Renyi Entropies, the Analytic Bootstrap, and 3D Quantum Gravity at Higher Genus
Matthew Headrick, Alexander Maloney, Eric Perlmutter, Ida G. Zadeh
BBRX-TH-6292
R´enyi Entropies, the Analytic Bootstrap, and3D Quantum Gravity at Higher Genus
Matthew Headrick , Alexander Maloney , Eric Perlmutter , Ida G. Zadeh Martin Fisher School of Physics, Brandeis University, Waltham, MA 02454, USA Department of Physics, McGill University, Montr´eal, QC H3A 2T8, Canada Department of Physics, Princeton University, Princeton, NJ 08544, USA
Abstract
We compute the contribution of the vacuum Virasoro representation to the genus-twopartition function of an arbitrary CFT with central charge c >
1. This is the perturbativepure gravity partition function in three dimensions. We employ a sewing construction, inwhich the partition function is expressed as a sum of sphere four-point functions of Virasorovacuum descendants. For this purpose, we develop techniques to efficiently compute correla-tion functions of holomorphic operators, which by crossing symmetry are determined exactlyby a finite number of OPE coefficients; this is an analytic implementation of the conformalbootstrap. Expanding the results in 1 /c , corresponding to the semiclassical bulk gravityexpansion, we find that—unlike at genus one—the result does not truncate at finite looporder. Our results also allow us to extend earlier work on multiple-interval R´enyi entropiesand on the partition function in the separating degeneration limit. a r X i v : . [ h e p - t h ] M a r ontents c CFTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Vacuum amplitudes from sewing . . . . . . . . . . . . . . . . . . . . . . . . . 15 c . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Operator modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3 The holomorphic bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3.1 Four-point functions: General case . . . . . . . . . . . . . . . . . . . 233.3.2 Four-point functions: Identical operators . . . . . . . . . . . . . . . . 26 C h ,h ( x ) . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2.1 C h, ( x ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2.2 C h, ( x ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.2.3 C h, ( x ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2.4 C , ( x ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 C h ,h ( x ) More details on the sewing construction 55
B.1 Operators and norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55B.2 Four-point functions of vertex operators . . . . . . . . . . . . . . . . . . . . 56
C Schottky parameters in the separating degeneration limit 58D The order- c part of the free energy and the sewing construction 61 Three-dimensional gravity has proven to be a fruitful testing ground for our ideas aboutholography. Perhaps the most interesting question is whether pure general relativity—atheory with only metric degrees of freedom—with a negative cosmological constant exists asa quantum theory in its own right. If this were the case, then one should be able to findits holographic dual for a given value of G N /R AdS , the Newton constant in AdS units. Thisappears to be an extremely difficult problem (see e.g. [1–3]). However, general relativityexists as a sub-sector of any theory of gravity in three dimensions. From the boundarypoint of view, it captures the dynamics of the Virasoro sector of any two-dimensional CFTwith central charge c = 3 R AdS / G N . This semi-microscopic interpretation is unavailable inhigher-dimensional AdS/CFT, where the stress tensor does not generate a closed symmetryalgebra.This perspective lends a universality to AdS /CFT that underlies, for example, thematching of the asymptotic symmetry group of anti-de Sitter space to the Virasoro algebra [4]and the matching of BTZ black hole entropy to the Cardy growth of states [5]. These arefeatures of any theory of three-dimensional gravity, and of any dual CFT. Recent work hasrevealed an even richer set of properties of two-dimensional CFTs that admit a large- c limitand are dual to weakly coupled bulk theories of gravity. These relate to aspects of suchtheories’ spectra and thermodynamics [6–10], entanglement [11–19], Virasoro blocks [20–24],modular geometry [25], and chaotic response to perturbations [26], among others.Despite this fascinating progress, much remains to be understood about basic conse-quences of Virasoro symmetry. To this end, in this paper we will focus on the computationof the partition function of three-dimensional gravity in a universe whose boundary is aRiemann surface Σ g of genus g . Schematically, this should be given by a bulk path integralover geometries M g which asymptote to Σ g : Z grav (Ω g ) = (cid:90) ∂ M g =Σ g D g e − S [ g ] . (1.1)This partition function is a function of the conformal structure moduli of the Riemann surface2 g , denoted Ω g . These partition functions contain vital information about the theory: forinstance, one can recover the correlation functions of a given CFT from its higher-genuspartition functions by pinching handles [27]. Thus, by tuning the moduli Ω g , one could inprinciple recover the correlation functions of the boundary CFT.Equation (1.1) is in general an extremely difficult object to compute. Moreover, it is not“universal” in the sense described earlier. In particular, in (1.1) we have written the bulkpath integral only over metric degrees of freedom; in more complicated theories of gravitymore degrees of freedom should be included. The partition function Z grav (Ω g ) written aboveis that of the CFT dual to pure gravity at a given value of Newton’s constant, if it exists. Inthis paper we will not be interested in the full partition function (1.1), but rather in an objectwhich is both easier to compute and universal: we will study the contribution to Z grav (Ω g )from a single saddle-point geometry M g , including perturbative quantum corrections. Thisrestricted partition function maps to the contribution of the Virasoro sector to the CFTpartition function on Σ g .This is easiest to see at genus one. In the semiclassical regime G N (cid:28) R AdS , the pathintegral (1.1) can be recast as a sum over saddle points of the Einstein action with solid torustopology, along with perturbative corrections. The simplest such saddle point is thermalAdS , the Euclidean geometry found by taking empty AdS and periodically identifying inEuclidean time, which contributes to the partition function as Z TAdS ( τ, τ ) = | q | − c/ ∞ (cid:89) n =2 | − q n | , q := e πiτ . (1.2)In this expression, we have included not only the classical action of thermal AdS (the factorof | q | − c/ ) but also all of the perturbative quantum corrections which come from loops ofgravitons in thermal AdS. With certain reasonable assumptions, all other saddle points aresimply SL (2 , Z ) modular transformations of thermal AdS, and the sum over geometries isa sum over SL (2 , Z ) transformations of (1.2). A direct calculation of Z TAdS does not yielda result consistent with an interpretation as a trace of the Hilbert space of a CFT [2, 7]. Nevertheless, (1.2) does have a natural interpretation as the Virasoro vacuum character of any
CFT with central charge c > SL (2 , R ) × SL (2 , R )-invariant ground state. Wenote that this object is not modular invariant, which reflects the fact that in (1.2) we havefocused on only one saddle out of the SL (2 , Z ) family. In the language of Riemann surfaces,(1.2) is a function not of the conformal structure of the boundary torus, but rather of theTeichm¨uller parameter τ .In the present paper, our goal is to compute the analog of (1.2) at higher genus. Any the- However, in the quantum regime G N ∼ R AdS , it was argued in [3] that at specific minimal-model valuesof c , not only can the sum be performed, but it agrees with the minimal-model partition functions. gravity contains solutions which are handlebodies—solid genus- g geometries—which have the Riemann surface Σ g as their conformal boundaries. These solutions are quo-tients of Euclidean AdS , much like thermal AdS. The contribution of a given handlebody tothe path integral—including graviton loop corrections—has a universal CFT interpretationfor any value of c , as the contribution of the states in the vacuum representation to theCFT partition function on Σ g . We call this quantity Z vac . Z vac is a function not just ofthe conformal structure of Σ g , but rather of the Teichm¨uller parameters that parametrizethe universal cover of the moduli space. In other words, to compute Z vac we must specify amarking of the Riemann surface Σ g that fixes a choice of contractible and non-contractiblecycles of the handlebody (A- and B-cycles, respectively). Thus Z vac is not invariant underthe modular group; a modular-invariant partition function could be obtained only, for ex-ample, by summing over bulk saddles that describe the different ways a boundary Σ g can be“filled in” by bulk geometries. One of our goals in this paper will be to give a direct CFTcomputation of Z vac , which can then be interpreted gravitationally.There has been a recent resurgence of interest in higher-genus partition functions of two-dimensional CFTs. This interest is partly motivated by the study of entanglement entropies(EEs). The computation of EEs via the replica trick involves evaluating entanglement R´enyientropies (EREs), which in turn are equal to certain higher-genus partition functions. Aparticularly interesting line of research uses calculations of EREs to test the Ryu-Takayanagi(RT) classical EE formula and understand the quantum corrections to it [11, 13–15, 29].The partition functions relevant for EREs have been computed in holographic CFTs intwo ways: in gravity, by explicitly finding the relevant saddles and evaluating their classicalactions and one-loop determinants, and in field theory, by computing twist-operator corre-lators in certain cyclic orbifold CFTs and then expanding the results in powers of 1 /c . Inevery case where the computation has been carried out on both sides, agreement has beenfound. This is a check of our basic understanding of AdS /CFT duality. Further, in manycases the computation using one technique gives results that are not practically computableusing the other, thereby giving new information about both three-dimensional gravity andlarge- c CFTs. For example, by expanding the results of the twist-operator computation tohigher orders in 1 /c , one determines higher-loop quantum corrections on the gravity sidethat would be exceedingly difficult to obtain by direct computation. These results hint at asurprising novel structure, which we describe below. This was called Z fake in [28], and the correspondence with the bulk saddle point partition function waswritten as Z fake = Z saddle . vac ∞ x p p p h i p h j ∑ i,j where we have defined r = x . Using (4.5), the functions C h ,h ( x ) are found to be C h ,h ( x ) = X i , i hi g =2 Y i =1 G i i ⌧ g =2 Y i =1 V out ( i , r i ) V in ( i , a i ) (4.14)= X i , i hi G G ⌧ V out ( , ) V out ( , x ) V in ( , V in ( , . These two formulae apply in general. For our purposes of computing Z vac , we only allowVirasoro descendants of the identity to be inserted at the boundary circles of the handles.Henceforth, we refer to (4.13) with the understanding that we compute Z vac specifically. vac (4.15) As mentioned earlier, in the orthogonal basis of states that we choose, the Zamolodchikovmetric is diagonal and so the outgoing and ingoing vertex operators are the same up toM¨obius transformations. Consequently, we can (and will) define the norm of the states as N ⌘ G .Let us now define the vertex operators needed in (4.14), starting with those at (0 , ).The functions C h ,h ( x ) are invariant under the map a i ,r i ! a i ,r i t , where t ( z ) = tz , t C . For the handle with its two ends located at a = 0 and r = we consider a M¨obiustransformation of the form a ,r /r and find a ,r r = z zr r !1 = z, a ,r r ˆ = 1 z + r r !1 = 1 z . (4.16)This therefore gives the identity map for a = 0 and the inverse map for r = . The vertexoperator at the origin is simply V in ( ,
0) = V ( ,
0) = (0) . (4.17)The vertex operator at infinity follows from usingˆ ( z ) = z , ˆ ( z ) = 2 z , (4.18)26 ( ∞ ) i Internal comments - We need to remember to be clear about including, or not, the anti-holomorphic side of thepartition functions.
Notes on notation • O ( c n ) versus O ( y n ) or O ( p n ), etc. • Free energy is F = log Z , and the large c expansion is F = X ` =0 c ` F ` (0.1)(Using S ` conflicts with the notation for R´enyi entropy.) • Period matrix is ⌦. • N O = norm • Generic operators are O . There are essentially two ways to treat three-dimensional pure gravity with negative cosmo-logical constant in the context of holography. One is to understand it as a quantum theoryin its own right, and find its holographic dual for a given value of G N /R AdS , the Newtonconstant in AdS units. The other is to understand it as a part of a larger bulk theory, suchas string theory, that captures the dynamics of the Virasoro sector of any two-dimensionalCFT with central charge c = 3 R AdS / G N . The latter, semi-microscopic interpretation isunavailable in higher-dimensional AdS/CFT, where the stress tensor does not generate aclosed symmetry algebra.This perspective lends a universality to AdS /CFT that underlies, for instance, theoriginal matching of asymptotic symmetries and BTZ black hole entropy to the Virasorosymmetry and Cardy growth of states, respectively, obeyed by all two-dimensional CFTs[xx bh, strom]. Recent work has revealed an even richer set of properties obeyed by familiesof two-dimensional CFTs that admit a large-central-charge limit, especially those that areholographic. These theories have a sparse spectrum of “light” operators of conformal dimen-sion less than of order c , dual to perturbative bulk states below the black hole threshold.Above the threshold, the Cardy formula dictates the growth of states even away from theasymptotic limit ! 1 [xx hartman keller]. Their vacuum Renyi entropies are given by the2 ( x ) j Internal comments - We need to remember to be clear about including, or not, the anti-holomorphic side of thepartition functions.
Notes on notation • O ( c n ) versus O ( y n ) or O ( p n ), etc. • Free energy is F = log Z , and the large c expansion is F = X ` =0 c ` F ` (0.1)(Using S ` conflicts with the notation for R´enyi entropy.) • Period matrix is ⌦. • N O = norm • Generic operators are O . There are essentially two ways to treat three-dimensional pure gravity with negative cosmo-logical constant in the context of holography. One is to understand it as a quantum theoryin its own right, and find its holographic dual for a given value of G N /R AdS , the Newtonconstant in AdS units. The other is to understand it as a part of a larger bulk theory, suchas string theory, that captures the dynamics of the Virasoro sector of any two-dimensionalCFT with central charge c = 3 R AdS / G N . The latter, semi-microscopic interpretation isunavailable in higher-dimensional AdS/CFT, where the stress tensor does not generate aclosed symmetry algebra.This perspective lends a universality to AdS /CFT that underlies, for instance, theoriginal matching of asymptotic symmetries and BTZ black hole entropy to the Virasorosymmetry and Cardy growth of states, respectively, obeyed by all two-dimensional CFTs[xx bh, strom]. Recent work has revealed an even richer set of properties obeyed by familiesof two-dimensional CFTs that admit a large-central-charge limit, especially those that areholographic. These theories have a sparse spectrum of “light” operators of conformal dimen-sion less than of order c , dual to perturbative bulk states below the black hole threshold.Above the threshold, the Cardy formula dictates the growth of states even away from theasymptotic limit ! 1 [xx hartman keller]. Their vacuum Renyi entropies are given by the2 (0) i Internal comments - We need to remember to be clear about including, or not, the anti-holomorphic side of thepartition functions.
