Conformal blocks, entanglement entropy and heavy states Shouvik - PowerPoint PPT Presentation


 Conformal blocks, entanglement entropy and heavy states Shouvik Datta Institut für Theoretische Physik ETH Zürich Workshop on topics in 3d gravity, ICTP Trieste

arXiv:1601.06794 Higher-point conformal blocks & entanglement entropy of heavy states with Pinaki Banerjee (IMSc) and Ritam Sinha (TIFR)

Why conformal blocks? AdS/CFT –– universal features of CFTs find natural analogues in gravity. (eg. Cardy’s formula, entanglement entropy) [Cardy; Cardy-Calabrese; Ryu-Takayanagi;…] Finding correlation functions by decomposition into conformal blocks is a powerful and minimalistic approach to extract information to study CFTs. Conformal bootstrap anomalous dimensions of operators, bounds on central charges, … [Rattazi-Rychkov-Tonni-Vichi; ElShowk-Paulos-Poland-Rychkov-SimmonsDuffin-Vichi;…] AGT correspondence Liouville and Toda conformal blocks are related to 4d SCFTs N = 2 [Alday-Gaiotto-Tachikawa; Wyllard; Nekrasov;…] Holography bulk locality, graviton scattering, … [Heemskerk-Penedones-Polchinski-Sully; ElShowk-Papadodimas; Fitzpartick-Kaplan-Walters; Jackson-McGough-Verlinde;…]

What are conformal blocks? Consider a p -point correlation function h O 1 ( z 1 ) O 2 ( z 2 ) O 3 ( z 3 ) · · · O p − 1 ( z p − 1 ) O p ( z p ) i This can be rewritten as a sum of conformal blocks upon inserting ( p– 3) resolutions of the identity. X h O 1 ( z 1 ) O 2 ( z 2 ) | α ih α |O 3 ( z 3 ) | β i · · · h ζ |O p − 1 ( z p − 1 ) O p ( z p ) i α , β ,..., ζ [Ferrara-Grillo-Gatto; Zamolodchikov; Dolan-Osborne;…] The sum is over all operators of the theory – complete set of states. A typical term in the above sum is referred to as the conformal block F ( p ) ( z i , h i , ˜ h j ) = h O 1 ( z 1 ) O 2 ( z 2 ) | α ih α |O 3 ( z 3 ) | β i · · · h ζ |O p − 1 ( z p − 1 ) O p ( z p ) i These are the building blocks of CFT correlators.

Conformal blocks and holography Recently an intriguing relation has been found between conformal blocks and geodesics in AdS. [Fitzpatrick-Kaplan-Walters; Asplund-Bernamonti-Galle-Hartman; Hijano-Kraus-Snively;…] An important object, in this context, is the correlator between two heavy and two light operators. The corresponding conformal block can be reproduced from the bulk from worldline configurations in a conical defect background. This result is known for 4- and 5-point blocks. [Fitzpatrick-Kaplan-Walters; Alkalaev-Belavin] We generalise these results to blocks with an arbitrary number of light operator insertions, both from CFT and holography. This result can be used to calculate entanglement entropy and mutual information of heavy states.

Heavy-light conformal blocks Consider a 2d CFT at large central charge. We shall focus on correlators with an arbitrary number of light operators and two heavy operators. h O H ( z 1 , ¯ z 1 ) O L ( z 2 , ¯ z 2 ) O L ( z 3 , ¯ z 3 ) · · · O L ( z m +1 , ¯ z m +1 ) O H ( z m +2 , ¯ z m +2 ) i The conformal dimension of these operators are ✏ H = 6 h H = O(1) c ✏ L = 6 h L ⌧ 1 c Example of light operators : twist operators to implement the replica trick while calculating entanglement entropy

Heavy-light conformal blocks Consider a 2d CFT at large central charge. We shall focus on correlators with an arbitrary number of light operators and two heavy operators. h O H ( 1 ) O L (1) O L ( x 3 , ¯ x 3 ) · · · O L ( x m +1 , ¯ x m +1 ) O H (0) i We focus on OPE channels in which the light operators fuse in pairs. x  1 x  x  x  x  x m x m +1 � L � L � L � L � L � L � L � L ˜ ˜ ˜ ˜ � p � p � p � p 0 ∞ ˜ ˜ � H � H � Q � R

The monodromy method At large central charge, the conformal blocks are expected to exponentiate − c � � F ( z i , h i , ˜ h i ) = exp 6 f ( z i , � i , ˜ � i ) [Zamolodchikov^2] One can insert a field which obeys the null-state condition at level 2 inside the conformal block. Ψ ( z, z i , h i , ˜ h j ) = O 1 ( z 1 ) O 2 ( z 2 ) | α α | ˆ ψ O 3 ( z 3 ) | β · · · ζ |O p − 1 ( z p − 1 ) O p ( z p ) Insertion of this, multiples the conformal block by an overall wavefunction Ψ ( z, z i ) = ψ ( z, z i ) F ( z i , h i , ˜ h i ) [Fitzpatrick-Kaplan-Walters] ˆ The null-state condition on inserted within the conformal block can then ψ be translated into an ODE.

