JHEP01(2019)025 1. For Springer c > January 3, 2019 October 24, 2018 : with : December 11, 2018 2 : CFT Received Published Accepted general if the total Liouville momentum 1 − 12 c (or the crossing kernel). Applying this Published for SISSA by https://doi.org/10.1007/JHEP01(2019)025 fusion matrix . This might be interpreted as a black hole formation . 3 BTZ threshold 1810.01335 The Authors. 1, which was unknown until now. In this study, we computed it in general c Conformal Field Theory, AdS-CFT Correspondence, Field Theories in Lower
The light cone OPE limit provides a significant amount of information regard- , → [email protected] z . 3 We also provide a universal form of the Regge limit of the Virasoro conformal blocks As another application of our light cone singularity, we studied the dynamics of entan- E-mail: Center for Gravitational Physics,Kyoto Yukawa Institute University, for Kitashirakawa Oiwakecho, Theoretical Physics Sakyo-ku, (YITP), Kyoto 606-8502, Japan -th Renyi entropy after a local quench and out of time ordered correlators. Open Access Article funded by SCOAP Dimensions ArXiv ePrint: from the analysis ofn the light cone singularity. This Regge limit isKeywords: related to the general exceeds beyond the in AdS glement after a global quench andvaried. found a We Renyi also phase investigated transition the as dynamics the replica of number the was 2nd Renyi entropy after a local quench. the limit by studying the pole structureresult of to the the lightthe cone universal bootstrap, binding energy) we of obtainedwe two found particles the that at universal the a total total large twist twist angular is momentum. (or saturated by In equivalently, the particular, value Abstract: ing the conformal field theoryfunction. (CFT), We like started the with high-lowthis the temperature purpose, light limit cone we of bootstrap needed the in an partition the explicit asymptotic form of the Virasoro conformal blocks in Yuya Kusuki Light cone bootstrap inentanglement general from 2D light CFTs cone and singularity JHEP01(2019)025 , 6 20 conformal bootstrap 18 CFTs c c 13 21 31 27 ]. The conformal bootstrap equation originates 12 – 1 – 4 33 – 27 l 1 33 10 1 23 5 spectrum of twist 25 l One important tool to determine which CFT data are consistent is A.1 Leading termA.2 in light cone Sub-leading limit terms in light cone limit 3.2 Dynamics of3.3 Renyi mutual information Renyi in mutual3.4 large information beyond large 2nd Renyi entropy after local quench 2.1 Lowest twist2.2 operator at large Large 3.1 Mutual information in light cone limit been devised to classifyclassified; CFTs in and other only words, a wewith do few the not modular models know invariance. (for how example, to minimal identify models) which CFT are or data equivalently, the are crossing consistent symmetry [ try, the so-called Virasoro symmetry.only specified This symmetry by leads aexpansion to central (OPE) the charge, fact coefficients spectrum that of ofsimplification, 2D primary the 2D CFTs operators, are primary CFTs and operators offerare operator also in ideal product investigated the avenues as the for spectrum.AdS/CFT key exploring correspondence. to quantum Owing understanding However, quantum field to mechanics despite theories. this in several AdS decades by They of using the studies, no criterion has 1 Introduction & summary Two-dimensional conformal field theories (2D CFTs) have an infinite dimensional symme- B Light cone modular bootstrap C Zamolodchikov recursion relation 5 Discussion A Light cone limit from fusion matrix 4 Regge limit universality 3 Entanglement and light cone limit Contents 1 Introduction & summary 2 Light cone bootstrap JHEP01(2019)025 , ) z (1.2) (1.3) (1.1) v, u ( , where high-low , which ) p t z 12 ) , ,l ]. Unlike z with two − ( ,α p 12 τ s (1 K 3 19 g α K 0 is related to – ≥ π d F p 12 , − → ,l 0 as the 11 p ¯ 12 z τ ] → = e , P z, ) ¯ z 1. The method for β z 12 p , − z, , 0 X − . In this way, the light ≥ 11 c > 1 (or equivalently 1 12 | Z t ∆ ]. α ∈ ]. The most difficult point ]. Apart from this famous h → ∞ 10 ( 5 n v 13 12 l – 20 x x 41 32 6 -channel conformal blocks. F t # 0 as u 0) relates high-energy physics (i.e. 12 1 4 ∆ are the conformal blocks and their 0 through a map e α α → and → ) 2 3 s ¯ z → 24 12 α α ij,kl τ,l x x β ) for any z, n, l " P ( . Consequently, we can explicitly provide t 2 n, l 2 ), ,α ( AB s Q γ α +∆ AB – 2 – u, v in the following). At present, there is no exact 1 F ( t γ ∆ we can evaluate, for example, the density of states , as pointed out in [ α ) light cone limit 3 1 + 1 d , ij,kl ≥ τ,l 23 S n g x = 1 + 6 1 of the Viraosoro conformal blocks (which we will Z 14 c + 2 x → ( ) = B z 0 (or equivalently, z | )= s ) and α → i , there must exist operators with twist [ + ∆ h In 2D CFTs, the task of solving the light cone bootstrap is much u, v v α B ( A ( φ 21 − 34 p − 22 , ,l 1 F p Q = ∆ 11 τ ( u, and g i τ p α 22 A , ,l φ p 0, the light cone limit relates the vacuum blocks to the OPE at large spin. 11 = τ . P i → h ¯ p z are the cross ratios, and X high-low temperature duality light cone limit Virasoro blocks z, 2 1), has attracted considerable interests in higher-dimensional CFTs [ u, v 2∆ 34 the light cone singularity of the conformal blocks. We then summarize the results, are invertible fusion transformations between where is to evaluate thecall limit asymptotic formula for the lightform. cone limit of However, Virasoro in blocks,the this let light study, alone cone their we limit explicit Virasoro accomplishedaccomplishing blocks this to is in to provide any study the unitary the CFT structure asymptotic of with form poles in of the fusion matrix more difficult than that in CFT x 1 In this paper, we give the following results: Recently, another kinematic limit, 1 ) is the elliptic integral of the first kind. For this reason, we will call the limit The conformal bootstrap equation of a particular four-point function in the limit ¯ z 1 z ( 2∆ 12 − x the modular bootstrap inK the high-low temperaturetemperature limit limit cone limit provides substantial information about the spectrum. Light cone singularity In fact, the lightscalar cone operators bootstrap imposes a condition such that in the CFT and in particular, the anomalous dimension consequence, we also have several results from this duality1 [ the limit Therefore, the light cone bootstrap reveals the structure of the OPE in the large spin limit. requirements for CFT dataparticular, can the limit be obtainedinformation using about the the spectrum conformal at large bootstrapFrom conformal this dimensions) equation. to the In vacuum contribution. at large conformal dimensions, the so-called Cardy formula [ where coefficients. This equation relates different OPE coefficients, and therefore, nontrivial from the OPE associativity. JHEP01(2019)025 (1.4) (1.5) (1.6) 0 of the . 1) → , , ¯ z A B → z, ¯ , z | < α < α 2 Q (0 . , B A < α α AA BB . B F α , which are defined as 0) and and + . → 4 4 A Q Q α z | < < if otherwise (0 A B , . Therefore, we expect that this α α B AA c BB h 24 , if if otherwise − F ABBA blocks B , A α h A (0) OPE channel, A ) = − α 1 A Qα 2 , − 2 24 c O − B , BTZ − ) ) ) B z B α z z light cone limit ( Qα h h 2 2 2 – 3 – A − − − − − − O 1 B A light cone limit Q (1 (1 − h h 24 ( c 2 4 i −−−−−−−−−→ ( ) ) ) z z z BTZ (0) ≡ ≡ 1 in the − − − A α ) ) → . The other type is i z x x O −−−→ | | (1 (1 (1 ) = p p ) ¯ z (0) h h z | ( ( A z, s captures some important properties of holographic CFTs. In ( 1 α O BTZ AA BA ) BB A BA h h → z ( z F F O ( −−−→ A ) AA BB (1) z O | AABB blocks F B s α O (1) h ) , where 1. In fact, it is given by the ( B 2 Q ∞ O ( ) → BA BA = B z F ∞ O ( h B BTZ O that the transition point ofα the AABB blocks islight characterised cone by transition the BTZfact, threshold one gravity interpretationbootstrap of as this we transition will can explain be later. obtained by the light cone to find a quantitylimit that can be evaluated by only one block contribution and in the Therefore, we can detect the transition in the light cone limit. We want to emphasize This is one of thewe main can results find presented a intransition transition this in paper. of the the One blocks. interesting behaviouronly point of one However, is (or in a that few) general, four-pointt-channel block we function expansion. contribution cannot because appears If observe one the if this wants considering approximation to the find by limit this transition in a correlator, then one has The asymptotics of the AABB blocks are given by The light cone singularities for the ABBA blocks is given by which we call types of Viraosoro conformalh blocks. One is the Virasoro block for the correlator but before that, we will introduce some notations. In this paper, we consider two JHEP01(2019)025 . ]; . 4 23 . In = 1; A.1 A entan- n ]. In this 49 , , we evaluate 48 ] and proven in increases beyond 3.4 CFTs but also for 22 B , c α 21 + is always above , we consider the Renyi A ∗ α n 3.2 can also be found in its Renyi ∗ n ]. In section 1 and without extra conserved current. 47 c > duality (for example, one concrete question 2 – 4 – ) holds true not only for large /CFT 1.4 3 , unlike the Zamolodchikov monodromy method [ . ), ( 2 Q 1. Note that our formula breaks down for the CFT data if the total Liouville momentum 1.5 → ∞ = 1 , we investigate the other poles (sub-leading poles) in the c c > − 12 c BTZ A.2 In the main sections of this paper, we will apply this explicit form α . Therefore, the transition at ]. We intend to emphasize that the fusion matrix approach does not 3.1 1 carefully. Note that the transition point 23 , we first consider the Renyi entanglement entropy for a special setup ]. Similarly, it is expected that the light cone singularities revealed in → Apart from the light cone bootstrap, the light cone limit Viraosoro blocks 46 3.1 n – using the explicit asymptotic form of the light cone limit Virasoro blocks. 