Notes on notation • O ( c n ) versus O ( y n ) or O ( p n ), etc. • Free energy is F = log Z , and the large c expansion is F = X ` =0 c ` F ` (0.1)(Using S ` conflicts with the notation for R´enyi entropy.) • Period matrix is ⌦. • N O = norm • Generic operators are O . There are essentially two ways to treat three-dimensional pure gravity with negative cosmo-logical constant in the context of holography. One is to understand it as a quantum theoryin its own right, and find its holographic dual for a given value of G N /R AdS , the Newtonconstant in AdS units. The other is to understand it as a part of a larger bulk theory, suchas string theory, that captures the dynamics of the Virasoro sector of any two-dimensionalCFT with central charge c = 3 R AdS / G N . The latter, semi-microscopic interpretation isunavailable in higher-dimensional AdS/CFT, where the stress tensor does not generate aclosed symmetry algebra.This perspective lends a universality to AdS /CFT that underlies, for instance, theoriginal matching of asymptotic symmetries and BTZ black hole entropy to the Virasorosymmetry and Cardy growth of states, respectively, obeyed by all two-dimensional CFTs[xx bh, strom]. Recent work has revealed an even richer set of properties obeyed by familiesof two-dimensional CFTs that admit a large-central-charge limit, especially those that areholographic. These theories have a sparse spectrum of “light” operators of conformal dimen-sion less than of order c , dual to perturbative bulk states below the black hole threshold.Above the threshold, the Cardy formula dictates the growth of states even away from theasymptotic limit ! 1 [xx hartman keller]. Their vacuum Renyi entropies are given by the2 ( ) j Internal comments - We need to remember to be clear about including, or not, the anti-holomorphic side of thepartition functions.
Notes on notation • O ( c n ) versus O ( y n ) or O ( p n ), etc. • Free energy is F = log Z , and the large c expansion is F = X ` =0 c ` F ` (0.1)(Using S ` conflicts with the notation for R´enyi entropy.) • Period matrix is ⌦. • N O = norm • Generic operators are O . There are essentially two ways to treat three-dimensional pure gravity with negative cosmo-logical constant in the context of holography. One is to understand it as a quantum theoryin its own right, and find its holographic dual for a given value of G N /R AdS , the Newtonconstant in AdS units. The other is to understand it as a part of a larger bulk theory, suchas string theory, that captures the dynamics of the Virasoro sector of any two-dimensionalCFT with central charge c = 3 R AdS / G N . The latter, semi-microscopic interpretation isunavailable in higher-dimensional AdS/CFT, where the stress tensor does not generate aclosed symmetry algebra.This perspective lends a universality to AdS /CFT that underlies, for instance, theoriginal matching of asymptotic symmetries and BTZ black hole entropy to the Virasorosymmetry and Cardy growth of states, respectively, obeyed by all two-dimensional CFTs[xx bh, strom]. Recent work has revealed an even richer set of properties obeyed by familiesof two-dimensional CFTs that admit a large-central-charge limit, especially those that areholographic. These theories have a sparse spectrum of “light” operators of conformal dimen-sion less than of order c , dual to perturbative bulk states below the black hole threshold.Above the threshold, the Cardy formula dictates the growth of states even away from theasymptotic limit ! 1 [xx hartman keller]. Their vacuum Renyi entropies are given by the2 Figure 1: A depiction of the sewing construction as applied to Z vac , the contribution of theVirasoro vacuum representation to a genus-two CFT partition function. The coordinates p and p represent the widths of the two handles in a Schottky uniformization of the Riemannsurface. The handles are replaced by a sum over pairwise operator insertions, where weinclude all Virasoro descendants of the identity, O ∈ H vac . This recasts Z vac as a sum ofsphere four-point functions, weighted by powers of p and p . The operators O i and O j haveholomorphic conformal weights h i and h j , respectively. A detailed description of the sewingconstruction is presented in section 4 (see equations 4.12 and 4.13). In this paper we will directly compute Z vac at genus two, for arbitrary values of c >
1, usingCFT techniques. We will use a sewing construction, represented schematically in figure 1. Westart with a Riemann surface Σ that has been constructed by Schottky uniformization andreplace the handles of Σ with a sum over states propagating along these handles. The resultis a weighted sum over four-point functions of local operators on the sphere. If we wereto include all possible operators in this sum, we would obtain the full, modular-invariantpartition function, as a function of the pinching parameters p and p , that describe thewidths of the handles, and a third modulus x , which is the cross-ratio of the four-pointfunctions on the sphere. The universal contribution Z vac is computed by summing only overoperators in the Virasoro vacuum block. The four-point functions of these operators aredetermined completely by conformal Ward identities. Thus Z vac is in principle completelydetermined in terms of the central charge. We will compute the answer perturbativelyin p and p but exactly in x . We will mostly assume that the CFT has no extendedsymmetry algebra beyond two copies of Virasoro, although as we will see in section 5.1.1 itis straightforward to extend our results to higher-spin theories.Conformal bootstrap methods play an important role in our computation of Z vac , sinceour computation requires us to sum over all four-point functions of Virasoro descendantson the sphere. These correlation functions—and indeed the correlation function of anyfamily of chiral operators—can be efficiently computed using a holomorphic version of the5onformal bootstrap. The essential idea is that these correlation functions can be regardedas a meromorphic functions on M , , the moduli space of four marked points on a sphere,with poles only when the operators coincide. A meromorphic function on a compact space isdetermined entirely by its polar behaviour. For chiral operators of finite conformal dimension,this polar part is determined by a finite number of three-point function coefficients. Theresult is an exact expression for the correlation function in terms of a finite number ofthree-point function coefficients. This should be contrasted with the usual approach, wherea four-point function involves a sum over an infinite number of intermediate states, so iswritten in terms of an infinite sum of OPE coefficients. Similar ideas have been advancedin [30–32]. When the chiral operators are Virasoro descendants of the identity, we showusing free bosons that all connected n -point functions have polynomial dependence on c .This implies that, when expressed in terms of c , bulk scattering of graviton states in AdS is purely classical, in analogy to the one-loop exactness of the torus partition function.Our result for Z vac will hold for a general Riemann surfaces Σ, but for certain values ofthe moduli—those corresponding to the so-called replica surface—our results can be usedto compute genus-two EREs. We mainly consider the case of two disjoint intervals in thevacuum of a CFT; the replica manifold has genus two when n = 3, and is denoted R , . Ourresults extend previous results in [16,33], which were obtained from the twist-field four-pointfunction. Those works employ a short-interval expansion in which the conformally invariantcross-ratio, which we call y , is taken to be small. The sewing technique is well-suited tocomputation to higher orders in y ; [33] worked through O ( y ), and we extend this to O ( y ).In fact, the authors of [33] found a quite remarkable result: their O ( y ) term in log Z vac exhibits a two-loop truncation in the expansion in 1 /c at large c .To understand why that result is interesting, let us consider the bulk AdS interpretationof our results. Our computation of Z vac is not limited to large c ; it is a truly quantum resultfor the saddle-point partition function for a genus-two handlebody of three-dimensional puregravity, applicable even when G N /R AdS is of order one. Expanding our result at large c is equivalent to making the semiclassical approximation in the bulk. More precisely, theexpansion of the “vacuum free energy,” F vac := − log Z vac , (1.3)at small Newton constant (large c ) is the loop expansion of three-dimensional AdS gravity: F vac = ∞ (cid:88) (cid:96) =0 c − (cid:96) F vac; (cid:96) (1.4)where F vac; (cid:96) denotes an (cid:96) -loop contribution. In the bulk, no computations have been done6eyond one-loop order. At one loop, there is a closed-form expression for the gravitonhandlebody determinant [28, 34]. Our result for F vac;1 is a computation of this determinantin a new regime of moduli space, not described by previous computations [15, 28].What about at higher loops? At genus one, the expansion (1.2) truncates at one-looporder: higher-loop contributions only renormalize the value of c [2]. It is natural to askwhether the higher-genus partition function obeys an analogous truncation. Indeed, theresults of [33] imply that that F (cid:96)> ( R , ) vanishes through O ( y ), perhaps suggesting thatthe partition function at genus two truncates at two loops. One motivation for this paper wasto investigate whether this truncation really occurs for the full partition function Z vac . Ourconclusion is that the truncation does not occur, and that the cancellation observed in [33] isan artifact of the small- y expansion. Indeed, we will show that on the replica manifold R , there are nonzero contributions to the free energy at all orders in the 1 /c expansion. Thesefirst appear at O ( y ) in the short-interval expansion, explaining why these corrections werenot found in [33].More generally, we will show that the genus-two partition function Z vac of pure three-dimensional gravity does not truncate at any order in 1 /c . The same is true of pure higherspin gravity. Explicit contributions to the all-loop terms F vac; (cid:96) are given in section 5.1. Toour knowledge, these are the first all-loop results beyond genus one for a Riemann surfacewith three independent moduli. We show that in the regime of small p and p , the onlypoint in the moduli space at which the loop expansion (1.4) truncates is the separatingdegeneration point, at which Σ degenerates into the union of two tori.This paper is organized as follows. In section 2 we recall the relationship between R´enyientropies and higher-genus partition functions, and review the sewing construction of highergenus partition functions as a weighted sum over sphere-correlation functions. In section 3we describe techniques to compute these correlation functions, including an analytic versionof the conformal bootstrap. In section 4 we apply these techniques to compute Z vac , thecontribution to the partition function from the Virasoro descendants, at genus two. Wediscuss the large central charge limit of this result in section 5, which allows us to understandthe nature of quantum corrections to the higher-genus partition function of three-dimensionalgravity, as well as applications to R´enyi entropies, before concluding in section 6. Appendices On general grounds, such a truncation might seem to conflict with the pole structure of CFT correlationfunctions, regarded as analytic functions of c . Let us make the argument at genus two for concreteness.In the sewing construction, a genus-two partition function is written as a sum over four-point functions.The statement of truncation becomes the statement that at each order in the sewing expansion, the totalcontribution from all four-point functions truncates at order 1/c. As argued by Zamolodchikov [35], theconformal block decomposition of a given four-point function contains poles at minimal model values of c where the exchanged operators become null. Unless these poles cancel against the poles in the other four-point functions contributing at a given order in the sewing expansion, the partition function will not truncatein a 1 /c expansion. This sort of cancellation at every order in the sewing expansion seems highly unlikely.Indeed, our computations bear out this conclusion. In this section we will review some relevant background material, and explain the methodol-ogy and philosophy behind our computations. In subsection 2.1, we briefly review previouswork on R´enyi entropies in the vacuum of 2D CFTs. Using these R´enyi entropies as a guide,we explain how pure 3D quantum gravity naturally computes the universal contribution ofthe Virasoro identity block to CFT partition functions on generic Riemann surfaces. Then,in subsection 2.2, we will explain the sewing construction, which we will apply in section 4to the computation of higher-genus partition functions.
