Holographic description of interfaces in 2-d CFTs - Physics

Holographic duals of 2d interface theories 

M. Gutperle UCLA 

Holographic duals of two dimensional interface theories 

Rockwall Trail 

Kootenay National 

Park, Alberta, Canada 

Great Lakes/SPOCK meeting, Cincinnati, March 2010



Based on: 

“Half-BPS Solutions locally asymptotic to AdS 3 × S 3 and interface conformal 

field theories” by M. Chiodaroli, M. Gutperle and D. Krym arXiv: 0910.0466 

“Open worldsheets for interfaces” by M. Chiodaroli, E. D’Hoker and M. 

Gutperle arXiv: 0912.4679 

Work in progress with J. Hung, B. Shieh, M. Chiodaroli and D. Krym 

Other work by: 

E. D’Hoker, J. Estes, M.G., D. Krym, P. Sorba 

J. Gomis, C. Römmelsberger, F. Passerini, S. Yamaguchi, O. Lunin, J. Kumar, A. 

Rajaraman



Plan of the talk 

• Introduction 

• Ansatz for interface theories 

• Local solution of the BPS equations 

• Global regularity conditions 

• Half-BPS Janus solution 

• Multi-Janus solutions 

• Many boundaries 

• Conclusions



AdS 3 /CF T 2 

Introduction 

duality is one of the best studied examples of AdS/CFT 

• Near horizon limit of D5/D1 or NS5/F1 bound states (or SL(2,Z) 

orbits) 

• type IIB solution: AdS 3 × S 3 × K 3 (T 4 ) with self-dual NS-NS or R-R 

three form flux 

• Vacuum preserves 16 supersymmetries 

• CFT: N=(4,4) 2 dimensional SCFT 

• Important for entropy counting of (near) extremal BH, Hawking 

radiation etc. 

• NS-NS vacuum can be described using standard worldsheet 

techniques



Introduction 

Defects and interfaces are interesting in 2 dim CFT for many reasons: 

• Folding trick allows to use machinery of boundary CFT 

• defect/interface entropy analog of boundary entropy 

• For higher dimensional interfaces/defect interesting interplay 

between Chern-Simons action, Theta-angles, supersymetry 

• Many interesting new developments: topological interfaces, fusion 

of interfaces etc 

• Application to condensed matter: Kondo effect, quantum wires 

Goal: holographic description of interfaces and defects in 

AdS 3 × S 3 × K 3 (T 4 )



D5 

D1 

D3 

Introduction 

• Intersecting branes (1/4 BPS before near horizon 

limit) near horizon limit enhances supersymmetry 

0 1 2 3 4 5 6 7 8 9 

0+1 defect 

• Probe D-branes in AdS and sphere. Neglect backreation. 

Supersymmetric embedding: κ-symmetry of Born-Infeld action 

Example: D3 brane with AdS 2 × S 2 in AdS 3 × S 3 

AdS 2 

S 2 

Karch, Randall; 

Bachas Petropoulos; 

Erdmenger et al... 

AdS 3 S 3



Introduction 

Janus solution: holographic description of a codimenion 1 conformal 

interface in preserve SO(2,d-1) of SO(2,d) symmetry: Use 

slicling of 

AdS d 

AdS d+1 

AdS d+1 

ds 2 = dx 2 + f(x)ds 2 AdS d 

φ = φ(x)



Introduction 

Janus solution: holographic description of a codimenion 1 conformal 

interface in preserve SO(2,d-1) of SO(2,4) symmetry: Use 

slicling of 

AdS d 

AdS d+1 

AdS d+1 

ds 2 = dx 2 + f(x)ds 2 AdS d 

φ = φ(x) 

as x → ±∞ 

(asymptotic AdS region) 

lim 

x→±∞ ds2 = dx 2 + e2|x| 

z 2 

In Poincare coordinates the spatial 

section of the boundary consists of 

two three dimensional half planes 

joined by a two dimensional interface. 