24 CFTs [ c In addition, we discuss the sub-leading terms of the blocks in the light cone The detailed derivations of these singularities are provided in appendix 1) of the minimal model, as explained in the final paragraph of appendix c < that in section entanglement entropy. We also predictthe that lowest bound maximal on scrambling the ishand, light characterized cone the by singularity quasi of a particlelight particular cone picture correlator; limit on comes the gives from other us the information upper about bound. scrambling. In It section means that the that when we trythe to limit evaluate entanglementtherefore, using this the Renyi Renyi phase entropy, transitionholographic entanglement we does entropy must not formula [ contradict use entanglement with entropy the for derivation of doubled the setup, CFTs, the which light was cone introduced limit by non-trivially [ appears in the calculation, but is similar to glement In section with a Lorentz boostedIn this interval case, and the anwe light found unboosted cone a interval, limit Renyi as naturally phase shown appears transition in in as figure the the calculation. replica number As was a varied. result, This implies by the BTZ threshold the BTZ threshold also appear in many different situations. In the rest of this paper, we discuss ments pertaining to conformalCFTs blocks [ helps uncover informationthis paper about also holographic lead tofound some that interesting there predictions must inat holographic be large CFTs. a spin In universal for fact, long-distanceParticularly, we any interaction we unitary found between CFT that two with objects the total twist of two particles at large spin is saturated of light cone singularitytives to is study to the understand lightis the cone how AdS bootstrap. Virasoro One blocks of can the be main derived objec- from AdS gravity). Many recent develop- ( limit. In appendix fusion matrix and identifysingularities them perfectly as match the thelight sub-leading HHLL limit. terms. Virasoro blocks’ In singularities fact, in the the sub-leading heavy- fact, this conclusion islarge supported by numerical computationsrely [ on the assumption therefore, the conclusion ( any unitary CFT with Entanglement Light cone bootstrap JHEP01(2019)025 B (2.1) ], which -th Renyi 55 n – 50 , the asymptotic in the following. B . BTZ O have large anomalous C and general B > α 2 and O . B 3 unitary CFTs, the large 0 A α ]. ≥ ≥ O + and Z 60 d – A A ∈ 56 α O n , the 2nd Renyi entropy is saturated. 1 − 32 captures information about holographic c 1 and no extra currents, the heavier the 1. It might give a key to understanding for any . 3 c > L – 5 – c n h + 2 B τ Interestingly, our result from the bootstrap equation . From this result, we can predict, for example, the late + 4 A τ light cone transition Using the Zamolodchikov recursion formula, we can obtain the ), which cannot be observed in the test mass limit. We expect = ] using the light cone bootstrap. In the light cone limit, we can calculate Virasoro conformal deficit angle in AdS n 3 τ 61 , 12 , multiple 11 The light cone singularity undergoes a transition at the BTZ thershold. suggests that inville a CFT particular (see regime, figure that 2+1 this gravity gives new is insights non-trivially into the related relation between to Liouville Liou- CFT and 2+1 gravity. This suggests that this CFTs in some sense,tained like by the the light Hawking-Page cone transition. bootstrap.) (One interpretation is ob- blocks beyond the testdynamics of mass limit time behaviour of out ofentropy time after ordered a correlators local (OTOCs) quench. upper bound of the Reggesingularity. limit singularity In in particular, the above Virasoro theform blocks BTZ using threshold of the light the cone Reggewill limit explain this Virasoro in blocks section can be given by a universal formula. We prediction that in aoperator unitary used CFT with to createhowever, a if its local dimension quench, exceedsThis the the might larger value the be 2ndreveled related the Renyi to growth entropy of instability becomes; the and entanglement entropy thermalisation. after a local (See quench in also other [ setups.) the dynamics of the 2nd Renyi entropy after a local quench. The result leads to the Besides these main contents, we give the light coneAt modular the bootstrap end in of appendix this section, we would like to emphasize the interesting points of our The time evolution of OTOCs at late time is also discussed in [ 2 primary operators in the OPE have a twist l This is proved in [ Let us consider aIn unitary general, CFT the with operators appearing twodimensions in primary originating the operators from OPE its between interactions.the However, large the spin crossing limit symmetry simplifies implies the that structure of the OPE. In 2 Light cone bootstrap BTZ thershold Liouville CFT & 2+1 gravity results. Beyond test mass limit and the details of the Zamolodchikov recursion relation in appendix Regge limit universality JHEP01(2019)025 with (2.5) (2.2) (2.3) (2.4) in the ij l ∆ 1 2 , v ) 2 τ ¯ z -dependence v u − 1 | and spin , p ) 3 l ¯ h ( − 41 v, u 32 ( F , these two particles are ) l τ,l 0, which means that the z g = ∆ , ]; however, this study focused on τ,l − τ ≥ 2 P 1 20 ). | τ p τ,l 23 13 h 2.1 X ( x x . 12 41 14 24 32 ) have the singularity ∆ x x , F 1 2 p − u, v 23 ( = 0 u = ) C are the cross ratios defined by l p B ij,kl τ,l if otherwise 14 g , +∆ u, v C -channel expansion) in the bootstrap equation 2 A , , v t – 6 – 2 l 2 p − (∆ 2 − X 1 2 d d are orbiting a common centre with a large angular − 34 13 l v ( x x ) = 1 ¯ z ≥ | 12 24 p x x v τ ∼ ¯ h correspond to states with large angular momentum in AdS, ( l u 21 34 = ) F B u ) , and parameters 0. In particular, in the light cone limit, the identity provides the z | B +∆ p A O → h ( (∆ 1 2 21 34 . In this equation, it can be observed that, to match the and u, v − F j u p A x at 34 O − j C i p ). ∆ x 12 2 − C = i ij p x X Before moving on to the light cone bootstrap, we will interpret this statement in terms However, in 2D CFTs, the twist bound is given by The key point is the existence of a twist gap between the vacuum and minimal twist. = ∆ In fact, the light cone bootstrap in 2D CFTs has been studied in [ 3 ij only the semiclassical limit asgeneral the blocks study in relied the on light using cone HHLL Virasoro limit, blocks. we will Now consider that we more have general the unitary most CFTs and external operators. (see figure 2.1 Lowest twist operatorIn 2D at CFTs, large the conformal bootstrapblocks equation as can be given in terms of Virasoro conformal of AdS. The operators at large and thus two-particle states atmomentum. large At this stage,well it separated, is as naturally the expected interactionsmomentum. between that It two at means objects large that become the negligible anomalous at dimension large of angular the two-object state should vanish to the zero-twist part,blocks; we thus, can using incorporate thein the Virasoro 2D contributions algebra, CFTs. into we From the can thisthe Virasoro investigate background, light the conformal it cone large is bootstrap spin interesting in to structure 2D investigate even CFTs. what We is will predicted discuss from it in this section. with of both sides, there must exist the operators withidentity twist contribution ( cannot beabove separated process from does the not other work contributions. in 2D Therefore, CFTs. the Even though there are many contributions where the sum isOPE taken between over the primary operators of twist the bootstrap equation as∆ the global blocks dominant contributions to one handas follows: side ( The unitarity imposes the following bounds on a twist spectrum (except for vacuum): This leads to the separation of the identity conformal block from the other contributions in JHEP01(2019)025 l L ¯ h (2.7) (2.8) (2.6) ) can 2 − 2.5 1 lead to ! ¯ δ ) → ¯ , z ) z − ¯ z ¯ δ (1 z, on the left-hand ( − dependence of the L 1 τ,l h z g 2
, − ) τ,l 1) z z P − ¯ δ √ . ( l τ,l L X ¯ (2 h on the left-hand side, there ) B ¯ z ¯ h AB 1) − − ∆ − A ¯ δ (1 ¯ ( h K L l − L π ¯ h ) h ) ¯ z 2 1 of this equation. In the following, we ¯ z r , − − 1) − AB − δ (1 ∆ ) are modified Bessel functions (see more ¯ ¯ δ ) z ) are conformal blocks, which are usually . For simplicity, we assume the external ( 1 2 . When comparing the x z B z i L − ( z | H h ¯ ¯ h h − p 1 2 τ ¯ h δ ∆ ) − is the conformal block coefficient. In this limit, h ) ¯ z – 7 – (1 c A ( ¯ K 24 z h = 2 − − ij kl τ,l − i 1 − z − ) P (1 F τ ]. z 1 L (1 l 20 h − , and 1) q 2 ]. The HHLL Virasoro blocks in the limit ¯ ≡ j − +2 − δ ) τ (1 ∆ 20 z ( = global blocks, which are given by a simple approximated z 2 | L l ' p h − 1 ¯ δ ) ∼ h ) i z ( ¯ ) z ¯ z | ¯ z ji − kl − 1 (0 and F z, = ∆ ( , and therefore z AA i H BB −−−−−−→ ij τ,l ¯ h h F g ) = (1 ) ¯ c z 24 | ≥ z , ∆ | i ¯ h (0 − h (0 1 + ) is the global block and LL . This is discussed in [ HH AA BB ¯ ]). z h q are OPE coefficients and H F F ) τ z, 11 1. Although no exact closed form of the Virasoro blocks is available, if we restrict z ( = | + ijk τ,l (0 → δ L C g z δτ To proceed further, we have to determine the behaviour of the Virasoro blocks in the LL HH F → left- and right-hand sides,contributions one on can the find right-handside. that side there to Moreover, must reproduce to bemust the reproduce an be singularity infinite the a number contribution singularityτ of from (1 an large infinite number of operators having increasing spin with where operators to be limit ¯ ourselves to the heavy-light limit,bootstrap the equation HHLL discussed Virasoro in blocks [ make it possible to study the where ∆ = details in [ where we are interested inform the as large be approximated using the vacuumuseful Virasoro to block. illustrate As the theexpress global asymptotics the block right of expansion hand the is sidelight right more using cone hand global limit side blocks. reduces in to As a the result, light the cone bootstrap limit, equation we in the re- We are interested in the lightassume cone limit that there are nosymmetry. additional continuous Under global this symmetries assumption, apart from the the light Virasoro cone limit of the left hand side of ( expressed using the Feynman diagram as follows: where JHEP01(2019)025 . , i → τ A 1 ]. In − 12 (2.9) c (2.11) (2.10) 23 2 − because 1 lowest τ l q − , 1 2 Q Q . This is just the ). Considering 1 < = − ) in appendix 24 increases beyond c i B , 2.10 B α α = τ 2 A.17 + ¯ . 