Two-dimensional CFTs provide perhaps the simplest arena in which to investigate entan-glement entropies (EEs) in field theories. In this subsection, we will briefly review somecalculations of these quantities, with particular attention to their dependence on the centralcharge c of the theory. The simplest quantity one can consider in this context is the vacuum EE of a single interval[ u, v ] on the line. The corresponding R´enyi entropy is given in terms of the partition functionon the surface R ,n , which is the plane branched n times over the interval [36]: S ( n ) ([ u, v ]) = − n − Z ( R ,n ) . (2.1)This partition function is in turn related to the two-point function on the plane of twistoperators in the orbifold theory C n / Z n (where C is the original CFT) [37]: Z ( R ,n ) = (cid:104) σ ( u )˜ σ ( v ) (cid:105) C n / Z n . (2.2)It will be convenient to work in terms of the free energy, which we define on any surface X as F ( X ) := − ln Z ( X ) . (2.3) In general, we will denote the plane branched n times over a set of N intervals by R N,n ; this surface hasgenus ( N − n − The partition function on a genus-zero surface is defined, for a given theory, up to a multiplicativeconstant independent of the metric. We are choosing that constant so that Z ( C ) = 1; otherwise the argumentof the logarithm in (2.1) would be Z ( R ,n ) /Z ( C ) n . F ( R ,n ) is proportional to c and otherwise independent of the theory: thesurface R ,n has genus zero, so the free energy is given entirely by a Liouville action multipliedby c ; alternatively, in the twist-operator language, the two-point function depends only ontheir dimension, which is proportional to c . The result is [36, 37] S ( n ) ([ u, v ]) = c (cid:18) n (cid:19) ln (cid:18) v − u(cid:15) (cid:19) , (2.4)where (cid:15) is an ultraviolet-cutoff length scale; its presence reflects the divergence in the par-tition function due to the presence of conical singularities in R ,n . This gives rise to thewell-known formula for the EE [36, 37], S ([ u, v ]) = S (1) ([ u, v ]) = c (cid:18) v − u(cid:15) (cid:19) . (2.5)The above result is easily generalized to the case of a single interval on a circle at zerotemperature or on the line at finite temperature [37]. In either case, the branched coversurface continues to have genus zero, and therefore the EREs and EE continue to be propor-tional to c . The simplest cases where higher-genus partition functions appear are a singleinterval on the circle at finite temperature and two intervals on the line at zero temperature;the corresponding branched-cover surfaces have genera n and n −
1, respectively. This im-plies that the ERE will depend on the full operator content of the CFT, not just its centralcharge [38]. In rest of this subsection we will focus on the two-interval case, which is thebest-studied one.For two intervals [ u , v ] ∪ [ u , v ], it is convenient to work with the mutual (R´enyi)information, which is ultraviolet-finite, hence conformally invariant and dependent only onthe cross-ratio y of the four endpoints [38]: I ( n ) ( y ) := S ( n ) ([ u , v ]) + S ( n ) ([ u , v ]) − S ( n ) ([ u , v ] ∪ [ u , v ]) , y := ( v − u )( v − u )( u − u )( v − v ) . (2.6)We have I ( n ) ( y ) = 11 − n F ( R ,n ) + subtractions . (2.7)The subtractions, given by the EREs of the individual intervals, soak up the divergences in F ( R ,n ), leaving an unambiguous finite value for I ( n ) ( y ). The partition function on R ,n canbe expressed as a four-point function of twist operators in the orbifold theory: Z ( R ,n ) = (cid:104) σ ( u )˜ σ ( v ) σ ( u )˜ σ ( v ) (cid:105) C n / Z n . (2.8) In the literature, this cross-ratio is often denoted x ; however, we will use x for a different cross-ratio inwhat follows. u v u v A B
Friday, January 9, 15
Figure 2: The n -sheeted replica surface R ,n , which is the branched covering surface of theplane with two intervals and has genus n −
1. On each sheet, there is a cycle separating thetwo intervals called the A -cycle, and another cycle encircling the two points v and u , calledthe B -cycle. There are n − R ,n has genus n −
1, so the partition function depends on the full operatorcontent of the theory and not just its central charge. However, it contains a universalcontribution that only depends on c . To define this part, it is useful to first set up somenotation regarding the topology of the surface R ,n .A useful basis of cycles on R ,n can be described as follows. On each sheet, there is acycle that separates the two intervals. We will call these A-cycles. The sum of all n of themis trivial, so there are n − v , u , crossing each cut once, which we call B-cycles; again, there are n − A i , B j can be constructed with intersection numbers A i · B j = δ ij , butthis will not be necessary for our purposes.) It is also useful to visualize the surface R ,n as two spheres connected by n tubes. This can be related to the branched cover by cuttingeach sheet along a small ellipse surrounding the interval [ u , v ] and another one surroundingthe interval [ u , v ]. Each interval then becomes a sphere with n holes, while each sheetbecomes a tube connecting one sphere to the other. Each A-cycle wraps a tube, while eachB-cycle runs along one tube and back along another. (See figure 3.) R ,n enjoys a Z n “replicasymmetry”, which cyclically permutes the sheets, and hence also the tubes.The universal part of Z ( R ,n ) to which we alluded above is defined as the contributionin which only Virasoro descendants of the vacuum appear on the A-cycles. In other words,if for any circle C we define P vac ( C ) as the projection operator onto the conformal family ofthe vacuum of the Hilbert space of C on C , then we define the vacuum partition function as10 n B Figure 3: An alternate depiction of the surface R ,n in figure 2. R ,n can be visualized as twospheres connected by n tubes. The two spheres, one for each interval, are made by cuttingsmall holes around each pair of intervals on all n sheets. The tubes connecting the holes onthe two spheres represent the sheets. In this picture, the A -cycles wrap the n tubes and the B -cycles run through two different tubes.the path integral with projectors P vac ( A ) · · · P vac ( A n − ) inserted: Z vac ( R ,n ) := (cid:104) P vac ( A ) · · · P vac ( A n − ) (cid:105) Z ( R ,n ) . (2.9)With only vacuum descendants on A , . . . , A n − , the cycle A n (which is a linear combinationof the others) is automatically guaranteed to host only such descendants as well. Note thatthe choice of representative of any given cycle A i is unimportant; any representative can bemapped to any other by a holomorphic diffeomorphism, which acts on the Hilbert space bythe Virasoro group, under which conformal families don’t mix. In the orbifold description,the vacuum partition function can be written Z vac ( R ,n ) = (cid:104) σ ( u )˜ σ ( v ) P orb vac ( A ) σ ( u )˜ σ ( v ) (cid:105) C n / Z n , (2.10)where P orb vac is the projector onto states of C n / Z n composed of descendants of the identity of C , and A is a circle enclosing [ u , v ]. Note that, unlike the full partition function, Z vac ( R ,n )is not a modular invariant quantity, due to the distinguished role of the A-cycles.As we will see in the next subsection, the vacuum partition function is particularly well- To see this, cut along all n A-cycles, leaving two sphere n -point functions (on the left and right spheresof figure 3). For each n -point function, n − n th one is not a vacuum descendant, then the n -point function vanishes, and hence does notcontribute to the vacuum partition function. This set of states includes more than just Virasoro descendants of the vacuum of C n / Z n . Rather, itincludes all descendants of the vacuum under the larger algebra consisting of ( Z n -symmetric) products ofVirasoro generators acting on the different copies of C . c CFTs. c CFTs
We are interested in families of CFTs, such as holographic ones, that admit a large- c limit.In such theories, all of these quantities—the free energies, entanglement (R´enyi) entropies,and mutual (R´enyi) informations—admit an expansion in 1 /c starting at order c . We thuswrite, for example, I ( n ) ( y ) = ∞ (cid:88) (cid:96) =0 c − (cid:96) I ( n ) (cid:96) ( y ) , F ( R ,n ) = ∞ (cid:88) (cid:96) =0 c − (cid:96) F (cid:96) ( R ,n ) . (2.11)In a holographic CFT, the parameter 1 /c is proportional to the bulk Newton constant,1 c = 2 G N R AdS , (2.12)so the expansion in 1 /c is a loop expansion in the bulk (hence the index (cid:96) ).From a CFT perspective, the loop corrections ( (cid:96) ≥
1) are “cleaner” than the classicalone ( (cid:96) = 0), in the following sense. First, F (cid:96) ≥ is unambiguous, since the scheme dependenceof the free energy is due to the Weyl anomaly, which is proportional to the central charge.Second, it is finite even on a singular surface such as R ,n , since the Weyl transformationthat smoothes out those conical singularities shifts the free energy by c times a Liouvilleaction. Third, since it is Weyl-invariant, it depends only on the complex structure of R ,n ,hence only on the cross-ratio y , not the positions of the endpoints themselves. Finally, sincethe subtractions present in (2.7) are proportional to c , we simply have I ( n ) (cid:96) ( y ) = 11 − n F (cid:96) ( R ,n ) for (cid:96) ≥ . (2.13)These properties will all be useful when we study the loop corrections below.The RT formula makes a strikingly simple prediction for the classical part of the mutualinformation [11]: I ( y ) = , y ≤ / /
3) ln( y/ (1 − y )) y ≥ / . (2.14)It is interesting that this formula does not depend on the field content or other specifics ofthe dual theory. On the other hand, the loop corrections do depend on the field content,although they always include certain “universal” terms due to the gravitational sector, aswe will explain below.Significant effort has gone into testing the prediction (2.14) and computing the loop cor-12ections using the replica trick. Two strategies have been followed to compute the relevantfree energies. The first is to find the dominant gravitational saddle point whose conformalboundary is R ,n ; the terms in the 1 /c expansion of F ( R ,n ) are then given by the classi-cal action, the one-loop determinant of the fields about that background, and so on. Thesecond strategy is to compute the four-point function of twist fields (2.8) using CFT tech-niques such as the conformal-block decomposition. The RT prediction for the classical partwas successfully confirmed, modulo some assumptions, by both methods, in [14] and [13]respectively.Consider the calculation of F ( R ,n ), starting with the gravity method. In [14], two grav-itational saddles were constructed with conformal boundary R ,n . Both are handlebodies; inone, which we will call H A , the A-cycles are contractible, while in the other, H B , the B-cyclesare contractible. H A has a smaller action for y < / H B for y > /
2. These are theonly solutions that preserve the replica symmetry of R ,n , and are also believed to be theonly type of solution that exists uniformly for all n . Their actions are analytic functions of n ; when continued down to n = 1, they reproduce precisely the RT prediction (2.14) for theEE. In [13], an analysis of conformal blocks in the C n / Z n theory at large c —again, imposingthe replica symmetry—led to the same result.An important subtlety regarding these calculations is as follows. For general n and y , itis not clear whether the dominant gravitational saddle is always H A or H B , and thereforewhether their actions indeed give the correct free energy and R´enyi entropy. However, forsmall y , the tubes are very thin (as we will see in section 5.2 when we discuss the periodmatrix for R ,n ), so the dominant saddle must indeed be the handlebody that fills them in,namely H A . This is important for our purposes because the calculations we will describefrom here on will always be done in an expansion in y , and therefore we can safely ignorethis subtlety and assume that H A is the dominant saddle.We now turn to the one-loop correction to the free energy, F ( R ,n ), which as notedin (2.13) directly gives the one-loop correction to the mutual R´enyi information, I ( n )1 ( y ). F ( R ,n ) is proportional to the sum of the logs of the fluctuation determinants of all thefields propagating on the relevant gravitational saddle. In any theory of gravity, this includesthe metric fluctuations. Their one-loop determinant on the handlebody H A was computedin an expansion in y to order y for all n in [15], and to order y for n = 1 in [39].As we will now explain, this contribution to the free energy is simply the O ( c ) part of thevacuum free energy Z vac ( R ,n ) defined in the previous subsection. In fact, more generally,consider the partition function obtained from the classical action and loop corrections toall orders of perturbative pure gravity on H A . We will now argue that this quantity is Even if this is not the case, one can argue that these are the relevant saddles to consider for the purposesof analytically continuing the ERE down to n = 1 to find the EE. Z vac ( R ,n ). In the genus-one case, this was shown in [2], and we can adopt theirargument here. In a Hilbert-space interpretation, we can choose to think of the A-cycles asdefining a spatial direction and the B-cycles a (Euclidean) time direction. This is convenientbecause the states defined on the A-cycles are perturbative pure quantum gravity stateson an AdS background, since the A-cycles are contractible and the handlebody is locallyEuclidean AdS . Since the creation operators for metric fluctuations are, from the CFTviewpoint, Virasoro generators, these states are thus Virasoro descendants of the vacuum.Thus the perturbative pure gravity partition function on H A is precisely Z vac ( R ,n ). Theexact correspondence between the perturbative quantum gravity partition function and theuniversal identity block contributions to CFT partition functions was articulated and testedin [28]. We will extend that work in section 5.3.We now return to computation of R´enyi entropies specifically. Having established theCFT interpretation of Z vac ( R ,n ), we can see that reproducing the results of [15,39] using thetwist-field method requires including only descendants of the vacuum as intermediate statesin the conformal-block decomposition of the 4-point function (2.8), since the intermediatestates are precisely those living on the A-cycles. More precisely, one should include states ofthe orbifold theory C n / Z n that are made up of descendants of the vacuum of C ; these includemore than just the descendants of the vacuum of C n / Z n . It is easy to see that the term oforder y h in I ( n ) ( y ) is given by descendants at level h .These calculations were carried out to order y by Chen et al. in [33]. Expanding theirresult in powers of 1 /c , the one-loop (order c ) term matched the bulk metric one-loopdeterminant computed earlier in [15]. Their c − (cid:96) term started at order y (cid:96) +2 , so their resultscould access (cid:96) ≤
3. In other words, they not only reproduced the one-loop determinant,but effectively computed two-loop and three-loop free energies, which would presumably bequite challenging from a direct bulk perturbative calculation. The coefficient at each orderin 1 /c and y is a rational function of n . We will not reproduce these rather complicatedfunctions here for general n . However, let us note the following pattern in the n -dependenceobserved by [33]: F vac ,(cid:96) ( R ,n ) = ( n − n − · · · ( n − (cid:96) ) (cid:32) (cid:88) m =2 (cid:96) +2 α m,(cid:96) ( n ) y m (cid:33) + O ( y ) . (2.15)The α m,(cid:96) ( n ) are functions of n ; some of them have zeroes at positive n , but none of thesezeroes coincide, unlike those shown in (2.15).There are some notable features of this formula. First, F vac; (cid:96) ≥ ( R ,n ) carries a factor of n −
1. The fact that it vanishes at n = 1 can be understood from the fact that the genus-zero free energy is given entirely by a Liouville action multiplied by c ; it is also necessary,14iven (2.13), for the mutual R´enyi information to have a smooth limit as n →
1. Second, F vac; (cid:96) ≥ ( R ,n ) carries an overall factor of n −
2. The fact that it vanishes at n = 2 can beunderstood from the fact that the contribution of the identity family to the genus-one freeenergy is one-loop exact: aside from a classical (order- c ) part, it is given by − ln χ vac ( y ),where χ vac ( y ) is the character of the identity family, which is independent of c .Perhaps surprisingly, the y term of F vac; 3 ( R ,n ) computed by Chen et al. carries anoverall factor of n −
3. (Recall that [33] only computed through O ( y ) in the y -expansion.)If the pattern (2.15) were to hold to all orders in y , this would imply a truncation in the loopexpansion around handlebodies asymptotic to R ,n with appropriate cycles contractible. Onthis basis, Chen et al. were led to suggest that the genus- g free energy might be g -loop exactfor all g , at least for the replica manifolds R N,n . One might even wonder whether this couldbe true for all genus- g manifolds. One of the main purposes of this paper is to test thisintriguing idea. To do this, we will calculate Z vac on generic genus-two Riemann surfaces, Σ,using a different technique that we describe now; this complementary approach will providea gateway to applications to R´enyi entropy and 3D quantum gravity. In section 4, we will compute Z vac via the sewing construction. We heuristically explainthis method here with the help of figure 4; the method applies to computation of the fullpartition function Z of C , but can be specialized to computation of Z vac . The basic ideais to replace each handle of Σ by a sum over local operator insertions at its ends. Thisframes the computation of Z as a weighted sum of sphere four-point functions. As stressedearlier in this section, computing Z vac as opposed to the full partition function of C meansthat we only allow Virasoro vacuum descendants to propagate along the handles. Thisconstruction is perturbative in the width of the handles. There are many parameterizationsof a given surface Σ; we use the Schottky construction, which forms Σ as a quotient ofthe Riemann sphere by a discrete subgroup of P SL (2 , C ), the M¨obius group. The genus-twoSchottky space is parameterized by coordinates { p , p , x } . Roughly speaking, these describethe width of the two handles and the sphere coordinate of the lone endpoint not fixed byconformal symmetry, respectively. The computation of Z vac is then a double power series in p and p , where the powers are the left-moving conformal weights of the operators insertedat the endpoints.In order to make eventual contact with R´enyi entropies and the work of [28], we will alsoneed to express Z vac in terms of the period matrix of Σ, denoted Ω. That is, we need toperform the coordinate map { p , p , x } (cid:55)→ Ω. This is known in closed form, but is complicated15 ∞ x p p p h i p h j ∑ i,j ( ∞ ) i Internal comments - We need to remember to be clear about including, or not, the anti-holomorphic side of thepartition functions.
