JHEP10(2017)147 Springer October 12, 2017 October 20, 2017 September 1, 2017 : : : Accepted Published Received Published for SISSA by . We investigate the entanglement and https://doi.org/10.1007/JHEP10(2017)147 3 a [email protected] , and Sridip Pal b [email protected] , . 3 1708.04242 Shouvik Datta The Authors. a -series are shown to precisely agree with the universal corrections to R´enyi c Conformal Field Theory, AdS-CFT Correspondence, Field Theories in Lower q

The Monster CFT plays an important role in moonshine and is also conjectured , [email protected] Institut f¨urTheoretische Physik, Eidgen¨ossischeTechnische Hochschule Z¨urich, Wolfgang-Pauli-Strasse 27, 8049 Z¨urich,Switzerland E-mail: Department of Physics, University9500 of Gilman California Drive, San La Diego, Jolla, CA 92093, U.S.A. b a Open Access Article funded by SCOAP Dimensions, Discrete Symmetries ArXiv ePrint: entropies at low temperatures.computations Furthermore, of entropy these R´enyi using results the one-loop arealso partition shown explore function some to on features match handlebodies. of with entropies R´enyi We bulk of two intervals onKeywords: the plane. to be the holographic entropiesR´enyi dual of to this pure theory gravity along insingle with AdS interval other on extremal the CFTs.in torus The the are entropies expansion R´enyi of evaluated contains a using closedterms the form in short expressions the interval in expansion. the modular Each parameter. order The leading Abstract: Diptarka Das, Monstrous entanglement JHEP10(2017)147 ] (or its chiral 8 [ 3 20 6 22 11 15 4 22 28 9 16 29 26 , as its automorphism group. The moonshine – 1 – M 26 ]. Apart from playing a key role in proving the = 72 3 , c 24 2 12 7 14 -series of entropy R´enyi -series of entropy R´enyi q q 2 14 ≥ 6 = 48 and k orbifolding [ c 2 Z 1 ]. One of the crucial players in the story is that of the moonshine module 7 – 1 7.1 Second entropy R´enyi 7.2 and mutual information Constraints on7.3 second R´enyi On higher genus partition functions 3.5 Intermezzo: other vignettes 4.1 Partition functions 4.2 Leading terms in the 3.1 The Monster3.2 CFT Torus 1-point3.3 functions and R´enyi entanglement3.4 entropy in the SIE Leading terms in the , which has the Fisher-Griess monster, \ V module can be realizedlattice, as along the with CFT a ofMonstrous 24 Moonshine free conjecture, bosons this CFT compactifiedalso (along on been with the its proposed self-dual ‘extremal’ Leech generalizations) to has be a candidate dual for pure gravity in AdS 1 Introduction Monstrous moonshine was born froma a set mysterious of observation, profound and it connectionstheory eventually between grew modular [ into forms, sporadic groups and conformal field A Eisenstein series and Ramanujan identities B Extremal CFT at 8 Conclusions and future directions 5 entropy R´enyi from the gravity dual 6 Towards a closed formula for7 entanglement entropy entropies R´enyi of two intervals on a plane 4 Extremal CFTs of 3 entropies R´enyi of the Monster CFT on the torus Contents 1 Introduction 2 Short interval expansion for entropy R´enyi JHEP10(2017)147 , ]. 8 M -series q ]. It is 19 – . in the short ) 14 8 , and tempo- \ L V /L 8 ` + O( ). This will allow us to m 2 are the Eisenstein series of . In more physical terms, these  4 , ` L πiτ 6 ]. One can hope that studies 2 πiβ/L e  E 2 ] for previous works. Moreover, 22 = m – 34 ≡ q are some constant coefficients. An  – 20 τ m 23 and non-perturbative in the length of 2 2 [ 744 ) u q τ − ≥ ( ]. This is essentially the next-to-leading see [ ) j k 1 τ 38 ( j ) ) – 2 – τ τ ( ( 6 4 -series; -invariant. q E E j  is the ultraviolet cutoff. m ]. Fundamentally, since the Hilbert space of the Monster 2 . Nevertheless, one can expand these results both in the  u q 12 – ] are perturbative in =1 3 X 10 m 38 ; the modular parameter is + β ) is the Klein  ` ]. In recent years, entanglement entropy has served as a powerful τ ( j 13 , 6 ]. As we shall demonstrate, the SIE is fully determined to some order by ) = 4 log ` 37 -series. Note that, the regimes of investigation of our present work and that ( – q E 35 S ]). In this work, we begin an exploration into the information theoretic properties 9 is the interval length and ` ] are somewhat different from each other which makes the agreement rather non- is the nome, which is related to the modular parameter by In this paper, we study the entanglement and entropies R´enyi of extremal CFTs. In Entanglement entropy captures how quantum information is distributed across the 38 q 1 the entangling interval. On thelength other hand, and our non-perturbative expressions in are perturbative in the interval encode the corrections owing tois the equivalent finiteness to of a the low/high both temperature the expansion. spatial and temporal directions. The important check of our resultcorrections constitutes of to a the precise entropy, agreement R´enyi term with derived the in in universal the thermal [ of [ trivial. The expressions in [ Here, weights 6 and 4 and expectation values can, inother turn, words, the be entanglement fully and entropies R´enyi for determinedfirst a few by single orders Ward interval in on identities interval theFor on length, instance, torus, turns the at to out the torus. the to order beinterval In we expansion completely have reads determined calculated, by the the entanglement spectrum. entropy of extract finite-size corrections from the this unravels the piecesthe special short to CFTs interval of expansionthe this (SIE), torus kind. which [ requires The information entropiesexpectation shall of values be one-point of evaluated functions quasi-primaries in on within the vacuum Virasoro module alone. These of entanglement and its informationand theoretic the inequalities (arising like) from may unitarity, shed causality further light on thisthe issue first by setup, filtering out theral/thermal the CFTs periodicity, allowed shall theories. be put on the torus (of spatial periodicity, is also expected to haveThis an is incarnation something in which the hasalong microstates not of these been the lines understood conjectured [ fully bulktool yet; dual in see [ reconstructing although the fortherefore bulk developments natural geometry to and explore equations thisregarding of quantity motion the in of existence this gravity context. of [ extremal Moreover, ambiguities CFTs still for exist of this theory. Hilbert space of the systemCFT [ has an enormous groupexpect of manifestations symmetries of (i.e., this of in the studies largest of sporadic entanglement group), entropy. one The might huge symmetry, version [ JHEP10(2017)147 2 ]. ]. ]. 8 55 44 – 51 by the Schottky 3 . The analysis is then AdS 3 dual of the CFT replica . We reproduce the CFT 3 4 ]. The aspects of entanglement and AdS 41 we review the short interval expan- 2 ]. The 40 , 39 – 3 – . An attempt to determine the entanglement entropy ] and has been a subject of renewed interest [ ] has been proposed in [ 5 8 50 – 45 ]. The mapping of the replica surface to the torus and the knowledge of the 43 , 42 2 [ / The outline of this paper is as follows. In section We also analyse the entropies R´enyi of two intervals for these CFTs on a plane. In Yet another striking confirmation of our results comes from holographic computations To investigate the SIE at higher orders, the calculations will involve the one-point A modification to the conjecture of [ 2 = 1 extended and generalized toresults from other the extremal bulk dual CFTs in in section section entropies,R´enyi which we study here,modifications. are robust in the sense that they hold true with or without of the evaluated in a seriesAlthough of we works do [ not delve entropy inR´enyi of this two direction, disjoint this intervals. information can be used to obtain thesion third for calculating entropy R´enyi onCFT the on torus. the torus We and specialize provide to details the of case the calculations of in the section Monster some interesting featuresx of the crossingextremal symmetric partition point, functions i.e., facilitatesfor when a this non-perturbative the quantity, (in cross whichFor the ratio the allows cross-ratio) third is for expression entropy, R´enyi an the explicit genus 2 verification partition of function unitarity is constraints, necessary. [ This has been particular, we shall consider themanifold, second entropies. R´enyi via The resulting the genusare of Riemann-Hurwitz the exactly formula, replica known is forsecond 1. these R´enyis. One theories, can Since this also the information extract torus can the partition be mutual functions directly information. R´enyi used We to shall find uncover the CFT results for entropy R´enyi only (to does the this order concurrence constitutebut which as we also a have serves powerful calculated confirmation asAlthough in of a this the our agreement novel CFT SIE). is verification calculations, arbitrary Not perturbative, substantiating genus it both the on does holographic the require conjecture CFT details and of of gravity [ sides. partition functions of of entropy R´enyi using themanifold is techniques given of by [ group. handlebody Upon solutions. evaluating the These treeintegral are level on quotients and of these one-loop contributions geometries, to we the have gravitational been path able to show remarkable agreement with our spectrum and the information ofan three interesting point line OPE of coefficients investigation which ofthe is the torus sensitive Monster to 1-point CFT. finer functions This detailsthat, of is about the to primaries theory. transform some Since as controllableproperties cusp order, alone. modular these forms, one-point Wenear one have future. functions may not hope may attempted be to fixed do using modular so here and hope to address it in the verify consistency of the results.the We details shall of also the see operatorinto that content Monster the of entropies R´enyi irreps. are the sensitive theory, i.e., to it realizes the McKay decompositions functions of primaries on the torus. A necessary ingredient for this would be both the interval length and the nome (which is the ‘region of overlap’ of these two analyses) and JHEP10(2017)147 is r K a and cor- A (2.2) (2.3) (2.4) (2.1) n L Z , ´alathe n . ) for a chiral CFT ` + 1) . y (0) − K x + 1) . In this section, we briefly ]. The twist operators have Φ . The entanglement entropy x r ρ 11 0 ∂ defined by partially tracing out Γ( r A , iβ/L + + 1)Γ( n . ] . A K ] y h n A ) z A n ρ 1 Γ( [ ρ ! -th entropy R´enyi of a subsystem Z r K -sheeted Riemann surface, glued along Z r n A a ( n = log 0 y x ≥ A log r C X ρ log Tr n K , is therefore – 4 – d n 1 τ Tr[ − contains our conclusions and avenues for future 1 ) can be written as the partition function − 1 − . The general expression for the OPE of the twist K X . The second entropy R´enyi of two intervals on the where, 8 , i.e., with spatial and temporal perodicities 1 1) = = 2.1 1 β σ 6 n − h n S 2 E = n 2 c 24 n S S ( z n × c , S 1 L 1 1 = S − − r . Section ˜ r σ + (0) = ≡ + h is the normalization of the twist operators. The coefficients 7 σ ), i.e., the von Neumann entropy of the corresponding reduced K 2 K )˜ n h in the above equation is over all the quasi-primary operators in r = h r c 2 T z ( 2.1 C σ C K σ h ≡ ], can be obtained as a correlation function of twist and anti-twist operators r K 37 a n – ]. Upon spatial bipartitioning, the Z 35 37 1 limit of ( is the reduced density matrix of the region → A ρ n , inserted at the endpoints of the entangling interval [ ˜ respectively. The modular parameter, σ The summation over the replicated CFT and ar defined as operators is [ In most cases, σ, conformal dimension In the path integral representation,responding ( to the ‘replicathe manifold’, entangling interval. which In isRiemann-Hurwitz the a theorem. present context, The entropy the R´enyi can replica be manifold rewritten is as of genus the complementary region inis the the full densitydensity matrix, matrix. Tr the work of [ given by, where, We are interested inon the entropy R´enyi a of rectangular a torus, singleβ interval (of length review the formalism to evaluate entropy theof as R´enyi a short perturbative interval expansion length. in the limit Much of this analysis for a generic CFT has been advanced by plane is studied inresearch. section A number of technicalare details referred and appropriately extensions are from relegated the to main the text. appendices and 2 Short interval expansion for entropy R´enyi in closed form is discussed in section JHEP10(2017)147 . . ] ).