 

− dt 2 + dx 2 1 + ···+ dx 2 d−2 + dz 2 

x 

z =0 

x → −∞ x → +∞



Ansatz for interface theory AdS 3 × S 3 × M 4 

Construction proceeds analogous to constructions of Susy Janus solutions: 

• AdS 3 × S 3 has global bosonic symmetry SO(2, 2) × SO(4) 

•1 dim conformal defect preserves SO(2, 1) corresp. to AdS 2 

• Superconformal defect preserves 8 of the 16 supersymmetries: 

type II solution should have 8 unbroken susys 

• SO(4) R-symmetry is reduced to SO(3) corresp. to S 2 express 

three sphere as a fibration of a two sphere over an interval 

Susy-Janus ansatz depends on two coordinates x,y 

y ∈ [0, π] 

• Internal moduli associated with four torus or K3 are not turned on 

Ansatz: 

fibration over Riemann surface 

AdS 2 × S 2 × K 3 Σ 2



Ansatz for interface theory AdS 3 × S 3 × M 4 

Bosonic fields of type IIB supergravity depend only on 

metric: ds 2 = f1 2 ds 2 AdS 2 

+ f2 2 ds 2 S + f 2 2 3 ds 2 K 3 

+ ρ 2 dzd¯z 

i =0, 1 j =2, 3 k =4, ··· , 7 a =8, 9 

dilaton/axion: Q = q a e a , P = P a e a 

complex 3-form: 

Self-dual 5 form: 

G = g a (1) e a01 + g a 

(2) 

Σ 

e a23 

AdS 2 S 2 

AdS 2 × S 2 K 3 

F 5 = h a e a0123 + ˜h a e a4567



Local solution of BPS-equations 

Susy variation of dilatino and gravitino vanishes for unbroken susy 

δλ = i(Γ · P )B −1 ε ∗ − i (Γ · G)ε 

24 

δψ M = D µ ε + i 

480 (Γ · F (5))Γ µ ε − 1 

96 (Γ µ(Γ · G) + 2(Γ · G)Γ µ ) B −1 ε ∗ 

Expand 

in terms of Killing spinors on AdS 2 × S 2 × K 3 

= η 1 ,η 2 

χ η1 ,η 2 ,η 3 

⊗ ξ η1 ,η 2 ,η 3 

2 dim spinors on Σ 

Use discrete symmetries of the equation to reduce BPS equations to a 

single 2 dim spinor ξ 

Solution of reduced BPS equations give 8 unbroken susys




α 

Spinor ξ = 

β 

BPS equations become 

 

dilatino: 4P z α ∗ − g z (1) + ig z 

(2) β =0 4¯P z β − 

 

ḡ z (1) + i ḡ z 

(2) α ∗ =0 

gravitino AdS: 

gravitino sphere: 

1 

α + 2D zf 1 

β − 2h z β − 

f 1 f 1 

1 

β ∗ − 2D zf 1 

α ∗ − 2h z α ∗ + 

f 1 f 1 

ν 

α + 2D zf 2 

β − 2h z β + 

f 2 f 2 

ν 

β ∗ + 2D zf 2 

α ∗ +2h z α ∗ + 

f 2 f 2 

3 

4 g(1) z 

1 

4 g(1) z 

3 4ḡ(1) 

z 

1 4ḡ(1) 

z 

− i 4 g(2) z 

− i 4ḡ(2) 

z 

− i 3 4 g(2) z 

− i 3 4ḡ(2) 

z 

 

α ∗ =0 

 

β =0 

 

α ∗ =0 

 

β =0



gravitino K3: 

gravitino Σ: 


2D z f 3 

β +2h z β + 

f 3 

2D z f 3 

α ∗ − 2h z α ∗ + 

f 3 

1 

4 g(1) z 

1 4ḡ(1) 

z 

 

D z + i 2 ω z − iq 

z 

2 + h z α − 1 4 

 

D z − i 2 ω z − iq 

z 

β − 1 2 8 

 

D z − i 2 ω z + iq 

z 

α ∗ − 1 2 8 

 

D z + i 2 ω z + iq 

z 

2 − h z β ∗ − 1 4 

+ i 1 4 g(2) z 

+ i 1 4ḡ(2) 

z 

 

α ∗ =0 

 

β =0 

 

g z (1) − ig z 

(2) 

 

 

g z (1) + ig z 

(2) 

 

ḡ z (1) + iḡ z 

(2) 

α ∗ =0 

 

β =0 

 

ḡ z (1) − iḡ z 

(2) 

β ∗ =0 

 

α =0




• Dilatino, grav-AdS, grav-S, grav-K3 are used to express metric factors 

axion dilaton an five form in terms of spinors 

• 2 of 4 grav-Σ can be used to solve spinor in terms of two 

holomorphic functions 

• The remaining equation and the Bianchi identity of the five form can 

be solved in terms of two harmonic functions. 

