Holographic duals of interface theories and junctions of CFTs

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1 Holographic duals of interface theories and junctions of CFTs y Σ 2 CFT (2)... CFT (n) x H,1 x H,2 x H,3 x x A,1 x A,2 x A,3 x A,4 x A,5 CFT (1)

2 Plan of the talk Interfaces and junctions in CFT Holographic interface solutions Half-BPS Janus solution Calculation of boundary entropy Multi-Janus solutions Conclusions 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: Boundary Entropy of supersymmetric Janus solutions by M. Chiodaroli, Ling-Yan Hung and M. Gutperle, arxiv: Work in progress by M. Chiodaroli, M. Gutperle, Ling-Yan Hung, D. Krym and Brian Shieh

3 Interfaces and junctions in CFT A conformal interface is a junction betwen two CFTs preserving part of the conformal symmetry. Gluing conditions need to be scale invariant. In 2d continuity of T xt across the interface is sufficient. The folding trick relates interface CFT s to boundary CFT: conformal boundary conditions (open string description) conformal boundary state B (closed string description) Affleck and Ludwig; Bachas, de Boer, Dijkgraaf and Ooguri

4 Toy model: compact boson with a jump in the coupling S =2r 2 x<0 Interfaces and junctions in CFT dtdx a φ a φ +2r 2 φ φ +2π x<0 dtdx a φ a φ After rescaling the bosons have a jump in the compactification radius φ φ +2πr ± 2πr + After folding one has 2 bosons φ 1 φ 1 +2πr + φ 2 φ 2 +2πr φ 1 0 φ 2 2πr Conformal boundary conditions correspond to a diagonal D1 brane in a 2 torus spanned by φ 1,2 Bachas, de Boer, Dijkgraaf and Ooguri

5 Interfaces and junctions in CFT Basic quantities which can be calculated for interface CFT s: g-factor g = 0 Band boundary entropy S bd =lng which measures the ground state degeneracy of the interface g = 1 2 r+ r + r r + Reflection and transmission coefficients for scattering off the interface R = r2 r 2 + r 2 + r 2 + Casimir energy between two interfaces Affleck and Ludwig Bachas, de Boer, Dijkgraaf and Ooguri

6 Interfaces and junctions in CFT Generalizations Interfaces between 2 CFTs can be generalized to junctions of 3 or more CFTs, for example a star graph: CFT (2) φ 3 r 3... CFT (1) CFT (n) conformal boundary conditions i T (i) xt =0 n bosons φ i with radius r i : junction of n quantum wires (with different LL couplings) Bellazinni, Mintchev and Sorba Affleck, Chamon and Oshikawa r 2 φ 2 0 φ 1 r 1 conformal boundary conditions: p dim D brane in n-torus g-factors calculated in BCFT

7 Interfaces and junctions in CFT Applications of interface CFT: Impurity problems in 1+1 dimensions. Interaction of bulk CFT with an impurity which preserves conformal invariance Kondo effect Topological interfaces, classification, fusion of interfaces Compact bosons give the effective description of interacting fermions systems in 1+1 dimensions via the theory of Luttinger liquids (Radius is related to the coupling constant in the LL). Hence interface CFT describe junctions of (critical) quantum wires. Questions like calculation of conductance, tunneling etc can be answered using CFT. Affleck et al. Goal: Provide a holographic dual description of interface CFT s using AdS 3 /CF T 2 Bachas and Brunner

8 Holographic interface solutions 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 φ(x) ds 2 = dx 2 + f(x)ds 2 AdS d φ = φ(x) as x ± the dilaton φ φ ± x conformal boundary: divergence of conformal factor of the metric x ± ds 2 = dx 2 + e2 x ξ 2 dt 2 + dx dx 2 d 2 + dξ 2 +o(e 2 x ) Divergence in three limits x ± and ξ 0