Notes on notation • O ( c n ) versus O ( y n ) or O ( p n ), etc. • Free energy is F = log Z , and the large c expansion is F = X ` =0 c ` F ` (0.1)(Using S ` conflicts with the notation for R´enyi entropy.) • Period matrix is ⌦. • N O = norm • Generic operators are O . There are essentially two ways to treat three-dimensional pure gravity with negative cosmo-logical constant in the context of holography. One is to understand it as a quantum theoryin its own right, and find its holographic dual for a given value of G N /R AdS , the Newtonconstant in AdS units. The other is to understand it as a part of a larger bulk theory, suchas string theory, that captures the dynamics of the Virasoro sector of any two-dimensionalCFT with central charge c = 3 R AdS / G N . The latter, semi-microscopic interpretation isunavailable in higher-dimensional AdS/CFT, where the stress tensor does not generate aclosed symmetry algebra.This perspective lends a universality to AdS /CFT that underlies, for instance, theoriginal matching of asymptotic symmetries and BTZ black hole entropy to the Virasorosymmetry and Cardy growth of states, respectively, obeyed by all two-dimensional CFTs[xx bh, strom]. Recent work has revealed an even richer set of properties obeyed by familiesof two-dimensional CFTs that admit a large-central-charge limit, especially those that areholographic. These theories have a sparse spectrum of “light” operators of conformal dimen-sion less than of order c , dual to perturbative bulk states below the black hole threshold.Above the threshold, the Cardy formula dictates the growth of states even away from theasymptotic limit ! 1 [xx hartman keller]. Their vacuum Renyi entropies are given by the2 ( x ) j Internal comments - We need to remember to be clear about including, or not, the anti-holomorphic side of thepartition functions.
Notes on notation • O ( c n ) versus O ( y n ) or O ( p n ), etc. • Free energy is F = log Z , and the large c expansion is F = X ` =0 c ` F ` (0.1)(Using S ` conflicts with the notation for R´enyi entropy.) • Period matrix is ⌦. • N O = norm • Generic operators are O . There are essentially two ways to treat three-dimensional pure gravity with negative cosmo-logical constant in the context of holography. One is to understand it as a quantum theoryin its own right, and find its holographic dual for a given value of G N /R AdS , the Newtonconstant in AdS units. The other is to understand it as a part of a larger bulk theory, suchas string theory, that captures the dynamics of the Virasoro sector of any two-dimensionalCFT with central charge c = 3 R AdS / G N . The latter, semi-microscopic interpretation isunavailable in higher-dimensional AdS/CFT, where the stress tensor does not generate aclosed symmetry algebra.This perspective lends a universality to AdS /CFT that underlies, for instance, theoriginal matching of asymptotic symmetries and BTZ black hole entropy to the Virasorosymmetry and Cardy growth of states, respectively, obeyed by all two-dimensional CFTs[xx bh, strom]. Recent work has revealed an even richer set of properties obeyed by familiesof two-dimensional CFTs that admit a large-central-charge limit, especially those that areholographic. These theories have a sparse spectrum of “light” operators of conformal dimen-sion less than of order c , dual to perturbative bulk states below the black hole threshold.Above the threshold, the Cardy formula dictates the growth of states even away from theasymptotic limit ! 1 [xx hartman keller]. Their vacuum Renyi entropies are given by the2 (0) i Internal comments - We need to remember to be clear about including, or not, the anti-holomorphic side of thepartition functions.
Notes on notation • O ( c n ) versus O ( y n ) or O ( p n ), etc. • Free energy is F = log Z , and the large c expansion is F = X ` =0 c ` F ` (0.1)(Using S ` conflicts with the notation for R´enyi entropy.) • Period matrix is ⌦. • N O = norm • Generic operators are O . There are essentially two ways to treat three-dimensional pure gravity with negative cosmo-logical constant in the context of holography. One is to understand it as a quantum theoryin its own right, and find its holographic dual for a given value of G N /R AdS , the Newtonconstant in AdS units. The other is to understand it as a part of a larger bulk theory, suchas string theory, that captures the dynamics of the Virasoro sector of any two-dimensionalCFT with central charge c = 3 R AdS / G N . The latter, semi-microscopic interpretation isunavailable in higher-dimensional AdS/CFT, where the stress tensor does not generate aclosed symmetry algebra.This perspective lends a universality to AdS /CFT that underlies, for instance, theoriginal matching of asymptotic symmetries and BTZ black hole entropy to the Virasorosymmetry and Cardy growth of states, respectively, obeyed by all two-dimensional CFTs[xx bh, strom]. Recent work has revealed an even richer set of properties obeyed by familiesof two-dimensional CFTs that admit a large-central-charge limit, especially those that areholographic. These theories have a sparse spectrum of “light” operators of conformal dimen-sion less than of order c , dual to perturbative bulk states below the black hole threshold.Above the threshold, the Cardy formula dictates the growth of states even away from theasymptotic limit ! 1 [xx hartman keller]. Their vacuum Renyi entropies are given by the2 ( ) j Internal comments - We need to remember to be clear about including, or not, the anti-holomorphic side of thepartition functions.
Notes on notation • O ( c n ) versus O ( y n ) or O ( p n ), etc. • Free energy is F = log Z , and the large c expansion is F = X ` =0 c ` F ` (0.1)(Using S ` conflicts with the notation for R´enyi entropy.) • Period matrix is ⌦. • N O = norm • Generic operators are O . There are essentially two ways to treat three-dimensional pure gravity with negative cosmo-logical constant in the context of holography. One is to understand it as a quantum theoryin its own right, and find its holographic dual for a given value of G N /R AdS , the Newtonconstant in AdS units. The other is to understand it as a part of a larger bulk theory, suchas string theory, that captures the dynamics of the Virasoro sector of any two-dimensionalCFT with central charge c = 3 R AdS / G N . The latter, semi-microscopic interpretation isunavailable in higher-dimensional AdS/CFT, where the stress tensor does not generate aclosed symmetry algebra.This perspective lends a universality to AdS /CFT that underlies, for instance, theoriginal matching of asymptotic symmetries and BTZ black hole entropy to the Virasorosymmetry and Cardy growth of states, respectively, obeyed by all two-dimensional CFTs[xx bh, strom]. Recent work has revealed an even richer set of properties obeyed by familiesof two-dimensional CFTs that admit a large-central-charge limit, especially those that areholographic. These theories have a sparse spectrum of “light” operators of conformal dimen-sion less than of order c , dual to perturbative bulk states below the black hole threshold.Above the threshold, the Cardy formula dictates the growth of states even away from theasymptotic limit ! 1 [xx hartman keller]. Their vacuum Renyi entropies are given by the2 Figure 4: A picture of the sewing approach to computing a genus-two partition function, Z . The mechanism was explained in figure 1. To compute Z rather than Z vac , one simplylets the sum run over all operators in the CFT Hilbert space.(see, e.g., [28, 40]). If we define multiplicative periods q ij := e πi Ω ij (2.16)then q ij admits a power series in p and p of the following form: q = p ∞ (cid:88) n,m =0 p n p m n + m (cid:88) r = − n − m c ( n, m, | r | ) x r ,q = x + x ∞ (cid:88) n,m =1 p n p m n + m (cid:88) r = − n − m d ( n, m, r ) x r q = q ( p ↔ p ) . (2.17)The c ( n, m, | r | ) and d ( n, m, r ) = d ( m, n, r ) are coefficients given in Appendix E of [40]through m = n = 7.Thus, in order to compute Z vac via sewing, we must compute four-point functions ofoperators in the Virasoro identity representation. We turn to this now, by way of the moregeneral subject of computing four-point functions of arbitrary holomorphic operators. In this section, which may be read independently of the rest of the paper, we discuss methodsfor computing correlators in 2D CFTs. We will focus on four-point functions. A standardway to compute a four-point function is to do an OPE expansion of pairs of operators. Thisyields a power series in the cross-ratio x of the four points. However, we wish to calculate16he correlator at finite values of x . We will describe two methods to do this. The first,described in subsection 3.2, is via direct manipulation of operator modes, and may be familiarto CFT practitioners. The second, described in subsection 3.3, is an analytic realizationof the conformal bootstrap that applies specifically to correlators of only holomorphic (oronly anti-holomorphic) operators. The upshot is that the combination of holomorphy withfundamental properties of the OPE and crossing symmetry yields a result that is determinedby a finite number of OPE coefficients. In the case that all four operators have identicalholomorphic dimensions, the solution of crossing symmetry leads to an especially simplealgorithm.Before turning to those methods, we first describe a simple approach that applies specif-ically to correlators of descendants of the identity, and use the results to discuss the powersof the central charge c that appear in such correlators. c As explained in subsection 2.2 (and illustrated in figure 1), Z vac , the universal part of thegenus-two partition function, can be constructed from four-point functions on the plane ofdescendants of the identity. In this subsection, we will discuss general properties of suchcorrelators.The first property to note is that they are independent of the rest of the field content ofthe theory, and depend only on its central charge. This follows from the fact that the identityVerma module is closed under fusion. Since in this paper we are particularly interested inthe powers of c that appear in Z vac , in this subsection we will focus on the question of whatpowers of c can appear in such correlators. We will first use a simple counting argumentin a free field theory to show that the powers of c are highly constrained. In particular, ifone thinks of 1 /c as a coupling constant, then it appears that these correlators are tree-levelexact. We will relate this classicality to the fact that the sphere partition function, for anyCFT, has a particularly simple c -dependence, and then discuss its bulk interpretation forholographic CFTs. Finally, we will discuss the generalizations of these statements at highergenus.The fact that correlators of descendants of the identity are independent of the theory,except its central charge, implies that we can compute them in any convenient theory witha variable central charge. One simple choice is the theory of c free bosons; by writing therelevant operators in terms of elementary fields, it is in principle straightforward to computetheir correlators using free-field Wick contractions. This procedure is fairly tractable forcalculating, for example, the four-point function of the stress tensor, but it rapidly becomesunwieldy when applied to higher-point functions or higher-level descendants, and for thesecalculations the methods described in the following subsections are far more efficient.17onetheless, the free-boson method gives a fast way to answer the important question ofwhat powers of c appear in a given correlator. For example, the stress tensor is T = − c (cid:88) µ =1 : ∂X µ ∂X µ : . (3.1)An m -point function of stress tensors (cid:104) T ( z ) · · · T ( z m ) (cid:105) includes indices µ , . . . , µ m . The 2 mX fields appearing can be contracted in various ways, linking the different T ’s, and thereforethe different µ ’s, to each other. For example, a contraction between X µ and X µ leads toa factor of δ µ µ . In the connected part of the correlator, they are all linked in one group,so a non-zero contribution occurs only when all the µ are equal: µ = · · · = µ m . Hence theconnected part of the correlator is linear in c , independent of m . Disconnected parts givehigher powers of c ; for example, the stress tensor four-point function has a term quadratic in c , from contractions in which the T are linked in two separate pairs (see (3.8) for the explicitform).All descendants of the identity can be written as normal-ordered products of derivativesof stress tensors. The connected part of any correlator—regardless of the number and type ofoperator—again comes from terms in which all the µ ’s are equal, and is therefore again linearin c . This also follows from the fact that the generating function for connected correlatorsof the stress tensor is the sphere free energy as a functional of the metric, which is simply c times the Liouville action. Thus, if we think of 1 /c as playing the role of (cid:126) , any CFT onthe sphere is purely classical in this sector, in other words its correlators are effectively givenentirely by tree-level contributions. To translate this into the usual field-theory language, ifwe normalize T by a factor of c − / (so that its two-point function is 1), then a connectedcorrelator with a total of P factors of T is proportional to c − P/ , just like a tree-level diagramwith P external legs in a field theory with coupling constant 1 /c .We now turn to holographic theories, where 1 /c ∼ G N is indeed the bulk couplingconstant. An operator made of p stress tensors corresponds to a state containing p gravitons.This again leads to c − P/ for a tree-level bulk process involving a total of P gravitons.From this point of view, the absence of loop corrections may seem mysterious, given thatthere certainly exist Witten diagrams in the bulk containing loops, which make non-zerocontributions to such a correlator. However, in 3D gravity, all terms in the effective actionwhich depend only on the metric (even those generated by loops of other fields) can beabsorbed in the Einstein-Hilbert term [41]. Hence, the full quantum effective action for themetric is simply the classical action, with a renormalized value of the Newton constant.Since it is the renormalized Newton constant that enters in the relation 1 /c = 2 G N / R AdS ,when working in terms of c , the theory appears to be entirely classical. This property is not directly related to the absence of propagating degrees of freedom in pure 3D gravity. /c , and these higher-order terms depend onthe full operator content of the theory. So one would not expect correlators to be purelyclassical. Similarly, from a bulk point of view, gravitons and other particles can propagatein loops that wrap non-trivial cycles of the bulk, giving corrections that cannot be capturedby a local effective action.Nonetheless, as noted below (2.15), at genus one, the vacuum free energy F vac ( T ) = − ln Z vac ( T ) does have the special property that it is one-loop exact, in other words containsonly terms linear and constant in c (in any CFT). Z vac ( T ) is defined as the path integralwith the insertion of the operator P vac ( A ) that projects onto vacuum descendants on somefundamental cycle A of the torus. Derivatives of the free energy with respect to the metricgive connected correlators of the form (cid:104) P vac ( A ) O O · · ·(cid:105) con , where the O i are descendantsof the identity. Such correlators are therefore also one-loop exact (contain only terms linearand constant in c ). We will confirm this property by explicit calculation in subsection 4.2below. To begin, we will compute vacuum correlators of the form (cid:104)O ( ∞ ) T (1) T ( z ) O (0) (cid:105) (3.2)where (cid:104)·(cid:105) := (cid:104) |·| (cid:105) . The operator O is allowed to be an arbitrary, non-holomorphic operator,not necessarily primary or quasi-primary. As is conventional, we leave its anti-holomorphicdependence implicit in what follows. We define mode expansions T ( z ) = (cid:88) n ∈ Z L n z n +2 , O ( z ) = (cid:88) n ∈ Z O n z n + h (3.3)where the stress tensor modes obey the Virasoro algebra,[ L m , L n ] = ( m − n ) L m + n + c n ( n − δ m + n, . (3.4) To demonstrate this, one can consider the correlator of four spin- s currents with s >
2. As we will show byexample later in section 3.3.2 (see (3.41)–(3.42)), these correlators do not truncate in a 1 /c expansion. Thisimplies a non-trivial loop expansion for bulk four-point scattering of spin- s gauge fields in pure 3D higherspin gravity, even though these theories also lack propagating modes.