Local solution: All bosonic fields are expressed in term of 

2 holomorphic functions A(z),B(z) 

2 harmonic functions H(z, ¯z),K(z, ¯z) 

Satisfies all equations of motion and Bianchi identities of type 

II B supergravity



axion/dilaton: e 4Φ = 1 4 


2 

(B + ¯B) 

A + Ā − 

A + 

K 

χ = 1 B 2 − ¯B 2 

− A + 

2i K Ā 

Ā − 

(B − ¯B) 

2 

K 

 

metric: 

f 2 1 = ce−2Φ 

2f 2 3 

f 2 2 = ce−2Φ 

2f 2 3 

|H| 

K 

|H| 

K 

f3 4 = 4c 2 e2Φ K 

A + Ā 

ρ 4 = e 2Φ K |∂ wH| 4 

H 2 

(A + 

(A + 

A + Ā 

|B| 4 

Ā)K − (B − ¯B) 

2 

Ā)K − (B + ¯B) 

2




3 form AST can be written as a total derivative 

where 

b (2) = −i 

5-brane Page-charges supported on various three spheres 

q NS5 = 

 

f 2 2 ρe −Φ Re(g (2) 

z )=∂ w b (2) 

f 2 2 ρe Φ Im(g (2) 

z )+χf 2 2 ρe −Φ Re(g (2) 

z )=∂ w c (2) 

H(B − ¯B) 

(A + Ā)ĥ − (B − ¯B) + ˜h 1 , ˜h1 = 1 2 2i 

c (2) H(A 

= − 

¯B + ĀB) 

(A + Ā)ĥ − (B − ¯B) + h 2, h 2 = 1 

2 2 

S 3 e −Φ Re(G), q D5 = 

 

 

∂w H 

B 

+ c.c. 

A 

B ∂ wH + c.c. 

S 3 

e +Φ Im(G)+χe −Φ Re(G) 

similar expressions exist for D1 brane and fundamental strings charges



self dual five form AST is given by 


f 4 3 ρ˜h z = ∂ w C K , C K = i 2 

B 2 − ¯B 2 

A + Ā 

+ 1 2 ˜K 

Where ˜K is the conjugate harmonic function, i.e. 

K(z, ¯z) =k(z)+¯k(¯z), ˜K(z, ¯z) =−i 

k(z) − ¯k(¯z) 

 

D3-brane charge: 

 

C×K 3 F 5 = 

 

dzf 4 3 ρ˜h z + cc



Global regularity conditions 

How does AdS 3 × S 3 with RR flux look like ? 

Σ is infinite strip 

w = x + iy, 

x ∈ [−∞, ∞] 

y ∈ [0, π] 

f 1 →∞ 

H →∞ 

π 

f 2 → 0 

H → 0 

f 2 → 0 

0 x 

H → 0 

f 1 →∞ 

H →∞ 

f 2 1 = cosh 2 x, f 2 2 =sin 2 y, ρ =1, f 3 = const 

H = −i sinh(w)+c.c. K = i 

A = i 

1 

sinh w , 

B = icosh(w) sinh w 

1 

sinh w + c.c.