9 Holographic interface solutions ds 2 = dx 2 + e2 x ξ 2 dt 2 + dx dx 2 d 2 + dξ 2 +o(e 2 x ) as the boundary is spanned by x ± R 1,d 2 R + t, x 1,,x d 2 and ξ [0, ] as ξ 0 the boundary is and distances in x shrink to zero R 1,d 1 x In Poincare coordinates the spatial section of the boundary consists of two d-1 dimensional half planes joined by a d-2 dimensional interface. (complicated) map to Fefferman- Graham coordinates possible Skenderis and Papadimitriou ξ 0 x x +

10 Holographic interface solutions Solution of type IIB superstring theory locally asymptotic AdS 3 S 3 K 3 Ansatz: fibration over Riemann surface AdS 2 S 2 K 3 Σ 2 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 express three sphere as a fibration of a two sphere over an interval Susy-Janus ansatz depends on two coordinates x,y S 2 y [0, π]

11 Holographic interface solutions Bosonic fields of type IIB supergravity depend only on consistent with SO(2,1) x SO(3) symmetry Σ and are metric: ds 2 = f 2 1 ds 2 AdS 2 + f 2 2 ds 2 S 2 + f 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) e a23 AdS 2 S 2 F 5 = h a e a h a e a4567 AdS 2 S 2 K 3

12 Holographic interface solutions 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 in terms of 2 harmonic and 2 holomorphic functions on Σ

13 Holographic interface solutions 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 metric: f 2 1 = e 2Φ 2f 2 3 χ = 2 (B + B) A + Ā A + K B 2 B 2 A + 2i K H (A + Ā)K (B B) 2 K (A + Ā)K (B + B) 2 f 2 3 = e 2Φ 2f 2 3 H K f 4 3 = 4e2Φ K A + Ā With similar formulae for ρ and the AST fields Ā (B B) 2 K

14 Holographic interface solutions Globally regular solutions: All metric factors and physical fields are real, no curvature singularities. Imposes conditions on the harmonic functions Riemann surface Σ has boundary the metric factor f 2 0 Σ where sphere closes off, i.e. Harmonic functions satisfy Dirichlet boundary conditions on the boundary, except for simple poles K =(A + Ā) =(B + B) =H = 0 on Σ Singularities (simple poles) of harmonic functions only at the boundaries Several other conditions (see arxiv: )

15 Holographic interface solutions Counting of moduli of regular solution on the half plane H = i A = i n i=1 2n 2 i=1 c H,i u x H,i + c.c. c A,i u x A,i + ib n 1 i=1 B = B (u x H,i) 2 0 2n 2 i=1 (u x A,i) uh K(x, y) =K(z)+K( z) by H,A,B. n-3 positions of poles, n residues: SL (2,R) fixes three positions 2n-2 positions of poles, 2n+2 residues, one constant 1 overall constant is determined y Σ 2 Dual harmonic function K(x, y) =i(k(z) K( z)) one extra constant contains number of moduli: n-3 + n + 2n-2 +2n =6n-4 x H,1 x H,2 x H,3 x A,1 x A,2 x A,3 x A,4 x A,5 x

16 Holographic interface solutions y Σ 2 Expansion near pole of H z x H,i = re iθ r 0 gives asymptotic boundary region x H,1 x H,2 r θ x A,1 x A,2 x A,3 x H,3 x A,4 x A,5 x n = number of poles of H = number of asymptotic AdS regions ds 2 R 2 AdS (i) 1 r 2 dr 2 + ds 2 AdS 2 + dθ 2 +sin 2 θds 2 S 2 + o(r 2 ) Exponential map r = e x produces Janus asymptotics ds 2 R 2 AdS (i) dx 2 + e 2x dξ2 dt 2 ξ 2 + ds 2 S 3 + o(e 2x )