37 `/L 2.4 (2.8) (2.9) (2.5) (2.6) (2.7) +47 . When 2 T n by defining nd K = cutting-sewing d ]. For example, + 2) , equation ( T ` b can be on any of n 37 -sheeted Riemann n . = K 16( d z B −
]. ). , where the replica index, + 29 p 24 . 93 . ` 56 c 3 , 1 ...j +11
2 6 n, 2 70 j 1 36 n 1 n R j T T ...T i − + d )
2 T, z p 1 T 4 ( n 4 coefficients that we will require [ ∂ orbifold of the bosonic string compactified survive. As a result, the contribution of the h h + 864299970 – 6 – 2 T hDi 2 k φφ b + ( Z q D b b ...T 12 j ) orbifolding, which removes the states at level 1. The φ 4 + T φ 1 + 4 i h h 2  2 T ϑ 2 ) b Z 3 ` L σ 1 limit. In particular, in the identity Virasoro module, ) hBi ϑ h σ ( B h 2( φ  b → X 2( L + 21493760 ` 24 n n 1 + + q c η = 1 extremal CFT or the moonshine module 2 k ] = + n ) A 24 ’ come with higher powers of ρ η + 196884 Leech 2 [( q Θ 1 ... A ]. = 4 ) = [ = tr τ ( n M Z Z -series display ‘moonshine’, i.e., they can be decomposed in terms of dimensions of q The 196884 states at level 2 have the following decomposition in terms of irreps of An equivalent depiction of this is that of an 4 196884 = 196883 + 1,trivial which representation and is the the smallestthe McKay’s non-trivial conformal equation. representation field of This theory, these means are it dimension decomposes 2 into operators a on the which Leech constitute lattice. the lightest of the irreducible representations of the Fisher-Griessshown monster, that automorphism group ofis indeed the vertex operator algebra of the moonshine module In the first equality, theorbifold whilst first the term other terms is are the from the contribution twisted from sector. the As untwisted is well-known, sector the of coefficients the plicitly constructed by Frenkel, Lepowskyinvolves and 24 Meurman free [ bosons compactifiedThis is on followed 24-dimensional by self-dual an thepartition asymmetric function Leech lattice, of the theory reads 3 entropies R´enyi of the Monster3.1 CFT on the The torus Monster CFT The Monster CFT (the simplification occurs in the only the coefficients ofidentity Virasoro the module form, to thethe entanglement torus entropy and is completely its determined higher by powers. We shall return to this feature in section where contributions from the primariesmation and and their the descendants ‘ are contained in the we have the following, Collecting the leading order terms in the SIE for the JHEP10(2017)147 , D i , ) (3.5) (3.3) (3.4) (3.2) s B v , ]. This ( A T 58 , T, ··· 57 ) 2 v ( T ) 1 v ( . T h  ], is as follows  j v 60 , ∂ . ) i i j ) 59 ) . v . s s 1 v v  η ( ( 3.4 T singular terms] T 744 Z. j ··· ) + 2 ··· − i − j ) ) ) v log 2 j v v τ ( +1 ( −  j T v T 6 v ) ( ) ( πi∂ – 7 – u E 1 4 ζ T ( 2 v ) E ( π T 1 = 2 ) and upon using the identity for the derivative of h 4 T − [ i j ) + ( v ih 3.1 v 1 T = ) ( h η → v i lim u T ( T T ). We shall now elaborate how to evaluate the one-point h ··· ) + 2 i ≡ ) j ) 2.7 1 v v i − h v ) ( , the short interval expansion for the thermal entropy R´enyi − ), ( :( s T v v h 2 ( ( ) 2.6 TT j T ℘ : v h 2( ), we have ···  − ) are the Weierstraß elliptic functions. Once the correlators are evaluated ) v v 2 s =1 ( A.6 ( v j X 00 ζ ( ℘ T + ) c τ 1 12 v ) and ( s v =1 πi∂ T j ( X 2 ) ℘ v  -invariant ( + ( The expectation values of normal ordered products of the stress tensor can be found Since we are dealing with a chiral CFT, there is a possible dependence on the conformal j T = h using the above, we find theby expectation taking values of the normal coincident ordered limit powers of ofFor these the instance, operators stress tensor and subtracting out the OPE singularities. Here, as follows. We can firstof use multiple the stress-tensors. Ward identity The on Ward the identity, derived torus in to [ find the correlation function As expected, the modularconformal weight weight. of We shall the remark one-point on function this of further a below. quasi-primary equals its Substituting the partition functionthe ( requires the one-point functions ofVirasoro operators on module. the The torus. quasi-primaries Letas contributing us given first in in focus the on equations SIE,functions the ( till identity of level these 6, quasi-primaries. are from the The partition one-point function function itself: of stress tensor can be found ignore these contributions since theydealing are with. not specifically important for the theories3.2 we are Torus 1-point functions As discussed in section dition, there are 196883representation primary shall fields play an of important weight role 2. in section Thisframe decomposition in of which the the entropies reducible R´enyi results are being in calculated an owing to extra anomalies term [ which is frame-dependent. In what follows, we shall however states of the theory. The trivial representation corresponds to the stress tensor. In ad- JHEP10(2017)147 A of a (3.8) (3.9) (3.6) (3.7) 5 (3.10) i T . h i  T 2 τ h ] 1 π∂ η 2 τ − . τ  πi∂ i∂ . ) 6 ). Note that the leading 1 modular forms of weight 744) η E ) + 8 j . 6 1 2 6 , transforms as − c A.5 η 24 π E 27 h + 4 j 8 − ( i i , 3 4 h + 8 ) for the Monster CFT in the = T  q E i h i 3 . i 27344 T 3.1 τ | 3( h i  j Φ 744) | )( 2 6 O π i 1 h 15381 meromorphic h − E η h , g ) j i + + 2 4 -invariant and the Eisenstein series were 40 ( d X ] j 2 2 4 + 4 E g + i 4 E 37 = τ ). We thank the anonymous referee for clarifying 16) 3 π T , 744) i cτ h 4 + − , c 24 πi∂ – 8 – 3.10 − i j = − = ( j T c + [( 240) 0 60 24) h 2 b d L i 744) ) + − + q − ) in ( + τ T 0 744) − h j cτ aτ j Φ 1 ( ( 2 i j h η − 6 4 93(1823 15381( πi∂ ( g 2 O j 15 , g  E E h ( 2 . These are actually cg 6 6 0 + 2 , E 31 q π 836) E i = Tr 2 1 +  6 0 η π T 175 i − π 3 4 h + 29) 496 g 8 c Φ 3 π 15 c h 4 + 4 − − = 4 cg 61 25 i , (primary or quasi-primary) of weight 1 9 = = = + 29) η ) has a pole in the upper half-plane at the point where the partition T − O + 22) h c − 2 24(70 c c -series for the 1-point functions of the quasi-primaries within the vacuum 3.3 3 hBi hAi hDi q + ( (5 are the Eisenstein series with different normalizations (see also appendix cg c 70 5(70 (42 2 3 9 3 , 2 cg 120 + 744 vanishes.) On the other hand, the unnormalized expectation value − − g − = = = term) in the above expressions can also be derived by finding the appropriate j and 0 q 1 hBi hAi hDi η It is easy to see that the 1-point functions of the quasi-primaries are modular functions Here, ‘unnormalized’ refers to the feature that the the quantity is not divided by the torus partition 5 . (For example, ( function. Thiscomments is on denoted this paragraph. by the prime ( Furthermore, the Virasoro module starts out at h function primary, Φ, on the torus is given by the plane to the cylinder. with their respective conformalvalue weights. of an Under operator modular transformation, the expectation In the above expressions,simplified the by derivatives making of repeated use the ofterms the ( Ramanujan identities ( Schwarzian derivatives for each of the quasi-primaries under conformal transformation from Upon substituting the formabove formulae, of we the have the partition following function expressions. ( Here, for further details). values of the quasi-primaries at level 4 and 6 [ Performing this procedure, we end up with the following expressions for the expectation JHEP10(2017)147 κ κ,i 2 P almost (3.12) (3.13) (3.14) (3.15) (3.11) . The ρ . It is a well with coefficients h = 1 − τ ]) is an almost modular ) appearing below 6 τ ), implies that the 4 τ for further details). ( The functions /E 3.9 Z . 6 A . ), the primary fields of . , 7  E . of weight ) κ 2  6 2 , q . This is 2 6 , E  χ 2.10 4 ` P L P Z  6 Z and 744. ) 4 E κ τ 4 ) E − ( ) are modular functions of weight cusp form E − τ κ ) τ ( 2 − ( 1 τ and higher. The 3-point coefficients , κ ( 1 Z 6 , 2 j M , 4 24 1 P , ) M 2 P 3 3 6 4 =1 ∞ 2 2 6 4 κ X E E P `/L E E  6 4 + + (higher powers of  E E  ` – 9 – c 24 , the holomorphicity breaks down at 5
− + 1) 3 n log χ + 1) πi 3 n n + 1) n h 2 ( n q n ( i -series, therefore, starts out with a positive power of ( q χ + 1) 2 6 | n exp 4 3 π π n Φ π ) are non-zero. We thank Matthias Gaberdiel for pointing this out. | ≡ 1080 60840 − 2( χ ρ h 3.10 = . This reduces to the standard modular form when the polynomial is of = ) = ) = ) = τ ) and the results of the one-point functions of the quasiprimaries ) = 0 τ τ τ τ are holomorphic modular forms, whilst the ratio ` i ( ( ( ( 6 2 4 6 , Φ n 4 h 2.10 S E M M M 2). The above ), ( ]! Hence, from the short interval expansion ( ≥ 2.3 h 61 becomes 0 at , 4 is, in fact, a generalization of modular forms and is polynomial in (Im[ 47 E ), the short interval expansion can be organized as follows. 3.8 The Eisenstein series This statement can be refined even further by using the Monster symmetry. It is possible that primaries 7 6 . As mentioned earlier, we have calculated this till the sixth order. Explicitly, they are κ . This fact combined with the modular transformation property ( of weights even higherstructure than constants 12 appearing are in the ( ones which contribute.form. This can Since be pinnedmodular down form by finding which being holomorphic function of degree zero. Notice that the structure‘almost of modular the forms’ above built expressions of are the in Eisenstein terms series of a basis of weight 2 The leading term is the2 universal area-law term. given by We now have the necessary ingredientspansion. to Using write ( the entropy R´enyi infrom the ( short interval ex- is the partition function of the moonshine module, do not play atrum. role, This feature, until arising thatof purely entropy order, R´enyi from for modular the and first properties, the few greatly entropies R´enyi orders simplifies depend in the3.3 the only analysis short on interval and the expansion. R´enyi entanglement spec- entropy in the SIE q unnormalized torus 1-point functionknown of fact a no primary cusp is formsAs a of a weight consequence, less all than torus12 12 1-point exist is functions (see zero of appendix [ primariesthe with Monster conformal weight CFT less contribute than only at order ( to this torus 1-point function arises from the lightest primary, Recall that in themension moonshine (and module, all primary operators are of integer conformal di- In the second equality, the sum is over all operators of the theory. The leading contribution JHEP10(2017)147 , )— 2
(3.21) (3.18) (3.19) (3.20) (3.16) (3.17) 1 2.9 − , 2 # , ) n 8 24) . 4 . 4 /L ` # 8 L −