19n terms of modes, the four-point function is (cid:104)O ( ∞ ) T (1) T ( z ) O (0) (cid:105) = z − (cid:88) n ∈ Z z − n (cid:104)O h L − n L n O − h (cid:105) . (3.5)To proceed, we break up the sum into the n = 0 mode term, and two sums over positiveand negative integers (denoted Z + and Z − , respectively). Using the fact that L O − h | (cid:105) = h O − h | (cid:105) , the n = 0 mode contributes a term z − h N O , where N O = (cid:104)O h O − h (cid:105) is the normof O . Using the Virasoro algebra, and relabeling n → − n , we can rewrite the sum over Z − in terms of a sum over Z + as (cid:88) n ∈ Z − z − n (cid:104)O h L − n L n O − h (cid:105) = (cid:88) n ∈ Z + z n (cid:16) (cid:104)O h L − n L n O − h (cid:105) + (cid:16) nh + c n ( n − (cid:17) N O (cid:17) (3.6)= (cid:88) n ∈ Z + z n (cid:104)O h L − n L n O − h (cid:105) + (cid:18) c z (1 − z ) + 2 h z (1 − z ) (cid:19) N O . The quantity in angle brackets is simply the squared norm || L n O − h | (cid:105)|| .There is a further simplification of this sum: it truncates on account of vacuum invariance.Suppose O is a level N descendant of a primary field O (cid:48) of holomorphic dimension H = h − N .Then O − h can be written as a linear combination of “lexicographically ordered” operators, L − n · · · L − n k O (cid:48)− H (3.7)where n ≥ n ≥ . . . ≥ n k and N = (cid:80) ki =1 n i . This implies that the sum in (3.6) truncatesat n = N , because L n O − h | (cid:105) = 0 for n > N by definition of a primary.Taking this into account and adding (3.6) to the other pieces, the full correlator is (cid:104)O ( ∞ ) T (1) T ( z ) O (0) (cid:105) = z − (cid:32) N (cid:88) n =1 ( z n + z − n ) (cid:104)O h L − n L n O − h (cid:105) + (cid:18) c z (1 − z ) + 2 h z (1 − z ) + h (cid:19) N O (cid:33) . (3.8)A pleasing feature of the expression in parenthesis is its manifest invariance under z → /z ,which is simply invariance under crossing symmetry corresponding to exchange of the twostress tensors.If O is quasi-primary, then the n = 1 term of the sum vanishes. If O is primary, theentire sum vanishes. That leaves us with a very simple expression: O primary : (cid:104)O ( ∞ ) T (1) T ( z ) O (0) (cid:105) = z − (cid:18) c z (1 − z ) + 2 h z (1 − z ) + h (cid:19) N O . (3.9)20 useful check of (3.8) is to take O = T , which yields the stress tensor four-point function.Using the Virasoro algebra to compute (cid:104) L L − L L − (cid:105) = c /
4, we find (cid:104) T ( ∞ ) T (1) T ( z ) T (0) (cid:105) = z − (cid:18) c (cid:18) z + z (1 − z ) (cid:19) + 2 c (1 − z + z )(1 − z ) (cid:19) . (3.10)This agrees with previous results (e.g. [42]). We may define the conformal cross-ratio as x := z z z z = z , (3.11)in which case (3.10) has the correct form of a four-point function as determined by conformalsymmetry: (cid:104) T ( ∞ ) T (1) T ( z ) T (0) (cid:105) = z − F ( x ) . (3.12)We will reproduce this result in the next subsection in a more efficient way. One can easily generalize this analysis to correlators where T is replaced by a differentoperator. For simplicity, we consider the four-point function of two pairs of holomorphicquasi-primaries O a and O b , of dimensions h a and h b , respectively. (Their modes are definedas in (3.3).) In this case, the resulting expression is (cid:104)O a ( ∞ ) O b (1) O b ( z ) O a (0) (cid:105) = (3.13) z − h j h a (cid:88) n =1 ( z n + z − n ) (cid:104)O ah a O b − n O bn O a − h a (cid:105) + (cid:104)O ah a O b O b O a − h a (cid:105) + (cid:88) n ∈ Z + z n (cid:104)O ah a [ O bn , O b − n ] O a − h a (cid:105) To understand why the first sum truncates at n = h a , we need to examine the OPE betweenquasi-primaries: in terms of modes,[ O am , O bn ] = (cid:88) c C abc P ( m, n ; h a , h b ; h c ) O cm + n + G ab δ m + n, (cid:32) m + h a − h a + h b − (cid:33) . (3.14) G ab is the Zamolodchikov metric, O c are also quasi-primary, C abc are OPE coefficients, andthe P ( m, n ; h a , h b ; h c ) are known functions encoding the contribution of the full globalconformal family of O c . All modes O bn with n > − h b annihilate the vacuum. This enablesus to write O bn O a − h a | (cid:105) = [ O bn , O a − h a ] | (cid:105) for n > − h b ; the OPE (3.14), combined with theunitarity bound h ≥
0, ensures that modes with n > h a give vanishing contribution. This Following our discussion in subsection 3.1, note that this could be derived for all c from a 3D gravitycomputation at large c , by thinking of T as a single-graviton state: the O ( c ) part is the free-field Wickcontraction, and the O ( c ) part is the connected bulk correlator of four gravitons expressed in terms of therenormalized Newton constant. See e.g. equation (3.4) of [31]. All P ( m, n ; h a , h b ; h c ) are finite in unitary CFTs for operators of finitedimension. O b is made ofcurrent modes alone, its modes may be given by infinite sums over products of the L n , whichare difficult to manipulate. More generally, the four-point function appears to depend onthe full holomorphic operator content of the theory, due to the presence of the commuta-tor [ O bn , O b − n ]. In fact, this latter point belies the true structure of the result. We nowdemonstrate this explicitly as we turn to a much more powerful method of computation forcorrelators of holomorphic operators. We will now describe a general method to compute the correlation functions of chiral oper-ators using crossing symmetry. We will see that any correlation function of chiral operatorswhich obey a closed operator product algebra may be determined uniquely by a finite num-ber of three-point function coefficients. This is in contrast to the typical situation, where theOPE allows us to determine correlation functions only in terms of an infinite sum over inter-mediate states. In many cases, such as for the correlation functions of Virasoro descendantsof the identity, this leads to an extremely efficient computational algorithm.Let us recapitulate our conventions for chiral operators. We make no further reference tothe mode notation of the previous subsection. We will consider a family of chiral operators O a ( z ), with integer dimensions h a , and ¯ h a = 0. We will take the basis O a to be quasi-primaries and assume that the O a satisfy a closed OPE O a ( z ) O b ( z ) ∼ (cid:88) c C abc O c ( z ) z h a + h b − h c + (descendants) (3.15)The two point functions (cid:104)O a ( z ) O b ( z ) (cid:105) = G ab z h a + h b (3.16)and three-point functions (cid:104)O a ( z ) O b ( z ) O c ( z ) (cid:105) = C abc z h a + h b − h c z h a + h c − h b z h b + h c − h a (3.17)are fixed, up to constants, by conformal invariance.22 .3.1 Four-point functions: General case Conformal invariance constrains the four-point function to take the form (cid:104)O a ( z ) O b ( z ) O c ( z ) O d ( z ) (cid:105) = (cid:32) z h a + h b z h c + h d (cid:18) z z (cid:19) h ab (cid:18) z z (cid:19) h cd (cid:33) F abcd ( x ) . (3.18)where we define the cross ratio x as in (3.11), x = z z z z , − x = z z z z . (3.19)We will use the notation H = (cid:80) a h a , h ab = h a − h b , z ab = z a − z b , etc.Our starting point is the observation that the four-point function (3.18) depends analyt-ically on the z i and has poles only when the points z i coincide. Thus F abcd is a meromorphicfunction of x with poles only at x = 0 , , ∞ . So F abcd is a rational function of x , whichis uniquely completely determined (up to a constant piece) by its polar behaviour at thesepoints. As we will see, this polar behaviour is fixed by only a finite number of three-pointfunction coefficients.We begin by considering the expansion of F abcd near x →
0. This can be found byinserting the O a O b and O c O d OPE into the four-point function (3.18). The result is a sumover intermediate operators O e . The contributions from the descendant states of a givenquasi-primary are given by a rigid (i.e. SL (2 , R )) conformal block. The rigid conformalblocks were written in terms of hypergeometric functions in [35]. The result is F abcd ( x ) = (cid:88) e ( C abe C cde ) x h e F ( h e − h ab , h e + h cd ; 2 h e ; x ) (3.20)From this we see that F abcd is finite as x →
0. The constant term as x → F abcd ( x ) = G ab G cd + . . . (3.21)where . . . denotes terms that vanish as x → x → x → ∞ . To do this wewill use the transformation properties of the four-point function under crossing symmetry.The crossing symmetry conditions can be derived by considering how the correlation function(3.18) transforms when the z i are permuted. In particular, let us consider a permutation π ∈ S of four elements. We have (cid:104)O a ( z ) O b ( z ) O c ( z ) O d ( z ) (cid:105) = (cid:104) O π ( a ) ( z π (1) ) O π ( b ) ( z π (2) ) O π ( c ) ( z π (3) ) O π ( d ) ( z π (4) ) (cid:105) (3.22)23his relates F abcd ( x ) to F π ( abcd ) ( π ( x )), where the permutation π acts on the cross-ratio as π ( x ) ≡ z π (13) z π (24) z π (12) z π (34) (3.23)One just needs to determine how the permutation π acts on the prefactor in parenthesis inequation (3.18). Some permutations have π ( x ) = x ; these give identities for the F abcd ( x ) withfixed x . One can verify that these identities follow immediately from the conformal blockexpansion (3.20), using properties of the hypergeometric function identities and symmetriesof the three-point function coefficients. Other permutations act on x , and give non-trivialinformation about four-point functions. In particular, the permutations π = (14) and π =(24) give the crossing symmetry equations F abcd ( x ) = ( − H x h a + h d F dbca (1 /x )= ( − H x h c + h d (1 − x ) − h c − h b F adcb (1 − x ) (3.24)These crossing equations strongly constrain the allowed form of the three-point functioncoefficients. Since F abcd ( x ) is finite as x →
0, we see that F abcd has a pole of order h a + h d at x → ∞ and a pole of order h b + h c at x → x → ∞ we insert the conformal block expansion (3.20) into the first crossing symmetryequation to get F abcd ( x ) = ( − H (cid:88) e C dbe C cae x h a + h d − h e F ( h e − h db , h e + h ca ; 2 h e ; 1 /x ) (3.25) ∼ h a + h d (cid:88) n =1 α n x n + . . . as x → ∞ . (3.26)Here . . . denotes terms that are finite at x → ∞ . The important point is that, because thehypergeometric function is finite as x → ∞ , the only terms that contribute to the pole arethose with h e < h a + h d . In particular, the power series expansion of the hypergeometricfunction at x → ∞ gives an explicit formula for the α n in terms of the three-point functioncoefficients C dbe C eca with h e < h a + h d . We find α n = ( − H h a + h d − n (cid:88) h e =0 C dbe C cae ( h e − h db ) h a + h d − h e − n ( h e + h ca ) h a + h d − h e − n ( h a + h d − h e − n )!(2 h e ) h a + h d − h e − n . (3.27)24imilarly, near x → F abcd ( x ) = ( − H (cid:88) e C ade C cbe (1 − x ) h e − h c − h b F ( h e − h ad , h e + h cb ; 2 h e ; 1 − x ) x h c + h d ∼ h b + h c (cid:88) n =1 β n (1 − x ) − n + . . . as x → . (3.28)where . . . denotes terms that are finite as x →
1. Again, the hypergeometric function hasa simple power series expansion at x →
1, giving an explicit formulas for the coefficients β n in terms of the three-point function coefficients with h e < h b + h c . The formula for the β n is a bit more complicated than that for α n , since we must expand x h c + h d in powers of1 − x as well as the hypergeometric function, so we will not write it explicitly. However, theimportant point is that there is a completely explicit (albeit complicated) expression for the β n in terms of the three-point function coefficients C ade C cbe with h e < h b + h c .The four-point function F abcd is now completely fixed. It is the unique rational functionof x which is finite everywhere except at 1 and ∞ , where its polar behaviour given by (3.26)and (3.28), and whose value at x = 0 is given by (3.21): F abcd ( x ) = G ab G cd + h a + h d (cid:88) n =1 α n x n + h b + h c (cid:88) n =1 β n (cid:2) (1 − x ) − n − (cid:3) . (3.29)We see that F abcd depends on a total of H = h a + h b + h c + h d coefficients, α n , β n , which aredetermined by combinations of a finite number of three-point function coefficients. This isa consequence of crossing symmetry applied in a holomorphic setting; for non-holomorphicoperators, there is no simple formula for a four-point function in terms of a finite number ofoperators.This has a remarkable consequence for the conformal bootstrap program, where crossingsymmetry is used to place constraints on the three-point function coefficients. This is espe-cially true for chiral CFTs. In a typical CFT, the bootstrap results in equations involving aninfinite number of three-point function coefficients, which can only be solved by truncatingor approximating the crossing symmetry equations in some way. For a chiral CFT, the con-straints are all written in terms of a finite number of equations. For example, by comparing(3.20) with the expansion of (3.29) around x = 0 we can obtain explicit formulas for all ofthe coefficients C abe C cde , for all e , in terms of the coefficients C dbe C cae with h e < h a + h d and C ade C cbe with h e < h b + h c . Of course, our results also apply to chiral operators in non-chiralCFTs.Moreover, we note that (3.29) is not the unique way of writing the four-point function. Inwriting (3.18) we chose to separate out a particular combination of z ij to define a meromor-25hic function. This choice led to a meromorphic function depending on H coefficients whichwere determined by three-point function coefficients C dbe C cae with h e < h a + h d and C ade C cbe with h e < h b + h c . Other ways of separating out a meromorphic function will lead to differentexpressions which in some cases may be more useful. For example one particular interestingway of imposing the crossing symmetry relations is to write the four-point function as F abcd ( x ) z h a + h b − H/ z h a + h c − H/ z h a + h d − H/ z h b + h c − H/ z h b + h d − H/ z h c + h d − H/ (3.30)where F abcd ( x ) = x H/ (1 − x ) h/ − h b − h c F abcd ( x ) (3.31)The function F abcd is convenient because it treats the four points democratically – whichmakes the crossing symmetry equations very simple – but does so at the price of introducinga branch cut in F abcd coming from the fractional powers of H/ F abcd has singularities oforder H/ x = 0 , , ∞ ; the crossing equations determine F abcd tobe F abcd ( x ) = (cid:98) H/ (cid:99) (cid:88) n =0 (cid:0) a n x n − H/ + b n x H/ − n + c n (1 − x ) n − H/ (cid:1) (3.32)where the a n , b n , c n are determined by the three-point