Structure Constants from Modularity in Warped Conformal Theories - PowerPoint PPT Presentation

Introduction Known results of WCFT Structure constants of WCFT Summary and outlook Structure Constants from Modularity in Warped Conformal Theories Jianfei Xu (Southeast University) collaborate with Prof. Wei Song TSIMF, Sanya Workshop January 11, 2019

Introduction Known results of WCFT Structure constants of WCFT Summary and outlook Contents Introduction 1 Known results of WCFT 2 Structure constants of WCFT 3 Summary and outlook 4

Introduction Known results of WCFT Structure constants of WCFT Summary and outlook Introduction The holographic dualities, which relate a quantum theory of grav- ity to a quantum field theory without gravity in fewer dimensions, play essential roles in theoretical physics. The benchmark of the holographic dualities is the AdS/CFT cor- respondence established based on string theory [J. Maldacena, 1998] .

Introduction Known results of WCFT Structure constants of WCFT Summary and outlook However, the existence of holographic dualities is not contingent on the validity of string theory. The Asymptotic Symmetry Group (ASG) method is successfully applied to AdS 3 /CFT 2 without invoking string theory [J. D. Brown, M. Henneaux, 1986] . The study of holography also goes beyond the standard AdS/CFT correspondence. The main reason behind these expectation is that the entropy of black holes is given by the area instead of volume in a general form, S BH ∼ Area . (1) ℓ d − 1 p

Introduction Known results of WCFT Structure constants of WCFT Summary and outlook It is necessary to extend the idea of holography to non-AdS case in order to understand the quantum gravity on a complete level. The efforts come from many aspects: The scaling limit of near horizon geometry of Kerr black holes has enlarged SL ( 2 , R ) L × U ( 1 ) R isometry [J. Bardeen, G. Horowitz, 1999] . Kerr/CFT claims that the extremal Kerr black holes are described by a chiral half of a two dimensional CFT [M. Guica, T. Hartman, W. Song, A, Strominger, 2009] . c L = 12 J T L = 1 � , 2 π . (2) The enhancement of the U ( 1 ) R isometry to the full Virasoro alge- bra. The perfect match between the black hole entropy formula and the Cardy entropy.

Introduction Known results of WCFT Structure constants of WCFT Summary and outlook The part of the geometry (at fixed polar angle) that appears to play the key role in the duality is a warped AdS 3 (WAdS) factor. � − dt 2 + dy 2 � + d θ 2 + Λ( θ ) 2 ( d φ + dt da 2 = 2 GJ Ω( θ ) 2 y ) 2 y 2 (3) The structure of WAdS is that of a fibration (with warping factor multiplying the fiber metric) of a real line over AdS 2 . The warping factor along the fiber breaks the SL ( 2 , R ) L × SL ( 2 , R ) R isometry group of AdS 3 down to SL ( 2 , R ) × U ( 1 ) . In topological massive gravity (TMG), the AdS 3 vacua is unstable due to the negative energy of massive excitations ( G > 0 , µ > 0) [S. Deser, R. Jackiw, S. Templeton, 1982] .

Introduction Known results of WCFT Structure constants of WCFT Summary and outlook For generic µℓ , WAdS as possibly stable vacua and various type of warped black holes are found in TMG [D. Anninos, W. Li, M. Padi, W. Song, A. Strominger, 2009] . For µℓ > 3, the WAdS is said to be stretched and there exist reg- ular black holes which are asymptotic to WAdS with a spacelike U ( 1 ) . These regular black holes are shown to be discrete quotients of WAdS just as BTZ black holes are discrete quotients of ordinary AdS 3 . Further more, given the left and right moving temperature dur- ing quotients, the warped black hole entropy matches the Cardy entropy provided 15 ( µℓ ) 2 + 81 12 µℓ 2 c L = G (( µℓ ) 2 + 27 ) , c R = G µ (( µℓ ) 2 + 27 ) . (4) So µℓ > 3 TMG is conjectured to be dual to a 2D CFT.

Introduction Known results of WCFT Structure constants of WCFT Summary and outlook The asymptotic symmetry analysis for spacelike stretched WAdS and the consistent boundary conditions are also presented [G. Comp ` ere, S. Detournay, 2008,2009] . The asymptotic algebra they get is a Virasoro algebra and a cur- rent algebra, which indicate that dual field theory would have symmetry other than conformal symmetry. The Virasoro Kac-Moody algebra also shows up in the asymptotic symmetry analysis for AdS 3 with mixed chiral boundary condi- tions (CSS B.C.) [G. Comp ` ere, W. Song, A. Strominger, 2013] .