The monodromy method The ODE is d 2 ψ ( z ) + T ( z ) ψ ( z ) = 0 dz 2 p � � c i � i where, . � T ( z ) = ( z − z i ) 2 + z − z i i =1 � i = 6 h i /c Accessory c i = − ∂ f ( z i ) parameters ∂ z i ∂ c i = ∂ c j Integrability condition : ∂ z j ∂ z i

The monodromy method The ODE is d 2 ψ ( z ) + T ( z ) ψ ( z ) = 0 dz 2 p � � c i � i where, . � T ( z ) = ( z − z i ) 2 + z − z i i =1 � i = 6 h i /c Accessory c i = − ∂ f ( z i ) parameters ∂ z i We need to determine these accessory parameters. This can be done by studying monodromy properties of , ψ ( z, z i ) around contours containing the operator insertions.

The monodromy method The monodromy method The ODE is d 2 ψ ( z ) + T ( z ) ψ ( z ) = 0 dz 2 p � � c i � i where, . � T ( z ) = ( z − z i ) 2 + z − z i i =1 � i = 6 h i /c Accessory c i = − ∂ f ( z i ) parameters ∂ z i Monodromy around a contour = info about the resultant operator γ k which arises upon fusing the operators within γ k � � e + π i Λ � 0 � M ( γ k ) = − Λ = 1 − 4˜ , � p e − π i Λ 0

The monodromy method The monodromy method The ODE is d 2 ψ ( z ) + T ( z ) ψ ( z ) = 0 dz 2 p � � c i � i where, . � T ( z ) = ( z − z i ) 2 + z − z i i =1 � i = 6 h i /c Accessory c i = − ∂ f ( z i ) parameters ∂ z i ψ ( z ) = ψ (0) ( z ) + ψ (1) ( z ) + ψ (2) ( z ) + · · · , Perturbation theory in T ( z ) = T (0) ( z ) + T (1) ( z ) + T (2) ( z ) + · · · , ✏ L = 6 h L /c (heavy-light limit) c i ( z ) = c (0) i ( z ) + c (1) i ( z ) + c (2) i ( z ) + · · · ,

Choice of contours / OPE channels Choice of monodromy contour Choice of OPE channel We choose the contours such that each of them contains a pair of light operators within. This is equivalent to looking at the OPE channel in which light operators fuse in pairs. [Hartman; Faulkner; Headrick] This choice is geared towards entanglement entropy calculations. 1 x 3 x 4 x 5 � L � L � L � L γ 2 γ 1 = x 3 1 x 5 x 4 ˜ ˜ � p � p 0 ∞ � H � H

Choice of contours / OPE channels Choice of monodromy contour Choice of OPE channel We choose the contours such that each of them contains a pair of light operators within. This is equivalent to looking at the OPE channel in which light operators fuse in pairs. [Hartman; Faulkner; Headrick] 1 x 3 x 4 � L � L γ 1 γ 2 = � L 1 x 3 x 4 ˜ � p 0 ∞ � H � H

Seed solutions [Alkalaev-Belavin] Idea : Use the accessory parameters of the lower point blocks as zeroth order solutions for the higher-point ones. We work to the leading order in the light-parameter . ✏ L = 6 h L /c Solutions to the ODE to the linear order in can be found ✏ L by the method of variation of parameters.

Accessory parameters The monodromy conditions for all the contours form a coupled system of equations for the accessory parameters. Performing the exercise for 5- and 6-point blocks provides sufficient intuition to guess the solutions. For light operators located at x p and x q living with a contour, the corresponding accessory parameters are p ( α + 1)) + ( x p x q ) α / 2 α ˜ c p = − � L ( x α q ( α − 1) + x α � a x p ( x α q ) p − x α q ( α + 1)) + ( x q x p ) α / 2 α ˜ c q = − � L ( x α p ( α − 1) + x α � a x q ( x α p ) q − x α These obey the integrability conditions.

� � Factorisation of higher-point blocks The accessory parameters can now be used to obtain the conformal block c i = − @ f ( p ) ( z i , ✏ i , ˜ ✏ j ) − c h i F ( p ) ( z i , h i , ˜ h j ) = exp 6 f ( p ) ( z i , ✏ i , ˜ ✏ j ) @ z i Even-point conformal blocks The ( m +2) -point block factorises into a product of m/ 2 4-point conformal blocks − c � � � F ( m +2) ( { x i } ; � L , � H ; ˜ � a ) = exp 6 f (4) ( x p , x q ; � L , � H ; ˜ � a ) { ( p,q ) } Ω i → � = F (4) ( x p , x q ; � L , � H ; ˜ � a ) { ( p,q ) } Ω i → OPE channel (pairings of the light operators)

� � Factorisation of higher-point blocks The accessory parameters can now be used to obtain the conformal block c i = − @ f ( p ) ( z i , ✏ i , ˜ ✏ j ) − c h i F ( p ) ( z i , h i , ˜ h j ) = exp 6 f ( p ) ( z i , ✏ i , ˜ ✏ j ) @ z i Odd-point conformal blocks The ( m +2) -point block factorises into a product of ( m –1)/2 4-point conformal blocks and a 3-point function − c � � � � a ) = ( x s ) − � L F ( m +2) ( { x i } ; � L , � H ; ˜ exp 6 f (4) ( x p , x q ; � L , � H ; ˜ � a ) Ω A { ( p,q ) } → i � = ( x s ) − � L F (4) ( x p , x q ; � L , � H ; ˜ � a ) Ω A { ( p,q ) } → i x α / 2 + x α / 2 � � x α i − x α � � j j i where, f (4) ( x i , x j ; � L , � H ; � p ) = � L (1 − α ) log x i x j + 2 log + 2˜ � p log 4 α x α / 2 − x α / 2 α j i