functions of operators with h e ≤(cid:98) H/ (cid:99) . Note that, since n = 0 , . . . , (cid:98) H/ (cid:99) we now have 3 (cid:98) H/ (cid:99) coefficients to deter-mine. It is reasonably straightforward, though tedious, to write explicit expressions for thesecoefficients in terms of the three-point functions. The advantage of this approach is thatit will, in principle, require the computation of fewer three-point function coefficients. Forexample, if the number of operators in the chiral algebra increases rapidly with dimension(as in the case of the Virasoro algebra) then this expansion would be much more efficient. Let us now simplify to the case where the four operators O a are identical operators O ofweight h , which is of interest for our computation of the higher genus partition function.In this case the above procedure simplifies considerably. The four-point function F ( x ) = F abcd ( x ) is a meromorphic function with poles only at x = 1 , ∞ which obeys the simplifiedcrossing symmetry equation F ( x ) = x h F (1 /x ) = x h (1 − x ) h F (1 − x ) (3.33)26n fact, the space of such functions is a vector space of dimension 1 + (cid:98) h/ (cid:99) . To see this,consider the function a ( x ) = (1 − x + x ) (1 − x ) (3.34)which obeys (3.33) with h = 1. The function F ( x ) a ( x ) − h is invariant under the anharmonicgroup generated by x → − x and x → /x . Moreover, this function is analytic everywhereon the Riemann sphere with the exception of a pole of order 2 h at x = e πi/ , along with amirror image pole at x = e − πi/ . These points are order-three fixed points of the anharmonicgroup, so when expanded around x = ± e πi/ , only cubic powers may appear. The function k ( x ) = x (1 − x ) (1 − x + x ) (3.35)is the unique meromorphic function invariant under the anharmonic group that has a poleof order 3 at x = ± e πi/ . We can therefore expand F ( x ) a ( x ) − h in integer powers of k , toobtain F ( x ) = (cid:98) h/ (cid:99) (cid:88) n =0 c n x n (1 − x + x ) h − n (1 − x ) h − n (3.36)To implement this way of computing four-point functions, we note that to determine thecoefficients c n , we now simply expand this function in powers of x and use the OPE todetermine these coefficients as products of three-point functions.It is instructive to phrase our conclusions in the language of modular functions. Equating x with the modular lambda function x = λ ( τ ) ≈ q / − q + O ( q / ) , (3.37)where q = e πiτ , gives a map from M , (the moduli space of four marked points on thesphere) to M , (the moduli space of a torus). Accordingly, SL (2 , Z ) transformations of τ induce anharmonic group transformations of x : specifically, x → − x and x → /x areinduced by the S and T ST transformations, respectively. The problem of finding a functioninvariant under the anharmonic group therefore maps to finding a modular function, withdesired polar structure in q determined by the poles in x via (3.37). Given the identification(3.37), our function k ( x ) in (3.35) is just (256 times) the inverse of the J function: k ( x ) =256 /J ( τ ). So the construction of the four-point function F ( x ) is literally identical to thatof torus partition functions of holomorphic CFTs, as in [1]. Likewise, (3.36) implies aRademacher expansion for OPE coefficients of higher dimension operators.27e now treat some useful examples. For h = 1 we have F ( x ) = c (1 − x + x ) (1 − x ) (3.38)This is exactly the four-point function of a spin-1 current, j . The coefficient c = k isdetermined by the first (trivial) OPE coefficient jj
1, where k is the level of the currentalgebra.For h = 2 we get two possible functions, F ( x ) = c (1 − x + x ) + c (1 − x ) x (1 − x + x )(1 − x ) (3.39)This is the stress tensor four-point function. Matching the small x expansion with OPEcoefficients of T T
T T T , we find c = c / c = c (2 − c ), where we used thecanonical norm for the stress tensor, N T = c/ (cid:104) T T T T (cid:105) as computed in(3.10).For h = 3 we have F ( x ) = c (1 − x + x ) + c (1 − x ) x (1 − x + x ) + c (1 − x ) x (1 − x ) (3.40)This is the four-point function of a spin-3 current, call it W , which was first worked out in [43]in the context of CFTs with W symmetry. Matching the small x expansion with OPEcoefficients determines the c i . In a theory with W symmetry, the first three quasi-primaryoperators appearing in the exchange channel of (cid:104) W W W W (cid:105) are 1 , T and the level-four quasi-primary Λ, which is the normal-ordered product of T with a derivative term subtracted:Λ := ( T T ) − ∂ T . (3.41)This operator has norm N Λ = c (5 c + 22) /
10. The OPE coefficient
W W W vanishes, as itdoes for W being any odd-spin chiral quasi-primary. Computing OPE coefficients using theVirasoro algebra and matching to the small- x expansion of (3.40), we find c = N W , c = N W (cid:18) − c ) c (cid:19) , c = N W (cid:18) c − c − c (5 c + 22) (cid:19) (3.42)where (cid:104) W W − (cid:105) ≡ N W ∝ c is the norm of W . As explained in section 3.1, (cid:104)
W W W W (cid:105) We believe our result actually corrects a sign error in [43]: the parameter µ there should be a sum, nota difference, of two terms. If the chiral algebra contains a spin-4 current too, as in the case of the W ∞ [ λ ] algebra appearing inthe context of higher spin AdS/CFT [44], this current will also appear at level four with nonzero OPE /c expansion.For h = 4 we get F ( x ) = c (1 − x + x ) + c (1 − x ) x (1 − x + x ) + c (1 − x ) x (1 − x + x ) (1 − x ) (3.43)An example of an h = 4 chiral operator whose four-point function we will need in the sewingconstruction of Z vac is the quasi-primary Λ introduced in (3.41). Again computing the c i bymatching to OPE coefficients, we find c = N , c = N (cid:18) c − (cid:19) , c = N (cid:18) c + 590 c + 3704)5 c (5 c + 22) (cid:19) (3.44)In closing, one interesting comment is that certain general consequences can be immedi-ately read off from (3.36). For example, when h is not a multiple of 3 every contribution tothe four-point function F ( x ) includes a factor of 1 − x + x . Thus if h is not a multiple of3, the four-point function vanishes when x = ± e πi/ . We are now ready to compute Z vac at genus two, which captures the contribution of theVirasoro vacuum module to the partition function of a CFT on a genus-two Riemann surface.We begin this section by reviewing the Schottky uniformization of generic genus- g Riemannsurfaces and the sewing construction of the genus- g partition function. We then turn tothe actual computation of Z vac at genus two using sphere four-point functions of low-lyingVirasoro vacuum descendants. The final result can be found by substituting the results ofsubsection 4.2 into equation (4.12). We focus on the holomorphic part of Z vac henceforth. A non-singular genus- g Riemann surface can be constructed by cutting out 2 g disks on theRiemann sphere and identifying pairs of boundary circles to form g handles. The Schottkyuniformization of the Riemann surface entails the identification of pairs of circles throughM¨obius transformations γ i , i ∈ { , , · · · , g } , which are elements of P SL (2 , C ). The maps γ i form the generators of the Schottky group, Γ. There are three parameters { a i , r i , p i } associated with the i th handle: ( a i , r i ) are the locations of the centers of the boundarycircles, and p i determine the width of the handles. A global conformal transformation can coefficient, and will change the value of c . This generalization is simple to compute using the W ∞ [ λ ]algebra; one instead finds c = 3 N W ( λ (25 c − c + 546) − c − c − / (5 c (5 c + 22)( λ −
29x the positions of three boundary circles on the sphere and thus a genus- g Riemann surfacehas 3 g − g Riemann surface following theconventions of [40] (see their Appendix C), where the locations of each pair of identifiedcircles are given by the M¨obius transformation γ a i ,r i ( z ) = r i z + a i z + 1 , (4.1)where γ a i ,r i (0) = a i and γ a i ,r i ( ∞ ) = r i . The generators of the Schottky group are given interms of this map as γ i = γ a i ,r i γ p i γ − a i ,r i , (4.2)where γ p i ( z ) = p i z . We note that identified circles have opposite orientations: for the i th pair the two boundary circles are given by the maps C i = γ a i ,r i γ R i C and ¯ C − i = γ a i ,r i ˆ γγ R − i C ,where C is the unit circle at the origin, R i and R − i are the radii of C i and ¯ C − i respectively,and ˆ γ is the inverse map ˆ γ ( z ) = 1 z . (4.3)The product of the radii of the two circles is R i R − i = p i . We refer the reader to AppendixC of [40] for more details on the Schottky parametrization.The partition function of a genus- g Riemann surface uniformised by the Schottky groupis given by the following power series expansion in p i [40]: Z g = (cid:88) h , ··· , h g p h · · · p h g g C h ,..., h g ( a , . . . , a g , r , . . . , r g ) . (4.4)In the sewing construction, a handle is replaced by the boundary states inserted at thecenters of the two disks. The functions C h , ··· ,h g are 2 g -point functions on the Riemannsphere and h i is the conformal dimension of the operators inserted at the i th pair of disks.A schematic picture of the sewing construction is shown in figure 4. In the above equationwe have fixed the positions a = 0, r = ∞ , and a = 1. The 2 g -point functions C h , ··· ,h g ,whose ingredients we will explain in the next paragraphs, are sums over products of vertexoperators of the form C h , ··· ,h g = (cid:88) φ i ,ψ i ∈H hi g (cid:89) i =1 G − φ i ψ i (cid:28) g (cid:89) i =1 V out ( ψ i , r i ) V in ( φ i , a i ) (cid:29) , (4.5) Actually, the formula (4.4) just gives the partition function up to a factor of the form e − cF ; in otherwords, in an expansion of the free energy F := − ln Z g in 1 /c , it only gives the order c and higher terms.The order- c term cF depends on the full metric on the Riemann surface, not just the complex structure.Its calculation within the context of the sewing construction is explained in appendix D. G is the Zamolodchikov metric defined below in (4.20), and H h i is the Hilbert spaceof states of dimension h i .These expressions for the vertex operators should be understood as follows. Under anyM¨obius transformation γ ( z ), the vertex operator V ( φ ( z )) transforms as [46] V (cid:18) U (cid:16) γ ( z ) (cid:17) φ, γ ( z ) (cid:19) = V (cid:18) γ (cid:48) ( z ) L e γ (cid:48)(cid:48) ( z )2 γ (cid:48) ( z ) L φ, γ ( z ) (cid:19) , (4.6)where γ (cid:48) ( z ) = dγdz , γ (cid:48)(cid:48) ( z ) = d γdz . (4.7)For the M¨obius transformation (4.1) we have γ (cid:48) a i ,r i ( z ) = ( r i − a i )( z + 1) , γ (cid:48)(cid:48) a i ,r i ( z ) = − r i − a i )( z + 1) , (cid:16) γ a i ,r i ˆ γ (cid:17) (cid:48) ( z ) = ( a i − r i )( z + 1) , (cid:16) γ a i ,r i ˆ γ (cid:17) (cid:48)(cid:48) ( z ) = − a i − r i )( z + 1) . (4.8)The “in” and “out” vertex operators in the sewing construction (4.5) then transform as V in ( φ i , a i ) = V (cid:18) U (cid:16) γ a i ,r i ( z = 0) (cid:17) φ i , γ a i ,r i ( z = 0) (cid:19) = ( r i − a i ) L e − L φ i ( a i ) , (4.9)and V out ( ψ i , r i ) = V (cid:18) U (cid:16) γ a i ,r i ˆ γ ( z = 0) (cid:17) ψ i , γ a i ,r i ˆ γ ( z = 0) (cid:19) = ( − L ( r i − a i ) L e − L ψ i ( r i ) . (4.10)For i = 1, i.e. for the handle with the two ends at (0 , ∞ ), one can perform an extra M¨obiustransformation under which the two maps at zero and infinity become the identity and theinverse map, respectively. This is described in the next subsection. We note that if φ i and ψ i are quasi-primaries, then the vertex operators are given by V in ( φ qp , a i ) = ( r i − a i ) h φqp φ qp ( a i ) ,V out ( ψ qp , r i ) = ( − h ψqp ( r i − a i ) h ψqp ψ qp ( r i ) . (4.11) We now specialize to genus-two Riemann surfaces. The partition function is given by Z g =2 = ∞ (cid:88) h ,h =0 p h p h C h ,h ( x ) , (4.12)31here we have defined r = x . Using (4.5), the functions C h ,h ( x ) are found to be C h ,h ( x ) = (cid:88) φ i ,ψ i ∈H hi G − φ ψ G − φ ψ (cid:28) V out ( ψ , ∞ ) V out ( ψ , x ) V in ( φ , V in ( φ , (cid:29) . (4.13)These two formulae apply in general. For our purposes of computing Z vac , we only allowVirasoro descendants of the identity to be inserted at the boundary circles of the handlesas in figure 1. Henceforth, we refer to (4.12) with the understanding that we compute Z vac specifically.Let us now define the vertex operators needed in (4.13), starting with those at (0 , ∞ ).The functions C h ,h ( x ) are invariant under the map γ a i ,r i → γ a i ,r i γ t , where γ t ( z ) = tz , t ∈ C ∗ . For the i = 1 handle with its two ends located at a = 0 and r = ∞ , we consider aM¨obius transformation of the form γ a ,r γ /r and find γ a ,r γ r = z zr (cid:12)(cid:12)(cid:12) r →∞ = z, γ a ,r γ r ˆ γ = 1 z + r (cid:12)(cid:12)(cid:12) r →∞ = 1 z . (4.14)This therefore gives the identity map for a = 0 and the inverse map for r = ∞ . The vertexoperator at the origin is simply V in ( φ ,
0) = V ( φ ,
0) = φ (0) . (4.15)The vertex operator at infinity follows from usingˆ γ (cid:48) ( z ) = − z , ˆ γ (cid:48)(cid:48) ( z ) = 2 z , (4.16)which yields V out ( ψ , ∞ ) = V (cid:18) U (cid:16) ˆ γ ( z ) (cid:17) ψ , ∞ (cid:19) = lim z →∞ ( − L z L e − z L ψ ( z ) . (4.17)For the handle with vertices at ( a , r ) = (1 , x ), we use (4.9)–(4.10) to read off V in ( φ ,
