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Flux formulation of Double Field Theory. Quantum Gravity in the Southern Cone VI Maresias , September 2013 Carmen Núñez IAFE-CONICET-UBA. Outline. Introduction to Double Field Theory Applications - PowerPoint PPT Presentation
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Flux formulation of Double Field Theory
Quantum Gravity in the Southern Cone VI Maresias, September 2013
Carmen NúñezIAFE-CONICET-UBA
Outline
• Introduction to Double Field Theory
• Applications
• Flux formulation
• Double geometry
• Open questions and problems
• Work with G. Aldazabal, W. Baron, D. Geissbhuler, D. Marqués, V. Penas
T-duality
• Closed string theory on a torus Td exhibits O(d,d) symmetry
• Strings experience geometry in a rather different way to point particles.• T-duality establishes equivalence of theories formulated on very different
backgrounds
• Is there a more appropriate geometrical language with which to understand string theory ?
kk
kjkikjkiijij
kk
kiki
kk
kjkikjkiijij
kk
kiki
kkkk
GGBBG
BBGGB
GBBGG
GGGBG
GG
,
,,,1
• DFT is constructed from the idea to incorporate the properties of T-duality into a field theory
• Conserved momentum and winding quantum numbers have associated coordinates in Td
• Double all coordinates• Every object in a duality invariant theory must belong to some representation
of the duality group. In particular, xi have to be supplemented with
XM fundamental rep. O(D,D)
• Raise and lower indices with the O(D,D) metric
• Introduce doubled fields and write
with manifest global O(D,D) symmetry
DMii
Mi
iM
xx
X 2,...,1,~,~
DOUBLE FIELD THEORY
Dixwxp iii
i ,...,1,~,
MNMN
0110
)~,( ii xx )~,(~ xxxxddS DD
DFT L
daxwxp aaa
a ...,,2,1,~,
ix~
Field content
• Focus on bosonic universal gravity sector Gij, Bij,
• Fields are encoded in a
2D × 2D GENERALIZED METRIC
, O(D,D) INVARIANT GENERALIZED DILATON
ijk
ijkiiD
sugra HHReGxdS12142
2 eGd
),( DDOBGBGGB
BGG
ljkl
ikijkj
ik
kjikij
MN
H MNQNPQ
MP HH
The generalized metric spacetime action Hull and Zwiebach (2009)
O. Hohm, C. Hull and B. Zwiebach (2010)
),(~ 2 dexxddS dDDDFT HR
NLKKL
MMN
KLNKL
MMN
NMN
MNMMNMN
NMNMMN ddddd
HHHHHH
HHHHH
21
81
444),(R
0~
ijk
ijkiiD
sugra HHReGxdS12142
• DFT also has a gauge invariance generated by a pair of parameters
• Gauge invariance and closure of the gauge algebra lead to a set of differential constraints that restrict the theory. In particular, the constraints can be solved enforcing a stronger condition named strong constraint
O(D,D) symmetry is manifest
)~( ,i
iM
Strong constraint
• All fields, gauge parameters and products of them satisfy
• It implies there is some dual frame where fields are not doubled• Strongly constrainted DFT displays the O(D,D) symmetry but it is not physically doubled• Gauge invariance and closure of gauge transformations weaker condition• Certain backgrounds allow relaxations of the strong constraint, producing a
truly doubled theory:
– Massive type IIA O. Hohm, S. Kwak (2011)– Suggested by Scheck-Schwarz compactifications of DFT G. Aldazabal, W. Baron, D. Marqués, C.N. (2011) D. Geissbhuler (2011)– Sufficient but not necessary for gauge invariance and closure of gauge algebra
M. Graña, D. Marqués (2012)– Explicit double solutions found in D. Geissbhuler, D. Marqués, C.N., V. Penas (2013)
0)]~,()~,([ xxBxxAMM
)'~,'( xx
Applications of DFT
• DFT has been a powerful tool to explore string theoretical features beyond supergravity and Riemanian geometry
• Some recent developments include:– Geometric interpretation of non-geometric gaugings in flux compactifications of
string theory G. Aldazabal, W. Baron, D. Marqués, C.N. (2011) D. Geissbhuler (2011)– Identification of new geometric structures D. Andriot, R. Blumenhagen, O. Hohm, M. Larfors, D. Lust, P. Patalong (2011, 2012)– Description of exotic brane orbits F. Hassler, D., Lust (2013) J. de Boer, M. Shigemori (2010, 2012), T. Kikuchi, T. Okada, Y. Sakatani (2012)– Non-commutative/non-associative structures in closed string theory R. Blumenhagen, E. Plauschinn, D. Andriot, C. Condeescu, C. Floriakis, M. Larfors, D. Lust , P. Patalong (2010-2012) – New perspectives on ‘ corrections, O. Hohm, W. Siegel, B. Zwiebach (2012,2013)– New possibilities for upliftings, moduli fixing and dS vacua, Roest et al. (2012)
10D string sugra
SS reduction
on twisted T6
4D gauged sugra T-duality 4D gauged sugra geometric fluxes in d-dim all dual (geometric & ab
c & Habc non- geometric) gaugings Moduli fixing & dS vacua
T-duality
in D-dimDouble Field Theory
SS reductionon twisted T6,6???
