universality in Renyi entropy
Before closing out this chapter let us mention an interesting application of this generalized
entropy. In [13, 14, 15, 16] it was shown that for spherical entangling surfaces in four dimen-
sions the Renyi entropy has a universal feature. Namely
∂nSn|n=1 ∝ cT .
In four dimensions cT ∝ c, the Weyl anomaly. If we use Eq. (4.15) and identify it as the
expression for Sn with the choice for b given below it4, then we indeed find that this is true!
This also works for the six and eight derivative examples. Thus this approach enables us to
check some information away from n = 1. Further, as a bonus, we can predict what happens
in the case of a cylindrical entangling surface where holographic results for the Renyi entropy
are not available. If we use Eq. (4.10) or its analog for the six and eight derivative examples,
we find that ∂nSn|n=1 ∝ cT still holds. It will be interesting to explicitly verify this in field
theory.
4In order to get the proportionality constant to work out, we will need to adjust ∂nβ(n)|n=1 in b.
REFERENCES 91
4.9 Discussion
In this chapter we computed the Wald entropy on the r = 0 = τ co-dimension 2 surface in
the Fefferman-Graham metric and found that it gives the correct universal terms for both
spherical and cylindrical surfaces. In order to get the expected results, we needed to choose
a surface dependent but theory independent regularization parameter. Recall that in bulk
AdS space, from the entropy functional way of computing EE in general theories of gravity,
one needed to use the entropy functional proposed in [6, 7] , which differed from the Wald
entropy functional by extrinsic curvature terms as shown in the Chapter.(2). These extrinsic
curvature terms are important to get the correct universal piece for any entangling surface
with extrinsic curvature.
Whether EE can be thought of as a Noether charge needs further investigation. Our
findings in this chapter seems to suggest that this may indeed be true. The Fefferman-Graham
metric is the analog of the Iyer-Wald metric used to compute the entropy for dynamical
horizons in [2]. Our conjecture then is that the Wald entropy (after appropriately fixing the
regularization) evaluated on the r = 0 = τ co-dimension two surface in the Fefferman-Graham
metric is going to capture the expected universal terms for any entangling surface.
Recently it was pointed out in the [17], that the ambiguities in the Nother charge method
can be fixed by demanding that the resulting entropy functional satisfies a generalized sec-
ond law [18] when evaluated on any arbitrary surface. Also it has been shown that, those
ambiguities correspond to the extrinsic curvature dependent terms of the proposed entropy
functionals as shown in Eq. (2.6) upto O(K2). This opens up the possibility of deriving these
holographic entropy functionals [6, 7] directly from the Noether charge method and that will
also solidify our conjecture of connecting EE with the Noether charge.
References
[1] R. M. Wald, “Black hole entropy is the Noether charge,” Phys. Rev. D 48, 3427 (1993)
[gr-qc/9307038].
V. Iyer and R. M. Wald, “A Comparison of Noether charge and Euclidean methods
for computing the entropy of stationary black holes,” Phys. Rev. D 52 (1995) 4430
[gr-qc/9503052].
[2] V. Iyer and R. M. Wald, “Some properties of Noether charge and a proposal for dy-
namical black hole entropy,” Phys. Rev. D 50 (1994) 846 [gr-qc/9403028].
[3] T. Jacobson and R. C. Myers, “Black hole entropy and higher curvature interactions,”
Phys. Rev. Lett. 70, 3684 (1993) [hep-th/9305016].
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(1994) 6587 [gr-qc/9312023].
[4] L. -Y. Hung, R. C. Myers and M. Smolkin, “On Holographic Entanglement Entropy
and Higher Curvature Gravity,” JHEP 1104 (2011) 025 [arXiv:1101.5813 [hep-th]].
[5] J. de Boer, M. Kulaxizi and A. Parnachev, “Holographic Entanglement Entropy in
Lovelock Gravities,” JHEP 1107, 109 (2011) [arXiv:1101.5781 [hep-th]].
[6] X. Dong, “Holographic Entanglement Entropy for General Higher Derivative Gravity,”
JHEP 1401, 044 (2014) [arXiv:1310.5713 [hep-th]].
[7] J. Camps, “Generalized entropy and higher derivative Gravity,” JHEP 1403, 070 (2014)
[arXiv:1310.6659 [hep-th]].
[8] A. Bhattacharyya and M. Sharma, “On entanglement entropy functionals in higher
derivative gravity theories,” JHEP 1410 (2014) 130 [arXiv:1405.3511 [hep-th]].
[9] A. Lewkowycz and J. Maldacena, “Generalized gravitational entropy,” JHEP 1308,
090 (2013) [arXiv:1304.4926 [hep-th]].
[10] A. Bhattacharyya, M. Sharma and A. Sinha, “On generalized gravitational entropy,
squashed cones and holography,” JHEP 1401 (2014) 021 [arXiv:1308.5748 [hep-th]].
[11] D. V. Fursaev, A. Patrushev and S. N. Solodukhin, “Distributional Geometry of
Squashed Cones,” arXiv:1306.4000 [hep-th].
[12] K. Sen, A. Sinha and N. V. Suryanarayana, “Counterterms, critical gravity and holog-
raphy,” Phys. Rev. D 85 (2012) 124017 [arXiv:1201.1288 [hep-th]].
[13] D. A. Galante and R. C. Myers, “Holographic Renyi entropies at finite coupling,” JHEP
1308, 063 (2013) [arXiv:1305.7191 [hep-th]].
[14] L. -Y. Hung, R. C. Myers, M. Smolkin and A. Yale, “Holographic Calculations of Renyi
Entropy,” JHEP 1112, 047 (2011) [arXiv:1110.1084 [hep-th]].
[15] E. Perlmutter, “A universal feature of CFT Renyi entropy,” arXiv:1308.1083 [hep-th].
[16] L. Y. Hung, R. C. Myers and M. Smolkin, “Twist operators in higher dimensions,”
JHEP 1410 (2014) 178 [arXiv:1407.6429 [hep-th]].
[17] S. Bhattacharjee, S. Sarkar and A. Wall, “The holographic entropy increases in
quadratic curvature gravity,” arXiv:1504.04706 [gr-qc].
A. C. Wall, “A Second Law for Higher Curvature Gravity,” arXiv:1504.08040 [gr-qc].
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[18] A. C. Wall, “Ten Proofs of the Generalized Second Law,” JHEP 0906 (2009) 021
[arXiv:0901.3865 [gr-qc]].
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are functions of Lovelock densities,” Phys. Rev. D 88 (2013) 044017 [arXiv:1306.1623
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94 REFERENCES
5 Constraining gravity using
entanglement entropy
5.1 Introduction
One can constrain gravity in several interesting ways by using quantum entanglement. In
this chapter we focus on one of them and in the next chapter we discuss another one. From
Chapter.(2) and Chapter.(3) it is evident that the extremal surface plays a crucial role in
evaluating entanglement entropy (EE), in fact the entropy functionals have to be evaluated on
the extremal surface to get the correct universal terms. In this chapter we will demonstrate
that in the context of higher derivative gravity theories, one can constrain the coupling
of the higher derivative terms by demanding the smoothness of the extremal surface. More
specifically, we will derive constraints on the Gauss-Bonnet (GB) coupling by demanding that
the entangling surface for sphere, cylinder and the slab close off smoothly in the bulk. The
slab case was considered before in the Ref. [1]. The GB action is given in the Eq. (2.8) with
λ1 = λ3 = λ and λ2 = −4λ and the corresponding entropy functional is the Jacobson-Myers
(JM) functional as shown in the Eq. (2.42). At the onset note that treating the truncated
GB gravity on its own leads to problems with entanglement entropy as was pointed out in
the Ref. [1]. In particular if we consider an entangling surface that topologically looks like
M2 × R, then the R term in the JM entropy functional becomes topological. Adding more
handles to the entangling surface will allow us to lower the entanglement entropy arbitrarily
if λ > 0. Since this particular sign of λ happens to arise in many consistent examples
in string theory (see for eg.[2]), this hints at a problem in interpreting GB gravity on its
own as a model for theories describing c 6= a–of course, there is no reason to suspect any
inconsistencies if this is just the first perturbative correction in an infinite set of higher
derivative corrections. We will not have anything to add to this observation. We will simply
focus on what constraints arise on the GB coupling by demanding smoothness and compare
the result with the causality/positive energy constraints [3, 4, 5] as shown in Eq.(5.1).
− 7
36≤ λ ≤ 9
100. (5.1)
To elaborate a bit more, we note at this point that the GB coupling λ is bounded. Following
the Refs. [3, 4] for the calculation of the three point correlation function of stress tensor one
95
96 5.2. SMOOTHNESS OF ENTANGLING SURFACE
needs to compute a energy flux which comes form the insertion of εijTij , where εij and Tij
are the polarization tensor and stress tensor respectively. Demanding the positivity of this
energy flux in the holographic set up we get the following three constraints and from those
we obtain bounds on λ . These coincide with the bounds arising from micro-causality [6].
Tensor channel : 1− 10f∞λ ≥ 0⇒ λ ≤ 9
100
Vector channel : 1 + 2f∞λ ≥ 0⇒ −3
4≤ λ ≤ 1
4
Scalar channel : 1 + 6f∞λ ≥ 0⇒ − 7
36≤ λ ≤ 1
4
(5.2)
From this we get
− 7
36≤ λ ≤ 9
100.
This is the same as the condition 1− 4f∞λ− 60f 2∞λ
2 ≥ 0.
Rest of this chapter is based on the work done with Dr. Shamik Banerjee, prof, Aninda
Sinha, Apratim Kaviraj and Kallol sen [7].
5.2 Smoothness of entangling surface
The general strategy we will adopt is the following. The entangling surface equation follows
from Eq.(2.43), coming from minimizing JM functional. We will consider 5 dimensional AdS
spacetime.
ds2 =L2(dz2 + dt2 + dΣ2)
z2(5.3)
and choose a constant time slice t = 0.
When we consider a spherical entangling , we parametrize dΣ in terms of spherical polar
coordinates.
dΣ2 = dr2 + r2(dθ2 + sin(θ)2dφ2) (5.4)
and the extremal surface is given by r = f(z). We will consider a cylindrical surface also.
For that we parametrize dΣ in terms of cylindrical polar coordinates,
dΣ = du2 + dr2 + r2dθ (5.5)
where u is the coordinate along the length of the cylinder and the extremal surface is charac-
terized again by r = f(z). For the slab, we simply write dΣ in cartesian coordinate and the
corresponding extremal surafce is given by x = f(z), where x is one of the coordinates of Σ.
Now let us assume that the surface f(z) closes off smoothly at z = zh inside the bulk
spacetime. Around this point, let us assume
f(z) =∞∑i=0
ci(zh − z)α+i . (5.6)
CHAPTER 5. CONSTRAINING GRAVITY USING ENTANGLEMENT ENTROPY 97
We need to determine α and ci’s. At z = zh, f′(z) → +∞ since the tangent to the surface
will be perpendicular at that point. This means that 0 < α < 1 and c0 > 0. Using these two
conditions, we will find that λ will be bounded.
Cylinder
Consider the cylinder case first. In cylindrical coordinates, assume the required hypersurface
to have the form r = f(z). From Eq. (6.72), we get the following equation,[zf ′′(z)
(6f∞λzf
′(z) + f(z)((4f∞λ+ 1)f ′(z)2 − 2f∞λ+ 1
))−(f ′(z)2 + 1
)(f ′(z)
(z(4f∞λ+ 1)f ′(z) + 3f(z)
(f ′(z)2 − 2f∞λ+ 1
))− 2f∞λz + z
)]= 0 .
(5.7)
We take the trial solution Eq. (5.6) and determine an appropriate α. We obtain α = 1/2, 3/2.
We will drop the second solution since this will lead to a conical tip. Expanding the eom in
powers of (zh − z) and setting the leading order term to 0, we get 4 roots of c0. We take the
two positive ones, √2
3
√zh(1 + 4f∞λ±
√1− 10f∞λ+ 16f 2
∞λ2) . (5.8)
With f∞ = (1−√
1− 4λ)/2λ, this puts some constraints on λ. Since the bottom sign vanishes
in the λ→ 0 limit, we will ignore this solution. For the other case, we have
λ ≤ 7
64. (5.9)
The quantities inside the square root have to be positive to make the root real. If we look
carefully we will find that 1 − 10f∞λ + 16f 2∞λ
2 has to be positive. This is almost same as
that of the tensor channel constraint except for the extra additional factor of 16f 2∞λ
2 . That
is why we get a bigger bound instead of λ< 9100
.