1) = ( x − L e − L φ (1) , (4.18) V out ( ψ , x ) = ( − L ( x − L e − L ψ ( x ) . (4.19)32he Zamolodchikov metric is defined in terms of the in and out vertex operators as G φψ = (cid:28) V out ( ψ, ∞ ) V in ( φ, (cid:29) . (4.20)We choose an orthogonal basis of states at each level by diagonalizing the Gram matrix. TheZamolodchikov metric is thus diagonal and the ingoing and outgoing vertex operators arethe same up to M¨obius transformations. Consequently, we can (and will) define the norm ofthe states as N φ ≡ G φψ .To summarize, the following is the prescription for constructing the genus-two partitionfunction: insert vertex operators (4.15), (4.17)–(4.19) and the Zamolodchikov metric (4.20)into (4.13) to evaluate C h ,h ( x ), and sum over these using (4.12). C h ,h ( x ) We will momentarily compute some of the functions C h ,h ( x ) defined in (4.13). Before doingso, it is useful to elucidate some of their general properties.First, these functions are symmetric under the exchange of the positions of the twohandles: C h ,h ( x ) = C h ,h ( x ) (4.21)When h = 0 or h = 0, C ,h ( x ) = C h, ( x ) = d ( h ) , (4.22)where d ( h ) is the degeneracy of operators at level h . This follows from the definition of thevertex operators and of the C h ,h ( x ) themselves. It can also be understood intuitively: re-placing either handle with two insertions of the identity reduces the genus-two partition func-tion to the torus partition function, the holomorphic half of which is Tr p L = (cid:80) h d ( h ) p h . Thus, (4.12) implies (4.22).Now we consider the x -dependence of C h ,h ( x ). For general x , they obey C h ,h ( x ) = C h ,h (1 /x ) (4.23)which is a consequence of modularity with respect to Sp (4 , Z ). Taking various limits in x corresponds to taking OPE limits of the (dressed) four-point functions defining C h ,h ( x ). We note that our formulae (4.10) and (4.17) contain an extra factor of ( − L comparing to the formulaein Appendix C of [40]. The reason is that we choose a different convention than that of [40]. In our convention G is the Zamolodchikov metric whereas in [40] their metric ˆ G is a metric on the space of states which isrelated to G via ˆ G φψ = G ( − L φψ . Note the c -independence of this quantity. When summing over the vacuum module only, the dimensionsof each state are fixed by conformal symmetry, and hence unrenormalized. This is the CFT statement of theone-loop exactness of the pure gravity partition function on a solid torus. x →
1, which describes the fusion of two ends of the same handle. Inthis case, C h ,h (1) = d ( h ) × d ( h ) . (4.24)This again follows by definition, and is necessary for the partition function (4.12) to factorizein the separating degeneration limit.More subtle are the equivalent OPE limits x → x → ∞ , which describes the fusionof two ends of different handles. These limits are singular. In Appendix A, we show that ina 1 /c expansion, the leading powers of x → /c as follows: lim c →∞ lim x → C h ,h − h ( x ) ∼ O ( x − h ) + 1 c O ( x − h +2 ) + (cid:32) ∞ (cid:88) n =2 c n (cid:33) O ( x − h +4 ) . (4.25)We have defined h = h − h , and are assuming h > h > C ,h ( x ) is constant. Weare ignoring h - and h -dependent coefficients at each order, and displaying only the leadingsingular behavior at each order in 1 /c . The last term means that at O (1 /c ) and all ordersbeyond, the leading divergence scales as O ( x − h +4 ).We now proceed to compute C h ,h ( x ) explicitly for low values of h and h . A word onnotation: henceforth, we denote the set of operators at level h above the ground state as {O ( i ) h } , where i = 1 , , . . . , d ( h ). C h, ( x )In this case, the identity operator propagates through one of the handles, so the four-pointfunctions reduce to two-point functions C h, ( x ) = (cid:88) φ,ψ ∈H h G − φψ (cid:28) V out ( ψ, ∞ ) V in ( φ, (cid:29) ,C ,h ( x ) = (cid:88) φ,ψ ∈H h G − φψ (cid:28) V out ( ψ, x ) V in ( φ, (cid:29) . (4.26)As discussed earlier, C h, ( x ) = d ( h ); from the definition of G φφ in (4.20), this is obviouslytrue. It is less obvious that C ,h ( x ) = d ( h ) by looking at (4.26). Hence, we find it instructiveto compute two C ,h ( x ) in detail, to illustrate how to use the method of [40] outlined above.In the first example we consider a quasi-primary state and in the second example we con-sider a secondary state. The latter is particularly useful, as secondary operators transformnontrivially under the M¨obius transformations, so care must be taken in computing theircorrelation functions. 34t level 2 ( h = 2) there is only one state, the stress tensor T = L − | (cid:105) , which is aquasi-primary with norm N T = c . From (4.18)–(4.19), the vertex operators at x and 1 are V out ( T, x ) = ( x − T ( x ) ,V in ( T,
1) = ( x − T (1) . (4.27)Then we have, as expected, C , ( x ) = N − T (cid:28) V out ( T, x ) V in ( T, (cid:29) = 1 , (4.28)where we used the stress tensor two-point function (cid:104) T ( x ) T (1) (cid:105) = c x − . (4.29)At level 3, there is again one state O = ∂T = L − | (cid:105) , this time a secondary state withnorm N O = 2 c . The vertex operators at x and 1 are now, from (4.18)–(4.19), V out ( O , x ) = − ( x − ∂T ( x ) − x − T ( x ) ,V in ( O ,
1) = ( x − ∂T (1) − x − T (1) . (4.30)Using (4.29) we again have, as expected, C , ( x ) = N − O (cid:28) V out ( O , x ) V in ( O , (cid:29) = 12 c (cid:28)(cid:16) − ( x − ∂T ( x ) − x − T ( x ) (cid:17) (cid:16) ( x − ∂ T (1) − x − T (1) (cid:17)(cid:29) = 1 . (4.31)It is also useful to see the vertex operator at infinity, which carries non-trivial dressing onaccount of O being a secondary operator: V out ( O , ∞ ) = lim z →∞ V (cid:16) ( − L z L e − z L O , z (cid:17) = lim z →∞ (cid:16) − z ∂T ( z ) − z T ( z ) (cid:17) . (4.32)We next move on to the computation of C h ,h ( x ) for h (cid:54) = 0 and h (cid:54) = 0. We compute therequisite four-point functions using the methods described in section 3. The transformationproperties of the vertex operators are evaluated following the same procedure shown in theabove examples; accordingly, the presentation here is streamlined, with some details of thevertex operator transformations relegated to Appendix B. There, we also list the operatorsand their norms through level six of the vacuum module.35 .2.2 C h, ( x )First, consider the four-point function with h = 2 which corresponds to the four-pointfunction of four stress tensors, each dressed with appropriate factors of x and z (recallequation 4.13). We find that C , ( x ) = 1 (cid:0) c (cid:1) lim z →∞ (cid:28) z T ( z ) ( x − T ( x ) ( x − T (1) T (0) (cid:29) = (cid:18) x − + ( x − x (cid:19) + 8 c ( x − (1 − x + x ) x , (4.33)We note that C , ( x ) is manifestly symmetric under x → /x , as required.We next compute C , ( x ). This can be done by taking the derivative of C , ( x ) withrespect to the sphere coordinates at the insertion points of O , or by direct computation ofthe four-point function using the Virasoro mode expansion formula (3.8). We obtain C , ( x ) = 1 x (cid:16) − x + 24 x − x + 4 x + x + 4 x − x + 24 x − x + 4 x (cid:17) + 1 c x − x (cid:16) x − x + x + 4 x (cid:17) , (4.34)We used (4.27) and (4.32) to write down the necessary vertex operators. One can explicitlycheck that C , ( x ) = C , ( x ), as required.We next move to level four and evaluate C , ( x ). There are two orthogonal states at levelfour, O (1)4 = Λ = (cid:18) L − − L − (cid:19) | (cid:105) , O (2)4 = L − | (cid:105) , (4.35)where Λ is the quasi-primary first recalled in (3.41), and O (2)4 = ∂ T / N O (2)4 = 5 c . We obtain C , ( x ) = 2 x (cid:16) − x + 31 x − x + 11 x − x + 3 x (4.36) − x + 11 x − x + 31 x − x + 5 x (cid:17) + 1 c x − x (cid:16) − x + 10 x − x + 10 x − x + 4 x (cid:17) . Note that the finite expansion in 1 /c is at first surprising because N − has an infinite 1 /c expansion; there are non-trivial cancellations between the two correlators. We will explainthis in a moment.For the remaining computations, we will be briefer. (We remind the reader of Appendix36 containing further details.) At level five, we find C , ( x ) = 1 x (cid:16) − x + 124 x − x + 40 x − x (4.37)+4 x − x + 40 x − x + 124 x − x + 20 x (cid:17) + 1 c x − x (cid:16) − x + 7 x + 5 x − x + 5 x + 7 x − x + 6 x (cid:17) . At level six, we find C , ( x ) = 1 x (cid:16) − x + 220 x − x + 67 x − x + 2 x − x + 12 x (4.38) − x + 2 x − x + 67 x − x + 220 x − x + 35 x (cid:17) + 1 c x − x (cid:16) − x + 10 x − x + 46 x − x + 46 x − x +10 x − x + 8 x (cid:17) . We observe that all the functions C h, ( x ) we have computed so far contain a term pro-portional to 1 /c and a term constant in c . In fact, this is true for all h . The reason is thatthe sum over h of C h, ( x ) is related to the two-point function of the stress tensor on thetorus with the insertion of a vacuum projector: (cid:104) P vac T (1) T ( x ) (cid:105) = c x − (cid:88) h C h, ( x ) p h (4.39)(where p = e πiτ ). This two-point function can in turn be obtained by differentiating thevacuum free energy F vac ( T ) with respect to the metric. This free energy was shown atthe end of subsection 3.1 to be one-loop exact, in other words to contain only terms linearand constant in c . Hence, for all h , C h, ( x ) must contain only terms constant and inverselyproportional to c . C h, ( x )We next compute the functions C h ,h ( x ) with h ≥ h = 3. The first function is C , ( x ) which corresponds to the four-point function of four ∂T ’s. This correlation functioncan be evaluated by taking the derivatives of C , ( x ) with respect to the sphere coordinates37t the location of the four operators, or by direct computation. We find C , ( x ) = 1 x (cid:16) − x + 191 x − x + 71 x − x (4.40)+3 x − x + 71 x − x + 191 x − x + 25 x (cid:17) + 1 c ( x − x (cid:16) − x + x − x + x − x + 18 x (cid:17) , Next, for C , ( x ) we find C , ( x ) = 2 x (cid:16) − x + 399 x − x + 219 x − x + 9 x (4.41)+ x + 9 x − x + 219 x − x + 399 x − x + 45 x (cid:17) + 1 c ( x − x (cid:16) − x + 68 x + 3 x − x + 3 x + 68 x − x + 64 x (cid:17) , and for C , ( x ) we find C , ( x ) = 1 x (cid:16) − x + 2380 x − x + 1530 x (4.42) − x + 100 x − x + 4 x − x + 100 x − x + 1530 x − x + 2380 x − x + 245 x (cid:17) + 1 c ( x − x (cid:16) − x + 201 x − x + 3 x − x + 3 x − x + 201 x − x + 150 x (cid:17) . Again, all of the functions C h, ( x ) evaluated so far truncate at order 1 /c in a large- c expansion. This is because all four-point functions in C h, ( x ) can be computed by takingderivatives of stress tensors in C h, ( x ), which do not affect the c -dependence of the correlators. C , ( x )The function C , ( x ) is a linear combination of four four-point functions: C , ( x ) = (cid:88) i,j =1 N − O ,i N − O ,j (cid:28) V out ( O ,i , ∞ ) V out ( O ,j , x ) V in ( O ,j , V in ( O ,i , (cid:29) . (4.43)By the same argument below (4.37) and (4.42), the terms which contain at least one pair ofthe secondary operator O (2)4 = ∂ T / /c in a large- c expansion. The onlyterm in C , ( x ) which could potentially contribute at higher orders in 1 /c is the four-pointfunction of four quasi-primaries Λ, defined in (3.41). Let us focus on this contribution, which38e call C , | Λ ( x ).Using the definitions of the vertex operators simply yields C , | Λ ( x ) = N − ( x − lim z →∞ z (cid:28) Λ( z ) Λ( x ) Λ(1) Λ(0) (cid:29) . (4.44)The norm of Λ was given in (B.4). Substituting this and the results (3.43)–(3.44) for thefour-point function obtained via the holomorphic bootstrap, we find C , | Λ ( x ) = (1 − x + x ) x + (cid:18) c − (cid:19) ( x − (1 − x + x ) x + 4 (3704 + 590 c + 125 c )5 c (22 + 5 c ) ( x − (1 − x + x ) x . (4.45)Crucially to what follows, we observe that C , | Λ ( x ) contributes to an infinite expansion in1 /c . This comes entirely from the inverse norms in (4.44).The collection of C h ,h ( x ) computed in this subsection, plugged into the partition func-tion (4.12), forms one of our main computational results: namely, the first several terms inthe Virasoro vacuum module contribution to the partition function of an arbitrary CFT ona genus-two Riemann surface, in the regime of Schottky parameters p , p (cid:28) Having derived the first handful of terms in equation (4.12) for all c , it is trivial to expand it atlarge c . As we have discussed, this large c expansion may be interpreted as the semiclassicalexpansion of the pure 3D quantum gravity partition function around genus-two handlebodygeometries with conformal boundary Σ specified by Schottky parameters { p , p , x } . Insubsection 5.1, we present some of our main results. First, we provide explicit contributions ofthe Virasoro vacuum representation to the CFT free energy at all orders in 1 /c , correspondingto all-loop free energies in the gravitational loop expansion. We also show that at least inthe perturbative regime p , p (cid:28)
1, the loop expansion does not truncate except when Σ isthe union of two tori.We then proceed to subsections 5.2 and 5.3, where we expand our general result (4.12)near two symmetric points in the genus-two moduli space: the replica surface R , used tocompute the R´enyi entropy S for two disjoint intervals in vacuum, and the point corre-sponding to the separating degeneration limit x = 1. Our results will extend those of [33]and [28], respectively. 39 .1 All-loop results in 3D quantum gravity We consider the 1 /c expansion of the vacuum free energy, F vac = − log Z vac : F vac = ∞ (cid:88) (cid:96) =0 c − (cid:96) F vac; (cid:96) (5.1)where (cid:96) denotes the loop order. Its moduli-dependence is kept implicit. We can read off theloop corrections F vac; (cid:96) from (4.12) upon expanding the C h ,h ( x ) in 1 /c . We likewise expandthese as C h ,h ( x ) = ∞ (cid:88) (cid:96) =1 c − (cid:96) C h ,h ; (cid:96) ( x ) . (5.2)To begin, note that in the small ( p , p ) expansion in which we work, both the one- andtwo-loop free energies are nonzero. This follows from the explicit results in section 4 andfrom the R´enyi entropy computation (2.15), but also from our general exposition of c -scalingof identity module correlators in section 3.1.More interesting is the question of whether there are higher-loop terms. For (cid:96) >
2, the C h ,h ( x ) that we have computed all obey C h ,h ; (cid:96) ( x ) = 0 except for C , | Λ ( x ), computed in(4.45), which clearly has an infinite expansion. Accordingly, the leading contribution to thethree-loop free energy F vac; 3 in a small ( p , p ) expansion can, and does, appear at O ( p p ): F vac; 3 = p p (cid:18) C ,
4; 3 ( x ) −
12 ( C ,
2; 2 ( x )) (cid:19) + O ( p p ) . (5.3)There is no cancellation: instead, our results yield F vac; 3 = p p x − (1 − x + x ) x + O ( p p ) . (5.4)For x ∈ R , this only vanishes at x = 1. This point in moduli space corresponds to the strictseparating degeneration limit. As the torus free energy is known to truncate at O ( c ) in alarge c expansion [2], the fact that F vac; 3 = 0 when x = 1 is required by consistency. Theinteresting result proven here is that F vac; 3 is nonzero everywhere else on the real line. In fact, the infinite 1 /c expansion of C , ( x ) and the finite 1 /c expansion of C , ( x )together imply that the (cid:96) -loop free energy F vac; (cid:96) is nonzero for all (cid:96) , at least in a small( p , p ) expansion. The reason is simply that the term at O ( p p ) cannot be cancelled byhigher order terms in p and p . For (cid:96) > O (1 /c ) and beyond— F vac; (cid:96) is given Note that (5.3) and (5.4) also hold when p = p , because the only contribution at O (1 /c ) to C h ,h ( x )for h + h = 8 comes from C , ( x ). On the other hand, if x is a function of p and p , higher order terms in(5.4) are not necessarily suppressed. We will encounter such a situation in our discussion of R´enyi entropy.