4 as ¯ + ) Q , i i A A 2 τ Q α = α τ h − < otherwise , if ¯ Q B , ( α i B α Virasoro conformal blocks using ¯ h + ¯ , ] and monodromy method [ − , the singularity arising from that = A B A i ¯ α α ¯ h 22 , , the result exactly matches A − if ¯ otherwise α 1 A lowest c using the twist 2¯ 21 , − h τ 24 general c . To reproduce this light cone singularity i − B i ) ) , h α ¯ ¯ h α ¯ ¯ z z b 1 the lowest twist in the OPE is given by a 1 A − 24 − − c α 4 + , 4¯ 2 − – 8 – b 2 (1 (1 Q 1 − at small A A = q h h = B − , as the calculation is complicated. 2 2 τ 1 − − , A z z Q + lowest leads to the conformal dimension 1 . We have to mention that although we use the notations τ BTZ ( − A 12 = 2 Q c ,Q α τ i 2 1 in = α ( Q A ¯ z = α − BTZ gives the lower bound for the twist in the OPE at large 1 and ¯ α i z . To explore this, let us consider the expression ( h −−−−−−→ = 1 + 6 lowest 1 c ) 24 τ lowest − 24 c ¯ z c τ | 2 > − . Moreover, this universal twist equals the BTZ mass threshold. In other (0 1 1 . Here, one might face a contradiction to the thermalisation of the HHLL H − Liouville momenta q 12 c h AA 1 BB − 1 as F ) i Q z α | = (0 . This twist i limit achieved using the recursion relation [ B α AA BB τ c An interesting point is that if the total Liouville momentum In the following, we introduce notations usually found in Liouville CFTs. In the above, we restricted our investigation to the HHLL limit because it is extremely F + The Liouville momentum 4 A Virasoro blocks. Ittwo-point is function well on known a that thermalBTZ threshold the background if HHLL the vacuum mass block of can the be heavy particle interpreted exceeds asBTZ the a mass threshold. universal value words, the total twist graduallyBTZ increases unless mass the threshold, total Liouvillethe and momentum total exceeds the the Liouville total momentumshown twist in increases is figure beyond saturated it. by the This value behaviour of of that the threshold total if twist is δτ if there exists antwist operator is with never twist reproduced lower by the than left-hand side ofthe the BTZ bootstrap momentum equation. threshold where we can re-express theIn Liouville particular, momenta if ¯ we expand where from the right-hand sidethe of OPE the with bootstrap twist. equation, we must always have the operator in in Liouville CFTs, we neverparameters use are relations introduced that for convenience. onlythe hold According light in cone to Liouville the singularity is result CFTs. given ( The by Liouville the detailed calculation in appendix We denote difficult to study the Virasoroingly blocks simplifies in the general. structure Nevertheless, oflarge the light the cone Virasoro limit blocks. interest- fact, We we can can find the alsothe simplification evaluate fusion in the matrix. the light Here, cone we limit will only of present the results from the fusion matrix and present JHEP01(2019)025 1 ). − 12 c B h 2.10 (2.12) 24 c-1 1.4 B CFTs but also c 1.2 <
1 c : ( 1 (and no conserved primary currents), the mutual n 12 A c , n 3 ( q 2 2 log log A α I 4 1 ( ], as our result for the Renyi entropy is analytic in the Q 2 ) − c > = 1 1 n n n is given by a more complicated form than limit. Following ( ¯ z 1 ( 47 ( ∗ c I − c − − n 1 = 2 Q 1 1 n z −−−−−−→ s = c c ) 12 12 n 2 as expected. The B . In many cases, to calculate the entanglement entropy (for ex- α ∗ : 1 A is given by ( →∞ and ) c ¯ z n −−−→ 2 n s − ( ∗ n > n 1 Q I n z −−−−−−→ = 1. ) n 1) and not the large B = 1+6 : nc A c > ( In particular, we obtain the limit We expect that this phase transition arises from only the light cone limit ) n ( I This suggests that ininformation CFTs in with the light cone limit vanishes as in holographic CFTs. On the other hand, which satisfies shows that the transitionentanglement point from the is Renyi located mutual between information 1 has to be done carefully. where The transition point for general as follows: and the function vicinity of (and can immediately obtain the Renyi mutual information for CFTs with finite ample, the replica method),However, we we implicitly find assume that anhave the exception to Renyi of consider entropy this this is exception assumption analyticentropy. if in in We we the emphasize use light the thatthe cone replica this Ryu-Takayanagi formula method limit. assumption in to does [ Therefore, evaluate the we not entanglement contradict with the derivation of This result is consistentphasize that with the the additional holographicinformation logarithmic calculation for divergent term as appears in in [ the In particular, the mutual information is given by taking the limit Inserting this vacuum block into the function JHEP01(2019)025 n n 2)-th (3.22) (3.21) 3.0 n > ] (see also ). One can B 49 : 2.5 = 2. A ( ∗ ) n n ( 1, as mentioned in I ) (A:B with c=1 2 n (n) I ∂ 2 c 2.0 n . c > 100 matches , 2 . 1, as explained in [ 1.5 50 → ∞ ) n vs. coef. of-∂ c n > B min-entropy : ) (A:B c > (n) I A and ( 2 n ) leads to the fact that the ( . ) if ∗ (2) 0.15 0.10 0.05 I n B )) is valid only if : 3.15 = * coef. of-∂ ) = A n 10 limit of the Renyi mutual information approaches ( n ) B A.17 n cn vs. – 17 – : ( 30 . A 5 → ∞ dependence of the coefficient of ( 6 . The classical limit collision entropy < I ) ∗ n n n ) n ) (or ( ( dependence of the coefficient of the Renyi mutual informa- I B 25 n : ]. A.16 →∞ A = 1, the mutual information becomes ill-defined. This is lim ( n 71 c – 20 (2) I 67 1 * ) (A:B with c=1 dependence of n (n) c 1.4 1.2 2.0 1.8 1.6 15 . ) becomes discontinuous at B 9 : are respectively known as ). We can also find similar Renyi phase transitions as the replica number ) A 10 ( n vs. coef. ofI . We attribute this to the same reason why the quasiparticle picture breaks ∞ ) ( n S ( 3.3 Figure 5 , I ) (A:B ). The right figure shows the A.1 2 n . The left figure shows the ). It can be observed that the (n) ∂ B and 3.2 : 3.15 0.044 0.042 0.040 0.038 0.048 0.046 (2) It would be interesting to point out that ( A S ( coef. ofI 9 (2) and in particular, These properties are depicted in figure varies in other situations [ Renyi mutual information ininformation as the follows: light cone limit is bounded by the 2nd Renyi mutual if we consider anatural CFT because with the equationappendix ( down if we assume nosection extended symmetry algebra and Figure 6 tion ( I see that JHEP01(2019)025 . We 2 (3.27) (3.23) (3.26) (3.28) (3.24) (3.25) , i i and CFT , ) ) of the insertion i . 4 4 ) 1 4) 4 ¯ ¯ 4) w w it ¯ , , , w iβ/ 4 4 , iβ/ 4) + 4 → t w , w + t ( ( w − t iβ/ 4) ( n n − L − n σ L σ t iβ/ )¯ )¯ σ CFTs +2 + + 3 3 )¯ − t L D D 3 ¯ ¯ c w w ( ( ( ( ¯ , , w π π π π . β β β β 3 2 2 2 2 3 , . 2 , ] 3 w w 2 ) and consider two disconnected 34 24 ( ( L w i 2 w w ( n = e = e = e = e n n n | 12 13 σ σ 1 2 3 4 , x . The twist operators are put on the ), ) + 2 σ w w 1 2 ¯ ¯ ¯ ¯ . 2 w w w w ) t i β i h 2 7 2 ¯ ) w n = 2 | ¯ , H w ¯ 2 z , w L, D H respectively exist in CFT 2 , + w 2 β 2 ( 2 w 1 + i − ( n w ) and ( e and H ( n n 1 σ , D | )¯ n [ σ i i – 18 – 1 )¯ n = , x σ ) ∪ 1 ¯ )¯ 1 X w ¯ z t 1 (0) ¯ 1 , w and ] tot ¯ n 1 = , w z, 1 σ 1 ( , , H w i i )¯ , ,L 1 ( w 4) G ¯ z n ( n 4) w | n ( n , σ h z, iβ/ h n 4 σ , ( ) iβ/ = [0 TFD 4) | − for operators in two different copies of the CFT. The time h | σ n n t − z 4) h 7 t h | A σ + 2 β iβ/ 2 h + | L i iβ/ + L 4 t (1) + ], entanglement memory could be characterised using light cone + +2 t w − n log log 3 D − L D σ ( ( ( ( 49 1 1 )¯ w π π π π β β β β 2 2 2 2 2 1 1 − − ∞ w ( n n 1 n = e = e = e = e w σ | = = h 4 1 2 3 with a shift n ) w w w w h n 8 ( A A ) = I ¯ z π β z, 2 ( G The Renyi entropy in a doubled CFT can be given by a four-point function with For realizing the objective stated above, we consider the Renyi entropy in a doubled where The Renyi mutual information can be obtained in a simpler manner as follows: where the insertion points are given by twist operators on a thermalendpoints cylinder of of periodicity dependence is obtained bypoints considering of the the twist analytic operators continuation (see the right of figure Let us choose the total Hamiltonian acting on the doubled CFT as Thus, the thermofield-double state (TFD) has a non-trivial time dependence. as an entangled state. Twolabel states the coordinates ofintervals (see each the CFT left as of ( figure conformal blocks to the calculation of the entanglement entropy. CFT. That is, we will consider the thermofield double state in the doubled system, The dynamics of quantum informationmunities. has In attracted this the sense, attentionnately, we of as are many discussed also research in interested com- [ singularity; in therefore, the we propagation might ofmal be entanglement. blocks able Fortu- in to the study light entanglement cone memory limit. using our In confor- this section, we explain this in detail and apply our 3.2 Dynamics of Renyi mutual information in large JHEP01(2019)025 , ) is 2 ¯ z (3.34) (3.30) (3.31) (3.32) (3.33) (3.29) ≡ z, . ( ∗ G n < n otherwise , if . , , 32 nc ) D < n − 0 or 1 h t . The function . 