Sphere
The eom reads,[zf ′′(z)
(12f∞λzf(z)f ′(z) + f(z)2
((4f∞λ+ 1)f ′(z)2 − 2f∞λ+ 1
)+ 6f∞λz
2)−(f ′(z)2 + 1
)(6f∞λz
2f ′(z) + 2zf(z)((4f∞λ+ 1)f ′(z)2 − 2f∞λ+ 1
)+ 3f(z)2f ′(z)
(f ′(z)2 − 2f∞λ+ 1
)) ]= 0 .
(5.10)
We get only α = 1/2 as a solution to the indicial equation. We get six roots of c0 from the
leading order of eom. Three of them are positive:
√2zh ,
√4f∞zhλ± 2
√2zh√f∞λ(−1 + 2f∞λ) . (5.11)
The positivity of the first root cannot give us any constraint on λ. The other two roots go
to zero as λ goes to zero so we will ignore them.
98 5.3. DISCUSSION
Slab
The eom reads,
− 3(1− 2f∞λ+ f ′(z)2)(f ′(z) + f ′(z)3) + z(1− 2f∞λ+ (1 + 4f∞λ)f ′(z)2)f ′′(z) = 0 (5.12)
We get α = 1/2, 1 which give non-zero c0. Arguing as before we will only consider α = 1/2.
Here we get the following positive solution for c0:
c0 =
√2
3
√zh + 4f∞zhλ . (5.13)
Demanding this to be positive, we get
− 5
16≤ λ ≤ 1
4. (5.14)
This agrees with [1]. Thus together with the constraints from the cylinder we have
− 5
16≤ λ ≤ 7
64. (5.15)
We can recast this inequality as one for a/c where a, c are the Euler and Weyl anomaly
coefficients respectively for a 4d CFT. This gives us
1
3≤ a
c≤ 5
3. (5.16)
Quite curiously, the lower bound 1/3 is precisely what appears in non-supersymmetric
theories in the Refs. [3, 4, 5], in particular for a free boson. The upper bound of 5/3 cor-
responds to a free theory with one boson and two vector fields. For a non-supersymmetric
theory, the bound on a/c worked out1 in the Refs. [3, 4] was 1/3 ≤ a/c ≤ 31/18. Just to
point out in words, the 1/3 came from the cylinder calculation while the 5/3 came from the
slab. The causality constraints on the other hand translates into 1/2 ≤ a/c ≤ 3/2.
We compare the different bounds on λ in Fig. (5.1). As is clear, the causality constraints
are the tightest.
5.3 Discussion
We have considered different entangling surfaces and demanded that these close off smoothly
in the bulk. In Gauss-Bonnet gravity, this led to the coupling being constrained. The spher-
ical entangling surface did not lead to any constraints on the coupling while the cylindrical
and slab entangling surfaces did. It will also be interesting to find if there are other en-
tangling surfaces which lead to a tighter bound and if the bounds are stronger than the
causality constraints. Moreover one can generalize this for arbitrary R2 theory and try get
the constraints on the coupling.
1Note 31/18 ≈ 1.72 while 5/3 ≈ 1.67.
REFERENCES 99
Figure 5.1: Comparison between the various constraints on the GB coupling. The length of
the line represents the range of allowed λ.
References
[1] N. Ogawa and T. Takayanagi, “Higher Derivative Corrections to Holographic Entan-
glement Entropy for AdS Solitons,” JHEP 1110 (2011) 147 [arXiv:1107.4363 [hep-th]].
[2] A. Buchel, R. C. Myers and A. Sinha, “Beyond eta/s = 1/4 pi,” JHEP 0903, 084
(2009) [arXiv:0812.2521 [hep-th]].
[3] D. M. Hofman and J. Maldacena, “Conformal collider physics: Energy and charge
correlations,” JHEP 0805 (2008) 012 [arXiv:0803.1467 [hep-th]].
[4] D. M. Hofman, “Higher Derivative Gravity, Causality and Positivity of Energy in a UV
complete QFT,” Nucl. Phys. B 823 (2009) 174 [arXiv:0907.1625 [hep-th]].
[5] A. Zhiboedov, “On Conformal Field Theories With Extremal a/c Values,”
arXiv:1304.6075 [hep-th].
[6] M. Brigante, H. Liu, R. C. Myers, S. Shenker and S. Yaida, “Viscosity Bound Violation
in Higher Derivative Gravity,” Phys. Rev. D 77 (2008) 126006 [arXiv:0712.0805 [hep-
th]].
M. Brigante, H. Liu, R. C. Myers, S. Shenker and S. Yaida, “The Viscosity Bound and
Causality Violation,” Phys. Rev. Lett. 100 (2008) 191601 [arXiv:0802.3318 [hep-th]].
A. Buchel and R. C. Myers, “Causality of Holographic Hydrodynamics,” JHEP 0908
(2009) 016 [arXiv:0906.2922 [hep-th]].
[7] S. Banerjee, A. Bhattacharyya, A. Kaviraj, K. Sen and A. Sinha, “Constraining gravity
using entanglement in AdS/CFT,” JHEP 1405 (2014) 029 [arXiv:1401.5089 [hep-th]].
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6 Relative entropy
6.1 Introduction
In this chapter we will discuss one more way of constraining gravity based on quantum
entanglement. Certain entanglement measures such as relative entropy [1], which roughly
speaking tells us how distinguishable two states are, need to be positive in a unitary theory.
The positivity of this quantity was studied in holographic field theories with two derivative
gravity duals in [2]–related work include [3]. In the context of quantum field theories with
holographic dual gravity descriptions, one can ask what this inequality translates into.
Let us begin by discussing relative entropy. Relative entropy between two states ρ and σ
is defined as
S(ρ|σ) = tr (ρ log ρ)− tr (ρ log σ) . (6.1)
As reviewed in appendix A, in quantum mechanics, this quantity is positive for a unitary
theory. In [2], relative entropy was discussed in the holographic context. The state σ was
chosen to be the reduced density matrix for a spherical entangling surface. In this case,
σ ≡ e−H/tr e−H with H being the modular hamiltonian. It can be easily shown that (see
Refs. [2, 4])
S(ρ|σ) = ∆H −∆S , (6.2)
where ∆H = 〈H〉1 − 〈H〉0 and ∆S = S(ρ) − S(σ) with S(ρ) = −tr ρ log ρ being the von
Neumann entropy for ρ and is the entanglement entropy for a reduced density matrix ρ.
Then the positivity of S(ρ|σ) would require,
∆H ≥ ∆S . (6.3)
Now we can calculate the modular hamiltonian for the sphere [5], from the formula,
H = 2π
∫r<R
dd−1xR2 − r2
2RT00 . (6.4)
Here Tµν is the d-dimensional field theory stress tensor and 00 is the time-time component.
We know how to compute Tµν in holography. The Ryu-Takayanagi prescription (and known
generalizations [6]) gives us a way to compute ∆S. Thus we can check if and how the
inequality ∆H ≥ ∆S is satisfied. In [2], many examples were considered and in each case
101
102 6.1. INTRODUCTION
it was shown that this inequality is respected in Einstein gravity. If we consider a small
excitation around the vacuum state then to linear order in the perturbation ∆H = ∆S. This
can be shown to be equivalent with the linearized Einstein equations [7, 8]. This equality has
been recently shown to hold for a general higher derivative theory of gravity in [9]. It is thus
very interesting to ask what constraints we get at the nonlinear order. We will address this
question for the special case of a constant stress tensor for the case where the holographic
entanglement entropy is given by the Ryu-Takayanagi prescription—in other words, we will
ask if even at non-linear level we get Einstein gravity. We find that the constraints arising
from relative entropy give us a larger class of models than just Einstein gravity. However,
we show that there exists matter stress tensor for which the bulk null energy condition is
violated everywhere except at the Einstein point. This in turn implies that relative entropy
can continue to be positive although the bulk null energy condition is violated. In fact we
can ask the question the other way round: are there examples where the relative entropy
is negative but the bulk null energy condition still holds? We will give an example where
this happens. Thus the connection between energy conditions and the positivity of relative
entropy, which in some sense is reminiscent of the connection between energy conditions and
the laws of thermodynamics, appears to be less direct than what one would have expected.
In order to get some intuition about what feature of gravity ensures the positivity of
relative entropy, we extend the calculations in [2] to higher derivative theories. In particular
we focus on Gauss-Bonnet gravity in 5 bulk dimensions [10, 11, 12] since in this context there
is a derivation [13, 14] of the corresponding entropy functional [15, 16, 17]. We find that in
all examples that we consider, the positivity of the two point function of the stress tensor
guarantees that the relative entropy is positive. In particular we show this for a constant
field theory stress tensor as well as for a disturbance that is far from the entangling surface.
At this point we should emphasize that, the inequality for relative entropy can only be
explicitly checked when the modular hamiltonian is known. Unfortunately, currently this is
not known for cases when the entangling region is a cylinder or a slab.
This chapter is organized as follows. In section (6.2), we consider constraints arising from
the positivity of relative entropy in a holographic set up where the entanglement entropy
is given by the Ryu-Takayanagi entropy functional. These constraints arise at a quadratic
order in a perturbation with a constant field theory stress tensor. In section (6.3), we turn
to the study of relative entropy in Gauss-Bonnet holography. In section (6.4), we investigate
the relative entropy for an anisotropic plasma which breaks conformal invariance. We find
that the relative entropy in this case is negative and we suggest some possible explanations
for this. We conclude in section (6.5). The appendix contains further calculations relevant
for the rest of the chapter. We will use capital latin letters to indicate bulk indices and greek
letters to indicate boundary indices. Lower case latin letters will indicate an index pertaining
CHAPTER 6. RELATIVE ENTROPY 103
to the co-dimension 2 entangling surface. 1
6.2 Relative entropy considerations
In this section we will use the results in [2] to derive certain constraints at nonlinear order
that arise due to the positivity of relative entropy. In Fefferman-Graham coordinates, the
bulk metric can be written as
ds2 =L2
z2dz2 + gµνdx
µdxν . (6.5)
For Einstein gravity, the bulk equations of motion allow us to systematically solve for gµν as
an expansion around the boundary z = 0 (see eg.[19]). The idea here is to see what mileage
we get if we do not know what the bulk theory is but we demand that the relative entropy
calculated using the Ryu-Takayanagi entropy functional is positive. We want to calculate the
quadratic correction to the entanglement entropy for the following form of boundary metric,
gµν =L2
z2
[ηµν + azdTµν + a2z2d(n1TµαT
αν + n2 ηµνTαβT
αβ) + · · ·], (6.6)
where a = 2d
`d−1P
Ld−1. This form is consistent with Lorentz invariance for a constant Tµν . We will
treat Tµν as a small perturbation to the vacuum. At linearized order, it has been shown in
Refs. [7, 8, 9] Einstein equations arise from the condition ∆H = ∆S. We wish to investigate
what happens at the next order. We will keep n1 and n2 arbitrary and derive constraints on
them arising from the inequality ∆H ≥ ∆S. Our analysis follows [2] very closely, the only
change being that we will not specify n1 and n2 to be the Einstein values. Since at linear
order (the argument will be reviewed in the next section) we have the equality ∆H = ∆S
and since T00 from the holographic calculation is just given by the coefficient of the zd term
in the metric, the inequality implies ∆S ≤ 0 at quadratic order. Thus our task is to calculate
∆(2)S, the quadratic correction to ∆S, as a function of n1, n2. The analysis below is valid
for d > 2.
We start with the Ryu-Takayanagi prescription for calculating entanglement entropy in holog-
raphy,
S =2π
`d−1P
∫dd−1x
√h . (6.7)
From Taylor expansion one can show that the quadratic correction to√h is,
δ(2)√h =
1
8
√h(hijδhij)
2 +1
4
√h δhijδhij +
1
4
√h hijδ(2)hij . (6.8)
1 The paper by Erdmenger et al [18] deals with a related idea of looking for pathological surfaces in certain
higher derivative theories of gravity and is of some relevance in this context. Interested readers are referred
to that.