Introduction Known results of WCFT Structure constants of WCFT Summary and outlook Known results of Warped CFT A two dimensional quantum field theory with two global transla- tional symmetries and a chiral global scaling symmetry have an extended local Virasoro plus U ( 1 ) Kac-Moody algebra [D. Hofman, A. Strominger, 2011] . x − → x − + a , x + → x + + b , x − → λ x − (5) A warped conformal field theory is characterized by the warped conformal symmetry. The global symmetries are SL ( 2 , R ) × U ( 1 ) , while the local symmetry algebra is a Virasoro algebra plus a U ( 1 ) Kac-Moody algebra.

Introduction Known results of WCFT Structure constants of WCFT Summary and outlook In position space, a general warped conformal symmetry trans- formation can be written as x ′− = f ( x − ) , x ′ + = x + + g ( x − ) , (6) where f ( x − ) and g ( x − ) are two arbitrary functions. Consider a WCFT on a plane, denote T ( x − ) and P ( x − ) as the Noether currents associated with translations along x − and x + , the conserved charges � � L n = − i P n = − 1 d xx n + 1 T ( x ) , d xx n P ( x ) . (7) 2 π 2 π form a Virasoro Kac-Moody algebra, [ L n , L m ] =( n − m ) L n + m + c 12 n ( n 2 − 1 ) δ n , − m , [ L n , P m ] = − mP n + m , [ P n , P m ] = kn 2 δ n , − m , (8)

Introduction Known results of WCFT Structure constants of WCFT Summary and outlook Some specific examples of WCFT: chiral Liouville gravity [G. Comp ` ere, W. Song, A. Strominger, 2013] Weyl fermion models [D. Hofman, B. Rollier, 2015] free scalar models [K. Jensen, 2017] CSYK as broken symmetry of WCFT [P. Chaturvedi, Y. Gu, W. Song, B. Yu, 2018] .

Introduction Known results of WCFT Structure constants of WCFT Summary and outlook Under the warped conformal transformation, a primary field trans- forms as an h -form under Virasoro and a scalar under U ( 1 ) , [W. Song, JX, 2017] � ∂ x ′− � − h φ ′ ( x ′− , x ′ + ) = φ ( x − , x + ) . (9) ∂ x − The correlation functions of WCFT are given by, � δ h 1 , − h 2 � j q j x + e i � j δ � � φ 1 φ 2 � = k q k ( x − 12 ) 2 h 1 � � C 123 j q j x + e i � j δ � � φ 1 φ 2 φ 3 � = k q k ( x − 12 ) h 1 + h 2 − h 3 ( x − 23 ) h 2 + h 3 − h 1 ( x − 31 ) h 3 + h 1 − h 2 � h 12 � � h 34 � � x − x − � G ( z ) j q j x + e i � j δ � 24 14 � φ 1 φ 2 φ 3 φ 4 � = 34 ) h 3 + h 4 . k q k x − x − ( x − 12 ) h 1 + h 2 ( x − 14 13

Introduction Known results of WCFT Structure constants of WCFT Summary and outlook The R ´ enyi entropy for an interval D can be written as, 1 − n log � Φ n ( x 1 )Φ † 1 − n log tr ( ρ n D ) n ( x 2 ) � C 1 1 S n = ( tr ρ D ) n = (10) . � Φ 1 ( x 1 )Φ † 1 ( x 2 ) � n C Here in the first equality, the R ´ enyi entropy is related to the n th power of the reduced density matrix ρ D for D . This can be real- ized as a path integral an a manifold R n which is made up of n decoupled copies of the original space R 1 . In the second equality, Φ n is the twist field inserted at the endpoints of the interval that enforce the replica boundary conditions on a plane C . X 1 , 2 are the endpoint coordinates of the interval D .

Introduction Known results of WCFT Structure constants of WCFT Summary and outlook The expectation value of the current T ( x ) and P ( x ) on R n can be calculated either using twist field ward identity or using Rindler transformation, � T ( x )Φ n ( x 1 )Φ † = � T ( x ( i ) ) � R n = tr ( T ( x ) ρ n = tr ( U † T ( x ) U ρ n D ) H ) n ( x 2 ) � C , � Φ n ( x 1 )Φ † tr ( ρ n tr ( ρ n D ) H ) n ( x 2 ) � C where U stands for a unitary transformation inspired by Rindler transformation, tanh π x − � ¯ � ¯ x − = tanh π ˜ β β − α � κ κ − α � x + + x − = ˜ x + + x − . β κ , ˜ tanh ∆ x − π β κ 2 β Compere two sides, we can get the expressions for the conformal dimension and charge of the twist field, � P vac � c 24 + L vac − i P vac α − α 2 k � − i k α � 0 0 0 h n = n , q n = n . n 2 2 n π 16 π 2 4 π n The R ´ enyi entropy, � β ¯ − 2 ( n + 1 ) L vac πǫ sinh πδ x − � β − α � � − i α � � δ x + + S n = − iP vac δ x − π P vac 0 + . log 0 0 β β n The R ´ enyi mutual information [B. Chen, P. Hao, W. Song, in coming] .