Towards a C -theorem in defect CFT Yoshiki Sato IPMU May 30, 2019 - PowerPoint PPT Presentation

Towards a C -theorem in defect CFT Yoshiki Sato IPMU May 30, 2019 based on arXiv:1810.06995 In collaboration with Nozomu Kobayashi (IPMU), & Tatsuma Nishioka, Kento Watanabe (Univ. of Tokyo)

Introduction Let us consider a RG flow triggered by a relevant operator O of dimension ∆ < d , � d d x √ g O ( x ) I CFT + λ We are interested in a monotonic decreasing function under the RG flow. C -function The C -function counts the effective degrees of freedom. The monotonicity provides nonperturbative constraints on the RG dynamics. C -theorem We want to generalize a C -theorem by adding boundary or defect. 1 / 15

Even dimensions C -function = the type A central charge of the conformal anomaly d = 2 : Zamolodchikov’s c -theorem d = 4 : a -theorem Odd dimensions (no conformal anomalies) d − 1 log Z [ S d ] C -function = the sphere free energy F ≡ ( − 1) 2 The conjecture has been extended to continuous d dimensions. Generalized F -theorem � π d ˜ � log Z [ S d ] is positive and does not increase along any RG flow F ≡ sin 2 This is the most general C -theorems proposed in arbitrary dimensions. 2 / 15

Entanglement entropy � R � � a log log , ( d = even) S (CFT) = A d − 2 ǫ d − 2 + A d − 4 ǫ ǫ d − 4 + · · · + a 0 , ( d = odd) ( R : typical length scale, ǫ : UV cutoff scale) Universal terms a log , a 0 are conjectured to be C -functions. This entropic version of the C -theorem looks different from the generalized F -theorem based on the sphere free energy. But the two C -function are the same at the fixed points due to the relation S (CFT) = log Z (CFT) for spherical entangling surface Free energy and EE are C -functions! without boundary or boundary 3 / 15

C -theorem in BCFTs & DCFTs g -theorem ( C -theorem in BCFT 2 ) S thermal = c π L β + log g g -function 3 ( c : central charge, L : size, β : inverse temperature) log g monotonically decreases under a boundary RG flow. The g -theorem can also be proved by using the equivalence of the g -function and the boundary entropy S bdy := S (BCFT) − 1 2 S (CFT) . b -theorem ( C -theorem in BCFT 3 & DCFT d with 2-dim defect) µ � = − 1 K ( α ) ab + d 2 W abcd ˆ g ac ˆ � g bd � b ˆ R + d 1 ˜ K ( α ) ab ˜ � t µ δ d − 2 ( x ⊥ ) 24 π When d = 3, the b -theorem implies the g -theorem in BCFT 3 . For d > 3 it yields a class of C -theorems in DCFTs. 4 / 15

Towards a C -theorem in DCFT We want to establish a C -function under a defect RG flow, � I = I DCFT + ˆ d p ˆ � g ˆ λ x ˆ O (ˆ x ) Two possibilities Defect free energy : log � D ( p ) � = log Z (DCFT) − log Z (CFT) 1 = additional contribution to the sphere free energy from the spherical defect Defect entropy : S defect = S (DCFT) − S (CFT) 2 = increment of the EE across a sphere due to the planer defect Question Which is a C -function? 5 / 15

Plan of my talk Introduction 1 Proposal for a C -theorem in DCFT 2 Wilson loop as a defect operator 3 Conclusion 4 6 / 15

Sphere partition function and EE in DCFT Let us clarify a relation between log � D ( p ) � and S defect . t D (1) A conformal defect D ( p ) respects SO (2 , p ) × R 1 , d − 1 SO ( d − p ) of the conformal group SO (2 , d ). Σ R D ( p ) = { X p = · · · = X d − 1 = 0 } A Using CHM map, R 1 , d − 1 → S 1 × H d − 1 The R´ enyi entropies are Z ((D)CFT) [ S 1 n × H d − 1 ] 1 S ((D)CFT) = 1 − n log � n n Z ((D)CFT) [ S 1 × H d − 1 ] � Around n = 1, the free energy can be expanded, n × H d − 1 ] = log Z (DCFT) [ S 1 × H d − 1 ] log Z (DCFT) [ S 1 − 1 � S 1 × H d − 1 δ g ττ � ( T DCFT ) ττ � (DCFT) S 1 × H d − 1 + · · · 2 7 / 15

The difference of the entanglement entropies becomes � S defect = log � D ( p ) � + τ � (DCFT) S 1 × H d − 1 � ( T DCFT ) τ S 1 × H d − 1 with the stress-energy tensor S 1 × H d − 1 d x µ ⊗ d x ν � ( T DCFT ) µν � (DCFT) a T � d − p − 1 − p + 1 � d τ 2 + d x 2 + cosh 2 x d s 2 sinh 2 x d s 2 � � = sinh d x H p − 1 S d − p − 1 d d Main result 2( d − p − 1) π d / 2+1 S defect = log � D ( p ) � − sin ( π p / 2) d Γ ( p / 2 + 1) Γ (( d − p ) / 2) a T This is a generalization of the result for p = 1 [Lewkowycz-Maldacena ’13] . For p = d − 1, the defect entropy is given by S defect = log � D ( d − 1) � 8 / 15