104 6.2. RELATIVE ENTROPY CONSIDERATIONS
The induced metric is,
hij = gij +L2
z2∂iz∂jz . (6.9)
This is evaluated at the extremal surface z = z0 + εz1 =√R2 − r2 + εz1. Hence, at 0-th
order, the metric and its inverse are,
hij =L2
z20
(ηij +
xixjz2
0
)and hij =
z20
L2
(ηij − xixj
R2
). (6.10)
In ∆(2)S, we get 3 kinds of second order contributions. To be systematic, we write,∫dd−1x δ(2)
√h = A(2,0) + A(2,1) + A(2,2) , (6.11)
where schematically, these are the (δg)2, z1δg and z21 contributions respectively. To calculate
the first term, we can set z1 = 0. Then
δhij = aL2zd−2Tij andδ(2)hij
2= a2L2z2d−2
0 (n1TiαTαj + n2 ηijTαβT
αβ) . (6.12)
This gives,
A(2,0) = Ld−1a2
∫dd−1x Rzd0
(Ti0T
i0
(n1
2+ (d− 1)n2 − n2
r2
R2
)+ (T00)2
(n2
2(d− 1)− n2r
2
2R2
)+TijT
ij
(n1
2+n2
2(d− 1)− n2r
2
2R2− 1
4
)− n1
2R2xixjTi0T
0j + xixjTikT
kj
(1
2R2− n1
2R2
)+
1
8
(T 2 − T 2
x − 2TTx))
,
where Tx = xixjTijR2 and T = T ii . The last two terms in (6.11) are same as they appear in [?]
. Quoting the result,
A(2,1) = Ld−1a
∫dd−1x
R
2z0
[T
(z1 −
z20
R2xi∂iz1
)+TijR2
(2z2
0xi∂jz1 − z1x
ixj − z20x
ixjxk∂kz1
R2
)],
(6.13)
A(2,2) = Ld−1
∫dd−1x
R
zd0
[d(d− 1)z2
1
2z20
+z2
0(∂z1)2
2R2− z2
0(xi∂iz1)2
2R4+
(d− 1)xi∂iz21
2R2
]. (6.14)
We can find z1 by minimizing A(2,1) + A(2,2), which gives,
z1 = −aR2zd−1
0
2(d+ 1)(T + Tx) . (6.15)
Plugging this and summing we get from Eq. (6.11),∫dd−1x δ(2)
√h = Ld−1a2
∫dd−1x
(c1T
2 + c2T2x + c3T
2ij + c4Ti0T
i0 + c5
xixjTikTkj
R2
+ c6
xixjTi0T0j
R2+ c7TTx
),
(6.16)
CHAPTER 6. RELATIVE ENTROPY 105
where unlike [2]2, the coefficients c1 · · · c7 are dependent on n1 and n2,
c1 =(R2 − r2)(d−4)/2
8(1 + d)2R
(−4(1 + d)2n2(r2 −R2)2(r2 − (d− 1)R2) (6.17)
+R2(2(d2 + 2d− 1)r4 + (1− 5d2)r2R2 + (2d2 − d− 1)R4)), (6.18)
c2 =(−r2 +R2)
12
(−4+d)((1− 5d2) r2R3 + (−3 + d(3 + 4d))R5)
8(1 + d)2, (6.19)
c3 =(−r2 +R2)
d/2(−2n2r
2 + (−1 + 2n1 + 2(−1 + d)n2)R2)
4R, (6.20)
c4 =(−r2 +R2)
d/2(n1R
2 − 2n2 (r2 − (−1 + d)R2))
2R, (6.21)
c5 =(d2 − (1 + d)2n1)R (−r2 +R2)
d/2
2(1 + d)2, (6.22)
c6 = −n1
2R(−r2 +R2
)d/2, (6.23)
c7 =(−1 + d)R3 (−r2 +R2)
12
(−4+d)((1− 3d)r2 + (1 + 2d)R2)
4(1 + d)2. (6.24)
Now we integrate the expression (6.16) over the (d− 2)-sphere on the boundary. We use the
trick,∫dd−1x f(r)xixjxkxl · · ·n pairs = N(δijδkl · · ·+ permutations)
∫dd−1x f(r)r2n , (6.25)
where N is some normalization constant. For n = 1, N = 1/(d − 1); and for n = 2,
N = 1/((d− 1)2 + 2(d− 1)). The final result comes out in the form 3,∫dd−1x
√h = a2Ld−1Ωd−2
(C1T
2 + C2T2ij + C3T
2i0
), (6.26)
with
C1 =2−3−dd (1 + 4 (d2 − 1)n2)
√πR2dΓ[d+ 1]
(d2 − 1) Γ[
32
+ d] , (6.27)
C2 =2−3−dd
√πR2dΓ[1 + d]
(d2 − 1) Γ[
32
+ d] (
−1− 2d+ 4(d+ 1)n1 + 4(d2 − 1
)n2
), (6.28)
C3 = −2−1−dd(n1 + 2(d− 1)n2)√πR2dΓ[1 + d]
(d− 1)Γ[
32
+ d] . (6.29)
2There appears to be an overall sign missing for c6 in [2].3The expression for C3 in [2] after substituting for n1, n2 is off by a factor of d/(d+2) although the overall
sign is correct. This appears to be related to the opposite sign used for c6. We have cross-checked our results
on mathematica for various cases and the notebook may be made available on request.
106 6.2. RELATIVE ENTROPY CONSIDERATIONS
Now we must demand that ∆(2)S ≤ 0. We can write ∆(2)S = V TMV with V being a
(d−1)(d+2)/2 dimensional vector with the independent components of Tµν as its components.
Then demanding that the eigenvalues of M are ≤ 0 will ensure ∆(2)S ≤ 0. This leads to
n1 + 2(d− 1)n2 ≥ 0 , (6.30)
2d+ 1− 4(d+ 1)n1 − 4(d2 − 1)n2 ≥ 0 , (6.31)
d+ 2− 4(d+ 1)n1 − 4d(d2 − 1)n2 ≥ 0 . (6.32)
Figure 6.1: (colour online) For d > 2 we get the allowed n1, n2 region to be the blue triangle
above for a generic stress tensor. The region above the blue solid line and below the blue
dashed and dotted lines are allowed from the relative entropy positivity. For d→∞ the region
collapses to a line 0 ≤ n1 ≤ 1 indicated in green. The Einstein value (n1, n2) = (12,− 1
8(d−1)) is
shown by the black dot. The region below the solid red line and above the dashed and dotted
red lines are allowed by the null energy condition. By turning on a generic component of
the stress tensor only the Einstein value is picked out. By switching off certain components
of the stress tensor, various bands bounded by the solid, dashed and dotted lines are picked
out.
We get the region indicated in Fig. (6.1) allowed by this set of inequalities. One interesting
CHAPTER 6. RELATIVE ENTROPY 107
observation is that when d → ∞, then the allowed region becomes the interval 0 ≤ n1 ≤ 1
with n2 = 0 coinciding with the Einstein result. The area of the triangle is given by
Aread =d2
8(d+ 1)2(d− 2). (6.33)
Notice that the (extrapolated) Aread=2 is infinity. This makes sense since in d = 2 we expect
constraints on only 2 eigenvalues (since T 2 and T 2ij are no longer independent) which will give
us an unbounded region. Further Aread→∞ → 0 which leads to a line interval for d→∞ as
shown in Fig. (6.1).
At this stage, we have a wider class of theories that are allowed by the inequality than
the Einstein theory. The other theories need extra matter in addition to Einstein gravity to
support them. As such we could ask if the matter needed satisfies the null energy condition.
As an example consider turning on a constant T01 in d = 4. Then we find
RAB −1
2gAB(R +
12
L2) = T bulkAB , (6.34)
with T bulkAB working to be
T bulkAB = 16z6T 201
[3
2(δn1 + 4δn2)δzAδ
zB + (δn1 + 6δn2)δ0
Aδ0B − (δn1 + 6δn2)δ1
Aδ1B
− 2(δn1 + 3δn2)∑i=2,3
δiAδiB
].
(6.35)
m Here δn1 = n1−1/2 and δn2 = n2 +1/24. Using this we find that the null energy condition
T bulkAB ζAζB ≥ 0 leads to
T bulk00 + T bulk22 = −δn1 ≥ 0 , (6.36)
T bulk00 + T bulkzz =5
2δn1 + 12δn2 ≥ 0 , (6.37)
with T bulk00 + T bulk11 = 0. These simplify to n1 ≤ 1/2 and n2 ≥ −1/24. Thus the region in
fig.1 that respects the null energy condition is smaller than that allowed by the positivity of
relative entropy.
For a general constant stress tensor in general d we proceed as follows. We note that for
a metric of the form in Eq. (6.5), with gµν a function of z only, we have [20]
Rµν = R′µν − (z∂zKµν +KKµν − 2KµκKκν ) , (6.38)
Rµz = 0 , (6.39)
z2Rzz = −gµνz∂zKµν +KµνKµν , (6.40)
R = R′ − (2zgµν∂zKµν +K2 − 3KµνKµν) , (6.41)
108 6.2. RELATIVE ENTROPY CONSIDERATIONS
where Kµν = 12z∂zgµν . Here ′ denotes a quantity computed with gµν . Using these it is
straightforward (but tedious) to compute (setting L = 1 for convenience, defining Sµν =
n1TµαTαν + n2ηµνTαβT
αβ and aborbing the factors of a into Tµν ; the raising and lowering of
indices on Tµν , Sµν are done with ηµν . Also we have used T µµ = 0.)
gµν = z2[ηµν − T µνzd + (T µαT να − Sµν)z2d] , (6.42)
Kµν = − 1
2z2(2ηµν − (d− 2)zdTµν − 2(d− 1)z2dSµν) , (6.43)
Kνµ = −1
2[2δνµ − dzdT νµ + dz2d(TµαT
αν − 2Sνµ)] , (6.44)
K = −1
2[2d+ dz2d(TαβT
αβ − 2Sαα)] , (6.45)
Kµν = −z2
2[2ηµν − (d+ 2)zdT µν + 2(d+ 1)z2d(T µαT
να − Sµν)] , (6.46)
zgµν∂zKµν =1
2[4d+ z2d(4d(d− 2)Sαα − d(d− 4)TαβT
αβ)] , (6.47)
KµνKµν = d+
z2d
4[d(d+ 4)TαβT
αβ − 8dSαα ] , (6.48)
KµκKκν =
1
4z2[4ηµν − 4(d− 1)zdTµν + z2d(d2T κµTκν − 4(2d− 1)Sµν)] . (6.49)
Using these we find
T bulkzz = −d(d− 1)z2d−2TαβTαβ(δn1 + dδn2) , (6.50)
T bulkµν = d2z2d−2[−δn1TµκT
κν + ηµνTαβT
αβ(δn1 + (d− 1)δn2)]. (6.51)
Here δn1 = n1−1/2 and δn2 = n2 +1/(8(d−1)) i.e., the deviations from the Einstein values.
Now we are in a position to ask if the matter supporting this bulk stress tensor satisfies the
null energy conditions or not. First we note that T bulk00 + T bulk11 ≥ 0 immediately leads to
− d2(−T 200 + T 2
ij)δn1 ≥ 0 . (6.52)
This leads to a definite sign for δn1 if and only if (−T 200 + T 2
ij) has a definite sign. But in
general, there is no reason for this combination to have a definite sign. So we are led to
suspect that for a generic stress tensor, δn1 = 0. To confirm this suspicion let us look at
T bulkzz + T bulk00 .
T bulkzz + T bulk00 =
−d[(d− 1)T 2
00(δn1 + 2dδn2) + T 2ij[(2d− 1)δn1 + 2d(d− 1)δn2]
+T 20i[(2− 3d)δn1 − 4d(d− 1)δn2]
]. (6.53)
As in the relative entropy analysis, we write the RHS as V TMV where V is a (d−1)(d+2)/2
dimensional vector whose non-zero independent components are the T00, Tij, T0i’s. Then we
CHAPTER 6. RELATIVE ENTROPY 109
demand that the eigenvalues of M are positive for the null energy condition to hold for a
generic constant traceless stress tensor Tµν . This yields
(3d− 2)δn1 + 4d(d− 1)δn2 ≥ 0 , (6.54)
(2d− 1)δn1 + 2d(d− 1)δn2 ≤ 0 , (6.55)
δn1 + 2(d− 1)δn2 ≤ 0 . (6.56)
Only for δn1 = δn2 = 0 are these inequalities satisfied for d > 2. Thus the null energy
condition picks out the Einstein value if we ask if for a generic constant stress tensor the
O(T 2) terms are supported by matter. Of course as we saw for d = 4 we can turn on T0i and
set everything else to zero, there would be a region in the n1, n2 parameter space where the
null energy condition and the positivity of the relative entropy would hold (this corresponds
to the region between the red and blue solid lines in Fig. (6.1). For the generic case, only
the Einstein value is picked out. To emphasis, that the Einstein value was picked out for
the generic case, relied only on the null energy condition analysis and did not rely on the
positivity of the relative entropy. To summarize, we found that there exists a larger class of
theories in the (n1, n2) parameter space than just the Einstein theory. However, except at the
Einstein point, we found that there always exists some matter stress tensor which violates
the bulk null energy condition.