2 0 0 otherwise t, → if 3.1 2 β 0 −−−→ , , 2 → − )) ) , ) β t −−−→ n ) 2 2 L n − ) . D D, − D (1 n 2 + 2 h n − − ( c t 4 1, and therefore, this limit corresponds n t 12 2 c D − max( 2 . t, | 24 − − 2 0 z ) ) ) t, | − ¯ ¯ z z 2 ¯ z t,t , the anti-holomorphic cross ratio is given by 0 L min( 0 − − − L − → − π +2 β L → β 2 2 −−−→ 1 +2 β L D (1 (1 – 19 – −−−→ # +2 D n n A 2 ) , D 1) h h I ¯ z 0 + 2 2) 2 2 min( − z − − − π n β z, 0 2 ( D n z z ( max( ( n − → − < t < G e β t c −−−→ 24 , the anti-holomorphic cross ratio approaches 0; thus, the + 2 D ) min( 1 ' L t L − π β ¯ z 2 2 +2 ¯ z +2 +2 0 in the following, which simplifies the four-point function as 1 n D − − D D ( ( 1 1 n 1 → π π − β β 2 2 z − 1 β −−−−−−→ − − 1 e e ) < t < ¯ z c 12 ' ' c 2 12 D z, " ¯ z z ( G ) becomes 1 ¯ z , as sketched in the right figure. z, ¯ z ( β − 1 . The left figure shows two intervals in the doubled CFT. The entanglement entropy G z −−−−−−→ ) In general, a four-point function non-trivially depends on the CFT data, and its explicit n ( A I approximated as and, therefore, the Renyi mutual information is In particular, this cross ratioto satisfies the lightdoubled cone CFT limit. reduces to Thus, the the same calculation form of as the given in Renyi section mutual information in the As a result, we obtain the Renyi mutual information as On the other hand, in the range Outside the range function form is not known, excepthigh-temperature for limit a few CFTs.it corresponds Therefore, to to proceed some OPE further, limits. we focus That on is, the the cross ratio is given by Figure 7 in this setup canperiodicity be given by a four-point function with twist operators on a thermal cylinder of JHEP01(2019)025 ). , n α (3.42) (3.41) (3.36) (3.37) (3.38) (3.39) (3.40) (3.35) A.1 − Q ( n α and not just = n n Z h / n . , n as ) h ¯ 2 z . | n n − . Z ) h ) (0 / , 2 ¯ z ¯ z n n n 1 is given in appendix | n , − ¯ σ σ h ) . − n n 2 2 (0 Vir ¯ Q z , ¯ σ σ n − n n c > n (1 F ¯ ) σ σ − , which satisfies h < Z n ¯ n z n 2 / h ) is bounded by σ ¯ σ n n − 2 (1 ¯ z − n F α 1 − n s c 1 Vir z, 1 z ) (1 ( ¯ z − → → F if 2 otherwise lim . ¯ z ¯ G z ) . −−−→ n , − z ) . ) | ) ) & ¯ z n ¯ z ¯ 1 z (1 | ) (0 α − z, c ¯ z n = 0 n 2 ( − | (0 1 ¯ σ σ nc n n A n 1 G n – 20 – (0 − → ¯ σ σ I σ ), the mutual information is obtained by taking ¯ σ z n n − n n −−−→ z, . ¯ σ σ ¯ Q F σ σ 1 ) ( n n n n − z ¯ , σ σ n 3.34 Z 1 | h q 2 n / s 2 α 4 (1 Z n Q α 2 − ¯ z / − h n G n − s ( Vir 1 ( 1 ), as follows: ) ¯ F current; therefore, we should approximate a correlator with Vir z = AA BB 1 ) = 2 z Q −−−−−−→ n F ¯ − z n F , we can generalize the calculation of the Renyi mutual infor- 3.13 → 1 ) Z s lim ¯ z = ¯ z z, (1 → ( lim ¯ z n n 3.1 z, h G α ( 2 is defined as − G n z under some assumptions. This is extensively discussed in this section. s and c 2 is the conformal block defined by current algebra Vir Q 1 as n Z / → n n = 1+6 Vir F nc From the crossing symmetry, we can obtain For simplicity, we again assume that there are no extra currents. Even in such a case, We have to mention that from ( Here, In conclusion, the light cone limit of a correlator where the function In addition, we can also deduce the lower bound as The light cone singularity of the Virasoro block with any Therefore, we have the upper bound of the singularity of where Virasoro algebra. As mentioned in section mation to general orbifoldisation leads to the twist operators, instead of ( Here, the mutual informationbles vanishes maximally, for which contradicts all with times.rational the CFTs. quasiparticle It behaviour means shown in, that for entanglement example, scram- 3.3 Renyi mutual information beyond large Therefore, one has to takeentropy using care the when Renyi trying entropy. to predict the behaviour ofthe the limit entanglement As a result, one can again see the Renyi phase transition as the replica number is varied. JHEP01(2019)025 . ) D )) as b (3.46) (3.47) (3.44) (3.45) (3.43) in the − t i 2 ] as 0 | t, 72 2 − , as shown in 0 provides UV L A +2 ) (through ( > c D , using the operators n Z ) to ( min( / a n π β 2 , . i M ) ) i n i ¯ n B 1 σ n 0 , ¯ | σ : n i ) on the CFT vacuum ( n ) σ 0 x ), and hence, their mutual infor- | 1+ σ A n ( ( ih A ( O. ) n ) n O S x ⊗ n ( ⊗ ( from the boundary of 3.43 c O 12 − I O l O n ) n i ⊗ ≤ ≤ ) ⊗ iHt ) t O ⊗ · · · ⊗ − h O h D O Ψ( H | − ( − t ⊗ ) – 21 – e log 2 n ( A O n t, N S 2 1 = − = − n 1 i L ) = ) is the ultraviolet (UV) cut off of the local excitations and should ) ⊗ t t ( = O ) +2 ) 3.44 n Ψ( ( n | A D ( A S in ( S ∆ ∆ min( is defined on the cyclic orbifold CFT π is separated by a distance β 2 n ⊗ O as O 1 n . For simplicity, we move the framework from ( 10 − M s 8 n − 1 n is the normalization factor. The infinitesimally small parameter 1+ N As a result, the Renyi mutual information in the doubled CFTs satisfies the following We would like to stress that c , which is defined by acting with a local operator 12 to be the half-space and induce excitation in its complement, thus creating additional i 10 Ψ The local excitation the left of figure be distinguished from the UV cut off (i.e. the lattice spacing) of the CFT itself. where the operator in the seed CFT In fact, this quantity can also be calculated analytically using twist operators [ where regularization as the truly localizedA operator has infinite energy.entanglements We between choose them. the subsystem The mainentropy quantity compared of to interest the is the vacuum: growth of entanglement the light cone limit also appearsentropy after if a one local investigates the quench. dynamics| The of process the is Renyi as entanglement follows:following we manner, consider the locally excited state information, in theory, saturates the loweris bound, the then main it conclusion shows of maximal this scrambling. section. This 3.4 2nd Renyi entropyAt after the local end of quench this section, we discuss another application of light cone singularity. In fact, In fact, rational CFTsmation saturate is the universal. upperbehaviour This bound can shows be ( that observedpear entanglement in does to rational saturate not CFTs. the scramble lower On and bound. the quasiparticle other In this hand, context, holographic we CFTs can ap- state that if the Renyi mutual inequalities: JHEP01(2019)025 z ¯ z − 1 ) is 1 − (3.49) (3.50) (3.51) (3.52) (3.48) 32 c 1 with ≤ = 2 in the O n h ¯ z z, − 12 ). (Right) The equiv- ) can be obtained using c 3.46 . ) ¯ z . We can approximate the light , . z, , . We focus on ( O t ) and ( n 1 . h 3.3 b G z 2 − 32 c − O ¯ = z h , ) log 4 < O 2 | 2 O t w h z O 2 4 α h 2 − − O ' if otherwise 1 s − | ) ¯ z , z ) O Q h ( O 4 − | O α , z 2 α | (1 2 4 2 t – 22 – − = 4 1 i Q − i ( ¯ z n →∞ , we can approximate the cross ratio as ¯ n , O l z, σ 1 t ¯ −−−→ 2 σ n − α 4 n 1 Q ) ' σ 2 −−−−−→ t σ 11 ( z ih t ) n ( ¯ n z ⊗ ) is NOT defined in an orbifold theory, but just in an ordinal theory. (2) A ⊗ ¯ using a conformal map z = z, O S ( O z, n 8 O ( n ∆ ⊗ G s G ⊗ O ) explains the relation between a correlator with twist operators in an O h b h is defined as . n O z s = ) and ( w a . (Left) The positions of operators in the replica computation ( Using this cross ratio, we can re-express the correlator as Actually, this limit is the double light coneIn limit, this which case, is the defined function by the limit 1 11 12 fixed. Therefore, we do not needcone to limit consider of the a difficulty correlator explained using in just section the Virasoro conformal block. Therefore, the growth ofgiven the by 2nd Renyi entropy after a light local quench ( where the function The light cone limit of the four-point function can be approximated as following. In the late-time limit which is just the light cone limit. alence between ( orbifold theory and a replicaa manifold. conformal map The equivalence between ( shown in the right of figure Figure 8 JHEP01(2019)025 (3.54) (3.53) O h 32 c-1 ). , and in the other phase, . We intend to emphasize O 1 and no extra currents, the ]. The growth for a heavy , as explained in the next h 9 3.54 n 1.4 73 ]. c > 21 0 after picking up a monodromy . . 1.2 t → t ¯ z log log z, 2 1.0 O 2 Q h (t) with c=10 1 and no extra conserved currents. That is, 4 (2) = 1 Renyi entropy); therefore, we could not A 1 0.8 1 and there are no extra currents (and discrete →∞ , the result reduces to c > n t ), as shown in figure – 23 – O −−−→ c O c h h ) ). This limit is obviously different from the light −−−−→ c > t z 0.6 ( 3.54 (2) A − (2) A S S (1 ∆ 0.4 ∆ vs. coef. ofΔS πi 2 O , then the entropy is saturated by ( − h e 1 (t) − 32 0.2 c (2) → A ) ) is more interesting, that is, it has the following universal form: z 1 = 2, the Renyi entropy after a local quench can be explicitly given − 32 − c 0.6 0.4 0.2 n dependence of the coefficient of the growth of the 2nd Renyi entropy after a ≥ O O h coef. ofΔS h = 1 as (1 . The z We can, therefore, conclude that the 2nd Renyi entropy after a local quench undergoes section. 4 Regge limit universality In 2D CFTs, the Reggearound limit is defined by the limit without other assumptions except that spectra or, equivalently,when compactness). studying the Unfortunately, entanglementdetermine we entropy how ( did to not relatein find this fact, saturation this to we saturation the can dynamics generalize of this the result entanglement. to Note any that, replica number a phase transition asourselves the to conformal a unitary dimension (compact) of CFTin the with one local of the quench phases, isit the varied, is entropy if is saturated monotonically we by increasing restrict thethat in universal at form least ( when These results are consistent with the numerical results in [ In particular, if expanding this at small This result in thelocal light quench limit ( is consistent with the result in [ Figure 9 local quench. This dependenceheavier the might operator imply used that todimension in create exceeds a a the CFT local value with quench, the larger is the entropy growth; however, if its JHEP01(2019)025 ). are (4.1) (4.2) (4.3) (4.4) (4.5) 1 4.5 − 32 c in ( . > ) 4 h 0 in terms of even + A,B , . 