) ` 1 2 j 8  − ). It takes a rather /L 1)( 2 8 + O( ` n − 4 6 744) , j 3.12 2 + 51663360  19 #  − n ) j 8 L j L + O( π`
π` . (

). This is given as follows . 6  /L  6 4 4  8  ,  5952
E E = 31( `  coefficients — equation ( 6 6 L  L 3.12 2 2 π` + 103 π` − ` , L L ··· π` 4 4 O 2 2835 j   -expansion). In order to facilitate a -invariant. The coefficients of these b  45 π 1 limit of ( n 
q j + + O( P
−
270723 3 6 sin − 4 q → 170  + O 3 2 − 2  1654 L π ` S n ] and with holography (to be discussed in 2 2 L − 6 L 6 π`

L n n ` π` − 4 L 90241 + 38 3  2 n 3  2 2 log −
n  q 744) 67 n j 2 2 1 j, – 10 – . These have the forms 90 + 65179081 n S −
15 n 2 n 237139 j 1 − + 1) 1753 744) + n ( j 2 12 n 0 − 6 4 − = 24), which is (  S 2 c E E 73834 j 128 c + 508617 + 6944 + 8 n ( 4 L 2 π` = 2 2 2 = n 8 6 4 3 − π j n j n  2 ). This is because the E E 2

S − 1 3  4 945  − 2 univ 107271470 n 10 6 − S -expansion of entropy R´enyi ( L  ` 2.10  π` − q π −  754757 4 + 310 L + 29760  315 4 π` 2835 2  ` 2 2 n n − is none other than the short interval expansion of universal contri-  j n  − 4
0 n S log 31 341 1) ) = 4 log ` " − − ( − 131256 21493760 2 4 E 2 44107429 " " n n S n 278954 + 1) n 19 79 n 8 64 n n + 1 + 1 j, ), we require the n n 2( + + = 31( = = = 5 2 1 1 1 = = = , , , , 6 2 4 6 As mentioned earlier, the SIE contains terms which are closed form expressions in the The entanglement entropy is given by the 3 2 0 S S S P P P P bution to the entropy R´enyi (with wherein, we have written down just the chiral/holomorphic contribution. The expression for where modular parameter of the toruscomparison (or, with all the orders universal in predictionssection the of [ simple form, since the 1-pointfirst functions few of orders; the equation stressfor tensor ( the is other the quasiprimaries only vanish contribution in at this the limit. Explicitly, the entanglement entropy is polynomials depend on the index R´enyi are modular functions, which are polynomials of the JHEP10(2017)147 (3.25) (3.26) (3.22) (3.24) (3.27) (3.23) ). . h . 2 3.22 q  ]. In such a n . 26 . −  n ) . i of weight ) ) L ) π` ]). . ψ  ··· β/L 38 below) 0 4 π`/L π`/nL + ( πh ( sin 2 h πhβ/L h 2 2 th moment of the reduced 2 − /L > h 2 . n − e ψ 0 n  e sin )
h 2
sin  1 n πβh 1 L n π` π` nL 2 ··· − + O( − − − h 1 ) h e + 2 h 1 | ) ψ 2 2 n ) ( n ψ   sin β/L sin ih n c ψ 18 h ψ g δS | 1 2 π`/L 1 − πh π`/nL ( + n ψ 2 + ( 1 h h

− 2

2 P = + L L ge π` π` π` nL sin 1 n 2 + – 11 – sin i i q | ) ) 1 4 4 0 sin  − (1 + ih h 1 sin sin 2 0 L L π −∞ π` −∞ | 4 n ( (