6.3 Relative entropy in Gauss-Bonnet holography
In this section we will calculate relative entropy for excited states in Gauss-Bonnet gravity.
For definiteness, we will consider d = 4 or 5-dimensional bulk. We will follow the conventions
in [11]. The total action is given by
I = Ibulk + IGH + Ict , (6.57)
where
Ibulk =
∫d5x√g
[R +
12
L2+λ
2L2(RABCDR
ABCD − 4RABRAB +R2)
]. (6.58)
The generalized Gibbons-Hawking term is given by [21]
IGH = − 1
`3P
∫d4x√γ[K − λL2(2GµνK
µν +1
3(K3 − 3KK2 + 2K3)
]. (6.59)
Here Gµν = Rµν−1/2γµνR made from the boundary γµν , K2 = KµνKµν and K3 = Kα
βKβγK
γα.
Kµν is the extrinsic curvature and K = Kαα . The counterterm action Ict is needed for the
cancellation of the power law divergences in Itot. For our case this works out to be [22, 23]
(L and f∞ are defined below)
Ict =1
`3P
∫d4x√γ[c1
3
L+ c2
L
4R], (6.60)
110 6.3. RELATIVE ENTROPY IN GAUSS-BONNET HOLOGRAPHY
where R is the four dimensional Ricci scalar and c1 = 1− 23f∞λ and c2 = 1 + 2f∞λ .
The equations of motion are given by
RAB −1
2gABR−
6
L2gAB −
λL2
2HAB = 0, (6.61)
where
HAB =1
2gAB(R2 − 4RCDR
CD +RCDEFRCDEF )− 2RRAB + 4RA
CRCB
− 2RACDERBCDE − 4RCDRCABD .
(6.62)
AdS5 given by
ds2 =L2
z2
(dz2 − dt2 + dx2
1 + dx22 + dx2
3
)(6.63)
where L = L/√f∞ with 1 − f∞ + λf 2
∞ = 0. The dual CFT is characterized by the central
charges c, a appearing in the trace anomaly[11, 24]:
c =π2L3
`3P
(1− 2λf∞) , a =π2L3
`3P
(1− 6λf∞) . (6.64)
The CFT stress tensor two point function is given by
〈Tµν(x)Tρσ(0)〉 =40c
π2(x2)4Iµν,ρσ(x) , (6.65)
where I is a function of x and the positivity of the two point function leads to c > 0.
We will need the formula for the holographic stress tensor (see [25])
Tµν =1
`3p
[Kµν − gµνK + λL2(qµν −1
3gµνq)]−
3
Lc1γµν +
L
2c2[Rµν(γ)− 1
2γµνR(γ)] , (6.66)
where
q = hµνqµν
qµν = 2KKµαKαν − 2KµαK
αβKβν + Kµν(KαβKαβ − K2) + 2KRµν + RKµν − 2KαβRαµνβ −
4Rα(µKν)α . The terms proportional to c1, c2 come from Ict.
We also note that the GB coupling λ is bounded. Following [26] for the calculation of the
three point correlation function of stress tensor one needs to compute a energy flux which
comes form the insertion of εijTij , where εij and Tij are the polarization tensor and stress
tensor respectively. Demanding the positivity of this energy flux in the holographic set up
we get the following three constraints and from those we obtain bounds on λ . These coincide
with the bounds arising from micro-causality [27].
Tensor channel : 1− 10f∞λ ≥ 0⇒ λ ≤ 9
100
Vector channel : 1 + 2f∞λ ≥ 0⇒ −3
4≤ λ ≤ 1
4
Scalar channel : 1 + 6f∞λ ≥ 0⇒ − 7
36≤ λ ≤ 1
4
(6.67)
CHAPTER 6. RELATIVE ENTROPY 111
From this we get
− 7
36≤ λ ≤ 9
100.
This is the same as the condition 1− 4f∞λ− 60f 2∞λ
2 ≥ 0.
6.3.1 Linear order calculations
We are interested in considering the excited state to be a perturbative excitation of the
ground state. At linear order in the perturbation ∆H = ∆S. Let us review the argument of
[2] why. Let ρ0 be a reference state. Now let ρ(α) be a continuous family of states dependent
on a parameter α that runs over all possible values. We choose the parametrization such
that ρ(α = 0) = ρ0. Now relative entropy vanishes for two states that are equal. So we must
have S(ρ(0)|ρ0) = 0 and also S(ρ(α → ε±)|ρ0)→ 0+ > 0 where ε is a small positive valued
number denoting a small perturbation from the reference state ρ0. This means at α = 0 we
must have, d(S(ρ(α)|ρ0))/dα = 0. Or equivalently at the linear order of the perturbation ε,
∆H = ∆S , (6.68)
which follows from Eq. (6.2). We can demonstrate this with a simple example4. Let ρ0 to be
the vacuum of the CFT4 whose holographic dual is the empty AdS5 (our linearized results
are a sub-case of the more general case worked out in [9]),
ds2 =L2
z2
(dz2 − dt2 + dx2
1 + dx22 + dx2
3
)(6.69)
We choose ρ1 to be the dual of a metric which is being perturbed around the empty AdS.
Following [2], we take the perturbation to be of the form,
δgµν =`3P
2L3z2∑n
z2nT (n)µν . (6.70)
To keep track of the perturbation we keep the components of T(n)µν proportional to a small
number ε. We compute the entanglement entropy from the Jacobson-Myers functional,
S =2π
`3P
∫d3x√h(1 + λL2R) +
4π
`3P
λL2
∫d2x√h K . (6.71)
Here, hab is the induced metric on the minimal surface and R and K are respectively the
intrinsic ricci scalar and extrinsic curvature evaluated on that surface. To simplify notation,
we will set L = 1 . The minimal surface equation is given by
K + λL2(RK− 2RijKij) = 0 , (6.72)
4The change in entanglement entropy for excited states in GB holography has been considered in [28].
112 6.3. RELATIVE ENTROPY IN GAUSS-BONNET HOLOGRAPHY
which was derived in [13, 29] following [30]. For the spherical entangling surface in the
unperturbed metric the following continues to be an exact solution
z = z0 =√R2 − r2 . (6.73)
In the perturbed case, it changes to
z = z0 + ε z1 . (6.74)
However note that we obtained z0 by extremization. Hence z1 can only contribute to a
quadratic order in ε and not at linear order. Thus at linear order we can set z1 = 0. Using
z = z0 to compute (6.71) and then extracting the terms proportional to ε gives us ∆S.
Now we can calculate the modular hamiltonian, from the formula in Eq. (6.4), where T00 is
obtained in holography using Eq. (6.66). Since T00 = 0, for empty AdS, this directly gives
∆H. Now we will demonstrate the equality in Eq. (6.68) by considering a special case (we
have checked that this holds in the other examples considered below as well).
Using Gauss-Bonnet eom, we can determine T(n)µν in terms of the lowest mode T
(0)µν . It
turns out they are all derivatives of T(0)µν . To keep it simple we take T
(0)µν to be a constant.
Also note that to satisfy GB eom, we must have traceless and divergenceless conditions on
T(0)µν ,
T (0)µ
µ = 0 and ∂µT(0)µ
ν = 0 . (6.75)
Consider an isotropic perturbation
T (0)µν =
(E , E
3,E3,E3
)(6.76)
Note that this satisfies the conditions in Eq. (6.75). However the holographic dual tensor Tµν
is not same as T (0)µν . We compute it from Eq. (6.66) as,
Tµν = (1− 2f∞λ)
(E , E
3,E3,E3
)(6.77)
Now using (6.4) one gets5,
∆H =8π2L3ER4
15`3P
(1− 2f∞λ) .
As discussed before, we can compute ∆S from (6.71) with z =√R2 − r2, and then take out
the ε order coefficients. We obtain,
√h(1 + λL2R) = −E (R2 (3 + 30f∞λ)− r2 (1 + 58f∞λ))
6f3/2∞ R
(6.78)
5There is a typo in Eq. (6.29) in [9] for 〈Tµν〉. There is a factor of 2 missing in front of the term proportional
to a1 in that expression. Taking this into account our expression agrees with their both for GB and for the
general R2 theory discussed in appendix C.
CHAPTER 6. RELATIVE ENTROPY 113
from which we calculate,
∆S =8π2L3ER4
15`3P
(1− 2f∞λ) . (6.79)
This demonstrates ∆H = ∆S for an isotropic perturbation.
6.3.2 Quadratic corrections
Now we turn to the more interesting case of quadratic corrections which lead to inequalities.
We take the following form for the boundary metric,
z2gµν = ηµν + zdTµν + z2d(n1TµαTαν + n2 ηµνTαβT
αβ) + · · · (6.80)
where compared to Eq. (6.6) we have absorbed a factor of a into the stress tensor. We need to
fix the numbers n1 and n2. By plugging into the GB equations of motion given by Eq. (6.61),
we find that
− 3(n1 + 4n2) + f∞(1 + 6n1λ+ 24n2λ) = 0 (6.81)
n1(9− 17f∞ + 25f 2∞λ)− 4(4f 3
∞λ− 3n2(1− 9f∞ + 17f 2∞λ)) = 0 (6.82)
Solving the two equations and using the relation 1− f∞ + f 2∞λ = 0 we get
n1 =1
2
1 + 2f∞λ
1− 2f∞λand n2 = − 1
24
1 + 6f∞λ
1− 2f∞λ. (6.83)
These results match with the λ = 0 case given in [2]6.
6.3.3 Constant Tµν
The next step is to calculate the second order change in ∆S. For a general but constant
stress tensor we can guess the following form of the second order correction of entropy from
Lorentz invariance,
∆(2)S = C1T2 + C2TijT
ij + C3T0iT0i (6.84)
where T denotes the trace of the spatial part of the stress tensor Tµν . The latin indices run
from 1 to 3, and denote the spatial part of a tensor. They are raised with ηij. Our task is to
identify the constants Ci’s for a non-zero λ. The only condition on the stress tensor is that it
6Notice a curious fact. If we demanded that n1 ≥ 0 and n2 ≤ 0, or in other words even in GB gravity
they have the same sign as in Einstein gravity then with c > 0, we would get
1 + 2f∞λ ≥ 0 , 1 + 6f∞λ ≥ 0 .
But these are nothing but the scalar and vector channel constraints in Eq. (6.67)! These leads us to wonder
if entanglement entropy knows about the causality constraints.
114 6.3. RELATIVE ENTROPY IN GAUSS-BONNET HOLOGRAPHY
is symmetric and traceless. To do the perturbative analysis we assume that the components
of the stress tensor are proportional to a perturbative parameter ε. Also we have absorbed
a parameter a in the stress tensor. The background metric will be changed in the quadratic
order as given in (6.6). Now, assume that the minimal surface z0 =√R2 − r2 is modified as
z = z0 + εz1 . (6.85)
z1 contributes at the quadratic order in the JM functional (6.71). So it is sufficient to
consider only the first order fluctuation to the entangling surface. Next we expand the
entropy functional upto quadratic order and then extract the terms proportional to ε2 which
gives the quadratic correction to the entropy. We vary it with respect to z1. This gives us
the equation of motion for z1. We find the solution and put it back to ∆(2)S. Since it was
shown that at linear order, ∆S = ∆H, at second order we must have ∆(2)S > 0. We get the
following equation for z1 ,
(1− 2f∞λ)
[∂2(z0z1)− xixj
R2∂i∂j(z0z1)− (R2 − r2)2 (T + 3Tx)
]= 0 , (6.86)
with the solution,
z1 = −R2z3
0
10(T + Tx) . (6.87)
Notice that the equation is the same as what appears in the Einstein case upto the overall
factor of (1− 2λf∞). The Gibbons Hawking term doesn’t contribute to the action when we
put in the solution. Alternatively, we could have taken the action and integrated all terms
involving z′′1 (x)’s by part and cast it in the conventional form. The surface term resulting
from this will cancel with the appropriate Gibbons-Hawking term. We have checked both
approaches and have got the same result. Integrating the resulting action over the volume
of the entangling region, we obtain the second order correction to the entropy,
∆(2)S = −8π3L3(1− 2f∞λ)
`3P
(C1T
2 + C2T2ij + C3T
2i0
), (6.88)
C1, C2, C3 are same as the Einstein values obtained in section 2.
Note that this is just a factor of (1−2f∞λ) times what is obtained in the Einstein gravity
(the Einstein result was manifestly negative). This can be cross-checked easily on a computer
by suitably turning on various components of the stress tensor and identifying various tensor
structures.