• Local solution solves BPS and equation of motion, solution can be 

singular, geodesically incomplete or real fields can be complex 

• Boundary of Riemann surface; Locus where 2 sphere shrinks to 

zero size: f 2 → 0 

• Asymptotic region of AdS 3 isolated points where AdS 2 blows up 

f 1 →∞ 

• Volume of 

K 3 and dilaton/axion must remain finite 

f1 2 f2 2 f3 4 = H 2 

Boundary of Σ: H vanishes apart form isolated poles






• Boundary of Riemann surface; Locus where 2 sphere shrinks 

tozero size: f 2 → 0 H → 0 


f 1 →∞ 


K 3 and dilaton/axion remains finite 

f1 2 f2 2 f3 4 = H 2 







• Boundary of Riemann surface; Locus where 2 sphere shrinks 

tozero size: f 2 → 0 H → 0 


f 1 →∞ 

H →∞ 


K 3 and dilaton/axion remains finite 

f1 2 f2 2 f3 4 = H 2 





At the boundary 

H → 0 

f 2 1 = ce−2Φ 

2f 2 3 

f 2 2 = ce−2Φ 

2f 2 3 

|H| 

K 

|H| 

K 

f3 4 = 4c 2 e2Φ K 

A + Ā 

ρ 4 = e 2Φ K |∂ wH| 4 

H 2 

(A + 

(A + 

A + Ā 

|B| 4 

Ā)K − (B − ¯B) 

2 

Ā)K − (B + ¯B) 

2 

All functions satisfy Dirichlet boundary conditions on ∂Σ 

K =(A + Ā) =(B + ¯B) =H = 0 on ∂Σ





H → 0 

f 2 1 = ce−2Φ 

2f 2 3 

f 2 2 = ce−2Φ 

2f 2 3 

|H| 

K 

|H| 

K 

f3 4 = 4c 2 e2Φ K 

A + Ā 

ρ 4 = e 2Φ K |∂ wH| 4 

H 2 

(A + Ā)K − (B − ¯B) 

2 

K → 0 

(A + Ā)K − (B + ¯B) 

2 

A + Ā 

|B| 4 


K =(A + Ā) =(B + ¯B) =H = 0 on ∂Σ





H → 0 

f 2 1 = ce−2Φ 

2f 2 3 

f 2 2 = ce−2Φ 

2f 2 3 

|H| 

K 

|H| 

K 

f3 4 = 4c 2 e2Φ K 

A + Ā 

ρ 4 = e 2Φ K |∂ wH| 4 

H 2 

(A + Ā)K − (B − ¯B) 

2 

K → 0 

(A + Ā)K − (B + ¯B) 

2 B + ¯B → 0 

A + Ā 

|B| 4 


K =(A + Ā) =(B + ¯B) =H = 0 on ∂Σ





H → 0 

f 2 1 = ce−2Φ 

2f 2 3 

f 2 2 = ce−2Φ 

2f 2 3 

|H| 

K 

|H| 

K 

f3 4 = 4c 2 e2Φ K 

A + Ā 

ρ 4 = e 2Φ K |∂ wH| 4 

H 2 

(A + Ā)K − (B − ¯B) 

2 

K → 0 

(A + Ā)K − (B + ¯B) 

2 B + ¯B → 0 

A + Ā → 0 

A + Ā 

|B| 4 


K =(A + Ā) =(B + ¯B) =H = 0 on ∂Σ




Using the explicit expressions for the metric factors one can show 

that the following conditions guarantee a globally regular solutions 

R1: A + Ā, B + ¯B,K have common singularities 

R2: No singularities in the bulk of 

Σ 

R3: The harmonic functions A + Ā, B + ¯B,Kcannot have any zeros 

in the bulk of 

Σ 

R4: The holomorphic function B und ∂ u H have common zeros 

R5: The harmonic functions satisfy the inequality: 

(A + Ā)K − (B + ¯B) 2 > 0



Simple deformation of 

Σ is infinite strip 

w = x + iy, 

x ∈ [−∞, ∞] 

y ∈ [0, π] 

Half-BPS Janus solutions 

H = −iL sinh(w + ψ)+c.c. 

2 cosh θ +sinhθ cosh w 

A = ik + ib 

sinh w 

B = 

cosh(w + ψ) 

ik 

cosh ψ sinh w 

ĥ = 

cosh θ − sinh θ cosh w 

i + c.c. 

sinh w 

AdS 3 × S 3 with RR flux 

f 2 → 0 

π H → 0 

f 1 →∞ 

H →∞ 

0 x 

f 2 → 0 

H → 0 

f 1 →∞ 

H →∞ 

k, b SL(2,R) transformations 

L 

size of AdS 

θ, ψ deformation parameter




axion and 

dilaton: 

e 4Φ = k 4 cosh2 (x + ψ)sech 2 ψ + cosh 2 θ − sech 2 ψ sin 2 y 

 

cosh x − cos y tanh θ 

2 

χ = − k2 

2 

sinh 2θ sinh x − 2 tanh ψ cos y 

cosh x cosh θ − cos y sinh θ 

− b 

Plot for ψ = 1 , θ =0,b=0,k =1L =1 dilaton jumps ! 