( 1 0 ]. This term leads to additional contributions to the the  n  ψ ψ 4 ) ) A n 38 log -series of entropy R´enyi = q ∞ ∞ 1 Tr sin − ( ( 1 h n ]. i i 1 ψ ) is derived by considering the low lying spectrum of operators ψ n ) ) 1) h in the denominator is the degeneracy of the lightest states. It h + 1) 26 Tr = ] that the leading finite-size correction in the low temperature 12 g − n ) ( 3.23 = ] that the lightest state is a primary. The first subleading term in −∞ −∞ 38 n c ψ 3 ( ( ( ( n n 38 2 n ) T T = ) ) δS A 18 n ρ ) and, in turn, in the formula for the entropy, R´enyi yields ( ∞ ∞ ] in the low temperature expansion (with S + 1) ( ( ] for more details) Therefore, if the lightest operator is (or the set of such operators includes) 38 T T n Tr( h h ( 38 3.24 8 c − = ) is the degeneracy of the lightest primary operator in the theory. The above equation T g ( n The above analysis requires an appropriate modification if the lightest state is the The above formula ( See e.g., equation (2.25) of [ δS 8 the stress tensor, its contribution to first finite-size correction is given by The second term arises fromtion) the which is Schwarzian not derivative (of considered entropy. R´enyi in the [ uniformization transforma- Using this in ( stress-tensor, which issituation, a we quasiprimary. have an additional This term (instead has of been equation (20) pointed of out [ in [ has been assumed in [ the above expression is thenoperator equivalent (see to [ the following two point function of the primary The summation in theprimaries) numerator of is the over theory; the lightest operators (primaries and/or quasi- Here, and the ones to follow below have been modifiedand for their the contribution case of todensity the matrix the chiral is reduced CFT. density matrix. The where, the universal thermal correction is from the lightest primary, It has been provedexpansion in for [ the entropy R´enyi following on form the [ torus is universal. The entropy R´enyi takes the 3.4 Leading terms in the JHEP10(2017)147 2 q (3.28) (3.29) (3.30) acting 3 . − ) we need 2 3 L q q .  2 ) for level 2). or n q # at each order in ) − 8 3.28 ∂T ` ) 4 ) /L  -series of the R´enyis. 8 ` q L π` π`/L π`/nL (  (
+ O( 2∆ 2∆ -expansion using this proce- 6 q -expansion, as given in equa- sin  sin q ). The SIE at the order in 1 90241 L ], we can also consider partition ) π` ]. For instance, at O( i − 3 4 ψ −  1 3.18 , ( n n 2∆ 2 1 38
, n n δS 15  28 n =1 i X 73834 − 196883 1 , and the stress tensor. The leading finite-size + 508617 8 -series of ( 196884 + i q 2 ) ψ − – 12 – + n T ( n 2 2 5 q  δS n = 24, we obtain  L π` c = L 754757 315 π`  n −  ) and the derivative of the stress tensor δS 4 4 ψ n 1 n ) with sin − using the procedure of [ L 131256 1) q " 3.27 − 278954 n + 1) ( 3 8 ) reads as ]. 2 ] which does it for the vacuum block. Since the low-lying spectrum of n n ) and ( 3 + 39 28 0 and check it against the 3.29 = ( + 1) 3.23 n n → 4( δS ). − `/L = see e.g., [ 3.20 n 9 It is possible to proceed further to higher orders in the As mentioned earlier, for the moonshine module, the McKay decomposition of 196884 We thank Justin David for pointing this out. δS 9 . As we have alluded to in the introduction, this is somewhat complementary to the SIE McKay-Thompson series and twisted entanglement entropies In the spirit offunctions the Monstrous and moonshine their conjecture entanglement [ entropies corresponding to the McKay-Thompson se- This feature, in some sense,The makes operators moonshine at more manifest various(depending in levels on the contribute whether it differentlypartition is as function a functions (which primary, treats of quasiprimary the or the states a interval appearing descendant),3.5 length at as each opposed level to on Intermezzo: an the other equal vignettes footing). tropies order by order in to consider the contributionthe from primaries 21296876 of primaries weight of 2on ( weight the 3, vacuum. 196883 This descendantstaken makes into of it account clear while that calculating the the McKay entropy R´enyi decomposition (as explicitly in needs equation to ( be This agrees precisely withtion the ( next-to-leading term in the dure; operators is explicitly known for the mooshine module, one can calculate the en- R´enyi Note that the aboveq result is non-perturbative inwe the have interval derived length in in thearound previous sub-section. Nevertheless, wefrom can equation expand the ( above result Substituting ( ment entropy in [ lightest operators is into 196883 primaries, correction therefore splits as The first term in the above equation was obtained in the context of holographic entangle- JHEP10(2017)147 ]. ). z 38 ( (3.31) (3.32) (3.33) (3.34) (3.35) (3.36) (3.37) gX # ) 7→ . ··· and can be 8 2 ) q τ + /L M  4 8 + n q ` ∈ z ´nientropyR´enyi is − ( g ) X ) + O( . 6 # )  twisted 8 π`/L π`/nL ( L π` ( + 10698752 # /L 4 ) 8 2∆  ··· 3 . 2∆ 8 `  q + sin L sin /L π` 2 104 3 8 4 1 q 2835 `  ) + O( −  − A
6 4 1 − L (2 2∆ π` 4   4 n S + O( 4 2   L  π` + 1240002 ϑ 6
+ 18733 4 4 n 2 L   π` 2 + q ϑ q −
−  4 3 L ) 2 π` 2 4372 6019 ϑ 1 A n 1 n  90 (2  2 2 − + 45
– 13 – n S 2 3 2 2 2 − 2 ϑ ϑ n q + 96256 2 45 + as an example. The twisted partition function reads 16477 q )   + 310217 A A 2 L L 4926 π` (2 π` 0 n + 33883 ) = 16 S 128   8 2 τ 4 4 1 3 ( n + 4372 = − n − A ) 4 q 2 1 2 − sin 2 A n 518702  Z 945  (2  n 1) can be performed analogously and the 50383 −  ` L S ) = π` 945 L π` ) expressions can be reproduced by expanding the following about 4 − c  − 24  n 3.3  n 4 − ( 3 0 n log 2 3.35 n L 3 " 3 q 8744 96256 A . The partition function actually admits a closed form expression involving 2 + 1) 217509 " " B g 18686 n + 1) n ) and ( 8 64 n n n 4( + 1 + 1 and therefore there is one such quantity for each of the 194 conjugacy classes of n 2( n − 1 + + 3.34 − ) = Tr( = = = = 0. These are essentially the partition functions twisted by an element τ ) ) ) n ( A A A hgh → A 10 The calculations of this subsection are simple generalizations of the preceding ones. 2 δS (2 (2 (2 4 0 2 . We consider the conjugacy class 2 10 Z S S S -functions Here, we have used4371 the primaries McKay and decomposition the for stress the tensor operators at this at level. level 2, i.e., there are It can be checked that theThat above is, expression ( agree with predictions`/L from universality of [ where The analysis of section found to be Apart from having theBaby Monster, genus-0 property, theϑ coefficients also display moonshine for the shown to be genus-0 Hauptmouduls.ary Physically, this conditions twisting of refers to generic the fieldsOwing change of in to the bound- cyclicity theory of along thetion the trace, temporal these cycle: twisted partition functionsM are defined upto conjuga- ries. JHEP10(2017)147 (4.4) (4.1) (4.2) (4.3) (3.39) (3.38) ]. The . ], which 62 [ ··· 38 1 + N 5 − k . It can also be ) τ 3 ( 3.3 − J k 4
) − τ k ( ) J τ . ( is given by the polynomial 1) n ) is defined as J q . q . τ k in section
− (  1  − 1 n k F . 1 N + 43578174 . bd − ) 1 k 2 τ + ) =2 + ∞ N d ) ( Y n Z J k sτ 0 r + 393770 − T  k q 7→ π`/L r (21493760 π`/nL ( 744 + − F ( 2 = Z 2 a − 1 − r − 2 =0 sin q ) b k d X sin =0 r − τ r – 14 – X th extremal partition function compactly using k a ( s 1 n | ) k j k d  τ X − 16947377322260 ( = 24 theories in which spin-1 currents are present. ) = n ∞ = X 1 J τ r c ) = 59009461038 − ( − ) = N τ 1) 2 k ( τ ≡ 1 − k ( Z 2 − ) 2 Z and demanding the non-existence of primaries below the F = τ k k . The expansion above can be found by starting with a 0 s ( k ≥ 0 n ) T τ vac k acting on a modular function δS ( χ 0 r J are the degeneracies of the Virasoro vacuum character T + 1. (196884 r a k − ] k 19381654728 4231777443840 ) ) of degree 63 τ τ ( ( 744) + + J J − ) ) = τ ( τ ( j k Z ) = There is another way to write the τ ( J and the Hecke operator Here, the coefficients conformal dimension Hecke operators [ and this seriespolynomial stops of at 4.1 Partition functions The form of the( extremal partition function for arbitrary 4 Extremal CFTs of only modification which wechecked need to to agree make withis is the now universal prediction for the thermal correction of [ We can also considerThe meromorphic partition function is then of the form It has been conjectured that there are 71 such theories with specific values of Presence of spin-1 currents JHEP10(2017)147 ) . 3.2 = 2 # (4.7) (4.8) (4.9) (4.5) (4.6) ) 3 k q = 48 and . -expansion c q + O(  2 8 8 q ` L .   also to have the ) ) ] that no 2 3  , q q k 21 # + 31 ) 3 + O 2 q + O( 6 n κ 6 2 ) + O( ` 2 L q q 3  i  + O( ` L ) T + 23) 3 2 log( h  + 66 q q k k ) 2  4 k κ τ k −   ) n ( ) τ κ 1 3 = + O( ( 2 2(210 q 2 945 k Z + 11) q M − 45360 Z 2 is given by k + 11) 4 -function contains the dimensions of the k − n + O( ≥ =1 ∞ 2 log J 2. (60 + 1) κ X 2 4 n 4 k q , k ` L − ≥ + 45  2 2 1)(60
 ` ) i k + 17) . 3 – 15 – n 3 , , for T q − + 20 k q  + 29) h k M : ) log k k 2 k 3 Z k  q + 9) q 1 (420 k + O(  + O 720 + 42(2880 + 1) 2 2 2 . It is worthwhile noting that the entanglement entropy k n q q 3 − (60 + O( n + 11) -series of entropy R´enyi 9(1680 2 2  ( 2 − 31 q 2 2 2 k π + 17)(48 k q ` ]. Nonetheless, it has been argued in [ L − 8 + 2 − k 8  k i (60 2 n 6 − T k k = 90 2835 3 1 + 4 h π (42 k  n π 6 1 6 2 − − − k S π π  " " 16 4 − 175 8 15 − 2 4 6 q -series [  ` π π π q = = = 4 = = i . In the following, we just consider the leading terms in the k n n n log T , since the Hecke-transformed + 1) + 1) + 1) Z hBi h B hAi hDi k is given as follows upto M n n n in its ( ( ( ) κ τ ) ( = = = ) = 4 M τ κ ( ` 2 )) ) ) ) ( 2 3 Z ) M τ τ τ ) ) E τ ( ( ( -series of the partition function, τ τ 3.6 ( S 4 6 2 q ( ( Using the partition function above, we can calculate the short interval expansion of Z Z Z M M M where The corresponding entropy R´enyi is then Consequently, the expectation valuesand of ( the quasi-primaries is given by (from eqs. ( 72 in appendix and investigate some universal features at 4.2 Leading terms inThe the up to the sixth order in the SIE is given by We have performed the analysis for entropies R´enyi of the extremal CFTs at symmetry of irreps of extremal CFT may admit symmetry of the entropy R´enyi as in section This implies that one may expect the partition function for higher JHEP10(2017)147 ] 1. 8 0), ≥ (5.1) ≥ (4.10) k ). 2 q 11  4.8 ]. n 69 − – ) ) eigenvalues ). For the case of 65 , 0 5.1 L 13 π`/L π`/nL ( ( 4 in equation ( 4 — the holomorphic par- 2 sin q N sin 3 G 1 2 n ) (5.2) /  q n = 3 1 n − q c arising from a sum over geometries, + O( 1 1 − n J + q 1 2 1 − q =2 ∞ 1 Y n  — which is related to the three dimensional k ]. Yet another recent proof of this using localization L =2 k π` − ∞ 744, the partition function for the theory with Y 70 n q – 16 –  k − 4 − j q = 24 ) = . The coefficients of the polynomial can in turn be ]. sin q c J ( = — referred to as the ‘Farey tail’ [ 0 69 ) = 1) = 1 theory, i.e., the holographic dual to the moonshine , 3 J Z q ( 68 ). Further support in favour of this partition function has k − k q n ( Z ( J 3 2 n 3 ) = limit, matches with the terms of order + 1) q ( n ( Z k `/L 4 ] with the goal of making contact with the CFT calculations of the − 9 , = 8 ) T ( n ]. For a CFT with central is given by polynomials of δS k 64 [ = ]. It can be shown that the partition function of extremal CFTs can be obtained from chiral gravity 9 We shall focus only on the The vacuum character also equals the one-loop partition function of the graviton in 3 n More concretely, our holographic calculations is relevant in the context of the chiral gravity conjec- = 1, this is simply = 24 11 δS -invariant. Using the definition The (holomorphic) partition function can be separated into the classical orture tree-level [ piece as a regularized sumtechniques has over also Euclidean been geometries provided [ in [ determined by demanding that thek partition function is ofbeen the provided form by ( interpretingwhich the are polynomials modular images in of AdS module. The analysis will proceed in a very similar manner for the theories with The modular function of thej above form exists andc is unique, and is given in terms of the To account for the presencethis of should black be holes modified in as the theory (which have AdS Newton’s constant by the Brown-Henneauxtition formula, function with the one-loop graviton determinant included is previous sections. Recall that oneis of that the major the grounds Virasoroinclude of vacuum the black character holographic conjecture hole alone of is statesbriefly [ not review and the modular construct stream invariant, of a that logic modular one here. needs invariant to partition function. Let us 5 entropy R´enyi from the gravity dual In this section, wegravity shall dual evaluate [ the entropy R´enyi of the moonshine module from the tensor and is given by which, in the small The leading correction to entropy R´enyi in low temperature comes solely from the stress JHEP10(2017)147 (5.5) (5.6) (5.7) (5.8) (5.3) ] and ], the Γ is as 71 / 2 y, y 3 q −  πy L ··· , it is generated 2 g +  Γ as its conformal = 24) 3 ) (5.4) 4 / q 4  c given by [ C q sin Γ. The Schottky group πy L A / 2 1) C + O(  − 2 3 2 q ··· n 3 and with cos  n + + 21493760 .  3 2 πy Γ which has L  q q 2 / + 1)( . Once Γ is found, we need to find πβ/L πy L 3 2 πy 2 L  n tree 3 , g − ( 2 2 e S  4 3 + const  4 ]. The procedure consists of finding the ··· + = 4 cos ,  − 2 39 q sin 2 .  q , sin β πy 1) 2 πy L tree 1) 2 boundary. 2 = 1 −  – 17 – S ) = 1 + 196884 − n i  2 ) of the words for each primitive conjugacy class, q + γ 2 4 + const n 3 ( q n 3 n ). For a Riemann surface of genus  ), which arises from the Schwarzian derivatives of the sin n tree C 0 loop log sin , S by the Schottky group Γ are handlebody geometries. β πy − =1 + 1)( 1) 1 k 2 3.29 + 1)( , where 3 n i = Z − (  n + 1) 12 L n ( 2 3 n n 3 4 3 16 tree n n 2( S − − ) contribution can be obtained from studying monodromies of 1 and their inverses upto conjugation in Γ. The final step consists c − log sin i = = = q + 1)( L n ( tree tree tree 0 2 3 + 1) ) = n S S S q 3 16 n ( ] for further details. For the entangling interval 2( − =1 tree ]. We need to find the set of representatives of primitive conjugacy classes k 39 = , Z ], which uses Schottky quotient of the BTZ black hole, we implicitly consider the regularized 72 26 , 39 tree n 39 S of the Schottky group, Γ. This is done by constructing non-repeated words from the The prescription to find the one-loop contributions to the quotients AdS The classical (order Unlike [ ∈ P 12 loxodromic generators of calculating the largest eigenvalues ( sum over geometries as ouris starting the point. weighted sum In over other geometries words, with the genus ‘gravity partition function of handlebodies’ follows [ γ uniformization transformation as explainedwith earlier. CFT results, For we the organize the sake finite of size eventual corrections comparison as follows with The first term isterm the is well-known exactly universal the contribution first to term the in entropy. R´enyi ( The second the torus differential equation andreader then to integrating [ the accessoryresult parameter. is We (in refer a the low temperature expansion with Γ is a discreteby sub-group the of loxodromic PSL(2 generators the partition function of theboundary. holographic The dual quotients on of AdS AdS The formalism for calculating entropy from R´enyi extended the bulk to has include been one-loop expoundedSchottky in contributions uniformization [ in of the [ replica geometry in the boundary and the one-loop contribution. JHEP10(2017)147 )    ) 5.8 ξ !# (5.13) (5.14) (5.11) ( (5.10) (5.17) (5.12) (5.15) (5.16) 1 2 2 ), ( − 1)  5.7 2 −  ) 2 1 ), ( n − y ( u ), and using the 5.6 5373440 sin − ). Therefore y  u − 5.10 ) 2 5.3 4 γ q   ) + (  1 ξ ) in ( − y , of the single-letter word is u + O( n +1) γ 3 n γ + + 1 5.11 ( q q ··· y ) (5.9) n  u γ ( + q n ( , , 3 sin .  q ) ) ) − /n 3 0 2 1  1-loop loop S S S × − y 3.20 3.21 3.19 ξ ( ( ( − u Z  1 1) 3 ), we obtain 1 + 21493760 n − ∼ = ∼ = ∼ = −
/n S 2 γ 1 y n y − log 2 ( ) q 2.3 u  u + ) ) n )  tree 2 0 1 loop loop 5.6 − ∈P , which will suffice for our purposes) − ( q X 2 γ − y S − − 1 5.13 5.14 / sin – 18 – 1 1 u 3 2 ( ( /n 1 − y 1 = S S q −   u loop 196884 − y y  − u + + u q 1 2  ( loop ) ) , S Re ) is trivial. Substituting ( n − − 49221 tree tree   1 2 3 n 5.7 5.8 = ) ( ( − +  S S ∈P ) 1 Z ) it can been be seen that these precisely match with the X γ ξ    ξ n 5.10 − y  (  u  4 loop = log /n ) 4 ξ n
+ 5.14 1 y − ξ y 1 y ( n u  sin . Combining the terms from the classical part ( sin u u 4 , we have 6 S loop ( y ), ( − − − n − πy L . sin  1 n 1 2 /n 4 . These eigenvalues are independent of the index of the conjugacy # 4 sin 5 1 = 2 − y Z n 5.13 n /n n 2 − y u ` ≡ 1 y   " u u 1) 3 ξ πy/L log 2   2 n − n e 1) 1) 2 1) 2
" /n − − y 2 n = 1 y − ( / u 2 21493760 n u 1 y 196884 ( n n − − ( ( u − + q 1 /n = = − y ≈ 1 u − y 2 / u 1 4 loop loop Here, − γ − −