Now from the discussions in the previous sections, it is clear that this quantity has to be
negative. The only constraint to ensure ∆(2)S < 0 is
1− 2f∞λ > 0 . (6.89)
CHAPTER 6. RELATIVE ENTROPY 115
This is equivalent to saying the central charge c > 0 which also is the condition needed for the
positivity of the two point function of the field theory stress tensor. The condition λ < 1/4
ensures that this holds. If this inequality on λ did not hold, the corresponding vacuum would
have ghosts [11].
6.3.4 Shockwave background
Up to this point we have only considered constant stress-tensor. It is interesting to ask if we
get non-trivial constraints for Tµν not constant. To explore a nontrivial case of non-constant
Tµν , consider the following 5 dimensional metric
ds2 =L2
z2(dz2 + dxµdx
µ + f(t+ x3)W (z, x1, x2)(dt+ dx3)2) (6.90)
where µ = 1, 2.
The above metric solves the GB equation exactly, given that W (z, x1, x2) satisfies the
following differential equation,
∂2zW + ∂2
x1W + ∂2
x2W = −3
z∂zW , (6.91)
with no constraint on f(t+x3). If f = δ(t+x3) then this is the shockwave metric considered
for example in [26] to derive constraints on higher derivative gravity theories. We will set
f = 1 and in a slight abuse of terminology continue to refer the metric as a shockwave.
W (z, x1, x2) is taken as
W (z, x1, x2) =L2z4
(z2 + (x1 − x′1)2 + (x2 − x′2)2)3. (6.92)
Here (x′1, x′2) represent the point where the disturbance is peaked. Since in our calculations
we perturb the background metric, we should choose x′1 and x′2 to be outside the entangling
region. With this in mind we proceed with the second order calculation. Next we consider
a shockwave disturbance localized just outside the entangling surface. We will set x′2 = 0 in
(6.92). We start with the following metric which is obtained by expanding W around z = 0
and retaining the first two terms in the expansion,
ds2 =L2
z2(dz2 + dxµdx
µ + (z4L2ε3
(x21 + (x2 − x′2)2)3
− 3z6L2ε4
(x21 + (x2 − x′2)2)4
)(dt+ dx3)2) (6.93)
The ε factors have been inserted to keep track of the order of the expansion and matches with
the power appear in the denominator. If we write the entangling surface as z = z0 + ε3z1
then the quadratic terms in z1 will involve ε6 which is at a higher order than the second
order term in the metric above. Thus we expect to see an inequality ∆H > ∆S with the
116 6.3. RELATIVE ENTROPY IN GAUSS-BONNET HOLOGRAPHY
above metric setting z1 = 0. We thus evaluate the entropy functional considering only the
unperturbed entangling surface and expand it upto ε4 and pick out the ε4 term which gives
the first leading order change in the relative entropy. The integrand is shown below,
∆(2)S =2π
`3P
∫dx3dx1dx2
3L5
2Rf5/2∞ (x2
3 + (x2 − x′2)2)6
[(x2
3 + x22 + x2
1 −R2)(40f∞(x23 + x2
2 + x21 −R2)
(4R2(x23 + (x2 − x′2)2)− 4(x4
3 + x23(x2
1 + 2x2(x2 − x′2)) + (x21 + x2
2)(x2 − x′2)2))λ+ 16f∞
(R2 − x23 − x2
2 − x21)(x2
3 + (x2 − x′2)2)(2R2 − 13x23 − 2x2
1 − 13x22 + 12x2x
′2)λ
− (x23 + (x2 − x′2)2)2(60f∞(x2
3 + x− 22)λ−R2(1 + 18f∞λ) + x21(1 + 18f∞λ))) .
(6.94)
Then we perform the integration over x3 which goes from −√R2 − r2 to
√R2 − r2 and
x1 = r cos(θ) , x2 = r sin(θ) . Now after some algebraic manipulation we can write the
integrand as,
∆(2)S =2πL5
2`3Pf
5/2∞ R(r2 + x′22 − 2rx′2 sin(θ))6
(f1 + f2 sin(θ) + f3 sin(θ)2) (6.95)
where f1, f2, f3 are some function of r and λ . Integral over θ goes from 0 to 2π and integral
over r goes from 0 to R. We first perform the θ integral. To perform the θ integral we have
used the following integral identity:∫ 2π
0
dθ
a+ b sin(θ)=
2π√a2 − b2
,
Finally we get,
∆(2)S =2π
`3P
∫ R
0
dr[− L5
240f5/2∞ R
(f1(−30 (8a5 + 40a3b2 + 15ab4) π
(a2 − b2)11/2)
+ f2(90b (8a4 + 12a2b2 + b4) π
(a2 − b2)11/2)− f3(
30 (4a5 + 41a3b2 + 18ab4) π
(a2 − b2)11/2))],
(6.96)
where, a2 = r2 + x′22 and b = −2rx′2 . Next we perform the r integration. The leading
contribution in ∆(2)S comes form the lower limit of the r integral which is shown below.
∆(2)S =π2L5
96`3Pf
5/2∞ R2
(1− 2f∞λ)f(x′2) , (6.97)
where, f(x′2) is a negative valued function given by
f(x′2) =
(√x′22 − 1 (−136 + 72x′22 − 56x′42 + 15x′62 )− 3 (32− 16x′22 + 36x′42 − 22x′62 + 5x′82 )Csc−1(x′2)
)(x′22 − 1)
9/2
(6.98)
CHAPTER 6. RELATIVE ENTROPY 117
and plotted in Fig. (6.2). To satisfy, ∆S ≤ ∆H we will get, 1− 2f∞λ ≥ 0 or in other words
c > 0. Note that in order for us to be able to expand in small z, the perturbation needs to
be located far away from the entangling surface. This is because in the denominator in W
we had z2 + x21 + (x2 − x′2)2. When we plug in z = z0, the maximum value for z is R and
this happens when x1 = x2 = 0. Thus we will need R x′2 for the expansion to be valid. It
will be interesting to see what happens as we move the perturbation closer and closer to the
entangling surface. However this appears to be a very hard problem.
Figure 6.2: Negative of the function f(x′2) is plotted which is a positive valued function
6.3.5 Correction from additional operators
In this section we consider perturbed states in which certain additional operators acquire
nontrivial vacuum expectation value. Our analysis will follow [2]. The holographic dual of
these operators will involve additional massive fields in the bulk. We will show that even for
such cases in Gauss-Bonnet gravity, the relation ∆H > ∆S will hold. Again we are in AdS5
with the bulk action given by,
I =
∫d5x√−G
(R +
12
L2+λL2
2
(R2 − 4R2
AB +R2ABCD
)− 1
2(∂φ)2 − 1
2m2φ2
), (6.99)
where we have added a massive scalar field which acts as a bulk dual of a scalar operator of
dimension ∆. When m2 = ∆(4−∆), the field φ behaves asymptotically as,
φ = γOz∆ . (6.100)
118 6.4. RELATIVE ENTROPY FOR AN ANISOTROPIC PLASMA
Now we can work out the stress tensor corresponding to this from the formula,
TAB =1
2∂Aφ∂Bφ−
1
4gAB((∂φ)2 +m2φ2) . (6.101)
This will result in the following change to the boundary metric boundary metric,
z2δgµν = azd∑n
z2nT (n)µν + z2∆
∑n
z2nσ(n)µν (6.102)
where we must have,
σ(0)µν = − γ2
12(1− 2f∞λ)ηµνO2 . (6.103)
in order to satisfy Gauss-Bonnet eom. The higher modes, namely σ(n)µν (n > 0) are composed
of derivatives of σ(0)µν . As in [2], we consider O to be slowly varying and, hence, neglect the
higher modes.
It is not necessary to find any correction to the entangling surface. There are two different
perturbations, both in their first orders, and using the z0 minimal surface to compute ∆S
will suffice. The correction to entropy will have two parts,
∆S = ∆TS + ∆OS . (6.104)
The first part, ∆TS comes from the holographic boundary stress tensor Tµν , and its the same
as what we calculated before for the linear order. The second part comes from the scalar
field and is obtained by calculating the area functional with the metric of Eq. (6.102).
∆OS = −π3/2R2∆γ2(−2 + 3∆)Γ[−1 + ∆]Ωd−2
48 aΓ[
12
+ ∆] O2 . (6.105)
Note that the result is independent of λ. Since the result is negative it seems the met-
ric already knows of the positivity of relative entropy even for Gauss-Bonnet provided the
unitarity bounds are respected.
6.4 Relative entropy for an anisotropic plasma
We now want to turn our attention to a holographic anisotropic plasma–there is going to be
a surprise in store. We consider the holographic dual of the deformed N = 4 SYM where
the deformation is generated by anisotropy along one spatial direction viz.
S = SN=4 +1
8π2
∫θ(z) Tr F ∧ F, (6.106)
CHAPTER 6. RELATIVE ENTROPY 119
θ is the field generating anisotropy along the z direction. The holographic dual is the Einstein-
dilaton-axion system given by
Sbulk =1
2`3P
∫M
√−g(R +
12
L2− 1
2(∂φ)2 − 1
2e2φ(∂χ)2) +
1
2`3P
∫∂M
√−γ2K, (6.107)
where φ is the dilaton and at the level of the solution is taken to be a function of the AdS
radius only and χ is the axion dual to the gauge theory θ-term, responsible for inducing
anisotropy, which is taken to be χ = ρx3. This model was proposed and studied in detail in
[31]. The low anisotropy regime corresponding to ρ/T 1 in this model is unstable [31].
The metric equations are given by (L = 1)
RMN −1
2RgMN − 6gMN = TMN , (6.108)
where the bulk matter stress tensor is given as
TMN =1
2∂Mφ∂Nφ−
1
4(∂φ)2gMN +
1
2e2φ∂Mχ∂Nχ−
1
4e2φ(∂χ)2gMN . (6.109)
The metric, φ and χ equations can be written as
RMN + 4gMN −1
2∂Mφ∂Nφ−
1
2e2φ∂Mχ∂Nχ = 0,
∇2φ− e2φ(∂χ)2 = 0,
∇2χ = 0 .
(6.110)
The metric in the FG coordinates is given by
ds2 =dz2
z2+
1
z2γµν(z, x
i)dxµdxν , (6.111)
where
γtt = −1 +ρ2
24z2 + . . . ,
γx1x1 = γx2x2 = 1− ρ2
24z2 + . . . ,
γx3x3 = 1 +5ρ2
24z2 + . . . ,
(6.112)
If we introduce a temperature, the modification to the metric will start at O(z4). Further,
the scalar field introduces a new scale which breaks scale invariance explicitly and the trace
of the boundary stress tensor is now non-zero. It needs to be checked if the null energy
condition is satisfied by the bulk stress tensor TMN given by Eq. (6.109). Contracting the
above with the null vectors ξµ we have
TMNξMξN =
1
2[(∂ξφ)2 + e2φ(∂ξχ)2], (6.113)
120 6.4. RELATIVE ENTROPY FOR AN ANISOTROPIC PLASMA
where ∂ξ(φ, χ) = ξM∂M(φ, χ) and ξMξNgMN = ξ2 = 0. Since the bulk scalar axion follows
the profile χ = ρx3 then
ξM∂Mχ = ρξx3 , (6.114)
whereas the dilaton field φ depends on the radial coordinate. The NEC for the bulk stress
tensor becomes by contracting with the null vectors Tµνξµξν as
Tx3x3 =ρ2
2e2φ(ξx3)2 =
ρ2
2e2φ ≥ 0,
Tuu =1
2(∂ξφ)2 ≥ 0 .
(6.115)
Thus we have explicitly verified that the bulk stress tensor satisfies the null energy condition.
We now want to verify the calculation for the relative entropy in this low anisotropy
regime. As mentioned before, the low anisotropy phase is thermodynamically unstable. We
can thus try to see what happens to the relative entropy in such a phase. Also note that we
are considering Einstein gravity for which the entropy functional is the Ryu-Takayanagi one.
Further in the low anisotropy regime, we are interested in, since we are expanding γµν upto
O(z2) (assuming a small entangling surface Rρ 1) and the stress tensor appears at O(z4),
we have ∆H = 0. Here the state σ is the vacuum state which corresponds to ρ = 0 and is
conformally invariant. Thus the modular hamiltonian will be the same as in Eq. (4). Thus
we only need to compute the change in the entanglement entropy. Furthermore, at leading
order in ρ we expect to see an inequality and as such we do not need to evaluate the change
in the entangling surface.