2




axion and 

dilaton: 

e 4Φ = k 4 cosh2 (x + ψ)sech 2 ψ + cosh 2 θ − sech 2 ψ sin 2 y 

 

cosh x − cos y tanh θ 

2 

χ = − k2 

2 

sinh 2θ sinh x − 2 tanh ψ cos y 

cosh x cosh θ − cos y sinh θ 

− b 

Plot for ψ =0, θ = 1 ,b=0,k =1,L=1 axion jumps ! 

2




metric factors are 

regular 

plot of solution with 

ψ = 1 2 , θ = 1 2 

b =0,k =1,L=1 

3 form AST charges q RR = πkL cosh θ cosh ψ, q NS =0 

5 form AST charge q F 5 =0




holographic interpretation: Two combinations of massless scalars 

e −2Φ f 4 3 and χ − k 2 C K 

coupling constant of 2d CFT ( α ) blowup mode of orbifold 

dual to ∆ =2operator 

O 0 dual to ∆ =2operator 

T 0 

Take different values in the two asymptotic regions 

φ = φ 0 − + φ 1 −(y)e x + ... for x → −∞ 

φ = φ 0 + + φ 1 +(y)e −x + ... for x →∞ 

In the dual 2dim CFT, the operators are added which jump 

across a 1dim interface: 

L 1 = L 0 + Θ(x ⊥ )c 1 O 0 + Θ(x ⊥ )c 2 T 0




What are the operators O 0 and T 0 ? N=(4,4) SCFT 

For simplicity consider (T 4 ) n /S n orbifold 

S = 1 

d 2 z ∂X i,a ¯∂Xi,a + ψ i,a ¯∂ψi,a + 

4π 

¯ψ i,a ∂ ¯ψ 

 

i,a 

i,a 

O 0 (h, ¯h) 

 

=(1, 1) descendant of ψa i ¯ψ a j (h, ¯h) =(1/2, 1/2) 

lim 

z→w 

O 0 = ∂X i,a ¯∂Xi,a + fermions 

i,a 

T 

operator with vanishing SU(2)xSU(2) R-symmetry 

0 (h, ¯h) =(1, 1) 

descendant of Z_2 twist field 

 

G 2 (z) ˜G 1† (¯z) − G 1† (z) ˜G 2† (¯z) Σ 1 2 , 1 2 (w, ¯w) = 

1 

(z − w)(¯z − ¯w) T 0 (w, ¯w)+··· 

a



Multi-Janus solutions 

Generalization of half-BPS Janus solution to an arbitary number of 

asymptotic regions: map strip to upper half plane u = e w 

Asymptotic regions are mapped to u =0,u= ∞ 

Harmonic function H singularity is a simple pole at u =0 

c0 

H = i 

u − c 

0 

+reg 

ū 

Works for A,K as well. Superimpose simple poles and solve for 

regularity condition Constraints on parameters



H = i 

n 

i=1 

c H,i 

u − x H,i 

+ c.c. 

Im(u) 


Σ 

f 2 → 0 

Vol(S 2 ) → 0 

Σ 

x H,1 x H,2 

x H,3 x H,4 

Re(u) 

f 1 →∞ 

asymptotic region 

corresponds to half spaces 

in the holographic dual



H = i 

n 

i=1 

c H,i 

u − x H,i 

+ c.c. 