1 1 q 3 2 ; these are then substituted as the arguments for one-loop determinants. The free energy S S In order to compareentangling with interval, the CFT expressions,and we those need of to 1-loop makeshort ( the interval identification expansion of calculated the from the CFT. This is where in terms of class and therefore the sumpath in integral ( definition of the entropy R´enyi ( For the quadratic andfrom cubic single orders letter which words appear isgiven in by sufficient. the (we The above have largest equation, retained eigenvalue the terms up contribution to then reads For the case at hand, the one-loop contribution is given in equation ( P JHEP10(2017)147 0. → `/L ] and the one- 8 The first and sec- 13 ) which are joined by τ ] respectively. The third 38 ). ` -expansion and is non-perturbative q ]. 70 (the nome) at each order in the interval length. q ), which indeed agrees with the corresponding partition functions of the CFT and the gravity – 19 – iL n 7→ . q iβ, β 7→ L ), is substantially specific to the details of the CFT under consid- -series from the short interval expansion. q 5.17 replicas of a torus (each with modular parameter which puts the CFT and holographic computations in a larger context and 1 n The moduli space in the present situation is, however, restricted to a plane — . Illustrating the accessible regimes of CFT and holographic calculations. The short 14 This holographic verification serves a two-fold purpose. Firstly, it confirms the cor- It is worthwhile to note that this agreement is not specialThis for was extremal speculated CFTs. as This an is optimistic possibility expected in to [ be 13 14 tubes of the same pinching parameter (the interval length true in a more genericthank setting Justin when David the for equivalence pointing of CFT this and out. AdS partition functions can be shown. We it also provides aloop non-trivial verification exactness of of the theperturbative, holographic partition is conjecture function that of in of [ ourtheory. arbitrary context. genus Note thatconsisting this of agreement, albeit eration. A high temperaturemodular expansion transformation for ( the entropy R´enyi canmodular also transformed be found upon the rectness of our CFT expressions derived via the short-interval expansion. And secondly, See also figure illustrates the regime where weond are equalities testing above the are bulk/boundary wellto results. expected the entropy since R´enyi and they is correspond theequality, universal to equation thermal the ( correction universal of contribution [ Figure 1 interval expansion (SIE) isOn non-perturbative the in other hand,in the the holographic interval length calculation at provides each the order in This agreement can be explicitly seen by Taylor expanding the l.h.s. above about JHEP10(2017)147 , L (6.3) (6.2) (6.1) ] can → (5.18) ` . 73

  .

) cannot be ] β π` # 23 2 6.2 L / π` 2 1 ) and, more re- q 2 . ! / 1   3.17  744 ), we get ) L π` 744 τ τ ) . − ( ( 744 )  ) j ) − j 4 /τ . τ τ − ) q ( 1 ( ) j cot j − ) = /τ τ ( ) ) ( 1 L j 1 1 τ τ π` j /τ + O( − thermal ) ) 1 ( − − 3 S − τ τ j ( ( q − ( ( 6 4 ( 1 ) ) 6 4 0 j  E E E E  = → /τ /τ 

1 1  . It is clearly visible that the universal L π`   − − ) " and ( (  ), obtained from holography agree in the  6 4 196884 sin = E E iβ/L sinh 2 cot ` 5.18 × 2 /  expansion of this yields ( iβ/L ( = / 1 2 1 S L – 20 – − ) can be obtained by a modular transformation. π` = − τ − + 4  τ ) by demanding that the universal term should be − τ − ! `/L ), which is quite promising. The above formula is

6.1 ) = 1   5.6 744  . One may conjecture that these are expansions of the ) 744  − L 5.18 ) π` τ 1 − ( τ L − ( )  j ) j 744 τ ) = 0 limit. ( τ τ ( sin j − ` ( j ( ) ) ) j → ) ) L τ τ 21493760 E τ π -series, equation ( 1 1 ( ( q τ τ (

S q 6 4 j × ) and recalling − − ) ) E E ( ( τ τ 6 4  6.1 ( ( + 6 E E 6 4 L E E π = 4 log

0 limit — figure

 E -expansion yields ( β → π q S

), and the ]. `/L 3.17 73 ? = 4 log 1 limit) as derived from holography is ? E = 4 log → e S 0 and Despite these niceties, it is rather unfortunate that the above formula ( The high temperature version of ( We finally note that the low temperature expansion for the entanglement entropy (i.e., n E e S → correct. It can bethe seen limit that when it the does sub-system not equals reproduce the the full thermal system. entropy in This the is limit the property [ which easily reproduces thesee universal e.g., formula at [ high temperatures in the cylinder limit; Substituting this in ( suited for a lowformula temperature for expansion, entanglement entropybe on recovered a in cylinder the (with strict periodic spatial direction) [ As we have noted earlier, It can bemarkably, checked the explicitly that the The results for entanglementpansion, entropy which ( we have obtainedq from the shortfollowing interval expression ex- where, we have fixed thegiven ‘const.’ by the in ( Ryu-Takayanagi formula. 6 Towards a closed formula for entanglement entropy the JHEP10(2017)147 ) 24 ) 6.7 (6.6) (6.7) (6.8) (6.4) (6.5) `/L . . .

k # .  T `

b 2 # / ` 1 1 744 ) i 2 2 / τ 1 − − T 1 ( h i 2 ) j n τ " T 1 ( h . j → " k sin ) ) n 2 . It can be seen that ( τ τ ` 2 2 ) expansion. We claim that ( ( / = lim sin 6 4 1 i k 2 /L 2 E E / 2 T `/L T 1 b ˜ h  π i 2  ). This leads to the cylinder-limit , since one-point functions of non- k cylinder T

− i h ) yields the following expression for = 4  vac T 6.8

h S k ). The one point function of primaries ) is in fact a re-summation of identity 744) T 2.10 where, = 1 L b ˜ S − 0 of ( = 4 log E 6.1 cylinder ) k S i × k = 4 log X τ 2 ), therefore, become relevant only at ( → ( T ,

) — are not manifestly doubly periodic nor ` R h j +  k 6.5 – 21 –  ` i 6.2 ··· `/L L π` T h +  k = log( k T 2 ) in which b ˜ ` Z sin ) and ( k k 6.6 ) = 4 log i L π X log `