Putting in the solution for the entangling surface f(x1, x2, x3) =√R2 − x2
1 − x22 − x3
3 we
have
√h =
1
48(R2 − x21 − x2
2 − x33)2R
[48R2 + (R2− x21− x2
2− x23)(3R2− 5x2
3 + x21 + x2
2)ρ2] +O(ρ4) .
(6.116)
The entanglement entropy then becomes
S =2π
`3p
∫dx1dx2dx3
1
48(R2 − x21 − x2
2 − x33)2R
[48R2+(R2−x21−x2
2−x23)(3R2−5x2
3+x21+x2
2)ρ2] .
(6.117)
In spherical polar coordinates x3 = r cos θ, x1 = r sin θ sinφ, x2 = r sin θ cosφ where (θ, φ) are
spherical polar coordinates we have
∆1S =2πρ2
`3p
∫(3R2 − 2r2 − 3r2 cos 2θ)
48(R3 − r2R)r2 sin θdθdφdr . (6.118)
Carrying out the (θ, φ, r) integrals we find (on reinstating L factors)
∆1S =π2ρ2R2L3
6`3p
(−5
3− log[
ε
2R]) . (6.119)
CHAPTER 6. RELATIVE ENTROPY 121
Here ε is a cutoff and r = R − ε (since ε → 0 corresponds to z → 0 it is related to the UV
cutoff). The log-divergence is due to the breaking of conformal invariance by the excited
state. However, notice that in the limit of ε→ 0, the result leads to ∆1S > 0 and hence the
positivity of relative entropy is violated.
Since the positivity of relative entropy in quantum mechanics depends on unitarity (re-
viewed in appendix A), this leads to the following possible interpretations:
1. There are additional contributions which we are missing and they are required for
the positivity of the relative entropy to hold in this case. One could speculate that
there are additional saddle points of the bulk gravity theory which contribute to the
entanglement entropy. It will be interesting to find out those saddle points and see if
they “unitarize” the problem7.
2. Holographic relative entropy positivity needs further conditions than just bulk uni-
tarity. It could be that the derivation of the positivity does not go through in any
straightforward manner to quantum field theory.
3. In the low anisotropy regime, may be there is a loss of bulk unitarity that is not
immediately apparent.
All possibilities need further investigation. Let us first briefly comment on the third possi-
bility. Expanding the linearized equations near the boundary and upto linear order in ρ we
have
( +2
L2)hij = 0, ( +
2
L2)hMx3 +
ρ
2LMx3χ1 = 0, (6.120)
φ1 − 2ρ∂x3χ1 = 0, χ1 = 0, (6.121)
where hMN , φ1 and χ1 are metric, φ and χ fluctuations respectively and i, j take values
apart from x3. ∇A is evaluated using the AdS5 metric. Here LMN ≡ δMx3∂N + δNx3∂M
is a linear operator. The coupling between the metric and χ fluctuation is of the form
Hh + Lχ = 0, Hχ = 0. But this form is similar to what arises in the context of logarithmic
conformal field theories which are non-unitary [33]. Thus one should check if there are log
modes in the fluctuations. We can do this following [34]. According to the arguments in [34]
log modes arise if the form of the equations is ( + a)2hµν = 0. Let us check what the form
of the equations are when we decouple them. Using 8 ∇Aχ1 = ∇Aχ1 − 4L2∇Aχ1, we find
7This is very similar to the resolution of information loss paradox in case of eternal AdS Black Holes as
formulated by Maldacena [32]. The exponentially small correlation as required by the unitarity arises form
the periodically identified Euclidean AdS, although this is not the dominant contribution to the canonical
ensemble.8Useful identities can be found for eg. in the appendices of [35]
122 6.5. DISCUSSION
that the decoupled equation for hMx3 takes the form
( +2
L2)( +
4
L2)hMx3 = 0 ,
while for φ we get
( +4
L2)φ = 0 .
Neither of the four derivative linear operator is of the form ( + a)2 and hence following
the arguments in [34] there are no log modes so the dual field theory is not a log CFT.
Naively it may appear that the propagator for say the φ field will look like 1/(p2(p2 +m2)) =
1/m2(1/p2 − 1/(p2 + m2)), and hence the theory is non-unitary. However, this is not true
since in addition to the decoupled form of the equations above, the relations in Eq. (6.121)
still have to hold–any loss of unitarity would have shown up in the asymptotic fall offs in the
field. Thus it appears that the other two possibilities become plausible.
In closing this section, we note that in [2] it was argued that the relative entropy should
increase as the radius of the entangling surface increases. In our case since ∂RS(ρ1|ρ0) =
−∂R∆1S ≈ π2ρ2RL3
3`3p(log[ ε
2R]) < 0 and hence this monotonicity would also appear violated.
6.5 Discussion
In this chapter we used holographic entanglement to constrain gravity in interesting ways.
First, we started with the Ryu-Takayanagi entropy functional (which holds for Einstein grav-
ity) and considered what constraints arise at nonlinear order on the metric by demanding
that relative entropy is positive. At linearlized level, it is now known that for the spherical
entangling surface ∆H = ∆S leads to linearized equations for any higher derivative theory
of gravity [9]. We considered a constant field theory stress tensor. At the next order, we
found interesting constraints on the terms allowed by the positivity of relative entropy. These
were more general than what arises from Einstein gravity. We analysed energy conditions for
matter that could support these additional theories. We showed that the additional theories
could be supported by matter that violates the null energy condition. In other words, holo-
graphic relative entropy can be positive although the bulk null energy condition is violated.
It is an important open problem to understand if this feature persists for a more general stress
tensor. We also gave an example of a model which corresponds to an anisotropic plasma,
where for small anisotropy, the relative entropy is negative. This occurred even though the
bulk stress tensor satisfied the null energy condition. We gave some possible explanations for
this. We will leave further investigations of similar models as an open problem.
Second, we analysed the inequality in Gauss-Bonnet gravity for a given class of small
perturbations around the vacuum state. We found that for all our examples, the positivity
CHAPTER 6. RELATIVE ENTROPY 123
of the stress tensor two point function ascertained that this inequality was respected. On the
bulk side this corresponds to metric fluctuations having positive energy. The simplicity of the
final result does cry out for a simpler explanation for our findings. Although the intermediate
integrals involved appeared very complicated, the final result was simply proportional to the
Weyl anomaly c. It would be nice to find a simple explanation for this finding. Some
preliminary studies of the general four derivative theory has been made in appendix B.
Another interesting open problem is to consider a disturbance close to the entangling surface.
We were able to consider a disturbance that was localized far from the entangling surface
and show that the relative entropy is positive. Whether the constraints change as one moves
the disturbance closer to the entangling surface is an open problem.
In the previous chapter we considered different entangling surfaces and demanded that
these close off smoothly in the bulk. In Gauss-Bonnet gravity, this led to the coupling
being constrained. Now, suppose we knew how to extend the relative entropy results for the
spherical entangling surface to other surfaces. Then the smoothness criteria above seems to
constrain the coupling of the higher derivative interaction. This suggests that implicitly the
relative entropy inequality knows about this. Since the positivity of relative entropy seems
to rely only on the unitarity of the field theory, this raises the question if there is any conflict
with unitarity if one is outside the allowed region for the coupling. It will be interesting to
investigate this question since apriori there does not appear to be any such conflict in the
dual gravity.
Appendix
A: Positivity of Relative entropy
Here we review the proof in quantum mechanics leading to the positivity of relative entropy.
This can be found in Nielsen and Chuang’s book listed in [1]. We define relative entropy as,
S(ρ|σ) = Tr(ρ ln ρ)− Tr(ρ lnσ) , (6.122)
where ρ and σ are the density matrices of two different states. Now consider their orthonormal
decomposition,
ρ =∑i
pi |i〉 〈i| and σ =∑j
qj |j〉 〈j| (6.123)
124 6.5. DISCUSSION
where |i〉 and |j〉 may not be the same set of eigenvectors. We can write,
S(ρ|σ) = Tr(ρ ln ρ)− Tr(ρ lnσ) =∑i
〈i| ρ ln ρ |i〉 −∑i
〈i| ρ lnσ |i〉
=∑i
〈i| ρ ln ρ |i〉 −∑i
∑j
〈i| ρ lnσ |j〉 〈j|i〉 =∑i
pi ln pi −∑i,j
pi 〈i| lnσ |j〉 〈j|i〉
=∑i
pi ln pi −∑i,j
pi ln qj 〈i|j〉 〈j|i〉 =∑i
pi ln pi −∑i,j
Pij pi ln qj . (6.124)
In the second line we just inserted 1 =∑
j |j〉 〈j|, and in the last line we have used the
notation Pij = 〈i|j〉 〈j|i〉. Note that we must have,∑i
Pij =∑j
Pij = 1 . (6.125)
Till here, all that we have used is the unitarity of the theory. Now, lnx is a concave function;
which means we must have,
ln(tx+ (1− t)y) ≥ t ln(x) + (1− t) ln(y) for 0 ≤ t ≤ 1 . (6.126)
It is easy to generalize this to,
ln (x1t1 + x2t2 + ...+ xmtm) ≥ t1 ln(x1) + t2 ln(x2) + ...+ tm ln(xm) (6.127)
wherem∑i=1
ti = 1 and 0 ≤ ti ≤ 1 ∀i ∈ [1,m] .
The equality follows if for some p, we have tp = 1. Using this, and (6.125) we can write,
−∑j
Pij pi ln qj ≥ −pi ln ri where ri =∑j
Pijqj . (6.128)
Hence we get,
S(ρ|σ) ≥∑i
pi ln
(piri
)= −
∑i
pi ln
(ripi
). (6.129)
Now note that, lnx ≥ x− 1. This gives
S(ρ|σ) ≥ −∑i
pi ln
(ripi
)≥ −
∑i
pi
(1− ri
pi
),
=∑i
(pi − ri) = 0 . (6.130)
Hence, S(ρ|σ) ≥ 0 and the equality follows when ρ = σ. To repeat, the only assumption that
went in the proof was the unitarity of the quantum theory. So, whenever we have a unitary
theory we can expect relative entropy to be positive.
CHAPTER 6. RELATIVE ENTROPY 125
B: Relative entropy for R2 theory in shockwave back-
ground
In this section we want to sketch the calculation for the relative entropy in shockwave back-
ground for a general R2 theory9 where the disturbance is located very far away from the
entangling surface. The action for this theory is shown below,
I =
∫d5x√G
(R +
12
L2+L2
2
(λ3R
2 + λ2RABRAB + λ1RABCDR
ABCD))
. (6.131)
In this case, f∞ satisfies 1− f∞+ 13f 2∞(λ1 + 2λ2 + 10λ3) = 0. We start of with the shockwave
metric as given in eq.(6.90) . We have explicitly checked that this is still a solution for the
R2 theory. Next we quote the area functional for this theory [6, 14, 36],
SEE =2π
`3P
∫d3x√h(1+
L2
2(2λ3R+λ2(RABn
Ai n
Bi −
1
2KiKi)+2λ1(RABCDn
Ai n
Bj n
Ci n
Dj −KiabKabi ))
).
(6.132)
Here i denotes the two transverse directions to the co-dimension 2 surface z = f(x1, x2, x3)
and t = 0 and Ki’s are the two extrinsic curvatures along these two directions pulled back
to the surface and a, b are three dimensional indices. Then we proceed in the same way as
before. We set z = z0 =√R2 − r2 . Also as before we set x′1 = 0 and without loss of any
generality and we will expand the integrand around x′2 = ∞ . First we expand upto O(ε3)
which is the linearized term and hence should yield ∆H = ∆S. The expression for ∆(1)S is
∆(1)S =16π2L5R4
15f5/2∞ `3
Px′62
(1 + 2f∞(λ1 − 2(λ2 + 5λ3)) . (6.133)
The λi dependence has packaged into being proportional to c for the general theory [?].
Using the results of [9] (eq.(6.29) in that paper with the typo mentioned in footnote 4 taken
into account), we find that ∆H = ∆S at this order as expected. Then we expand (6.132)
upto ε4 order and pick out the ε4 term which gives us the ∆(2)S . Note that for a general R2
theory the surface term is not known. So we can only do this calculation for the disturbance
located very far away from the entangling surface such that we do not have to consider the
perturbation to the entangling surface as this will contribute to some order higher than ε4.
Further since the extrinsic curvatures are each proportional to ε3 and hence the O(K2) terms
would be proportional to O(ε6), they will not contribute. The result before carrying out the
9The corresponding entropy functional will be useful in studying relative entropy in non-unitary log
CFTs–for recent applications for entanglement entropy in these theories, see [37].
