World-sheet conformal interfaces

and their applications

Yuji Satoh

(University of Tsukuba)

Introduction

● conformal (world-sheet) interface :

natural extension of conformal boundary

D-brane

cond. mat. w/ boundary??

CFT CFT1 CFT2

boundary interface (defect)

1

● originated from

・ condensed matter w/ defect

・ twisted partition fn.[Won-Affleck ’94; Oshikawa-Affleck ’96]

[Petkova-Zuber ’00]

● may play an interesting role in

・ conformal field theory (CFT)

・ condensed matter phys.

・ string theory

● they have interesting properties

In fact,

[discussed shortly]

2

● world-sheet conformal interfaces :

expected to play a fundamental role in string theory

● conformal invariance : a guiding principle

of world-sheet approach to string theory

cf. conformal boundary ⇔ D-brane

● interfaces for string theory : yet to be explored

3

In this talk, we

● review properties of world-sheet conformal interfaces

● discuss applications to

・ renormalization group (RG) flows

・ entanglement entropy (EE)

・ non-geometric backgrounds in string theory

...

w/ Sakai ’08

w/ Sugawara, Wada ’15

4

Plan of talk

1. Introduction

2. World-sheet conformal interfaces

3. Their properties

4. Their applications

5. Summary

5

2. World-sheet conformal interfaces

World-sheet conformal interfaces

● consider 1 dim. defect/interface in 2dim. world-sheet

● condition to keep conformal Ward id.

[along interface]

T1(z)� T̃1(z̄) ⇥ T2(z)� T̃2(z̄)

CFT1 CFT2

x

x

z

・ conformal trans. preserving shape of interface : I ⇥✏(z)T (z)dz � ✏(z̄)T̄ (z̄)dz̄

⇤

・ in deriving Ward id, contributions from broken

T � T̄segments cancel if is continuous

6

● when

・ called topological interface

� ・ interface : freely deformed

T1(z) � T2(z) , T̃1(z̄) � T̃2(z̄)

Topological conformal interfaces

7

Simple example 1 - topological interfaces in RCFT -

[Petkova-Zuber ’00]

● CFT1 = CFT2 = rational CFT

● consider partition fn. twisted by interface operator II

● consistency [analog of Cardy cond. for boundary states]

� Ia =X

k

S⇤ak

S0kP kk̄

projector to rep. (k, k̄) [analog of Ishibashi states]

S: modular S-matrix (0: vacuum)

: labeled by primary

P kk̄ =�

n,n̄

�|k, n� � |k̄, n̄�

���k, n| � �k̄, n̄|

�

● analog of Cardy/Ishibashi states

8

In fact,

● modular trans. of characters

● Verlinde formula

[ fusion coeff. (integer) ]�j ⇥ �k = N l̄jk �l̄

�

[integer multiplicity = “Cardy” condition]

N cab =

�

k

SakSbkS�ck

S0k

�k(�) =�

l �l

��1�

�· Slk

Ia

=�

j

S�aj

S0j�j(�)�j̄(�̄)

=�

j,k,l

S�aj

S0kSkjSlj �k

��1/�

��l

��1/�̄

�

=�

k,l

N akl �k

��1/�

��l

�� 1/�̄

�

Za(�)

9

Simple example 2

- permeable interfaces for c=1 CFT -[Bachas-de Boer-Dijkgraaf-Ooguri ’01]

● given free bosons, fermions

straight forward to construct interface operators

● a systematic way to connect

interfaces and boundary states : folding trick

● analog of coherent states

10

● take b.d. state in

s.t.

then,

(L1n � L̃1

�n)I = I(L2n � L̃2

�n)

I =P

i,j cij |Bi⇥1 · 2�Bj |

CFT1 ⌦ CFT2 [ ¯ : ⌧ ! �⌧ ]

satisfies

CFT1

CFT2CFT2CFT1

o�= 0�

|B⇥ =�

i,j

cij |Bi⇥1 � |Bj⇥2(�2

n, �̃2n)⇥ (��̃2

�n,��2�n)

(L1n + L2

n � L̃1�n � L̃2

�n)|B⇥ = 0

⇐ interface

[conformal]

⇒ interface

● “unfold” conformal boundary

(Un)folding trick

11

● unfolding D-brane in c=2 theory

“permeable” interface

�

e.g.

Y

n=1

exp

1

n

�M11�

1�n�̃

1�n �M12�

1�n�

2n �M21�̃

2n�̃

1�n +M22�̃

2n�

2n

��

[Bachas-de Boer-Dijkgraaf-Ooguri ’01]

・ boundary cond.