17 Holographic interface solutions y Σ 2 Expansion near pole of H z x H,i = re iθ x H,1 x H,2 θ x A,1 x A,2 x A,3 x H,3 x A,4 x A,5 x As r 0 two sphere and θ make three spheres in the asymptotic region ds 2 RAdS(dθ 2 2 +sinθ 2 ds 2 S 2 )+ three spheres carry four charges: D1, D5, NS5 and F1. 4 (n-1) independent Page-charges Q Page NS5 = H 3, Q Page D5 F3 + χh 3 M 3 S 3 Q Page D1 = e φ F 3 4C 4 H 3 S 3 K 3 Q Page F 1 = e φ H 3 χe φ F 3 +4C 4 dc 2 ) S 3 K 3

18 Half BPS Janus solutions Simplest case: n=2, i.e. two poles of H and two asymptotic AdS regions. Simple deformation of AdS 3 S 3 Exponential map takes Σ to infinite strip w = x + iy, x [, ] y [0, π] f 1 H H = il sinh(w + ψ)+c.c. 2 cosh θ +sinhθcosh w A = ik + ib sinh w cosh(w + ψ) B = ik cosh ψ sinh w K cosh θ sinh θ cosh w ĥ = i + c.c. sinh w with RR flux π f 2 0 H 0 f x H 0 f 1 H k, b SL(2,R) transformations L size of AdS θ, ψ deformation parameter

19 Half BPS Janus solutions 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

20 Half BPS Janus solutions 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

21 Half BPS Janus solutions 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 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

22 Half BPS Janus solutions 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 1. i,a i,a O 0 (h, h) =(1, 1) descendant of ψa i ψ a j (h, h) =(1/2, 1/2) O 0 = i,a X i,a Xi,a + fermions a Jump in coupling constant 2. lim z w T 0 (h, h) =(1, 1) operator with vanishing SU(2)xSU(2) R-symmetry 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)+ Z_2 twist field jump: deformation away from orbifold point

23 Calculation of Boundary entropy Divide system into subsystem A and complement B S A = tr HA ρ A log ρ A. ρ A = tr HB ρ is the entanglement entropy The entanglement entropy for a system with an interface S A = c 6 log L + log g holographic calculation of entanglement entropy in AdS/CFT where g is the g-function, i.e. boundary entropy of folded theory Cardy and Calabrese Minimal surface γ on the boundary in bulk ending A enclosing A entanglement entropy: S A = Area(γ) G d+1 N Ryu and Takayanagi

24 Calculation of Boundary entropy Minimal (static) surface for nonsupersymmetric Janus ds 2 3 = dx 2 + f(x) dξ2 dt 2 ξ 2 ξ AdS-slicing Fefferman-Graham r γ ξ 0 x x + γ x A z UV-divergence as x is regulated by cutoff. For BPS-Janus solution: trace over all states with R- charges in entropy is equivalent to integration over three sphere

25 For BPS Janus: minimal surface in AdS 2 A(θ, ψ) = dω 2 dx Calculation of Boundary entropy γ spans Σ, two sphere atz = z 0 dy (f 2 2 f 2 3 ) (ρ 2 f 2 3 )=V S 2 dxdy H2 ρ 2 A is divergent as x, and has to be regularized using a UV cutoff The boundary entropy is given by the difference of A and the area of the minimal surface in pure AdS with the same radius f 2 1 AdS: S bd = A(θ, ψ) A 0 4G N = 2c 3 ln cosh θ cosh ψ

26 For BPS Janus: minimal surface in AdS 2 A(θ, ψ) = dω 2 dx Calculation of Boundary entropy γ spans Σ, two sphere atz = z 0 dy (f 2 2 f 2 3 ) (ρ 2 f 2 3 )=V S 2 dxdy H2 ρ 2 A is divergent as x, and has to be regularized using a UV cutoff The boundary entropy is given by the difference of A and the area of the minimal surface in pure AdS with the same radius AdS: θ =0 S bd = A(θ, ψ) A 0 4G N = 2 3 c ln cosh ψ = 2c 3 ln cosh ψ For θ =0(pure radius deformation) one can compare this to the boundary entropy for n=2/3c bosons where radius jumps from r + to r S bd = 2 3 c ln 1 r+ + r r + CFT: where = e ψ 2 r r + r Complete agreement! Not true f 2 1 for non BPS Janus solution T.Azeyanagi, A.Karch, T.Takayanagi and E.G.Thompson