Application I: Missing gaugings in geometric compactifications (see Aldazabal’s talk)
Application II: New geometric structuresNon geometry, Generalized Geometry
• Diffeomorphisms of GR
and gauge transformations of 2-form are combined in generalized diffeomorphisms and Lie derivatives
; New term needed so that
• Gauge transformations
The action of the generalized metric formulation is gauge invariant because R(H,d) is a generalized scalar under the strong constraint
L
BLBBGLGG ijijijijijij
,,
ijjiijij BB ~~
)~( ,i
iM PM
PPMMP
PM AAA )(ˆ L
)(,ˆ 22 dMM
dMNMN ee HH L
RR MM
0ˆ MNL
DFT vs Generalized Geometry
• The double geometry underlying DFT differs from ordinary geometry. • DFT is a small departure from Generalized Geometry (Hitchin, 2003; Gualtieri, 2004)
• Given a manifold M, GG puts together vectors Vi and one-forms i as
V + TM T*M . Structures on this larger space The Courant bracket generalizes the Lie bracket
V and are not treated symmetrically
• DFT puts TM and T*M on similar footing by doubling the underlying manifold. Gauge parameters and then C-bracket
For non-doubled M the C-bracket reduces to the Courant bracket
)(21],[],[ 1212212211 2121
VVVV iidVVVV LL
)~( ,i
iM
PMPM
NNM
C ]21[]21[21 21],[
Geometry, connections and curvature
• The action was tendentiously written as It can be shown that the action and EOM of DFT can be obtained from
traces and projections of a generalized Riemann tensor RMNPQ
• The construction goes beyond Riemannian geometry because it is based on generalized rather than standard Lie derivatives
• The notions of connections, torsion and curvature have to be generalized
• E.g. the vanishing torsion and compatibility conditions do not completely determine the connections and curvatures, but only fix some of their projections I. Jeon, K. Lee, J. Park (2011), O. Hohm, B. Zwiebach (2012)
• Strong constraint was assumed in these constructions. Can it be relaxed?