S 3 

Im(u) 

Σ 

S 3 

Σ 

X A,1 

X A,2n−2 

x H,1 x H,2 

x H,3 x H,4 

Re(u) 

f 1 →∞ 

asymptotic region 

corresponds to half spaces 

in the holographic dual 

Additional 2n-2 moduli: position of 

poles and residues of A(u) 

A = i 

2n−2 

 

i=1 

c A,i 

u − x A,i 

+ ib



c B,i = 


The other two functions are completely determined by regularity 

R1 and R4: holomorphic B has zeros of 

R1 R2 and R4: K is determined 

2n−2 

ĉ i 

K = i 

+ c.c. 

u − x A,i 

where 

i=1 

B = 

lim (u − x A,i )B(u) 

u→x A,i 

∂ u H 

n−1 

i=1 (u − x H,i) 2 

∂ uH 

2n−2 

i=1 (u − x A,i) 

It can shown that with these choices 

dilaton, metric factors are real (R5 is 

automatically satisfied) 

ĉ i = c2 B,i 

c A,i 

and poles of A 

(A + Ā)ĥ − (B + ¯B) 2 > 0



3-pole soultion 

u → 0 

Simplest Case: H has 3 poles at 0,1 and ∞: 

c0 

H = i 

u + c 

1 

u − 1 − c ∞u 

+ c.c. 

Σ 

u →∞ 

CFT: Fold 3 CFT’s on half spaces dual to asymptotic AdS regions 

u → 1 

CFT 1 

CFT 2 

CFT 3 

Interface= Boundary in CFT 1 ⊗ CFT 2 ⊗ CFT 3




Summary 

• Solution with n asymptotic regions has n half spaces glued by an 

interface 

• Moduli space of solutions has dimension 6n-6 

• In each asymptotic region there is 3-sphere and NS-NS or R-R flux 

(or both) 

• Scalar fields take different asymptotic values: Generalization of Janus 

solution to many asymptotic regions. 

• Central charge of CFT can be different in different asymptotic regions 

• No five form charge (no D3 brane flux) 

• Many interesting things to calculate: Holographic calculation of 

correlation functions, interface entropy etc 

• Fusion of interfaces in holographic dual ? 

• Application to physical systems



Many boundaries 

Multi-Janus solution can be generalized 

M. Chiodaroli, E. D’Hoker and 

M. Gutperle arXiv:0912.4679 

• Riemann surface with h boundaries (and g-handles) 

• Find solutions with nonzero five brane charge 

• Degenerations of Riemann surfaces: what do they mean for 

interface theory ? 

Simplest example: annulus with two poles on the same boundary 

S 3 

x H,1 x H,2 

C 

Two nontrivial three cycles and charges 

˜K is not single valued around C: five form 

charge does not vanish 

S 3




Multi-Janus solution can be generalized 

M. Chiodaroli, E. D’Hoker and 

M. Gutperle arXiv:0912.4679 

• Riemann surface with h boundaries (and g-handles) 

• Find solutions with nonzero five brane charge 

• Degenerations of Riemann surfaces: what do they mean for 

interface theory ? 

Simplest example: annulus with two poles on the same boundary 

x H,1 x H,2 

˜K is not single valued around C: five form 

charge does not vanish 

Two nontrivial three cycles and charges 

Shrink one boundary: point on disk 

position of 3-brane probe in AdS 3 × S 3




annulus can be generalized to a surface with arbitrary number 

of boundaries, using the doubling trick and the machinery of 

higher loop string perturbation theory 

boundaries are 

fixed points of 

Involution I 

Σ = ¯Σ/I 

I(A i )=A i 

I(B i )=−B i 

Harmonic function expressed in terms of holomorphic differential, 

Prime forms etc. Non-contractible cycles support 3 brane charge



Conclusions 

•We have constructed the half BPS Janus solution for type IIB 

which is locally asymptotic to AdS 3 × S 3 × K 3 

• Holographic dual to interface CFT in 2dim D5/D1 CFT 

• Solution with more than two asymptotic regions: Both NS 

and RR 3 form charges, different central charges in AdS regions 

Backreacted solution of intersecting branes ? 

• No Five form charge present if Riemann surface Σ is the disk 

• Constructed solution where Riemann surface Σ is disk with n 

holes. 

• Noncontractible cycles support 5 brane charge 

• Candidate for backreacted solution of D3 brane probe with 

AdS 2 × S 2 worldvolume

Holographic description of interfaces in 2-d CFTs - Physics

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