T ( 6.1 τ h + ∂ k  ` ’ in equation ( T + b ˜ ··· cylinder Z E k S X ) = 4 log ` + = 4 log (  ` = log vac ) captures the contribution to entanglement entropy coming from identity S cylinder E 6.1 ⊃ S 1 limit, the short interval expansion ( thermal E S ) = 4 log S → ` ( ’ represent the contribution from the conformal families of non-vacuum primaries. n E S ··· In where, in the second equalityis we precisely have the used Taylor expansionof around the identity given in equationvacuum ( primaries (and their descendants) vanish on the cylinder. Furthermore, one can On the other hand, consideringtransformation the to two point the function cylinder, of one twist can operators arrive and at its conformal of the spatial directionare intact 0 — leading on to thecontributions cylinder. from the In vacuum other Virasoro words, module the alone. entanglement entropy on the cylinder receives The proof oftemperature the limit above which decompactifies is the simple temporal direction, in but the keeps cylinder the periodicity limit of the torus (this is the zero- As noted mentioned earlier, theweight torus less 1-point than function 12 ofextra vanishes all terms since the denoted cusp primaries by forms with ‘ and of conformal beyond. lower modular Hence, weightVirasoro do the module, not conjectured i.e., exist. formula ( The entanglement entropy the ‘ correct by looking at higherthe equation order ( terms in shortVirasoro interval module ( alone and doesof not non-vacuum primaries. capture contributions arising from conformal families This disagreement with the proposedfact non-perturbative that formula the ismodular plausibly expressions related invariant. — to ( In the fact, one can explain why the conjectured formula is not entirely The thermal entropy is, in turn, given by JHEP10(2017)147 ]. 43 (7.2) (7.3) (7.4) (7.1) regime 2) = 0. is cross- / c x (1 0 2 S ], where ), having conformal 2 or equivalently at z / ∞ ]. One can perform a ( , 2 , σ i ) , v = 1 2 x ( u x ] can now be obtained as a 0 2 σ ), implying that corrections smoothen out the ) ] and [1 . , v x )) by conformal transformation  2 ) ∞ , x /c . Undoubtedly, a better approach u ( − x . ] and [ . 6.2 σ L 1
) ) ) ) (1 − 2 1 2 z x 2
, v v a. As the four point function of twist u u (0) 1 S (1 − − . Therefore it does not reproduce the σ 2 , x u − x − ] and [ − twist operators, 2 1 1 8 1 2 v 2 2 , , 2 (1) 2 v ) = v v 1 3.2 1 2  , v )( σ Z , x 1 via, , )( 1 h )( )( ( 1 2 1 1 2 1 1 u u 2 , x log u u u S 1 2 log 1 − – 22 – c − − − F 24 2 − 1 2 2 1 z u F ] get mapped to [0 ( ( u v = 2 2 ( ( i ) + 2 = , v τ ρ = ( 2 = w u Z x τ , depicted in figure . is related to x 1 β a, where the maxima occurs at log Tr log τ S 2 − − ] and [ becomes of the same order as × 1 ) can also be traced back to the fact that the conjectured formula ` = = R , v ) takes into account the dependence on the conformal frame. Using 2 and beyond, when cusp forms corresponding to 1-point functions 1 6.3 S u 2. However, it can be shown that 1 7.3 24 ]. The prominence of the peak therefore happens in the large / ) 39 `/L = 1 16. The appropriately normalized four point function of these twist oper- using the four point function of x c/ x = σ h . We also observe that the entropy R´enyi develops a sharper maxima with increasing i The failure of the ( = . From the dual gravitational perspective, the entanglement entropy undergoes a phase This conforms to theτ figure k transition at phase transition [ The second term in ( the knowledge of the genus-1 entropy partitionR´enyi as functions a for extremal function CFTs, of weoperators evaluate the are second crossing symmetric, we have Here the modular parameter function of weight ators, in turn, canFor be a expressed chiral in CFT, this terms is of given the by torus partition function of the CFT [ so that the intervals [ ratio given by, The second entropy R´enyi for the intervals [ In this section, weof aim two to disjoint find subsystems outconformal given second transformation by entropy R´enyi the and intervals second [ mutual information which captures the entanglement entropy in all its entirety is7 desirable. entropies R´enyi of two intervals7.1 on a plane Second entropy R´enyi and mutual information to the thermal cylinder, is not true atof ( primaries contribute; asexpected discussed behaviour when in section obtain the high temperature version (with the ‘sinh’ in ( JHEP10(2017)147 = 2. 1.0 / k 1 (7.5) (7.6) (7.7) ≥ x 0.8 .  x artifact can be 2 , there isn’t any − 0.6 x c c 1 8 0.52 2 Cross-ratio  ]. 0.4 74 log [ increases, corroborating to 0.50 (the spectrum of primaries d c k , 24 k . − 0.2 ) log 2 3 τ 0 5 . 0.48 10 ( − Z ) . The inset in (Right) shows the smooth x π 0.0 2 followed by a sharp rise for − 0 / 50 + log 2

100 250 200 150

1 Second Mutual Renyi Information Renyi Mutual Second = log
increases, the second mutual informa- R´enyi ≤ πk i 2 ) k x , as evident form the zoomed-in version of the and vector models in 3 x 1.0 (3 log 2 ∼ = – 23 – 2 ( 0 d I k ) σ i 2 ) ( Z ∞ ∼ = ( 0.8 σ 60 as a function of log , b. max (0) 55 ] max 2 ] , σ 2 2) becomes more prominent as ], for a chiral CFT, is given by, 2 2 / S 50 [ S 0.6 [ (1) , , v 2 σ = 1 45 h u , σ x h 40 2 2. b, we observe that as Cross-ratio , / at x 0.4 2 35 x The fact that the phase transition is indeed a large , ] and [ = 1 1 30 x , , v 15 1 = log 0.2 25 u , 2 intuition. Nonetheless, since all the plots given are for finite I corrections are highly suppressed. Furthermore, the absence of light primary 20 + 1). c . (Left) Second R´enyi entropies and (Right) Mutual R´enyi Information for , k a, is approximately given by , becomes flatter and goes to 0 for /c 15 2 , 0.0 2 I 0 ≥ From the figure 10 500 100 200 300 400

This is related to fact that presence of too many light states wash out the Hawking-Page transition.

, - - - - - with respect to Second Renyi Second h 5 15 , 2 We provide numerical evidence in favour of this in figure This happens, for instance, in coset theories in 2 figure which arises from actual discontinuity in thegraphs derivative as of shown inset of figure Second maxima, R´enyi Curiously enough, the maximum value of entropy R´enyi of two intervals, as depicted in tion, The sharpness of the changeI (to be precise, thethe apparent large discontinuity in the derivative of seen from evaluating mutual information assubsystems well. [ The mutual information between the two behavior around when 1 states also contributes tois such a phase transition at large Figure 2 1 JHEP10(2017)147 , j (7.8) (7.9) ]; al- 65 60 17 , which is equivalent ]. is large. At the self- πk The partition function k 44 2 e . 16 2 50 ≈ ) i ( + log 2 k + log 2. Z -modular tranformations. From a 40 πk πk S

2 k

2 2 − and therefore isn’t a statement about )

∼ =

∼ = g lo q

) k , especially when + i

τ ( k # k 30 i + (˜

Z π 2 ) = 1728), we are therefore led to a rather = k i τ ( −  j ) is given by q ) τ = 1 has a much debated status. One might hope – 24 – ( ≈ 7.9 20 k J ) 0 r τ ( T k  r Z at which . This is in the spirit of the ‘Farey tail’ sum [ − . Verifying log i 10 a πi/τ = k =0 2 r X τ − " e 0 Figure 3 th extremal CFT in terms of the Hecke-operators acting on = log k q 0

400 300 200 100 log log Z(i) and ˜ , the partition function ( ). i πiτ = ), we have 7.7 2 τ e 4.2 = ) are sub-dominant for purely imaginary q and the BTZ black hole. Contributions arising from the other modular images of Z , At the self-dual point ( 3 We are grateful toWe Christoph thank Keller Tarun for Grover this for proof. discussions and for pointing out this reference sometime ago. 16 17 to eliminate their possiblewhether existence they (or perhaps satisfy save alltake them the a from small inequalities extinction!) step involving in by entropy. R´enyi this checking In direction using this the subsection, inequalities we derived in [ SL(2 dual point, to equation ( 7.2 Constraints onThe second existence R´enyi of extremal CFTs beyond Here, though the partition functionholographic is perspective, invariant the only above under equationAdS takes into account contributions from thermal This approximation is truelarge even central at charge asymptotics. low values Afor of crude the derivation is extremal as CFT follows. can be approximated as non-trivial (but approximate) mathematicaltition identity. function of Usingequation the the ( expression for the par- JHEP10(2017)147 ] 2 < | , v t | 2 , (7.11) (7.14) (7.10) (7.12) u 0 = [ y < B , (0) = 1 and n  respectively. ) on right — for F )] x ] and B ( 1 n 7.14 , v F can be anything from 1 and u , with y x log [ A ) while the second one is for = [ k . − and 00 (7.13) A )] . ) to its reflection ( x 1 ≥ ≥ ) u ¯ 2 B  ) on left and ( < y  − p , B ) p ) | 2 ( ) t ) ) x x v n | y z 7.13 z ( , S ( )( n − − 0 th entropy R´enyi satisfies the following ). The inequalities are not sensitive to 1 n − F n v t, F (1 ) xy ) + (1 − − ¯  ( y > A √  2 A ∂ 7.11 u + 1.0 ∂x ( − ( ] that n -th entropies R´enyi for two intervals obey the y x p – 25 – S 44 n ) ) −→ − Wedge y y reflection ≥ ( 1 log [ n − ) ) 0.8 √ denote the Lorentzian coordinates within the wedge, is a function of cross-ratio  F ), is given by ¯ B x (1 ], figure 1) 1 n t, y y n A ( p 7.2 − ( F ) ) 44 − n 1 x x S ( n and 0.6 √ 2 n −  t F th entropy R´enyi for two intervals x (1 c 12 n (1 +  ] to constrain the entropies R´enyi is the wedge reflection positivity. , as they get cancelled out from both sides. 0.4 1 2 44 xy/ 2 κ − κ √ ). Consequently, the n , as defined in ( x 45, (the curves from left to right are for decreasing x , = = 2 0.2 − are the wedge reflected analogues of the intervals 20 n z , ¯ 8, for the chiral case. In the second inequality, B S (1 45. n , 15 c/ , F is some constant and . Verifying the second inequalities R´enyi — ( 30 , 0.0 and 10 = 0 κ , 27 27 27 26 27 27 p ¯ 5 15 A , , ) = In a chiral CFT, the The tool used in [ x

). In particular, it has been shown in [ 2.5×10 2.0×10 1.5×10 1.0×10 5.0×10 3.0×10