126 REFERENCES
integrations is shown below,
∆(2)S =2π
`3P
∫dx3dx1dx2
[ 3L5
2f5/2∞ Rx′82
((x21 + x2
2 + x23 −R2)[R2(2f∞(23λ1 + λ2 − 10λ3) + 1)
− 60f∞λ1
(x2
1 + x22
)− x2
3(2f∞(23λ1 + λ2 − 10λ3) + 1)])].
(6.134)
Then we perform the integration over x3 which goes from −√R2 − r2 to
√R2 − r2 and
x1 = r cos(θ) , x2 = r sin(θ) . Now after some algebraic manipulation we can write the
integrand as,
∆(2)S = −48π2L5R6(1 + 2f∞(13λ1 + λ2 − 10λ3))
35f5/2∞ `3
Px′82
. (6.135)
Note that this is not proportional to c for this theory. Since for generic values of the couplings
λi, the bulk theory is non-unitary this may not be surprising. This may be indicative of the
fact that rather than depending only on the two point function of the stress tensor, the
higher point functions also contribute as in the second reference in [26]. The bulk theory will
make sense as an effective theory where the couplings are small. In this circumstance, we
can use field redefinitions to make the theory equivalent to Gauss-Bonnet with λ ∝ λ1. For
the Gauss-Bonnet value λ1 = λ3 = λ , λ3 = −4λ it reduces to,
∆(2)S = −48π2L5R6(1− 2f∞λ)
35f5/2∞ `3
Px′82
, (6.136)
which is proportional to c for the GB theory.
References
[1] M. A. Nielsen, I. L. Chuang. Quantum Computation and quantum Information, Cam-
bridge Univ. Press., Cambridge
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7 Coding holographic RG flow using
entanglement entropy
7.1 Introduction
In this chapter we will briefly discuss an application of holographic entanglement entropy
(EE). We have observed throughout this thesis that, this quantity is sensitive to UV physics
and the leading divergence obeys the area law. This indicates that the EE is not a well
defined observable in the continuum limit.
In 2d, Casini and Huerta [1] devised a method to extract the universal contribution to the
entanglement entropy. Liu and Mezei [2, 3] generalized this prescription to higher dimensions.
The resulting quantity, known as the renormalized entanglement entropy (REE), is UV finite
and local on the scale of the entangling region. The evolution of REE with respect to the size
of the entangling region can be used as a probe to realize the RG flow. Moreover, for vacuum
states the REE for a spherical entangling region provides a c-function parametrizing the RG
flow 1. We want to compute the REE for states that break Lorentz invariance due to the
presence of a finite charge density. While holographic RG flows for vacuum states correspond
to domain walls (i.e. solutions interpolating between same dimensional AdS) [12, 13, 14], the
flow for charged states can be described in terms of black holes/branes.
Specifically, we want to study the REE for BPS black solutions in N = 2, 4d FI gauged
supergravity. Starting with the work of Cacciatori-Klemm [15] these models have been stud-
ied extensively over the last few years [16, 17, 18], culminating with the full construction of
static BPS solutions for all symmetric models by [19, 20]. These are solutions that interpo-
late between AdS4 and AdS2 × Σk, where Σk is the surface of constant sectional curvature
with k = −1, 0, 1. Since these objects interpolate between different AdS spaces they are
interesting from the holographic perspective. Morever, for the STU model these solutions
have a M-theory realization through an embedding into the de Wit-Nicolai N = 8 theory
[21]. This chapter is organized as follows. In section (7.2) we outline the computation for the
REE for the black brane solutions. In section (7.3), we summarize the BPS black objects in
1 For more applications of REE in the context of holographic RG flows, interested readers are referred to
some of these references [4, 5, 6, 7, 8, 9, 10, 11] and the references there in.
131
132 7.2. RENORMALIZED ENTANGLEMENT ENTROPY
AdS4, followed by the computation for the REE in section (7.4). Finally, in Appendix (7.5),
we discuss the symplectic invariant in N = 2. This chapter is based on the work [22] done
with Dr. Shajid Haque and Dr. Alvaro veliz-Osorio.
7.2 Renormalized Entanglement Entropy
In this section we outline a general procedure to obtain the universal contributions to the
entanglement entropy for quantum systems that can be described holographically by a metric
of the form
ds2 = −a2dt2 + a−2dr2 + b2dΩ2k. (7.1)
In the above expression, dΩ2k is the line element of a surface of constant sectional curvature
k. Clearly we must demand AdS asymptotics. Hence, as r →∞ the metric takes the form
a→ r
l4b→ r
l4. (7.2)
We wish to compute the entanglement entropy for a subsystem A consisting of a disk Σ(R) of
radius R. From the Ryu-Takayanagi prescription [28], we know that this quantity corresponds
to the area of an extremal surface attached to ∂Σ(R) going into the bulk (see Fig. 7.1). For
the metric (7.1) this problem corresponds to the Plateau problem for the functional
S(R) =2π
l2p
∫ R
0
dρ ρ b2√
1 + e−2ψr2 eψ ≡ ab. (7.3)
In the subsequent calculations we will absorb the factor 2πl2p
into S(R). From the above
functional, it can be showed that the profile r(ρ) of the minimal surfaces can be found from
the ODE
r − ψ′ r2 +
(r
ρ− 2
b′
be2ψ
)(1 + e−2ψ r2
)= 0, (7.4)
where ˙ = ∂ρ and ′ = ∂r. This equation is supplemented with the boundary conditions
r(0) = r0 > 0 r(0) = 0. (7.5)
Solutions of equations (7.4) and (7.5) correspond to extremal surfaces attached to the bound-
ary of a disk at infinity and whose tip is at r = r0. Moreover, the depth of the tip can be
related to the size R of the entangling disk at the boundary, R corresponds to the value
of ρ for which r(ρ) → ∞. Therefore, each surface can be labeled either by the size of the
entangling region R or by the depth it reaches in the bulk r0 (see Fig. 7.1).
Once we have found the profile r(ρ) of the minimal surface, we are instructed to plug it
into the functional (7.3) in order to obtain the holographic entanglement entropy. However,
CHAPTER 7. CODING HOLOGRAPHIC RG FLOW USING ENTANGLEMENTENTROPY 133
Boundary
r
R3 R2 R1> >
r3
r2
r1Horizon
Ρ
Figure 7.1: Minimal surfaces in AAdS
one must be careful since the resulting quantity is divergent. We should, therefore, regularize
this integral first. We introduce a UV cut-off in the following way–let ε 1 and restrict the
values of r such thatl4b(r)
> ε. (7.6)
Using r(ρ), we can translate this bulk cut-off into a boundary cut-off, i.e., we must consider
only ρ < Rε, wherel4
b(r(Rε))= ε. (7.7)
Then we can compute the finite quantity
S(R, ε) =
∫ Rε
0
dρA(ρ,R), (7.8)
where A(ρ,R) stands for the integrand of (7.3) evaluated on the solution r(ρ).
In order to systematically obtain the universal contribution to (7.8), which we call here-
after renormalized entanglement entropy, we use the operator introduced in [1, 2]. In four
bulk space-time dimensions, the renormalized entanglement entropy is given by
S(R) ≡(Rd
dR− 1
)S(R, ε). (7.9)
This quantity can be alternatively written as
S(R) = A(R,Rε)
(R∂Rε
∂R
)+
∫ Rε
0
(R∂A∂R−A
). (7.10)
134 7.3. BPS BLACK OBJECTS IN ADS4
As an illustration let us briefly discuss the application of this procedure for pure AdS4.
For this geometry equation (7.4) yields
r(ρ) =l24√
R2 − ρ2. (7.11)
This corresponds to an extremal surface reaching into the bulk until r = l24R−1. Using equa-
tion (7.7) we find
Rε =
√R2 − ε2
2. (7.12)
For this simple case we can compute equation (7.10) explicitly. First we find
A(R, ρ) =Rρ
(R2 − ρ2)3/2, (7.13)
and then ∫ Rε
0
dρA(R, ρ) =R√
R2 −R2ε
− 1, (7.14)
which diverges as we take ε → 0. However, the additional terms in (7.10) contribute as
follows:
A(R,Rε)∂Rε
∂R=
R2
(R2 −R2ε )
3/2∫ Rε
0
dρ∂A∂R
= − R2ε
(R2 −R2ε )
3/2. (7.15)
So we find that for AdS4
S(R) = 1. (7.16)
This result is consistent with the interpretation of the REE as a c-function probing the
holographic renormalization group flow [14, 29, 30, 31, 32].
7.3 BPS black objects in AdS4
We wish to apply the techniques presented in the previous section to an interesting class of
solutions of the form (7.1), namely 14-BPS black objects in N = 2, FI gauged supergravity.
These correspond to zero temperature solutions supported by scalars and abelian gauge fields.
They are parametrized by two vectors of 2nv + 2 real quantities
Γ =
(pI
qI
)and G =
(gI
gI
), (7.17)
where pI and qI are the magnetic and electric charges of the gauge fields while gI and gI are
the parameters of the Fayet-Iliopoulos potential. In (7.17) the index I = 1, . . . , nv + 1, where
CHAPTER 7. CODING HOLOGRAPHIC RG FLOW USING ENTANGLEMENTENTROPY 135
nv is the number of vector multiplets considered. These parameters are not independent. In
fact, they are subject to the symplectic constraint
〈G,Γ〉 = k, (7.18)
where just as in (7.1), k labels the horizon topology, e.g., k = −1, 0, 1 indicates spherical,
flat and hyperbolic respectively.
In the following discussion we focus only on the properties of the spacetime metric and
leave the behavior of the scalars and gauge fields aside. In terms of the quantities (7.17), the
warp factors are given by [19, 20]
eψ = a b =(I4(G)1/4 r + 〈G,B〉
)r b =
1
2I4(H)1/4, (7.19)
where I4(V ) is defined in equation (7.33) in the Appendix, and H is a symplectic vector of
linear functions
H = Ar +B. (7.20)
Here, A and B are constant symplectic vectors. The former can be obtained directly from
the FI parameters
A =1
2I4(G)−3/4dI4(G), (7.21)
while the latter is given by a combination of the charges and FI parameters, dictated by the
algebraic equation1
4dI4(B,B,G) = Γ, (7.22)
where dI4 is defined in equation (7.35) in the Appendix. Moreover, physical consistency
requires to choose solutions of (7.22) that fulfill the constraints
〈G,B〉 > 0 I4(B) > 0. (7.23)
Therefore the construction of a BPS solution is reduced to a purely algebraic problem.
The warp factors (7.19) correspond to a metric that interpolates between AdS4 at infinity
(UV) and an AdS2×Σ2k near horizon (IR) geometry as r → 0. The AdS radii corresponding
to these UV and IR geometries are given by
l4 = I4(G)−1/4 l2 =1
2I4(B)1/4〈G,B〉−1. (7.24)
Moreover, the entropy (density for k=0) is proportional to
b2 IR−−−→ σ20 =
1
4
√I4(B). (7.25)
At this point we want to remind the reader that these solutions are also accompanied by
flowing scalars. Due to the attractor mechanism [16, 17] the scalars flow from constant to
constant. These scalars can be thought of as coupling constants and giving rise to the notion
of an attractive RG flow [23].
136 7.4. REE FOR BPS BLACK BRANES
7.4 REE for BPS black branes
In this section we compute the renormalized entanglement entropy as discussed in section
(7.2) for solutions of the kind presented in section 7.3. In the following computation we
restrict ourselves to black objects with flat horizons (k = 0), i.e., black branes. Furthermore,
we will consider solutions of the STU-model. This model captures the essential features
of extremal black holes in N ≥ 2, d = 4 theories [26]. In the STU-model, the structure
constants for the prepotential (see Appendix 7.5) are given by cijk = |εijk|. By plugging the
warp factors (7.19) into equation (7.4), we obtain an explicit, albeit complicated, differential
equation for r in terms of ρ. Now we need to set the parameters that will support the
solution i.e. charges/FI parameters. To each such charge/FI configuration we can associate
three symplectic invariant combinations, which correspond to the AdS length scales and
the entropy density. As we will see in the following our results depend only on these three
quantities. In order for the solution to be regular these quantities must not vanish and we
choose the charges/FI parameters accordingly.
Hereafter we will consider solutions supported by non-vanishing charges/FI parameters
(q0, pi; g0, gi) i = 1, . . . , nv + 1 (7.26)
or
(p0, qi; g0, gi) i = 1, . . . , nv + 1. (7.27)
In the following discussion we will display the results for the first configuration (7.26). For the
second configuration we have verified explicitly that we get completely analogous results. It is
straightforward to extend the following discussion to other configurations as well. Moreover,
the reader must keep in mind that the results that follow are invariant under symplectic
transformations of the kind discussed in [25, 26].