3 ) is a universal n, . h h → q δ q | + ¯ q p 2 h , h i, ) ( + ). For simplicity, we z ) 1 (whose explicit forms z − 21 34 h ( → ). Note that the series C c (1 K 4( q K − C π 1 2 − − c c, , and )) B q ( for AABB blocks for ABBA blocks ) = e , h and elliptic nome 3 z n , A θ , ( z ( # h n 3 q h 1 n unit circle − − 32 c ) in appendix 2 c 0 h light cone limit cone light , q ≥ − − ) Z 1 for large q ∈ X C.13 | − B 24 c n p h ) is odd, which is described by h ), ( ( z n ) = – 24 – 21 1 34 ], we determined that the series coefficients could − q n − | 32 , depending on H c p C.12 √ (1 ) 22 n , h 2 q A | ( − h e p ), ( 21 − α 21 A 34 h 1 n ( h h H n Regge limit Regge − 21 34 ξ C.11 " 1 ). Considering that the series coefficients for − ) using a series expansion form as 24 ∼ c )( q | sgn z n 10 p 1 c ) = Λ h × − 24 z ( c | . The relation between cross ratio C.10 p − 21 34 ) can be calculated recursively (see appendix are constants in h p even q ( h H | α ) , n, p 21 34 q δ 1 ] or ( h F ( 23 and 21 34 – Figure 10 = A H 21 ξ ) = (16 q ) is the elliptic integral of the first kind and the function Λ | z p = 1. In our recent studies [ ( h 0 ( K c 21 34 The key point is that the Regge limit corresponds to the limit For this purpose, we introduce the elliptic form of the Virasoro blocks as follows: Λ The values of are given in [ coefficients for the AABB blocks vanish if the elliptic nome (see figure where where be expressed as prefactor given by The function express the function cone limit conformal blocks. where cone limit; nevertheless, we can contribute to studies on the Regge limit using our light JHEP01(2019)025 ) ) ); > z A 4 4.8 , − (4.6) (4.7) (4.8) 3 , h , (1 2 , B 1 πi 2 h , h ) does not − B q e | . p , h 1 h → A ( h ) − 32 ]. Note that the 21 z 34 c − H 21 > ) or ( 4 , , B 3
, ) 2 , q , h | 1 B p . For this purpose, it might . h 0 with (1 ) h ( , h z if A → 21 34 z , h H (log
, A ) 2 O h z , we moved the framework from an h − ) − z 1 1 and (1 h 3.4 − πi − 0 2 ) = ( (1 → ¯ z 4 − → πi 2 e q lim 0 with z 2 h , h → − → − 3 ) e with z 1 z h → – 25 – − , h ) ) itself. In this calculation, we have to calculate − 2 z (1 ), and, therefore, the function 1 − − 24 , h c & (1 1 3.46 4.2 −−−−−−−−−−−−→ ). In section z
h ) ) q ) q | | z 3.46 p p − transition at the BTZ threshold ), one can find that the main singularity in the light cone h h ( (1 ( 1 corresponds to πi 1.5 21 21 34 34 0 2 , its growth can be analytically calculated using a four-point and − H → H → e
with z z 1 → ) 3.4 z → . This aspect is discussed in detail elsewhere [ lim z , we obtain the relationship between the light cone singularity and 1 − ) with ( i − 32 (1 −−−−−−−−−−−−→ c 4.2 ) → ¯ z > q z, O ( h G and universality of twist ∗ At this stage, the most important future works are to understand the bulk interpreta- One application of this result is the evaluation of the Renyi entropy after a local quench. . Comparing ( 1 − 32 c This fact leads tointerpretation a of richer the structure transition of is presently the unclear. spectrum attion large of spin; however, theneed physical to resolve the following problems: about 2D CFTs.vacuum Virasoro conformal For block example, through thebackground, the light an cone large-spin bootstrap important equation. spectrum taskblocks Based was can on to in this be examine general. the derivedblocks light In from by cone investigating this only singularity the paper, of fusion the singularity matrix the we undergoes (or Virasoro reveal a the the crossing phase light kernel). transition cone Interestingly, as light structure cone the of external general operator Virasoro dimensions are varied. Regge limit also appears in the evaluation of OTOCs. 5 Discussion The light cone structure of Virasoro conformal blocks allows us to access information orbifold CFT to a seed CFTalso using a calculate conformal the map to growth evaluatethe using the Regge growth; just however, limit ( we can ofleads a to four-point the function, conclusion insteadn of that > the the n light Renyi cone entropy limit. after a This local result quench ( shows universality if As explained in section correlator with twist operators ( This conclusion leads to the Regge singularity of a four-point function as follows: where we only focushowever, we on expect the that cases this ( relationship can belimit generalized only for comes any from pairs thecontribute with to prefactor the ( singularity. Therefore, we can conclude that corresponds to Regge singularity as follows: always non-negative and JHEP01(2019)025 ] 17 , 16 . It is not possible n , similar to the thermali- 1 and no conserved extra , when considering the Renyi c > n BTZ threshold – 26 – . Apart from the transition in ]), n 77 – owing to the strong interaction above the BTZ threshold. It would 74 1 − 12 c . Above the threshold, the Renyi entropy is saturated. We expect that it is c 32 currents, which is expected to be chaotic. sation of the HHLL Virasoro blocks. gravity (see [ The transition point is characterised by the poor understanding of the relationship between Liouville CFT and 2 + 1 dimensional The transition can only be found in a CFT with dynamics of multiple deficit angles. One interesting future work is to generalize the analytic bootstrap program [ The light cone singularity also reveals the entanglement structure. From our studies The light cone bootstrap equation suggests that there must be a universal binding We expect that this transition would be related to thermalisation or black-hole forma- i) i) ii) ii) Acknowledgments Especially, the author wouldhelpful like discussion. to We express are specialcomments. grateful thanks We to to also Tadashi Henry thank Takayanagi for Masamichi Maxfield his Miyaji, for fruitful Nilay a discussions Kundu, very and and Pawel Caputa for useful related to the saturationthis of topic entanglement in in future. some way. It would beto interesting to two-dimensional CFTs. explore Wewould believe be that interesting our to result explore may this contribute issue to further. this progress. It to physically explainentropy why is the analytic lightentropy in cone after limit a local destroys quench,the the we conformal assumption found dimension that that the ofexceeded the Renyi the entropy Renyi became operator large used on to increasing create a quench, unless the dimension be interesting to reproducegravity. this binding energy at large spin from theon entanglement calculation in in various setups, AdS weoften found caused that discontinuousness the of transition the of derivative the of light the cone Renyi singularity entropy in a black hole, and thermalisation. energy even at largeif angular the momentum. total Liouville Moreover,is momentum this saturated increases binding by beyond energy the becomes BTZ larger threshold and the total twist In the first place, theretransition is point. limited information We as intendblack to to hole. why examine the It if BTZ is the threshold interesting BTZ appears to threshold as explore the is the relationships related between to the the transition, creation creation of of tion. There are two reasons for this: The reason for consideringLiouville (i) momentum, is instead because of the the transition total point mass. is characterised by the total JHEP01(2019)025 , - t ) , Q ) 1 (A.3) (A.4) (A.5) (A.1) (A.2) + and α u ) ) s ( − ]. ] than the 4 )), which b 3 Q t ]). It would 1 S 83 R α 83 + ) , t α + 81 . , Q 2 α i − α , + crossing matrix u − ( ]as follows: 80 (sl(2 b ( 3 , b Qη q b 24 6 α ) U S 82 st )Γ = t s , ) − 2 , α t i α α 2 ) α . 78 α z − 23 ) α ) 2 4 i − ( s u α α α 3 ) b − ( α as α 3 b α − S 1 3 1 i α | ) − − S u ] (see also [ + α α t is called the α 2 ) 4 ( 1 α Q b t 79 Q α − ( Q α h ( S + ,α ( i ( ) t + ) Q ) s b + ] to 1 α 2 3 α 41 α 32 34 (2 13 )Γ α 23 s F b F = , α − , α ( 3 st 2 α b 2 3 i # α )Γ α S α 2 − 1 4 + , α ) , α − α u s t 2 α α − 4 ( u 3 α α α α b 2 3 ( (2 ( ( α b b , h − S α α − ( ) 1 S b N N 1 b s " )Γ s ) ) ) α S 1 α s 1 t + ) 12 Q − α . The kernel ,α + b α , α s , α appearing in [ Q – 27 – + 3 Q t (2 3 α ∞ ( i 13 b − α − = i b η F ( t u Γ , α , α t b ( 1234 + 4 4 α )Γ α b S 3 α α 2 α Q ) d + S ( ( α s 3 − S N α N − α Z to u ,Q 2 ( ]. In this study, we used the new expression derived in [ ) is 2 − + b 2 1 Q α = 2 2 Q S 82 ) = , − α ] as follows: α # , α u z 1 ( 2 | 1 4 78 + b d α 78 s 1 α α S α , α − α ) ∞ h 3 2 3 i ( ∞ ( Q i b α Q α α = 1 + 6 + ( 21 − S 34 (2 c Q " − runs from b 2 t Q N t F 2 Γ S α ,α Z s is the Racah-Wigner coefficient for the quantum group + . The explicit form of the fusion matrix is given in [ 2 α b 4 | b ) = ) F α t 1 t s ) α + α α t s , α 1 14 2 4 2 α (2 α α α α b ( 2 4 , α b S 1 3 ¯ 3 α α α α S α ×| 1. The key point is that there are invertible fusion transformations between 1 3 In this appendix, we show the asymptotic form of the conformal blocks in the limit ( A similar structure for a 1-pt functionPonsot-Teschner on have a derived torus a can more be symmetric found in form [ of the Racah-Wigner coefficient [ = fusion matrix α α N → 13 14 ( be interesting to parallel our discussion for atraditional 1-pt expression function found on in a [ torus. and is given by where the function where the contour or Note that we can relate the parameter z channel conformal blocks [ A.1 Leading termIn in the light following, cone we limit introduce the notations usually found in Liouville CFTs. Nikita Nemkov and Taro KimuraJSPS for fellowship. giving YK us is usefulthe grateful comments. to conference the “Recent YK Developments conference is in “Strings supported Gauge and by Theory Fields the and 2018” in StringA YITP, Theory” and in Keio U. Light cone limit from fusion matrix conversations about the subject of this study, and Jared Kaplan, Luis Fernando Alday, JHEP01(2019)025 . l ) ) ) s ) α s t t α α α + α , and (A.9) (A.6) (A.7) (A.8) k 0 (A.10) (A.11) − + − . + α ≥ 4 4 4 ) ) can be 4 ), we can Z + α α α Q j α 1.4 ∈ α + ). However, + + + A.3 + 3 + 3 1 1 u i 1.4 α α α ( ) α α ( b 4 n, m ) leads to b t − . + − S 1 = ( ) ) and ( ) and satisfies the )Γ b α Q ) into ( Q t =0 x for Q A.2 ( t ( b α 1.5 − )Γ − 1 ) and ( ijkl t ( , + b − A.5 b n t α u )Γ ) − α b ( s − 1.5 24 x 4 b )Γ ) + ) α + st s 2 α S . − x 4 ) and α α ω ( − t . )) − b α 2 k mb Q 3 4 z ) = Γ − 23 + n 1 Γ ( h α ( α − b x . α 4 α u − ( ) − + O + Γ 1 ( 2 b α − j b x − h − 1 α ( = α 3 S ( − − b ω u ) b 1 Q α ( 1 + x Γ ) = 3 (1 + b − i 24 Q n x 1 2 c )Γ α 2 (2 + S t α ( b ) h ) − b + ) + α z − − Q = 34 bx )Γ 1 ( x t s 13 bx h + b − ( Q α α ( st πb − 3 ijk b Γ( s )Γ 2 α (1 =0 α h α s − − ,S 2 √ )Γ z , α ∞ − 3 1 u s + ,n – 28 – ) X ). α 1 ( 1 α 1 u α 0 → 2 b − n − ( z − − −−−→ α S b ) has poles at → ) = 2 + + ) A.2 ) ) z b ∂ Q ) x S −−−→ s t α ∂t 2 s b, b 2 + ( ) b, b z | α α ) | | b α + α 12 = + 2 s Q z Q ): x Q | (2 (2 α x α ( 1 − s α + b b ( − 2 + h α b α 2 − 1 ( Γ ( ) = Q A.9 )Γ )Γ h Γ b Γ ( α 2 t ( u s 21 ) are defined as )Γ( 34 b ( ) is introduced such that Γ t α − 1234 α x b )Γ 21 F 34 , ω x ( ( 2 α 2 s )Γ α 1 ( S b b t ) = F b α ω S − + − α x | − )Γ ( 3 s − x b u Q Q − ( α α ( 2 Γ 2 3 b (2 (2 α b − b S + α ) and Γ Γ 2 2 − u x − ( α d log Γ α 2 1 | b 2 α ) # ∞ − t − α i 1 4 ∞ i α 1 − + Q α α + − α ( Q ) is the double gamma function, (2 Q ( b 2 3 2 Q 2 b Q b 2 Γ ( α α S (2 Γ b Z , ω b " 1 Γ Γ t × × × | ω | The key to understanding how the transition is derived from the expression of the The conformal blocks have the following simple asymptotic form: ,α = s x ( α 2 F reproduced using the transformation ( fusion matrix is that thetherefore, function the Γ fusion matrixsecond also lines has of poles. the The expression following ( shows the poles in the first and Naively substituting this asymptotics into the fusion transformation ( Interestingly, this reproduces a partwe of cannot the straightforwardly asymptotic understand formulas how ( the transition seen in ( following relationship: By substituting the explicit formsimplify of the the expression Racah-Wigner coefficients for ( the fusion matrix into Note that the function Γ The functions Γ Γ where we have used the notations ¯ JHEP01(2019)025 (A.12) (A.13) (A.14) can cross , (6) t 4 α h always satisfy , . − i and 2 2 1 Q Q α h of the integral over < < (5) t − is always positive by S α 4 3 1 α α is deformed as shown in m,n − 24 + + S ] and we are very much grateful c Q 1 2 α α 84 m,n if if = m,n m,n Q , 3 m,n Q Q 4 + α Q 1 − + m,n α Q − 2 Q Q Q − , κ m,n Q 3 . The real values of − 3 h 0 Q ) + 2 ) + ) + ) α − − ≥ 4 3 3 4 2 2 ) can cross the contour + Z α α α α h α , 3 3 4 2 − α 4 + + − − α ∈ α (6) t − 2 h 1 2 2 1 exists only in a particular domain as shown in + α α + – 29 – + 2 α α α α = 1 2 ) ( ( ( ( i 1 h − ( α t 2 ) ) − α − n, m α and z z < = = ( = = = = − − (5) t for α , κ (1) (2) (3) (4) (5) (6) t t t t t t (1 (1 . b n 4 t α α α α α α α α ( . From this figure, we can find that only 1 ) + i ( t 1 α 2 α → mb z < −−−→ − ) 3 = z h | . As a result of this crossing, the contour s 2 − Q α m,n h 2 ( < Q h 21 34 4 − α F for the operators in unitary CFTs, and the value 4 of the integral over h + 2 Q S 1 . The domain of + 15 α ≤ . . One can find that only 1 i h α 11 12 = Poles of the numerator This deformation leads to another possibility that the conformal block has the asymp- This deformation of the contour in a particular case was described in [ 1 when 15 ≤ < κ t If we set three parameters as to Henry Maxfield for pointing out this. totics as follows: 0 definition. Therefore, each valuefigure of α figure where we define Figure 11 the contour JHEP01(2019)025 . 1. 2 Q − c → ∞ < (A.15) (A.16) (A.17) 3 c by α c + 2 , , , , α , 1 is given by A limit. Therefore, B 2 2 2 Q Q Q , c → < < < 2 Q < α < α . z 4 3 4 B A < α α α (5) t . α α α B + + + α 1 2 1 α α α + and and . if if if A 4 4 Q Q α if otherwise < 1 by replacing the factor < ]. We intend to emphasize that the , , ) A B B 4 α α h 22 c > α , , − if if otherwise B A − 21 , α h 3 A A − α α 1 deformed by the pole 2 − Qα 24 , c − 2 − S ) ) B , − 2 – 30 – z z B B α Qα h h − − 2 2 2 − − − − 1 (1 (1 1 A B α − h h 24 c 4 2 ( ) ) ) − z z z 1 , , 0 0 → . The contour − − − )(1 z −−−→ 3 > > (1 (1 (1 α ) 2 2 z | − s 2 2 2 α Q Q 1 h ] and numerical results in [ α ( → − − Figure 12 z 23 − −−−→ 4 3 AA 4 BB ) α α z α F | + + s + α 1 2 1 h α α ( α BA BA = = ( = F 2 2 1 ] can be generalized to any unitary CFTs with κ κ κ One might doubt this result because it contradicts with the singularities appearing in 23 − − − 3 1 3 κ κ κ above calculation based on thewe fusion expect matrix does that not thein rely [ asymptotic on form the large derived from the monodromythe method minimal with models. In the first place, when one of the external operators corresponds Interestingly, these results exactlyodromy method match in our [ previous results obtained using the mon- Further, the asymptotics of the AABB blocks is given by Therefore, the leading contribution of the ABBA blocks in the limit then we can show the following inequalities: JHEP01(2019)025 . n, m n, m (A.22) (A.18) (A.19) (A.20) (A.21) fixed, the ) -channel is δ t ) /c z 4 − , h 1 (1 , h with non-zero − ) with non-zero 1 n as | (5) t , p α + 0 L can be approximated as h h A.12 0 with 2 m, − , ω b ) µ m,n p ) diverges, and therefore, the m → ω , , h + b p L − A.9 h h m,n 1 ( , and ( δ µ 0 1 ) + + F z is satisfied, the corresponding term m, 4 -channel in terms of the 2 b α s µ ) − 1 L 2 Q h α m 2 2 (1 mδ. − − < + m p , 3 0 h h 0 L P h − 0 → 2 m,n − b m,n δ -channel is decomposed as a discrete sum of the ≥ −−→ h ) ) ω m,n> s Z – 31 – Q − 0 z m µ ∈ X 4 − + h m m, + − L + as µ ; however, these divergences can be cancelled only if δ h (1 4 1 t 1 ( (1 h 1. If taking the limit δ α 0 α ) ) → z z − z m, −−−→ + < . Thus, we can approximate b 4 µ 1 n ) − 1 − ) h z | α + c p (1 (1 24 m,n 1) h m ( − Q − mb δ P ]. We have to mention that this never happens for unitary CFTs ( 1 0 L + LL HH ≥ h 86 q 4 = ) Z F , z ∈ α X in the denominator of the expression ( = 85 m − b + m,n δ µ 1 1 α → or Γ ( z −−−→ also contributes to the conformal blocks as sub-leading terms. Its singularity , we tried to derive the spectrum of twist. For this, we need the sub-leading 2 b with Q ) = (1 is a positive constant defined by S δ z | are some constants. In fact, the HHLL Virasoto blocks have the same form. ≡ 2.2 p 1 because if the central charge is larger than one, then the conformal dimensions − h m, n m m,n ( 1 P µ m,n c > ' ω LL HH We expect that the sub-leading terms come from the poles ( 0 takes particular values. As a result, the F t m, -channels, instead of an integral over the continuum spectrum. This concept is explained This leads to the sub-leading contributions to the conformal blocks as follows: where and sub-leading terms are givenω by where As an example, letin us the consider semiclassical the limit.labelled contributions from If the poles is given by In section contributions of the conformalgive blocks the in sub-leading thegeneralize terms light it. in cone limit. the semiclassical In limit this using appendix, the we first fusion matrix and then t in greater detail in [ with of degenerate operators are negative. A.2 Sub-leading terms in light cone limit not a continuum, butbe is written as discrete. adegenerate, linear That combination is, of in deltacrossing such kernel functions. vanishes a for In case, general fact,α the if crossing the kernel external needs operator is to to a degenerate operator, the decomposition of the JHEP01(2019)025 1 to , as 1 m,n c (A.23) (A.24) (A.25) (A.26) h µ m, n , , 2 Q 2 Q < < . B B α α + . Therefore, we can + , L A h A α 2 α − if otherwise ) if , x ) − B α , otherwise 0; therefore, an infinite number + A m,n α → µ , ( , + b 2 Q 2 Q B B as h − α 1 − < A 1 c A h α 0 h m 2 1, there are only finite poles satisfying the − + m, − 1 ) B Q − z 24 α c A ) c > + − – 32 – α satisfy the inequality z ) in the semiclassical limit. The condition 2 4 0 (1 − α − ) ≥ , z + Z B (1 A.22 m,n 1 h − ∈ P 1 − α is satisfied. Thus, an infinite number of poles contribute 2 A Q (1 → m h z 2 < Q −−−→ . Therefore, the conformal blocks can be expressed using a m − 0 2 1 ) P Q < ≥ m,n − z 24 0 | c Z 4 Q ≥ p < ) ∈ Z + α h z X where ∈ ( for any B + α m,n − m m,n 0 1 AA + BB Q m, P (1 A α F α Q + . Note that now we take the limit 4 1 + α 1 13 → 4 z → −−−→ α + z are some constants. Note that the above process can straightforwardly be −−−→ ) 1 + z ) α | . This figure shows from where the sub-leading contributions to the blocks are obtained. z 1 p | m,n α p h ( h P ( AA BB Many studies on the fusion matrix approach remain to be carried out. The most impor- For general unitary CFTs with finite AA BB F F where applied to the ABBAthe blocks. power of That its is, singularity. the result can betant obtained one by is just identifying adding the logarithmic corrections to the light cone limit of the conformal and inequality finite sum as to the conformal blocks as showncan in immediately ( be relaxed to non-perturbative shown in figure of poles when the inequality Figure 13 which can be easilyconclude checked using that the the series sub-leading expansion contributions of come (1 from the poles with non-zero JHEP01(2019)025 ), ), 0 > ]. (B.4) (B.1) (B.2) (B.3) (C.1) R 2.24 94 ∈ ). If we ] because ¯ h ¯ τ ]. An earlier h, τ, 89 92 , – 37 90 , , 86 1 with ) z ) , z dependence, the sum − ( c 24 (1 K τ − K , ¯ ¯ h τi π 1 ¯ q − ) ) − 24 c 1 τ τ . ( ) − ) i τ η 1 h τ q − 24 ) = e (¯ c z ¯ h , . To my knowledge, there are no ( − , which means that we must have χ 1 h )¯ 1 ) = ( − 12 τ − ¯ c 1 τ 24 τ + ( c ( τ h − h 1 24 , χ = , q ) ( ) 1 τ ¯ h q ¯ h πi | − 2 p , χ h, ] is one of the tools used to calculate the con- − ) h ( ( q ρ )e Z 0 88 – 33 – 21 ¯ h 34 , − 1 (or equivalently H ¯ h> h, 87 X ) ) = (1 ( h, q ¯ τ ρ 1 ) is approximated by | ) ¯ q p − 0 24 τ c τ, h ( 16 ( ) = − ( B.1 − 1, the Virasoro character has the following simple form, ¯ h> η ¯ τ X Z q 1 h, 21 34 τ, ( ' c > ) = Z 1 q τ 24 ) = Λ ( − z 0 ) is the degeneracy of primary operators of weight ( | ¯ q ) is the Dedekind eta function. ¯ χ h p c τ 24 h ( h, ( − η ( q 21 ρ 34 F dependence on the left-hand