Proposal for a C -theorem in DCFT Two candidates for a C -function in DCFT defect free energy log � D ( p ) � = log Z (DCFT) [ S d ] / Z (CFT) [ S d ] 1 ( − 1) p / 2 B log ǫ + · · · , � ( p : even) , log � D ( p ) � = c p ǫ p + c p − 2 ǫ p − 2 + · · · + ( − 1) ( p − 1) / 2 D , ( p : odd) . defect entropy S defect 2 ( − 1) p / 2 B ′ log ǫ + · · · , � S defect = c ′ ǫ p − 2 + c ′ ( p : even) , p − 2 p − 4 ǫ p − 4 + · · · + ( − 1) ( p − 1) / 2 D ′ , ( p : odd) . Our proposal In DCFT d with a defect of dimension p , the universal part of the defect free energy � π p � log |� D ( p ) �| ˜ D ≡ sin 2 does not increase along any defect RG flow ˜ D UV ≥ ˜ D IR . 9 / 15

Summary of the conjectured and proved C -theorems d = 2 d = 3 d = 4 d = 5 g -theorem Proof [Friedan- p = 1 Konechny ’91, Casini- Landea-Torroba ’16] b -theorem (bdy c -theorem) p = 2 Proof [Jensen-O’Bannon ’15] bdy F -theorem Proposal [Nozaki- p = 3 Takayanagi-Ugajin ’12, Gaiotto ’14] p = 4 Our proposal reduces to the known ones in the shaded regions & provides new ones in the region colored in blue. 10 / 15

Wilson loop as a defect operator We test our proposal for p = 1 using a circular Wilson loop operators � � � d x µ A µ W R [ A ] = Tr R exp i The Wilson loop can be regarded as an action localized on the defect. [Gomis-Passerini ’06, Tong-Wong ’14] W R [ A ] = Z q [ A ] with Z q [0] Z q [ A ] ≡ 1 � D χ † D χ χ a 1 (+ ∞ ) · · · χ a q (+ ∞ ) χ † , a 1 ( −∞ ) · · · χ † , a q ( −∞ ) e − I χ q ! � d t χ † ( i ∂ t − A ( t )) χ I χ = The defect theory can flow to the trivial theory without fermions, W R [ A ] → 1 under the mass deformation � d t M χ † χ , I M = − M → ∞ 11 / 15

U (1) gauge theory in 4d � � � d x µ A µ W = exp i e , e ∈ R [Lewkowycz-Maldacena ’13] log � W � = e 2 / 4 The defect free energy The defect entropy S defect = 0 The Wilson loop becomes trivial under a defect RG flow, log � W � → 0. = ⇒ This is consistent with our conjecture. On the other hand, the defect entropy vanishes at both the UV and IR fixed points. = ⇒ The defect entropy doesn’t capture degrees of freedom on the defect. 12 / 15

Free scalar field in 4d � � � d t φ ( x µ ( t )) W = exp λ , λ ∈ C The defect free energy log � W � = 0 This result does not contradict with our assertion. The defect entropy S defect = − λ 2 12 It can be negative for real λ at the UV fixed point. But it is supposed to be zero at the IR fixed point. = ⇒ This is a counterexample for the defect entropy being a C -function. 13 / 15

Conclusion We examine the defect free energy log � D ( p ) � and the defect entropy S defect as a candidate C -function. We find the relation with them 2( d − p − 1) π d / 2+1 S defect = log � D ( p ) � − sin ( π p / 2) d Γ ( p / 2 + 1) Γ (( d − p ) / 2) a T We propose a C -theorem in DCFTs. The defect free energy does not increase under any defect RG flow. We find in Wilson loop examples that the sphere free energy decreases but the EE increases along a certain RG flow triggered by a defect localized perturbation which is assumed to have a trivial IR fixed point without defects. 14 / 15

We also checked more field theoretic examples, Conformal perturbation theory on defect Wilson loop Chern-Simons theory 1 1/2-BPS Wilson loop in 4d N = 4 SYM 2 1/6-BPS Wilson loop in ABJM 3 U ( N ) N = 4 SYM with N f hypermultiplets in 3d 4 We also provide a proof of our proposal in several holographic models of defect RG flows. Domain wall defect RG flow 1 Probe brane model 2 Holographic model of defect RG flow 3 [Yamaguchi ’02] AdS/BCFT model 4 [Takayanagi ’11] 15 / 15

Thank you for your attention!

Stress tensor δ log Z (DCFT) [ g µν ] DCFT = − 2 Defenition : T µν = T µν CFT + t µν √ g δ g µν T µν DCFT is traceless and partially conserved DCFT = − δ D ( x ⊥ ) D i , ∂ µ T µ a ∂ µ T µ i ( T DCFT ) µ DCFT = 0 , µ = 0 The one-point function of T CFT � ⊥ x j � δ ij − x i CFT ( x ) � = d − p − 1 p + 1 a T a T | x ⊥ | d δ ab , � T ab � T ai � T ij ⊥ CFT ( x ) � = 0 , CFT ( x ) � = − , | x ⊥ | d | x ⊥ | 2 d d Note that � T µν CFT � = 0 for p = d − 1. � t µν ( x ) � = 0 since t µν is localized on the defect, t µν ( x ) = δ D ( x ⊥ ) ∂ x µ ∂ x ν x b ˆ t ab (ˆ x ) ∂ ˆ x a ∂ ˆ ˆ t ab (ˆ x ) is a defect local operator of dimension p and invariant under the translation, rotation and scale transformation on the defect. 16 / 15