Given the intricacy of the resulting ODE describing the extremal surfaces’ profile, we
proceed to solve it numerically. Moreover, in order to realize the program outlined in section
7.2 we are compelled to produce a large sample of such minimal surfaces ri(ρ) with i being an
index for the sample (see Fig. 7.2). Now by introducing a cut-off ε it is possible to compute
numerically (7.8), creating thus a list of regularized areas Si(ε) corresponding to each of
the extremal surfaces ri(ρ). Hence we are left with a list of points (Ri, Si(ε)), which can be
interpolated to find S(R, ε). Finally, from this function we construct the desired renormalized
entanglement entropy S(R) for a given set of charges/FI parameters. We must point out that
by construction S(R) is a cut-off independent quantity. This behavior is exhibited by our
numerical computations as we tune ε to ever smaller values.
The resulting REE is depicted in Fig. 7.3 for a particular example. However the observed
behavior is generic regardless of the values chosen for the parameters (7.26). First of all, as
CHAPTER 7. CODING HOLOGRAPHIC RG FLOW USING ENTANGLEMENTENTROPY 137
0.0 0.2 0.4 0.6 0.80.0
0.1
0.2
0.3
0.4
0.5
r
Ρ
R4 R3 R2 R1
r4
r1
Horizon
Asymptopia
r(Ρ)
:
Figure 7.2: Some extremal surfaces for a p1 = 2, p2 = p3 = 1 and −g0 = g1 = g2 = g3 = 1
black brane
S(R)→ 1 as R→ 0 in agreement with (7.16), the REE then decreases monotonically until it
reaches a minimum S∗ when the entangling disk has a radius R∗. After reaching that critical
value, S(R) starts to increase and approaches the value σ0 as we get closer to the horizon.
0.2 0.4 0.6 0.8
0.5
0.6
0.7
0.8
0.9
UV IR
S(R)
Σ0
RRmin
Smin
Figure 7.3: Renormalized entanglement entropy for a p1 = 2, p2 = 1, p3 = 1/2 and −g0 =
g1 = g2 = g3 = 1 black brane
138 7.5. DISCUSSION
We wish to explore how the values of S∗ and R∗ depend upon the charges/FI parameters
Γ and G. First of all, it is clear that these parameters must enter only through symplectic
invariant combinations. The warp factors (7.19) can be specified in terms of the invariant
quantities I4(G), 〈G,B〉, I4(A), I4(B). From equation (7.24, 7.25) we can identify them as
l2, l4, σ0, I4(A). But a closer inspection shows that I4(A) can be expressed in terms of l4.
Hence, there are three independent invariants upon which S∗ depends, namely the AdS radii
and the entropy density. In order to find a pattern, we start by identifying a subclass of
parameters for which one of the symplectic invariants is held fixed. One such family is given
by
g1 = g2 = g3 p1 = p2 = p3. (7.28)
The crucial point here is that for black branes with these kind of parameters the near-
horizon AdS2 radius l2 is independent of the value of p1. Let’s see how this comes about. For
charge/FI combinations of the form (7.28) the solution of equation (7.22), consistent with
the conditions (7.23), reads
B1 = B2 = B3 = λ1 sgn(g0)
√p1
g1
, B0 = λ2 |g0|−1√p1 g1. (7.29)
Here λ1 and λ2 are known positive constants, and the components of B that are omitted
vanish. The upshot is that B is proportional to√p1. Therefore, since l2 in (7.24) is invariant
under rescalings of B, it is clear that the p1 dependence drops out.
Now we fix the overall scale l4, and study the behavior of S∗. In this context, changing p1
gives rise to a one parameter family of solutions with constant l2 and varying σ0. Interestingly,
we find that S∗ is constant along this family, which implies that S∗ is a function of l2 only.
Notice that in the regime R = 0 to R = R∗ the REE decreases monotonically from the
pure AdS4 value to a constant which can be determined solely from the AdS2 radius. This
is reminiscent of the c-function discussed in [23]. It would be interesting to investigate this
connection further.
Then we explore how S∗ varies with l2. The variation is displayed in Fig (7.4). This plot
clearly shows that S∗ increases with l2. On the other hand, R∗ depends on both l2 and σ0.
Moreover, it increases with l2 and decreases with σ0.
7.5 Discussion
In this chapter we have computed the renormalized entanglement entropy, S for 4d, N = 2
BPS black brane solutions. These solutions interpolate between AdS4 in the UV and a space
with an AdS2 factor in the IR. Specifically, we have investigated the behavior of S as a
CHAPTER 7. CODING HOLOGRAPHIC RG FLOW USING ENTANGLEMENTENTROPY 139
0.07 0.08 0.09 0.10 0.11 0.12
0.30
0.35
0.40
0.45
S*
l2
Figure 7.4: Variation of S∗ with l2
function of the size of the entangling region. We have found that in this context S first
decreases monotonically with R, reaches an extremum and then increases again. This is the
key finding of our investigation.
We have already mentioned in the main text that all the parameters of the brane solutions
enter by three independent symplectic invariant combinations that can be identified as the
three independent scales of the system, namely l4, l2 and σ0. In our inspection we found
that starting from the UV the S monotonically decreases until it reaches a minimum that is
determined completely by the radius of AdS2. Then it starts to increase again and approaches
the black brane entropy density. Furthermore, as pointed out in the previous section, when
we increase the entropy of the brane this transition occurs closer to the UV. The fact that
the S decreases monotonically for that region in R resembles the behavior of a c-function for
vacuum states. Then the chemical potential starts to dominate once we go deeper in the IR.
Entanglement entropy measures the entropy due to tracing out part of the total system.
If the total system is in a mixed state this quantity receives contributions both from en-
tanglement and from the mixedness of the original system. Since black branes correspond
to mixed states we expect our computation to be influenced by both of these factors. In
light of that, we are inclined to interpret our result in the following way–the REE is driven
predominantly by entanglement close to the UV before reaching R∗, where contributions due
to the mixedness of the branes take over.
140 7.5. DISCUSSION
Appendix
A: Duality transformations for N = 2 gauged supergrav-
ity
Supergravity solutions can be easily written in terms of symplectic vectors. These are vectors
with 2nv + 2 components of which the first nv + 1 components are labeled with an upper
index and the remaining ones with a lower index, e.g.,
V =
(V I
VI
). (7.30)
I = 0, . . . , nv. These vectors are acted upon by symplectic transformations(V I
VI
)=
(U Z
W V
)(V I
VI
), (7.31)
where
UTV −W TZ = V UT −WZT = 1
UTW = W TU , ZTV = V TZ. (7.32)
We refer to these reparametrizations as duality transformations [25].
In N = 2 models, physical quantities must be invariant under duality transformations.
Duality invariant quantities can be succinctly expressed in terms of the symplectic quartic
invariant I4 [27]
I4 (V ) =1
4!tMNPQVM VN VP VQ
= −(V IVI
)2+
2
3V0 cijkV
iV jV k − 2
3V 0 cijkViVjVk
+ cijk clmnV iV jVlVm. (7.33)
In this expression and the ones to follow, the lower-case indices run from 1 to nv only. The
constant coefficients cijk encode the underlying special geometry prepotential
F = −1
6cijk
X iXjXk
X0. (7.34)
For future convenience, we also define
dI4(V ) = ΩMN∂I4(V )
∂VN, (7.35)
REFERENCES 141
with
ΩMN =
(0 I−I 0
), (7.36)
the canonical symplectic matrix. Moreover, given four symplectic vectors we define
I4
(V (1), V (2), V (3), V (4)
)= tMNPQ V
(1)M V
(2)N V
(3)P V
(4)Q . (7.37)
Notice the absence of the overall symmetrization factor with respect to (7.33). In practice,
we can obtain the t-tensor by hitting (7.33) with four derivatives. The black hole solution is
determined by a set of electric/magnetic charges and fluxes.
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8 Conclusions
In this thesis we have tried to understand the interplay between quantum information and
geometry through entanglement entropy (EE) in the framework of AdS/CFT correspondence.
It has provided us with a nice geometrical problem and has the merit to shed light on how
holography works.
Ryu-Takayanagi (RT) proposal has provided us a way to compute EE from holography
for Einstein gravity. In this thesis we have discussed various generalization of RT proposal
for higher derivative gravity theories. These higher derivative theories are interesting as they
extend the AdS/CFT dictionary beyond supergravity limit and incorporate the effect of finite
coupling. Recently Lewkowycz and Maldacena (LM) have proposed to set this calculation
in a self consistent way, solidifying its geometrical connection. Their idea stems from the
fact that there exists a striking similarity between EE and the black hole entropy. Using LM
technique one can formulate a proof for RT proposal and but unfortunately this cannot be
generalized for more general theories of gravity, it seems that it only works for certain class
of gravity theories. It will be nice find a way to overcome this difficulty in future.
On the other hand there is a well defined way of deriving the black hole entropy for any
diffeomorphism invariant theory based on the Nother charge method proposed by Iyer and
Wald. Based on this Noether charge construction Wald has proposed a general formula for
black hole entropy for any diffeomorphism invariant theory. For Einstein gravity both the
black hole entropy and EE follows area law. Now the question is, can one simply use Wald
entropy functional to compute EE for any general theories of gravity? The answer is no,
as Wald entropy functional is valid only for bifurcation surface and for general surface one
has to systematically modify it to get correct EE as predicted form the holography. Now it
seems that one can connect EE with this Noether charge method and in this thesis we have
partially achieved that, although still a rigorous proof is needed. Recently it has been shown
that indeed one can connect EE with the Noether charge construction by demanding that the
entropy functionals used to evaluate EE for general theories of gravity satisfy second law of
thermodynamics (a generalized version of it). So it will be very nice to make this connection
more rigorous in the future as it will provide a solid geometrical interpretation for EE .
Further we have used holographic entanglement to constrain gravity in interesting ways.
First, we have used positivity of relative entropy and analysed the inequality ∆S ≤ ∆H
145
146
for Gauss-Bonnet gravity for a given class of small perturbations around the vacuum state.
We found that for all our examples, the positivity of the stress tensor two point function
ascertained that this was respected. On the bulk side this corresponds to metric fluctuations
having positive energy. The simplicity of the final result does cry out for a simpler explanation
for our findings. Although the intermediate integrals involved appeared very complicated,
the final result was simply proportional to the Weyl anomaly c. It would be nice to find a
simple explanation for this finding. Unfortunately the relative entropy calculation is limited
to the spherical entangling surface as for this kind of surface only, the modular hamiltonian
(H) is known.
Finally, we also considered other entangling surfaces and demanded that these close off
smoothly in the bulk. In Gauss-Bonnet gravity, this led to the coupling being constrained.
The spherical entangling surface did not lead to any constraints on the coupling while the
cylindrical and slab entangling surfaces did. This leads to an interesting question. Suppose
we knew how to extend the relative entropy results for the spherical entangling surface to
other surfaces. Then the smoothness criteria above seems to constrain the coupling of the
higher derivative interaction. This suggests that implicitly the relative entropy inequality
knows about this. Since the positivity of relative entropy seems to rely only on the unitarity
of the field theory, this raises the question if there is any conflict with unitarity if one is
outside the allowed region for the coupling. It will be interesting to investigate this question
since apriori there does not appear to be any such conflict in the dual gravity. It will also be
interesting to find if there are other entangling surfaces which lead to a tighter bound and if
the bounds are stronger than the causality constraints.
Then at the end we have discussed how to code holographic RG flow using EE. We have
investigated the behaviour of renormalized entanglement entropy (REE) in the context of a
lorentz violating RG flow. We have found that REE first monotonically decreases and then
increases smoothly along the RG flow, thus exhibiting a minima. It would be interesting to
see if we can use the behavior of REE as a function of entangling surface in order to establish
an order parameter for the phase transition between the vacuum to vacuum flow and vacuum
to charged state flow. This is because the existence of the extremum for the REE tells us
that the system is transiting from its vacuum behavior at that point.
Form all this analysis it is quite evident that entropy functionals used to evaluate EE
play a central role. In recent times, many different tools of quantum entanglement like,
entanglement negativity, differential entropy, quantum error coding, relative entropy, cMERA
etc, have been used in an attempt to build geometry from the field theory data, is some
sense trying to prove the holographic principle. In the light of these recent advances, these
holographic entropy functionals play a crucial role and a thorough understanding of these
entropy functionals will teach us many important lessons about gravity.