27 For BPS Janus: minimal surface in AdS 2 A(θ, ψ) = dω 2 dx Calculation of Boundary entropy γ spans Σ, two sphere atz = z 0 dy (f 2 2 f 2 3 ) (ρ 2 f 2 3 )=V S 2 dxdy H2 ρ 2 A is divergent as x, and has to be regularized using a UV cutoff The boundary entropy is given by the difference of A and the area of the minimal surface in pure AdS with the same radius f 2 1 AdS: S bd = A(θ, ψ) A 0 4G N = 2c 3 ln cosh θ cosh ψ Prediction for boundary entropy for both radius mode jumps by and orbifold blowup mode jumps by T 0 This has not calculated been calculated in the CFT! Some evidence from conformal perturbation theory. O 0

28 Multi Janus solutions What is the holographic interpretation of the solutions with n>2? x H,1 x H,2 x H,3 x A,1 x A,2 x A,3 x A,4 x A,5 y Σ 2 x ds 2 1 r 2 dr 2 + dξ2 dt 2 = ξ 2 ds 2 S3 1 r 2 ξ 2 ξ 2 dr 2 + dξ 2 dt 2 + ξ 2 ds 2 S 3 z x H,i = re iθ r 0 z x H,1 0 n asymptotic AdS regions : n half lines ξ>0 glued together at a point ξ =0 A junction (star graph) of n CFTs defined half spaces z x H,3 0 ξ 0 z x H,2 0

29 Multi Janus solutions What are the CFT s? Counting of 6n-4moduli of BPS interface solution n 3-spheres: support D1,D5, NS5 and F1 charges: 4(n-1) Attractor mechanism: in each asymptotic region 2 scalars are fixed 4C K e φ f χ = Q F 1 + f C 2 e φ K Q D1 f3 4 e φ f 4 3 = Q D5 Q D1 + 4C K e φ f 4 3 χ Q NS5 Q D1 Q NS5 Q D1 2 scalars are free: 2n parameters 6n-4 parameters corresponding to 6n-4 moduli of BPS interface solution Junctions of n arbitrary (U-duals) of D1/D5 CFT with radius and orbifold deformations

30 Multi Janus solutions Are these solutions related to configurations of branes in flat space? Bound state of D5 and NS5 branes wrapped on K3 and D1 and F1 in six dimensions all preserve 1/2 Susy of six dim Sugra and are selfdual strings {Q a α,q b β} =(P + C (6) γ µ (6) ) αβ C(4) ab P µ +(C (4) Γ i ) ab Zµ i Just as in Sen s string network: junction of 1/2 BPS strings in six dimensions preserves 1/4 of susy if strings intersect in a plane and angles are correlated with the charges Near horizon limit: enhancement of supersymmetry and coformal symmetry

31 Conclusions Exact half BPS solutions which are locally asymptotic to n=2 solution supersymmetric Janus solution n>2 solution dual to junctions of n 1+1 dim CFT s AdS 3 S 3 K 3 Solution has 6n-4 moduli identified with physical parameters of the CFT (Brane charges and expectation values of marginal operators) Calculation of boundary entropy for radius jump for supersymmetric Janus solution gives excat agreement with BCFT calculation

32 Conclusions Calculation of reflection on transmission matrices for junctions using holographically Work in progress Calculation of boundary entropy for the junction Comparison to CFT for junctions with radius jumps Applications of holographic dual to junctions of quantum wires In the solutions presented one did not turn on moduli of K3 and fluxes on two cycles of K3: generalize to solutions of N=4b 6dim supergravity