side, there must be an infinite number of and τ πiτ 2 = e ] presents a good review of this concept and discusses the connections between various recursion q 93 For simplicity, we will decompose the conformal blocks into two parts as In the light cone limit The Zamolodchikov recursion relation is also used in the conformal bootstrap [ 16 study [ relations. A generalization of the recursion relation to more general Riemann surfaces is given in [ The Zamolodchikov recursion relation [ formal blocks numerically and hasit been effectively recently encompasses receiving the muchprovide attention conformal a [ blocks brief beyond explanation the herein. known regimes or limits. We the large-spin spectrum withphysical twist explanations accumulating for to this result.gravity aspect. It would be interesting to explore it, particularly, the C Zamolodchikov recursion relation where we assume thatreproduce there the are noterms extra in currents the apart sumon on from the the the right-hand right Virasoro side. hand current. side Moreover, must to To be match the dominated ¯ by where the modular bootstrap equation ( Here, the function limit ourselves to CFTs with where the torus partition function has the Virasoro character decomposition in CFTs, which also appears in higher dimensions. B Light cone modularThe bootstrap modular invariance imposes the following condition on the torus partition function, blocks. This correction is related to the other anomalous dimension different from ( JHEP01(2019)025 . ) p,q (C.9) (C.5) (C.6) (C.7) (C.8) (C.2) (C.3) (C.4) λ − 4 . λ ) 4 − h 3 . + λ , . 3 ) is a universal , h )( q 2 i | nb + λ p 2 p,q h h , + λ − ( ) + q 1 − b 1 | 21 34 . m h 4 ) λ 4( − 24 mn − + 1 2 c mn 1 3 2 + − λ = = c + c, i ]. It reveals the general solu- )( )) m,n q , m,n p,q for AABB blocks for ABBA blocks ( m,n k h 22 3 , λ ( q h n , k,l θ ) ( # − i ( λ 21 p 34 21 1 3 − h ) 1 h λ H +1 k ( c − n − 32 k n − c 2 c − Y 2 m,n h ( ]. One of the recent results involving the m,n = m,n , λ form of l − − 1 m,n h =1 ∞ for large 1 0) R h , k X )( B − 24 m,n − c h +1 − mn p,q ) p R 6=(0 q ) m z λ p h m − Y h – 34 – 1 − k,l − ( = 1 n − =1 32 k c λ i ) = 1 + =1 √ ,n , λ (1 q = ∞ Cardy-like A + 2 ,n | X − e 2 h p =1 X α 2 m,n λ =1 mn − h , h A m ( 1 ( n λ h m 2 h n 21 34 − ξ − " k 1 =1 b 1 H 2 i X +1 − , which is defined by 24 1 ∼ c n q − + sgn z n − b Y 1 n ) = 1 + b c 1 =1(mod 2) = ) = q × q − | 24 p n + c p h + b − h ( q ) can be calculated recursively using the following relation: p ( ) as k even q h | takes the simple c q , 21 | n, ) 34 p 1 4 p δ 1 q n h H h ( +1 = 1 + 6 = ( 1 21 34 m c − 21 34 Y − = m H m,n =1(mod 2) H = ξ is a constant in p ) = (16 h m ) is the elliptic integral of the first kind and the function Λ q + | for large x p p ( m,n h n ( R c K = 2 21 34 Λ m,n R where For this series expansionlution form, our previous numerical computations suggest that the so- the function and the corresponding recursion relation as Unfortunately, this recursion process isanalytically; too complicated however, for from calculating the thetion viewpoint Virasoro is of blocks much numerical more computations, usefulrecursion than this relation the recursion is BPZ rela- method presented in [ tions our to previous this papers recursion [ relation via numerical computations. For simplicity, we re-express In the above expressions, we used the notations where prefactor given by The function where JHEP01(2019)025 ]. (C.14) (C.10) (C.11) (C.12) (C.13) ]. SPIRE IN [ , SPIRE A IN , , [ 1 B − 32 > h c , , .h 1 1 1 A . (1973) 161 − − − > 32 32 32 3 2 c c c + B > h 76 α > > < . , .h 2 1 (1984) 333 1 1 A B B B − )) 32 , − − 32 32 c h h h h c c z 1 − 32 if if if if if − c < > , , Infinite Conformal Symmetry in ] and partly proven in a particular . B 241 B B A B > h h . In other words, our conclusions in α 22 , B A Qα if if Annals Phys. (log(1 .h α 21 , , , 2 A.1 2 A π + 2 A 4 8 h 4 +5 +9 + 2 c c A − if otherwise ) B h ) into this approximation form, we can repro- − − , , Nucl. Phys. z 4 h ) ) , – 35 – 8 4 , but it does not matter.) − +5 +9 − − B B c c , A h h 1 C.12 Tensor representations of conformal algebra and A A (1 − − − 24 h h c + + 2 ) ) ∼ > h ), ( − A A B B ) in appendix q n − B h h 1 h h q π , h 1 − 24 2 0 n 2( 4( ), which permits any use, distribution and reproduction in c C.10 + + − 24 √ ( c A.17 ( ( A A A q e h h A q π , = = α ), ( 2 π 0 2( 4( n are non-trivial, depending on the external conformal dimensions A α ( α n A.16 X = = 1, the summation can be approximated using CC-BY 4.0 A α and → This article is distributed under the terms of the Creative Commons A z : : are supported by numerical verifications and analytic calculations in a par- ]. A 23 ) using the fusion matrix approach. We plan to take it up as our future work. (For simplicity, we assume Two-Dimensional Quantum Field Theory conformally covariant operator product expansion In the limit A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, S. Ferrara, A.F. Grillo and R. Gatto, C.14 [1] [2] ABBA blocks AABB blocks References Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. Inserting the expression for duce the conclusions ( appendix ticular case. However,in we ( have not attempted to reproduce the logarithmic dependence case in [ These expressions are numerically conjectured in [ and the central charge. The parameters JHEP01(2019)025 ] 09 04 (2013) (2018) JHEP Nucl. , 11 11 , JHEP (2015) 101 , JHEP , ]. 11 Zh. Eksp. Teor. (2015) 083 JHEP JHEP , ]. Phys. Rev. Lett. , , , , Ising CFT 11 (2018) 026003 SPIRE d arXiv:1705.05855 JHEP CFT from numerical 3 IN [ SPIRE , ]. D IN ][ ]. 2 D 98 JHEP ][ , The Analytic Bootstrap and SPIRE IN SPIRE ]. (2017) 136 ][ IN A Cardy formula for off-diagonal ]. ][ 10 Phys. Rev. SPIRE , IN Bounding scalar operator dimensions in SPIRE ][ arXiv:1212.3616 IN ]. JHEP [ arXiv:1403.6829 , ][ Universality of Long-Distance AdS Physics [ ]. SPIRE Eigenstate thermalization hypothesis and modular A new handle on three-point coefficients: OPE ]. ]. IN – 36 – arXiv:1711.09913 ][ Large spin systematics in CFT [ SPIRE (2013) 004 IN (2014) 145 Convexity and Liberation at Large Spin Probing thermality beyond the diagonal arXiv:1608.03284v1 SPIRE SPIRE 12 ][ ]. [ Analytic bootstrap at large spin Universal anomalous dimensions at large spin and large ]. IN IN ]. 08 arXiv:0807.0004 ][ ][ Renyi entropy for local quenches in [ An Algebraic Approach to the Analytic Bootstrap ]. ]. ]. ]. arXiv:1504.00772 Anomalous Dimensions from Crossing Kernels (2018) 115 SPIRE JHEP SPIRE [ ]. A Cardy formula for three-point coefficients or how the black hole SPIRE , IN JHEP IN IN 01 , [ ][ SPIRE SPIRE SPIRE (2017) 160 ][ SPIRE IN IN IN SPIRE IN The Lightcone Bootstrap and the Spectrum of the (2008) 031 05 arXiv:1611.01500 [ ][ ][ ][ IN [ JHEP 12 (2015) 026 , Nonhamiltonian approach to conformal quantum field theory arXiv:1612.08471 [ (1986) 186 JHEP 07 Large Spin Perturbation Theory for Conformal Field Theories Operator Content of Two-Dimensional Conformally Invariant Theories arXiv:1510.08091 arXiv:1804.08899 , [ [ JHEP ]. , (1974) 23 [ B 270 JHEP arXiv:1807.05941 arXiv:1212.4103 (2017) 111601 , 66 CFT [ [ (2017) 086 SPIRE D arXiv:1502.01437 arXiv:1502.07707 arXiv:1804.09658 IN 089 from the CFT Bootstrap conformal blocks 03 (2017) 157 [ twist 119 140 [ arXiv:1804.07924 AdS Superhorizon Locality invariance of two-dimensional conformal[ field theories three-point coefficients; or, how the(2018) geometry 005 behind the horizon gets disentangled got its spots asymptotics from genus two[ modular invariance Fiz. 4 Phys. A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Y. Kusuki and T. Takayanagi, D. Simmons-Duffin, L.F. Alday and A. Zhiboedov, C. Sleight and M. Taronna, A. Kaviraj, K. Sen and A. Sinha, L.F. Alday, Z. Komargodski and A. Zhiboedov, L.F. Alday, A. Bissi and T. Lukowski, A. Kaviraj, K. Sen and A. Sinha, E.M. Brehm, D. Das and S. Datta, A.L. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Duffin, Y. Hikida, Y. Kusuki and T. Takayanagi, A. Romero-Berm´udez,P. Sabella-Garnier and K. Schalm, P. Kraus and A. Maloney, J.L. Cardy, A. Maloney and H. Maxfield, R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, J.L. Cardy, A.M. Polyakov, [8] [9] [6] [7] [4] [5] [3] [20] [21] [17] [18] [19] [15] [16] [12] [13] [14] [10] [11] JHEP01(2019)025 ]. 3 , ] , 07 02 ]. JHEP JHEP SPIRE Phys. , , , IN JHEP JHEP , ][ (2018) 042 , ]. SPIRE ]. IN ]. 08 ][ SPIRE ]. SPIRE IN arXiv:1410.1392 [ ]. SPIRE ]. IN [ JHEP IN ]. , ][ ][ Conformal Block in SPIRE solutions from Virasoro SPIRE IN SPIRE 3 D -point toroidal block in the IN 2 1 SPIRE IN ][ arXiv:1609.07153 ][ IN ][ [ (2015) 171 ]. ][ ]. Holographic Entanglement Entropy arXiv:1501.05315 02 ]. [ SPIRE SPIRE ]. ]. arXiv:1712.00885 IN Semiclassical Virasoro blocks from AdS , IN SPIRE (2017) 072 JHEP ][ arXiv:1510.06685 arXiv:1703.09727 , ][ [ IN Virasoro Conformal Blocks and Thermality [ 04 ][ Multi-centered AdS SPIRE SPIRE (2015) 200 ]. ]. A Numerical Approach to Virasoro Blocks and arXiv:1609.00801 IN IN arXiv:1412.5123 [ [ arXiv:1603.08440 11 superconformal torus blocks ][ ][ ]. – 37 – [ ]. arXiv:1612.03879 c JHEP [ SPIRE SPIRE Conformal Blocks from Recursion Relation , (2016) 367 IN IN c (2017) 102 Worldline approach to semi-classical conformal blocks Holographic Description of SPIRE JHEP ][ ][ SPIRE Holographic duals of large-c torus conformal blocks Large- Holographic interpretation of Monodromic vs. geodesic computation of Virasoro classical Classical conformal blocks via AdS/CFT correspondence , IN 09 AdS Field Theory from Conformal Field Theory On the Late-Time Behavior of Virasoro Blocks and a IN (2016) 110 Two-dimensional conformal field theory and the butterfly ][ arXiv:1508.04987 B 904 ]. ][ (2016) 183 [ arXiv:1610.06717 10 (2017) 129 (2015) 131603 [ arXiv:1801.08549 06 Conformal blocks from Wilson lines with loop corrections JHEP [ 03 , SPIRE 115 arXiv:1501.02260 arXiv:1504.05943 IN JHEP [ [ , ][ Virasoro Blocks from Monodromy Method beyond Known Limits JHEP (2015) 077 , c JHEP Nucl. Phys. (2016) 070 , , Many-point classical conformal blocks and geodesic networks on the hyperbolic arXiv:1806.04352 arXiv:1707.09311 12 [ [ 12 A view of the bulk from the worldline arXiv:1208.0337 arXiv:1804.06171 New Properties of Large- Large (2018) 086014 [ [ (2015) 131 (2015) 049 ]. CFT: Heavy States and Local Quenches JHEP 07 08 , JHEP Phys. Rev. Lett. d 2 , , D 97 (2017) 140 (2018) 161 SPIRE IN arXiv:1805.12585 [ Rev. conformal blocks the Information Paradox 10 Semi-classical Limit plane effect semiclassical limit conformal blocks from [ JHEP gravity JHEP from Classical Background Fields Classification of Semiclassical Saddles (2018) 010 08 (2013) 054 H. Maxfield, K.B. Alkalaev and V.A. Belavin, Y. Hikida and T. Uetoko, H. Chen, C. Hussong, J. Kaplan and D. Li, K.B. Alkalaev and V.A. Belavin, B. Chen, J.