⇤

n=1

exp�

1n

Mij�i�n�̃j

�n

⇥|B�0

�e

x

x

1

2

D1

(�in �Mij�̃

j�n)|B⇥ = 0

�� cos 2� � sin 2�� sin 2� cos 2�

⇥Mij =

I0

I =

|B� =

Permeable interfaces

12

● when

factorized D-branes

● when

L1n ⇡ L2

n , L̃1n ⇡ L̃2

n

topological

Lin � L̃i

�n ⇡ 0

�

�

�1n � ±�2

n , �̃1n � ±�̃2

n

� = ⇥k/2 (k � Z)

� = ⇥(1 + 2k)/4 (k � Z)

X

X

1

2

X

X

1

2

I � |B1⇤ · ⇥B2|

● generic � : “permeable”

↵1n + cos 2✓ · ↵̃1

�n � sin 2✓ · ↵2n ⇥ 0

↵̃2�n � sin 2✓ · ↵̃1

�n + cos 2✓ · ↵2n ⇥ 0

�in ± �̃i

�n � 0 , [ i = 1, 2 ]

13

3. Properties

Generating symmetry/duality of CFT

● topological interfaces generate symmetries/dualities

[Frohlich-Fuchs-Runkel-Scweigert ’04, ’07]

unless CFT1 and CFT2 are “almost the same”

topological cond. T1 ⇡ T2 , T̄1 ⇡ T̄2 may not be satisfied

● two subclasses of top. interfaces

・ group-like defects :

D · D̄ = 1

・ duality defects :

D · D̄ =P

k Dk

[group-like]

generate dualities

∃inverse s.t.

fuse into group-like defects

（ ）

generate symmetries

14

Example of Ising CFT

● 3 primaries: 1 (id.), ε(energy), σ(spin)

● 3 top. defects: D1(= 1) , D✏ , D�

● fusion of defects D✏ ⇥D✏ = D1

D� ⇥D✏ = D�

D� ⇥D� = D1 +D✏

[same as primaries]

● D✏ : group-like, generates Z2-symmetry � ! ��

● duality defect, generatesD� :� ! µ

[disorder field]Kramers-Wannier duality

15

Graphically,

=1!2

=1!2

=

!

!

!

!

!

!

!

!

µ

µ

!

!

µ

µ

µ

µ

!

!"

"

"

"

1

●

σ(z) -σ(z)=

●

σ(z) μ(z)

ε

ε ε σ σ

=

= …

・ repeating similar manipulations, εdefect can be removed

16

[adopted from Frohlich et al ’04]

Remarks:

● topological defects can form junctions

● topological defects end

● generally, ∃defect (changing) fields

[cf. boundary changing fields]D D’

με

+

-on disorder fields

● such understanding of symmetries is generalized,

[Gaiotto-Kapustin-Seiberg-Willett ’14]

leading to “generalized global symmetry”

17

● world-sheet top. interfaces in WZNW model

・ not domain wall in target G

・ but “bi-brane” or “bi-conjugacy class” in G x G

cf. D-brane in G : conjugacy class

Ih1,h2 =⇥

(g1, g2)�� ⇥x, y � G : g1 = xh1y

�1, g2 = xh2y�1

⇤

Bh =⇥

g�� ⇥x � G : g = xhx�1

⇤

[Fuchs-Schweigert-Waldorf ’07]

[Kato-Okada ’96, Alekseev-Schomerus ’98]

Target space interpretation

[compute 2-pt. fn. w/ interface, and read its support]

+

+

● relevance to double field theory (DFT) ?18

Transformation of D-branes

● interfaces transform a set of D-branes to another

CFT1 CFT2 CFT1

I B B ’

[Graham-Watts ’03, ...]

・ non-perturbative trans. in string theory

・ D-branes ≈ solution to string field theory

⇒ solution generating technique?

[ cf. Erler-Kojita-Masuda-Schnabl, ... ]

relation to Erler-Maccaferri construction?

19

Fusion of interfaces

● interfaces fuse to a new interface

● form “algebra”

● solution generating algebra of string theory?

[cf. Geroch group, U-duality group]

CFT1 CFT2 CFT3 CFT1 CFT3

[Bachas-Brunner ’07]

I1 I2 I3

20

4. Applications

Generating boundary RG flow [Graham-Watts ’03, ...]

I a

RG

B Bb d

RG

B Ba x b a x d

● if ∃ boundary RG flow Bb � Bd

Ba�b � Ba�d by acting w/ top. interface Ia

● confirmed conjectured flows

∃

21

Generating bulk RG flow

● another way to view conformal interface :

turn on relevant operator only on one side

Z

�>0d

2x�relevant

CFT CFT CFT CFTUVUV UV IR

[Gaiotto ’12]

[different central charge]

τ= 0

● interface represents full RG flow

22

● explicit proposal of RG interfaces for minimal models

Mp+1,p � Mp,p�1 , �relevant = �1,3

● IR-UV relation:

�IRi =

�

j

bij�UVj �

● in the folded picture

bij = �BRG | �̄IRi � �UV

j �

bij = ��IRi | IRG |�UV

j �

[“non-Cardy”]

● reproduced Zamolodchikov’s results for large p

UV

IR

Mp+1,p

Mp,p-1

23

c=1 permeable interfaces

X1, R1 X2, R2p(i)L,R |n, w; (i)� =

� n

Ri± wRi

�|n, w; (i)�

topological

● for later discussion, let us recall permeable interface

I0 =�

r,s�Z|k2r, k1s; (1)� �k1r, k2s; (2)|e.g.