(

1 Inequality of LHS - y = 1 = 1 n Here, 0 to 1, while the overall constant F following set of inequalities, derivable from ( with cross-ratio where − inequality Here i.e., outside the light cone, (see [ and takes operator sub-algebra of one wedge ( k k This acts as follows, where Figure 4 two intervals, for extremal CFTs with different values of central charges. The first plot is for JHEP10(2017)147 ]. . ∞ )] x x ( (7.15) = τ [ 2 ] or from Z , v ]. It is not 75 12 =1 c/ 76 2  ) u x =[ − B (1 extremal CFTs. It is x ) Siegel modular forms. th entropy R´enyi in the ) however does not yield k ) and the interval length 1 Z n 16 , ] and  x 2 7.14 6 real dimensions. There are 1 κ = 24 = − πiβ/L 1 c ]. One way of calculating these g , v ) = = 2 x 58 ) and ( , ( τ =0 2 1 ] entropies into requires R´enyi a proper 57 F u 7.13 50 ) by the , =[ 4 A 49 , where , see also [ 47 ]). – 26 – i – )] 18 44 45 x ( 2 ). F Riemann surface has 6 7.4 g ] and are given in terms of Sp(4 log [ 50 − , conformal block with fixed internal weights as in [ ] c ∞ 49 , x ) 47 x – − 45 ) we have, (1 x 7.3 log [ c 8 ) is given by equation ( h x , albeit non-perturbative) a two-dimensional surface in moduli space is probed — 2 ( 3 κ τ ). The interval length, temperature and the cross-ratios (in the other two cases) can = However, translating the results of [ The moduli space of a genus The numerical verification of the inequalities ( Without the loss of generality we choose, We are grateful to Ida Zadeh and Xi Yin for discussions on this topic. 2 18 . In (b) we have a 6-point function of twist operators and this is a 3-dimensional surface S `/L the case of a singleshort interval interval expansion, on verified the this torusand with we expectations have have also from considered provided universal the thermal aof corrections corresponding the holographic CFT. analysis which The is analysis consistent reveals with that that, for the Monster CFT, features of moonshine are 8 Conclusions and future directions In this work, we have studied the and R´enyi entanglement entropies of extremal CFTs. For handling of conformal anomalyfactors factors; is to find thethe large- Liouville action correspondingclear the to relevant us correlators whether asoption, closed for described form checking in results the [ can inequalities of be [ obtained (which is a necessity and not an section the moduli being given( in terms of thebe temperature related ( to elements of the period matrix of the Siegel modular forms. three presentations in which thethird genus-2 entropy partition R´enyi of functions two can intervals, be or(c) used. (b) the the These second second are entropy of (a) R´enyi R´enyi of adimensional the three single trajectory intervals, or interval in on the the 6-dimensionalx moduli torus. space, The parametrized first byin the scenario moduli cross (a) space ratio parametrized scans by a the one- cross ratios. Finally, in (c) (which is a special case of As mentioned in thecalculation introduction, of entropies. R´enyi one can Thebeen calculated use genus-2 in partition higher [ functions genus for partition extremal functions CFTs for have the any surprises. Theseworthwhile checking are whether satisfied the same (see holds figure true for the higher7.3 R´enyis. On higher genus partition functions From equation ( where JHEP10(2017)147 ] 24 87 ≤ c arises in 4 for M ≤ g 19 ]. One may also ]. , which have been 82 ]. ]. The contribution 81 M 83 31 – 78 2. There are also other ≥ ]. Since much remains to be k 86 – 84 ] are obeyed. This can then shed light on ]; see also section 5 of [ 44 30 , – 27 – 25 ] that higher genus partition functions with . 49 k . One may expect that further constraints from modular 6 ]. It would be intriguing to explore whether the finite piece of the left- ). The computational tractability for the on R´enyis the torus is rendered 77 1 1 limit is not known, since the classical part is given in terms of Riemann-Siegel → n Finally, it would be exciting to explore entanglement in BPS sectors of string compact- It is also worthwhile investigating how the symmetry of the sporadic group There are several other avenues to explore features of entanglement in extremal CFTs. Another possible approach which we have eschewed in this work, is to evaluate the This is clearly not the full story and we have just begun to scratch the surface (as We note that it has been shown in [ 19 sures which can make these aspects(or more suitable manifest. modifications The thereof) supersymmetric entropy [ R´enyi can turn out be a useful measure in this context. theories are shown to be uniquely fixed by the torus partition function alone. techniques and the proposalconsider for an quantum approach corrections based to on entanglement the [ bulk-boundary reconstruction of [ ifications which exhibit moonshine in theirdeciphered elliptic regarding genus moonshine [ in these setups, it is worthwhile to consider refined mea- right entanglement entropy has a topological interpretation as in [ the bulk dual. The symmetryof is the realized as CFT. an It automorphism may ofusing be the the vertex possible operator entanglement to algebra derive entropy an derived analogue here, of along this symmetry with in clues the from gravity dual bulk reconstruction the plane, and verifyconundrums whether regarding constraints of the [ existencebipartitionings of of extremal the CFTs Hilbertpartitioning for space left v/s we right canconstructed movers work in of with. [ D-branes having For symmetries instance, of we can consider a to the theta functions. Furthermore, it isfor not known extremal whether CFTs the FLM with construction arbitrary has analogues One direction of immediate interest is to evaluate the higher for R´enyis two intervals on ´nientropyR´enyi using the Frenkel-Lepowsky-Meurman construction ofule. the This moonshine will involve mod- an analysisof akin the to bosonic [ oscillatorsHowever, is dealing with fairly the easy classical to parttice coming handle is from as non-trivial. the it compactification A on just potential the gets Leech drawback lat- raised of this to approach the is power that of the 24. analytic continuation the problem would be welcomed;with although partial a resummation success, of in theinvariance section series can was provide attempted, a but resulta closed for and form R´enyi entanglement expression entropies of which the is, modular hopefully, parameter and the interval length. ´nientropy ofR´enyi two intervals on the plane,Riemann in surface which the can partition be function mapped of the to branched the torus partition function. shown in figure by the short interval expansion. Since this approach is perturbative, a better approach to manifested in the and R´enyi entanglement entropies. We have also studied the second JHEP10(2017)147 (A.4) (A.1) (A.2) (A.3) -series. q -series are q at the Aspen in the q . κ n 2 q , in particular, are algebraically ) ) , 6 n 2 6 E . nτ ( 1 1 E is mock modular. Their + − πiτ κ − 2 2 2 and 6 2 3 e 4 m 1728 3 4 σ ( E 4 E E E = E 0) Exact methods of low dimensional Statisti- -series). It can be written in terms of the − =1 , ∞ q q X n = 3 2). 4 (0 \ 1728 κ E κ 24 2 Λ ≥ ) is the divisor function. The Eisenstein series ) 4 ∈ X s B – 28 – ) τ κ ( ( p ) = in the η − σ τ m,n ( ( q j ) = Quantum Gravity and New Moonshines τ ) = ) = 1 τ τ ( ( ∆( κ κ 2 2 ´ Etudes Scientifiques de Carg`ese,for an opportunity to present E -series do not exist. E q modular forms (for in the κ q -invariant is a weakly holomorphic modular form of weight zero (i.e., it j , at the Institut d’ is the Bernoulli number and r B The cusp forms are modular forms having solely positive powers of The Klein The lowest weight cusp formgiven is by the discriminant (or the Ramanujan tau function). This is has finitely many negativeEisenstein powers series. of We use the standard convention for the nome, Here, fall under the categorynegative of powers holomorphic of forms i.e.,independent these and are generate modular the forms space for of which all non- modular forms. These are weight 2 given by A Eisenstein series andThe Ramanujan Eisenstein identities series are defined by the following lattice sums Miranda Cheng for her lecturesSwissMap, on Moonshine. funded The by work the ofthe SD support Swiss is provided National supported by by Science theagreement the US Foundation. DE-SC0009919. NCCR Department of DD Energy and (DOE) SP under cooperative acknowledge research and participants of theCenter workshop for Physics (supportedrelated by topics. NSF SD grant also thanks PHY-160761)cal the Physics for organizers interesting of discussionsthis on work. SP thanks the organizers, participants and lecturers of TASI 2017, especially SD thanks Sunil Mukhi,versations for which inspired his this seminardiscussions work. at and for ETH We helpful thank Zurich, suggestions Justin onlin, and the Tarun David draft. Roberto Grover, and Christoph It Volpato, Matthias Keller, is forEdward Gaberdiel a Wei Li, pleasure con- Witten for to Roji and thank Pius, Ida Alexandre Anne Be- Zadeh Taormina, for Roberto Volpato, valuable discussions. SD is grateful to the organizers Acknowledgments JHEP10(2017)147 0 q (B.1) (B.2) (A.5) (A.6) 2 . q 2 4  . Hence, the alone n E = 1) evaluation ∂T T − − 2 k
) 6 ) 3 E q 2 E O π`/L π`/nL ( ( = + 4 4 2 6 q sin sin E (i.e., the lowest weight is the -derivatives of the Eisenstein q 3 , but non-perturbative at each τ 6 1 ` n ) with the one coming from the E ).  6 , n 4 B.2 , 1 E and − , q ∂ 4 j. 1 6 6 4 E + E E E + 40491909396 2 − q q − 3 ) 4 = 72 E L ) + 159769 = π` 2 at low temperature comes from – 29 – τ ( c ( j 4 E n j q ) also enables one to write a simple expression for S sin = q ∂ 4 A.3 1488 1) of the torus. Subsequently, the matching is done upon E − τ − q limit of the equation ( 2 n ) ` + 1 + 42987520 3 ( = 48 and τ 2 n 2 ( 1 q j c 18 = + 1) , q ∂ ) = 4 n = 48 τ -invariant ( ( E c j c ). − − =2 12 k 2 2 = 48) extremal CFT on a torus is described by the partition function = 3.8 Z expansion of the result. This involves (as in the case for E c ) q T ( n = 2 δS ) and ( E = q = 2 (i.e. 3.3 expansion of the partition function reveals that the lowest non-trivial quasi-primary n q ∂ k q One can match the small The Ramanujan identities provide expressions for the , having conformal weight 2. There are 42987520 operators of conformal weight 3, δS T short interval expansion.expansion As provides elucidated us with earlier anorder for expression, in the perturbative the Monster in modular CFT,doing parameter a the small short interval of the torus one point function of quasi-primaries in the identity module. leading contribution to entropy R´enyi The is out of which, 42987519 are primary operators and the lone descendant is The B Extremal CFT at Extremal CFT at This leads to simplificationse.g., in the ( expressions of the 1-point functions of quasiprimaries These relations have been usedprimaries in on the the final expressions torus.the for derivative the This of expectation and the values ( of quasi- term can be killed bylowest common a multiple combination of of the powers of weights of generators series. These are and has weight 12. The lowest weight is 12 since it is the first instance by which the JHEP10(2017)147 . 6 , (B.9) (B.3) (B.4) (B.5) (B.6) (B.7) (B.8) (B.10)  = 4 ) m + 69936) j 2 and given by n for 372 κ − k + 159769)  2 j 1) j 2 j − (5 m, 1258104 k  2 ) 6 2 1) f . 2 j , − 1488 n 2 E  6 κ − 4 , the expectation values of − 2 2 =0 , − 2 k X n + 16368) 4 P m  2 2 2 n 1488 2 + 159769) ` j L P = Z ( n j 2 6 − 3 4  2 108448 Z 2 -invariant E ) E 4 j j κ m, τ ) E − 47616( (866760 ( 1488 + (5 τ P κ 1 2 ( 6 − − + 356919981172496) , 2 − − 2 , 6 2 E 1 2 , Z = 744 = 2(1488 = 4754725440( = = M 1 n j 4 +1 P , 20 1 1 1 2 1 ( k 2 , , , , , 3 3 6 4 + 164157881616) 2 4 j 2 4 4 4 6 P 1 2 0 1 4 =1 1 ∞ E E P 2 2 2 6 4 E κ X   6 4 n m, k E E + + E E f  – 30 – + 15657362 5 1  ` + 1 + 576132818) n − j =0 3 n + 1 2 k X n m + 1 n log n n . + 159769 n = 2 2 6 j , f 1 , f 3 π + 1597690 4 π 1) 2 + 159769 + 1) j π m, 624694566015520 n + 159769) j 1080 102060 − − n 461214060768 j 1488 − P 2 259500504 + 159769) 4( 4 − n − + 930) j − 443936764 n ) = ) = ) = 4 30504 1488 2 2 = 2 τ τ τ + 159769 744) n j n − ( ( ( and 23808 j − − n 12253436) j 2 4 6 4 − 2 2 S 1488 − n 1) − j j , f j , M M M 2 ( − 2 1023 j − j 31 1488 , f n +1 2 31) 2 k − j 714705  1) − ( j n (31 4 − 6 1  2 4 , n − 2 2 j k E 6 = 2 extremal CFT, the expectation values appearing in the short interval E 2 6 n f 6 E ) are the meromorphic weakly modular functions of weight 2 n k  π 175 6 E 4 τ (237 (106931498 (237752546256 2 =0 4 2 π , f ( 32 E k X π π − − − κ 30501 15 2 16 8 8 − = = = = (267774584843024 = 5317604101763520( = 2(5611004 = 62( = 319538( = = 1 = 2(19 M 1 = = = = , 1 1 1 1 1 2 2 1 1 1 , , , , , , , , , , 2 Using the short interval expansion as described in section For the i 6 6 6 6 4 4 4 6 2 4 3 2 1 0 1 2 0 5 2 3 f f f f f f f f f f T P hBi h hAi hDi The coefficients of the polynomials above are the following: where we have defined the following polynomials of the quasi-primaries lead to an expression for entropy R´enyi Here, expansion, are given by: JHEP10(2017)147 . T 2 = 3 ∂ . The k (B.16) (B.12) (B.13) (B.14) (B.15) , having ∂T and 2 T q  A n + 1 (4 in this ) (B.11) 3 − k q 196845633216) ( ) ) O − 21457012458 2 + n 2 −  π`/L π`/nL ( q j ( 4 + 153264) 4 2 sin n , sin 36867719 3 1 n at low temperature comes from −   j n 368677190 ! ! n S 36867719) 36867719 − 1 ) − (1165848 (212892831576 − 36867719 j j − ) j 1 j − − 36867719) j j − + 578130935767 + 36867719 ) = = − 2 + 12756091394048 τ 2 2 2 j j ( , , 2 extremal CFTs, the leading contribution − + 1069957 q q 36867719) 36867719) 6 6 j 3 1 j 2 ˜ ˜ f f  ≥ j − − = 72) extremal CFT on a torus is given by + 1069957 k L j j π` . c – 31 – + 1069957 + 18189269 2 + 1069957 s are given by + 1069957 + 11769527  2 j 2 2 2232 , 2 2 j j 4 2 6 j j j j ˜ − + 1069957 f sin 3 2 + 1069957 4854564288 , + 1069957 j j 4464 44640 2 2232 2 − 1) ) 53568 11160 j = 3 (i.e., − τ + 1069957 + 1069957 − 46872 3 − ( − − j 3 k 2 2 3 − j 2232 3 − j j j j 3 n , where j 3 + 1 + 2593096794 3 25526133361) 3 j ( 3 2 2232 − j , 2 n j 6 j − 3 (84 31 /q ˜ − 2232 f 2 j 6 18 2232 2232 2 (12 ( 3  (31 2 E n 6 1) − j 4 + 1) + 78648104) 4904541 − − = 72 + 1 6 , E 3 2 E . Nonetheless for the sake of completeness, we provide the torus one  3 3 n − 3 c )