-q. Wu and J.-j. Zhang, K.B. Alkalaev, O. and Hul´ık,T. Proch´azka J. Raeymaekers, D.A. Roberts and D. Stanford, K.B. Alkalaev and V.A. Belavin, K.B. Alkalaev and V.A. Belavin, C.T. Asplund, A. Bernamonti, F. Galli and T. Hartman, E. Hijano, P. Kraus, E. Perlmutter and R. Snively, E. Hijano, P. Kraus and R. Snively, A.L. Fitzpatrick, J. Kaplan and M.T. Walters, A.L. Fitzpatrick and J. Kaplan, K.B. Alkalaev and V.A. Belavin, Y. Kusuki, A.L. Fitzpatrick and J. Kaplan, Y. Kusuki, [39] [40] [41] [37] [38] [34] [35] [36] [32] [33] [30] [31] [28] [29] [25] [26] [27] [23] [24] [22] JHEP01(2019)025 ] , , N , d 2 W (2016) N (2013) 090 12 (2016) 069 , ]. 08 ]. (2016) 129 10 ]. 08 ]. JHEP SPIRE , arXiv:1405.5946 JHEP SPIRE IN (2014) 041701 [ SPIRE , JHEP ]. IN CFTs with multiple local ][ , IN SPIRE ][ JHEP ][ D , IN (2016) 113B06 D 90 ]. ][ SPIRE IN ]. ]. ][ (1 + 1) 2016 Entanglement Scrambling in SPIRE Evolution of Entanglement Entropy Out-of-Time-Ordered Correlators in Quantum dimension as entanglement (2014) 093B06 IN SPIRE ][ Liouville and Super-Liouville CFTs SPIRE Phys. Rev. IN ]. ]. , IN [ D ][ 2014 arXiv:1701.03110 2 The Bulk-to-Boundary Propagator in Black [ arXiv:1506.03772 Out-of-time-ordered correlators and purity in arXiv:1805.06464 [ [ SPIRE SPIRE Multi-centered higher spin solutions from arXiv:1703.09939 IN IN Entanglement of local operators in large- [ ][ ][ – 38 – arXiv:1809.01387 [ On the universality of late-time correlators in Generalized gravitational entropy ]. (2015) 110 (2018) 047 Time Evolution of Entanglement Entropy from Black Hole arXiv:1802.08815 (2017) 244001 Entanglement entropy in Prog. Theor. Exp. Phys. [ 09 arXiv:1303.1080 , 08 [ arXiv:1810.02436 SPIRE ]. ]. ]. , ]. ]. ]. ]. (2017) 046020 IN c-Recursion for multi-point superconformal blocks. NS sector (2018) 101 A 50 ][ Superconformal blocks from Wilson lines with loop corrections JHEP 11 Prog. Theor. Exp. Phys. JHEP , SPIRE SPIRE SPIRE , , SPIRE SPIRE SPIRE SPIRE (2018) 154 D 96 arXiv:1806.09563 arXiv:1806.05836 IN IN IN Fractional Statistics and the Butterfly Effect [ [ IN IN IN IN (2013) 014 [ [ ][ ][ ][ 05 J. Phys. ][ ][ JHEP , CFTs 05 , d Bounding the Space of Holographic CFTs with Chaos Scattering effect on entanglement propagation in RCFTs 2 JHEP Phys. Rev. , (2018) 112 (2018) 101 JHEP , ]. , n Out-of-Time-Order Correlation Functions for Unitary Minimal Models 08 08 Z Conformal Bootstrap to R´enyiEntropy in / arXiv:1610.06181v2 n ) [ 2 SPIRE T arXiv:1602.08272 arXiv:1602.06543 arXiv:1602.06542 arXiv:1403.0702 IN arXiv:1304.4926 [ [ arXiv:1809.07228 rational conformal field theories [ ( arXiv:1711.00624 excitations [ 061 in Orbifold CFTs conformal field theories [ entropy in two dimensional conformal field theories [ Interiors Conformal Field Theory conformal blocks Hole Microstate Backgrounds semi-classical JHEP JHEP Y. Gu and X.-L. Qi, R. Fan, P. Caputa, Y. Kusuki, T. Takayanagi and K. Watanabe, E. Perlmutter, S. He, W.-z. Guo, S. He and Z.-X. Luo, P. Caputa, T. Numasawa and A. Veliz-Osorio, T. Numasawa, P. Caputa, Y. Kusuki, T. Takayanagi and K. Watanabe, P. Caputa, M. Nozaki and T. Takayanagi, S. He, T. Numasawa, T. Takayanagi and K. Watanabe, T. Hartman and J.M. Maldacena, C.T. Asplund, A. Bernamonti, F. Galli and T. Hartman, O. Hul´ık,J. Raeymaekers and O. Vasilakis, H. Chen, A.L. Fitzpatrick, J. Kaplan and D. Li, A. Lewkowycz and J.M. Maldacena, Y. Hikida and T. Uetoko, V.A. Belavin and R. Geiko, S. Banerjee, J.-W. Brijan and G. Vos, [59] [60] [57] [58] [54] [55] [56] [52] [53] [50] [51] [48] [49] [45] [46] [47] [43] [44] [42] JHEP01(2019)025 , ]. ]. ] A , 12 (2017) SPIRE SPIRE ]. (2007) IN IN , Lisbon, 11 N model ][ ][ , in 11 ) JHEP J. Phys. , SPIRE N , ( IN gravity and JHEP Class. Quant. O (2000) 929 , , ][ JHEP D 4 (2018) 061 3 , arXiv:1405.5946v2 ]. 09 [ Liouville Action as ]. ]. ]. SPIRE arXiv:1401.0539 JHEP (2001) R153 arXiv:1704.08250 [ IN , SPIRE ]. [ ]. ][ SPIRE IN 18 SPIRE ]. IN ][ arXiv:1412.5205 IN ][ [ (2014) 093B06 SPIRE ][ SPIRE Quantum Regge Trajectories and the IN Charged Renyi entropies and holographic IN Phase transitions in [ SPIRE ][ IN ]. 2014 [ (2014) 111602 Adv. Theor. Math. Phys. (2017) 435401 ]. , Quantum Entanglement of Local Operators in Entanglement Entropy in the Holographic Entanglement Entropy on Generic (2015) 382 112 SPIRE Entanglement of local operators in large- Conformal Bootstrap, Universality and IN arXiv:1802.07275 A 50 – 39 – SPIRE Holographic Phases of Renyi Entropies ]. [ ][ arXiv:1407.5630 IN Class. Quant. Grav. Genus two partition functions and R´enyientropies of [ B 901 , ][ cond-mat/0503393 arXiv:1703.00915 [ hep-th/0308031 ]. [ SPIRE Modular Bootstrap Revisited Comments on operators with large spin ]. arXiv:0904.4477 ]. J. Phys. IN [ , ][ Evolution of entanglement entropy in one-dimensional systems Entanglement entropy and conformal field theory arXiv:1811.05710 (2018) 080 , SPIRE ]. ]. (2015) 059 Phys. Rev. Lett. SPIRE SPIRE IN , 05 Nucl. Phys. IN Prog. Theor. Exp. Phys. IN 01 ][ , (2017) 021 , ][ hep-th/0008253 ][ [ SPIRE SPIRE (2005) P04010 06 IN IN 14th International Congress on Mathematical Physics (ICMP03) arXiv:0905.4013 JHEP (2009) 115122 ][ ][ [ , JHEP , 0504 JHEP Holography and Riemann surfaces Three-dimensional gravity, point particles and Liouville theory Liouville theory revisited From Liouville theory to the quantum geometry of Riemann surfaces arXiv:1306.2640 , B 80 [ (2001) 1291 ]. 18 arXiv:0708.0672 arXiv:1706.07056 [ [ (2009) 504005 SPIRE hep-th/0104158 IN hep-th/0005106 arXiv:1608.06241v1 [ proceedings of the Portugal, 28 July–2 August 2003, Gravitational Scattering Path-Integral Complexity: From Continuous Tensor097 Networks to AdS/CFT [ [ Grav. fractal dimension Conformal Field Theories conformal field theories superconductors large c conformal field theories 42 Phys. Rev. (2013) 050 Time Slices J. Stat. Mech. 019 [ Virasoro Analytic Bootstrap J. Teschner, J. Teschner, S. Jackson, L. McGough and H. Verlinde, P. Caputa, N. Kundu, M. Miyaji, T. Takayanagi and K. Watanabe, K. Krasnov, K. Krasnov, M. Nozaki, T. Numasawa and T. Takayanagi, P. Caputa, M. Nozaki and T. Takayanagi, A. Belin, L.-Y. Hung, A. Maloney and S. Matsuura, A. Belin, C.A. Keller and I.G. Zadeh, X. Dong, S. Maguire, A. Maloney and H. Maxfield, M.A. Metlitski, C.A. Fuertes and S. Sachdev, A. Belin, A. Maloney and S. Matsuura, Y. Kusuki, T. Takayanagi and K. Umemoto, P. Calabrese and J.L. Cardy, P. Calabrese and J.L. Cardy, S. Collier, Y.-H. Lin and X. Yin, S. Collier, Y. Gobeil, H. Maxfield and E. Perlmutter, L.F. Alday and J.M. Maldacena, [78] [79] [76] [77] [74] [75] [72] [73] [69] [70] [71] [67] [68] [64] [65] [66] [62] [63] [61] JHEP01(2019)025 ]. (2017) SPIRE ]. IN -point ]. [ 1 C 77 ]. (2016) 068 , 3 SPIRE 10 SPIRE superconformal IN SPIRE IN ][ IN . ][ (2014) 527 (1984) 419 = 4 ][ (2015) 088 ]. JHEP N 96 , (2018) 113101 (2018) 150 08 Eur. Phys. J. 104 , SPIRE 09 IN 1811 (1987) 1088 JHEP ][ , 73 JHEP hep-th/0409258 , arXiv:1606.07458 [ arXiv:1511.04065 Bootstrapping the Spectral Function: On the [ Lett. Math. Phys. Closure of the Operator Product Expansion in , J. Stat. Mech. ]. , Commun. Math. Phys. (2005) 845 [ Recursive representation of the torus Entanglement entropy of two disjoint intervals and – 40 – , Analytic continuation formulae for the BPZ arXiv:0911.2353 (2016) 030 [ (2017) 126 ]. SPIRE B 36 11 IN Theor. Math. Phys. ]. [ symbols for the modular double, quantum hyperbolic 05 , ]. Recursive Representations of Arbitrary Virasoro Conformal j Bootstrapping Pure Quantum Gravity in AdS SPIRE ]. ]. ]. Bootstrap, universality and horizons ]. ]. IN JHEP SPIRE Liouville bootstrap via harmonic analysis on a noncompact [ (2010) 063 , JHEP SPIRE IN , IN ][ 01 SPIRE SPIRE SPIRE SPIRE SPIRE ][ IN IN IN Conformal symmetry in two-dimensional space: recursion Conformal symmetry in two-dimensions: An explicit recurrence formula IN IN CFT [ ][ ][ ][ ][ 3 Acta Phys. Polon. JHEP hep-th/9911110 K , , Virasoro conformal blocks in closed form , Analytic properties of the Virasoro modular kernel arXiv:1703.09805 , arXiv:1610.02000 [ arXiv:1702.00423 arXiv:1502.07742v2 arXiv:1805.05975 arXiv:1202.4698 arXiv:1604.01774 Blocks Uniqueness of Liouville and[ the Universality of BTZ [ arXiv:1610.05814 bootstrap of the representation of conformal block for the conformal partial wave amplitude the recursion formula for[ conformal blocks conformal block the Non-Unitary Bootstrap geometry and supersymmetric gauge[ theories [ conformal block 368 quantum group M. Cho, S. Collier and X. Yin, S. Collier, P. Kravchuk, Y.-H. Lin and X. Yin, E. Perlmutter, J.-B. Bae, K. Lee and S. Lee, Y.-H. Lin, S.-H. Shao, D. Simmons-Duffin, Y. Wang and X. Yin, A.B. Zamolodchikov, P. Ruggiero, E. Tonni and P. Calabrese, L. Hadasz, Z. Jaskolski and M. Piatek, I. Esterlis, A.L. Fitzpatrick and D. Ramirez, A.B. Zamolodchikov, J. Teschner and G. Vartanov, 6 C.-M. Chang and Y.-H. Lin, N. Nemkov, B. Ponsot and J. Teschner, L. Hadasz, Z. Jaskolski and P. Suchanek, [94] [92] [93] [90] [91] [88] [89] [85] [86] [87] [83] [84] [81] [82] [80]