),(~ 2 dexxddS dDDDFT HR
• Basic fields are generalized vielbeins EAM and dilaton
• EAM can be parametrized in terms of vielbein of D-dimensional metric
D-dimensional Minkowski metric
• Arrange the fields in dynamical fluxes:
• Field dependent and non-constant fluxes, that give rise to gaugings or constant fluxes upon compactification (e.g. Fabc=Habc)
Flux formulation of DFT W. Siegel (1993) D. Geissbhuler, D. Marqués, C.N., V. Penas (2013)
NB
ABMA
MNNB
ABMA
MN EEESE ;H
dE
dM
MAAN
NB
MM
BA
MBE
MCNC
NBM
MAABC
eedEEEEF
EEEEEF
A
A
22
][
2
3
LL
abjb
abia
ij seseG ,
ab
ab
ABi
aki
ka
ia
MA
ss
Se
BeeE0
0,
0
ABCABC
AA
AM
MA
CFBEADCFBEADDEFABCBABM
MA
AB
FFFFFE
SSSSFFFFFES
612
121
412
R
),(~ 2 dEexxddS dDDDFT R
Vanishes under strong constraint
Generalized metric DFT action modulo one strong constraint violating term
The action
The action takes the form of the electric sector of the scalar potential of N=4 D=4 gauged supergravity
This action generalizes the generalized metric formulation, including all terms that vanish under the strong constraint
PA
MPP
MMA
PP
MAM
MMM EEEdd ,
21
Generalized diffeomorphisms
• The closure constraints (generalized Lie derivatives generate closed transformations) take the form:
and they asure that FABC, FA transform as scalars and S is gauge invariant
• Imposing these conditions only requires a relaxed version of strong constraint the theory admits truly double fields
• Constraints can be interpreted as Bianchi identities for generalized Riemann tensor
M
MBB
CABC
NBN
AMM
CABC
BACABC
AB
CDE
ABEECDE
ABBCDAABCD
ED
dDEE
FFFDFDZ
FFFDZ
,02
2
043
43
][
][
][][][
Geometric formulation of DFT
• Define covariant derivative on tensors
• Determine the connections imposing set of conditions:
– Compatibility with generalized frame:
– Compatibility with O(D,D) invariant metric
– Compatibility with generalized metric
– Covariance under generalized diffeomorphisms: – Covariance under double Lorentz transformations: Lorentz scalar– Vanishing generalized torsion: Standard torsion non covariant
– Compatibility with generalized dilaton
Only determine some projections of the connections
KB
BMA
NA
KMN
KAM
KAM VVVV
BMA
NBK
ANMK
NMK
NAM EEE 0
MBAMABMPNMNP
ABM
NPM
00
00 ABMNKM SHC
ABPPC
AB M
AMV
ABCABCABC
PQMQP
M
F
VV
][3
)(
ABAB
MP
PM Fd 2
Generalized curvature
• The standard Riemann tensor in planar indices is not a scalar under generalized diffeomorphisms
• It can be modified adding new terms, leading to
• Projections with give and similarly EOM
• Bianchi identities
KLQ
QMNKLQ
QMNKLMNMNKLMNKL RR R
)(21
MNMNMNP H MNKLNLMK PP RR
ABCDECDE
ABBCDAABCD ZFFFD34
34
][][][ R
MKNLKL
MN P RR
Scherk-Schwarz solutions
• All the constraints can be solved restricting the fields and gauge parameters as
where and
quadratic constraints of N=4 gauged sugra
• For these configurations all the consistency constraints are satisfied.• The dynamical fluxes become:
• This ansatz contains the usual decompactified strong contrained case (U=1, =0, xi, i=1,…, D). It is a particular limit in which all the compact dimensions are decompactified.
)()(ˆ,)()(ˆ)( YxddYUxXE MI
IA
MA
)()(ˆ)()( YUxxX MI
IA
AM
),~(;),;~,~( yyYyxyxX
0
2
3
][
][
KLH
IJH
MM
IINJN
MM
JI
NKN
JMM
IIJK
ff
UUUUf
constUUUf
KC
JB
IAIJKABCABC fFF ˆˆˆˆ
Conclusions
• Presented formulation of DFT in terms of dynamical and field dependent fluxes.
• The gauge consistency constraints take the form of quadratic constraints for the fluxes, that admit solutions that violate the SC allows to go beyond supergravity
• Computed connections and curvatures on the double space under assumption that covariance is achieved upon generalized quadratic constraints, rather than SC, which can be interpreted as BI.
• Interestingly, this procedure gives rise to all the SC-violating terms in the action, which are gauge invariant and appear systematically
• This completes the original formulation of DFT, incorporating the missing terms that allow to make contact with half-maximal gauged sugra, containing all duality orbits of non-geometric fluxes (FABCFABC).
Open questions
• Some elements of the O(D,D) geometry have been understood, but it is important to better understand the geometry underlying DFT
• Can this construction be extended beyond tori? Calabi-Yau?
• ’ corrections. Inner product and C-bracket are corrected deformation of Courant bracket and other structures in GG
• Beyond T-duality? U-duality?
• Relation between DFT and string theory. Is this a consistent truncation of string theory? No massive states, but fully consistent
• Worldsheet theory?
THANK YOU