E ( 3 j j n 6 τ 2 6 6 40 4 c ( ( 10) /q ( π n 3 2

4 4 E E 4 81104 j ( − 175 6 2 4 − E E 48 E π π π = 1 alone. We will not repeat the matching here, since this has been shown 2 15 ≡ = 45621 − 8 4 4 + + n ) = ) 2 , T τ T 6 expansion, it is evident that the lightest non-trivial quasi-primary is = = = = j ( ( n 93( f i q =3 − δS T k hBi h hAi hDi Z = = (465242371 = (484996533859 = 2 2 2 n , , , In fact, this is a generic feature of all 6 6 6 4 2 0 ˜ ˜ ˜ f f f δS case; these are given by comes from generally in section point functions of quasi-primaries, appearing in short interval expansion, for the out of which, 2593096792 onesHere, are as well, the the primaries leading whilethe contribution the to stress entropy R´enyi rest tensor. of the two are From the conformal weight 2.extremal The CFTs only have operator, lightest appearingcase), primaries with by appearing conformal construction. with weight The conformal 3, number weight is of operators with conformal weight 4 is 2593096794, Extremal CFT at The partition function for the We define JHEP10(2017)147 (B.23) (B.17) (B.18) (B.19) (B.20) 1) 54157248) − − 2 . 2 n n . ; (B.22) 1) 2 + 51988821696)  ! ! − 2 n k k 2 562464) 1) , 2 j j 6 κ n n 2 2 2  − − , , 2 6 4 P 2 2 , k k  + 1065354212232184136122) 3 4 f f n n ` 2 L Z n P + 27681264 6 =6 =0 =3 =0  3 4 k k k k X X E ) + 9327348981409665780) Z n κ

τ 4 2 ) − ( n τ (241056 421848( 331809471( E κ 1 = = ( , 2 − − − 3 6 2 2 + 5121944013384702) − , , Z M 6 4 2 = 1069957 = = (62642091456 = 4733624882169960( = = 1 P , n 4 only and are given by: 1 1 1 1 2 2 3 3 6 4 (9119952 , , , , , , appears again and given by the following 1 P P , 2 4 4 4 4 4 0 4 2 0 2 0 =1 + 4209289143408) E E ∞ − 2 n P κ X k 2  2 2 6 4 = n P + , f E E 6 4 1 , , ;, (B.21) . 5  `  – 32 – 6 2 7 E E + 1 38207667042) n ! ! ! 1289597264020531187530200 1) n k k k − 3 log − j j j + 1 2 n − n + 1 1 1 1 4 , , , 2526894768482940564821 n 2 n 2 4 6 n k k k n , f 6 n − f f f 2 π + 1) 4 13165871496849172584 3 n 4 π π , f n =2 =0 =5 =0 =8 =0 n 32734292306379489 − 3240 1837080 k k k k k k − X X X , f 4 79354333120896 6( −

n 31908117426479) 4 − are defined as j j j + 25110) = ) = ) = ) = n 4 4464 − 2 j τ τ τ n n 34278659631 = = = , f ( ( ( 2 n − S 2 4 6 1 1 1 n − , , , 1) = 2 4 6 374056938) 4 M M M 1 − , f , n − P P P 2 2 27621 1 , f 2 n − = 3: n 279) 1) 4 k n − − 2 2 s are function of the index R´enyi n n i,j k (49955320651536 (580982466026811352729428 (6399 f , f − − − + 708614797993719834800772) = (6046225178420247156 = (1285155631058486117707 = = 3664876992152254105946040( = = (42500558241 = = (16917854014928979 = (57892482 = (18170421644291 = 279( = 105925743( = 3 = (171 1 1 1 1 1 1 1 1 1 1 2 2 1 1 Here The entropy R´enyi reads as follows: , , , , , , , , , , , , , , 6 6 6 6 6 6 6 6 4 4 4 4 2 4 1 0 3 2 8 6 5 4 3 1 3 1 5 2 f f f f f f f f f f f f f f where the polynomials of where the weak modularexpressions function for of weight 2 JHEP10(2017)147 . , A 42 109 Rev. , , (2008) ]. (1979) 308 04 ]. 11 J. Phys. ]. , SPIRE Commun. Num. JHEP SPIRE , IN 1217158713330) , (2015) 11 Invent. Math. 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Entanglement entropy and conformal field theory n quant-ph/0703044 , where SPIRE 2 [ [ 2 , n IN 6 j ˜ ][ f SPIRE + 1365872932) , SPIRE 2 IN 1) (2006) 181602 n IN [ ][ 10) − CC-BY 4.0 96 Monstrous moonshine and monstrous lie superalgebras 2 − (2011) 849 (2008) 517 arXiv:0905.4013 n This article is distributed under the terms of the Creative Commons 2 [ ( 5 n Moonshine beyond the monster: The bridge connecting algebra, modular forms Constructing holographic spacetimes using entanglement renormalization 80 , Cambridge University Press, Cambridge U.K. (2006). Three-Dimensional Gravity Revisited Elementary introduction to Moonshine ≡ . 2 , 6 9(72148332579073807 2511( j f − − arXiv:0801.4566 = = 9(77476036142988777 = = 9(6555493484 = 9(491636422030439 [ 2 2 2 2 2 , , , , , arXiv:1411.6571 6 6 6 6 6 1 0 6 4 2 Phys. Rev. Lett. Theor. Phys. arXiv:1209.3304 Mod. Phys. 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