M = ki � Z : “winding”

|k1k2| = 1 :|k1k2| �= 1 :

group-like

e.g.

duality defects

�� cos 2� � sin 2�� sin 2� cos 2�

�,

�

n=1

exp�

1n

(�1�n, �̃2

n) · M · (�̃1�n,�2

n)t

�I = I0

⇒ k1R1 = k2R2 ,

tan � = k2R2/k1R1

24

Entanglement through interfaces

● replica trick, w → z

[Sakai-Y.S. ’08]

[from Gutperle-Miller ’15]

ZK = Tr �K

ZK: torus partition fn. w/

K pairs of (I, I†) inserted

S = (1� �K) log ZK

��K=1

● entanglement entropy through permeable

interfaces: exactly calculable

ρ: reduced density matrix

w z

25

● exact result of EE through permeable interfaces

● c(θ) : expressed by dilog

→ 0 (factorized D-branes)

→ 1 (topological)

● can control entanglement

: analog of topological EE

L: size of system

usual CFT scaling

� = � log |k1k2|

S = 13c(�) log L + �

● application in laboratories?

26

● extensions/generalizations

・ off-critical (c=1, 1/2) [Peschel ’12]

・ critical (c=1/2, N=1 susy) [Brehm-Brunner ’15]

・ critical, topological [Brehm et al ’15; Gutperle-Miller ’15]

・ holographic derivation [Gutperle-Miller ’15]

c(�) = sin 2�

27

Toward string theory

Toward string theory

● we do not know how to integrate “moduli” of interfaces

� conformal interfaces are not “defined” yet

in string theory

● still, one may think of them, at least at fixed genus

● probably, interesting applications to SFT

cf. Kojita-san’s talk

28

Supersymmetric interfaces [Y.S. ’11]

● possible to construct interfaces in string world-sheet

● Green-Schwarz formalism is simpler in a sense

● light-cone gauge

⇒ “space-time susy” [cf. D-branes]

● imposing susy condition on coherent type operators

⇒ topological-type interfaces, factorized D-branes

generating T-duality,

29

instead of “conformal”

Type IIA Type IIB

transformation of D-branes ...

O(d,d) interfaces [Bachas-Brunner-Roggenkamp ’12]

● N=1 super conformal interfaces

● condition that space-time fermions are

not projected out ⇒ topological or factorized D-branes

● group-like defects :

・ form O(d,d;Z) under fusion

● duality defects:

・ generate T-duality

・ almost T-dual, but w/projection for zero-modes

・ form O(d,d;Q) under fusion

[space-time susy select these as before]

cf. earlier doubt

30

Non-geometric backgrounds

Non-geometric backgrounds

● non-geometric backgrounds in string theory

・ rather ubiquitous

・ important for string vacua, dualities, ...

● dualities are symmetries of string theory

⇒ transition fn. may involve duality trans.

“T-folds” in the case of T-duality[Dabholkar-Hull ’02 , Hull ’04]

● notion of Riemannian geometry: generally lost

[ “stringy” geometry ]

duality

31

● more generally,

this type of non-geometric BG: “monodrofolds”[Flournoy-Wecht-Willams ’04]

● they are mainly analyzed by

● T-folds lie at fixed points (submanifold) in moduli space

low-energy theory (sugra), Double Field Theory (DFT)

● beyond low energy analysis, need world-sheet approach

32

typically at string scale

● topological interfaces induce symmetries and dualities

⇒ (generalized) orbifolds by interfaces may

give exact partition fn. for T-folds (monodrofolds)

33

T-folds and cosmological constant

● twists/orbifolds by interfaces generating T-duality

⇒ exact world-sheet partition fn. for T-folds[ concrete form of interface may not needed... ]

Τ

Sb

Target Space

d

base circle

fiber torus

World-sheet

twist

I

[CFT1=CFT2 (self-dual)]

[Y.S.-Sugawara-Wada ’15]

according to windings

34

● an interesting feature:

this set up realizes simple non-supersymmetric vacua

w/ vanishing cosmological constant at 1 loop

[ cf. Kachru-Kumar-Silverstein ’98 .... ]

● subtleties (phases) in T-dual trans., consistency

⇑ modular invariance

35

Duality defects and monodrofolds

● duality defects generate “dualities” [not necessarily exact symmetries]

● twists/orbifolds similar to group-like case

⇒ exact world-sheet partition fn. for monodrofolds[ concrete form of interface may needed ]

“generalized orbifolds”[ cf. correlation fn. point of view: Frohlich et al ’09 .... ]

● various wrapping config. of interfaces in twisted sectors

modular invariance dictates how to sum up them

(or “moduli”)

[Y.S.-Sugawara ’15]

36

5. Summary

Summary

In this talk, we

● reviewed properties of world-sheet

● discussed applications to

・ renormalization group (RG) flows

・ entanglement entropy (EE)

・ non-geometric backgrounds in string theory

...

conformal interfaces

37

● world-sheet conformal interfaces :expected to play a fundamental role in string theory

● conformal invariance : a guiding principle

of world-sheet approach to string theory

● interfaces for string theory ?

・ SFT

・ DFT (Double Field Theory)

・ non-perturbative aspects of strings...

38