40o leading order in p , p as F vac; (cid:96)> = p p C , (cid:96)> ( x ) + O ( p p ) (5.5)with C , (cid:96)> ( x ) = ( x − (1 − x + x ) x · (cid:18) c + 125 c )5 c (5 c + 22) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c − (cid:96) . (5.6)determined by (4.45). From a technical standpoint, this non-trivial 1 /c expansion arisesfrom the inverse norms appearing in the sewing construction, cancelling the norms in thecorrelator (cid:104) ΛΛΛΛ (cid:105) .Interpreted as a CFT result, (5.5) is an exact expression for the contribution of theVirasoro vacuum module to the genus-two free energy of any family of CFTs that admitsa 1 /c expansion. Interpreted as a pure gravity result, (5.5) is an explicit formula for all-loop free energies on genus-two handlebodies. The loop counting parameter in the bulk is G N = 3 R AdS / c . In contrast to the one-loop exactness at genus one, the genus-two freeenergy is not exact at any loop order.Strictly speaking, we have so far established that the semiclassical expansion does nottruncate for any real x (cid:54) = 1. What about complex x ? In particular, the all-loop terms (5.4)and (5.6) clearly vanish at x = e ± iπ/ , the complex roots of 1 − x + x = 0. This followsfrom the same property of the four-point function of Λ, as discussed below equation (3.44).But this is a special feature of correlators of identical operators with h/ / ∈ Z , so it willnot persist to higher orders in the sewing expansion. For instance, C , (cid:96)> ( e ± iπ/ ) (cid:54) = 0, andlikewise at all higher levels. Therefore, we have shown the following statement: perturbativelyin ( p , p ) , the loop expansion does not truncate for any genus-two handlebodies except at theseparating degeneration point. We note that at fixed order in p and p , the 1 /c expansion for Z vac converges. Thisdoes not necessarily imply, however, that at a fixed point in moduli space (i.e. for fixedvalues of p and p ) the 1 /c expansion converges. Indeed, since there are presumably othersaddle point contributions to the path integral (coming from bulk handlebodies with differenttopology) one might expect that the series expansion of Z vac is asymptotic in 1 /c . However,in the genus one case the series expansion for Z vac converges—in fact, it truncates at order c . Thus it is an interesting open question whether the 1 /c expansion for Z vac converges athigher genus.Finally, let us comment on the holographic interpretation of our result for the one-looppartition function, Z vac; 1 . This can be viewed as a computation of the holomorphic half of41he graviton handlebody determinant, Z grav1 = (cid:89) γ ∈P ∞ (cid:89) n =2 | − q nγ | , (5.7)The novelty of our computation is that we work in the regime of p , p (cid:28) x . This regime has not been probed directly in existing computations of (5.7). In [15], (5.7)was computed for handlebodies asymptotic to replica manifold for two-interval R´enyi entropyin a short interval expansion; this has only a single modulus and requires p = p (cid:28) x (cid:29)
1. In [28], (5.7) was computed near the separating degeneration limit where Σ becomesthe union of two tori, which requires x ≈ p , p ). So far, we have restricted to the pure Virasoro sector of the CFT. The meaning and calcula-tion of Z vac are conceptually unmodified in the presence of higher spin currents. Along withthe stress tensor, these Virasoro primaries live in the vacuum representation of an extendedconformal symmetry, typically a W algebra. In the computation of Z vac by sewing, we nowallow these currents and their normal ordered products to propagate through the handles.The resulting Z vac is again of the form (4.12), only with different coefficients C h ,h ( x ).The holographic dual of Z vac in the presence of higher spin symmetry is the perturbativepartition function of pure 3D higher spin gravity. A bulk Chern-Simons theory with connec-tions valued in two copies of a Lie algebra G describes the vacuum sector of a CFT whose W algebra is the Drinfeld-Sokolov reduction of G [47]. Accordingly, the 1 /c expansion of Z vac for such a CFT yields the semiclassical loop expansion of the G × G
Chern-Simons higherspin theory.As a simple example, consider a CFT with W symmetry, which contains a single higherspin current of spin three, W . Its presence will modify most of the C h ,h ( x ) coefficients,starting with C , ( x ) and C , ( x ). We can easily compute these using the correlators ofsection 3. The interesting term is C , ( x ). Denoting the contribution to C , ( x ) from the W The first product in (5.7) runs over primitive elements γ ∈ P ⊂ Γ, defined as those elements that cannotbe written as γ = β m for β ∈ Γ and m >
1. The eigenvalues of γ are eig( γ ) = q ± / γ , and we do not count γ and γ − as distinct elements. The generating function of quasi-primaries containing at least one W current is given in Appendix Bof [48]. δ W C , ( x ), we find, using (3.40) and (3.42), δ W C , ( x ) := N − W ( x − lim z →∞ z (cid:104) W ( z ) W ( x ) W (1) W (0) (cid:105) = (1 − x + x ) x + 6(3 − c ) c (1 − x ) (1 − x + x ) x + 3(5 c − c − c (5 c + 22) (1 − x ) x . (5.8)Expanded at large c , this yields an infinite series of loop corrections to the free energy of G = SL (3) higher spin gravity: F SL(3)vac; (cid:96)> = p p (1 − x ) x · (cid:18) c − c − c (5 c + 22) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c − (cid:96) + O ( p p ) . (5.9)This is nonzero for all x (cid:54) = 1, so we conclude that the loop expansion does not truncateaway from the separating degeneration limit for small ( p , p ). This is true for all higher spinalgebras G . We will return to the topic of higher spin theories in the Discussion. As discussed in section 2, there are three R´enyi entropies that involve genus-two replicamanifolds without punctures (i.e. for CFTs not in excited states). These are the N = 2 , n = 3and N = 3 , n = 2 R´enyi entropies for a CFT on the plane, and N = 1 , n = 2 for a CFTon the torus. We mostly focus on the N = 2 case, with replica manifold R , . Our resultsin section 5.1 are sufficient to rule out the truncation of the 1 /c expansion of F vac even inthe case of the replica manifold Σ = R , introduced in section 2.1. We now exhibit this indetail; the final results can be found in (5.17) and (5.18). Our goal is to express the free energy in terms of the coordinate y parameterizing the intervalspacing, defined in section 2. To do so, we need only to express the Schottky coordinates { p , p , x } in terms of y . One way to proceed is by using the period matrix Ω( y ) for the replicamanifold R , , which is known [38]. Thus, we will perform the map { p , p , x } (cid:55)→ { q ij ( y ) } ,where q ij ( y ) = exp[2 πi Ω ij ( y )] are the multiplicative periods, in the regime of y correspondingto small ( p , p ). Plugging into (4.12) gives F vac ( R , ) for arbitrary c ; we then proceed tostudy this result at large c . 43or two disjoint intervals and arbitrary n , the period matrix is [38]Ω ij ( y ) = 2 in n − (cid:88) k =1 sin (cid:18) π kn (cid:19) cos (cid:18) π kn ( i − j ) (cid:19) F (cid:0) kn , − kn ; 1; 1 − y (cid:1) F (cid:0) kn , − kn ; 1; y (cid:1) , (5.10)Specializing to n = 3, the period matrix is given byΩ( y ) = 2 i √ F (cid:0) , ; 1; 1 − y (cid:1) F (cid:0) , ; 1; y (cid:1) (cid:32) − − (cid:33) . (5.11)This is a highly symmetric genus-two Riemann surface: there is only a single modulus y , asopposed to the 3 g − y , we need to invert the power seriesexpansion given in (2.17). The fact that q = q implies that in Schottky coordinates, R , has p = p ≡ p , as a quick inspection of (2.17) reveals. Our results are applicable when p (cid:28)
1, so (2.17) forces us to take q (cid:28) y (cid:28)
1, often studied in the context of 2D CFT R´enyi entropy: taking y (cid:28) q (cid:12)(cid:12) y (cid:28) = y
729 + 10 y y O ( y ) ,q (cid:12)(cid:12) y (cid:28) = 27 y − − y − y − y O ( y ) . (5.12)Finally, we obtain the series expansion of p and x in terms of y by inverting (2.17) using(5.12) and the explicit results for the coefficients c ( n, m, | r | ) and d ( n, m, r ) given in [40]. Theresult is p ( y ) = y
729 + 2819683 y + 2619683 y + 57684782969 y + 4742943046721 y + 1058284410460353203 y + O ( y ) ,x ( y ) = 27 y − − y − y − y − y + O ( y ) . (5.13)Note that x ( y ) diverges linearly for small y . We can now compute the vacuum free energy F vac ( y ) = − log Z vac ( y ), and hence the We note that p ( y ) is nothing but the square of the larger eigenvalue of the Schottky generators themselves:eig( L i ( y )) = p ( y ) ± , where L ( y ) = L ( y ) are the Schottky generators in the y (cid:28) k = 1 , n = 3. One can then find x ( y ) using the Schottky relations. Such an algorithm is an alternative to that presented in the text. y (cid:28) Z vac ( y ) = ∞ (cid:88) h ,h =0 p ( y ) h + h C h ,h ( y ) , (5.14)We will further expand this result at large c and compare to those of [33].In order to perform this expansion, we need to be a bit careful: because powers of x introduce inverse powers of y , it is not manifest in (5.14) that the short interval expansioncan be meaningfully organized in powers of p ( y ). We need to know something about how C h ,h ( x ) scales with large x , and hence small y . Fortunately, we can read this off from (4.25).Keeping terms to leading order in y → /c , and ignoring coefficients, (4.25)and (5.13) imply that for h > h > c →∞ lim y → p ( y ) h C h ,h − h ( y ) ∼ O ( y h ) + 1 c O ( y h +2 ) + (cid:32) ∞ (cid:88) n =2 c n (cid:33) O ( y h +4 ) . (5.15)Therefore, we can indeed ignore higher order terms in the sum over h when we expand insmall y .Without further ado, the results are as follows. At (cid:96) = 1 , F vac; 1 ( y ) = y y y y O ( y ) F vac; 2 ( y ) = 8 y y y y O ( y ) . (5.16)Comparing to the results of [15, 33], we find agreement through O ( y ). [33] only computedthrough O ( y ), so our term at O ( y ) is new.At three-loop order, (5.4) implies a nonzero result. Evaluating (5.4) for p = p = p ( y )and x = x ( y ), we find F vac; 3 ( y ) = y (cid:18) · · (cid:19) + O ( y ) . (5.17)As discussed around (2.15), the authors of [33] computed F vac; 3 through O ( y ) only, andfound zero. We now see that the first contribution appears at O ( y ). It is remarkablethat a computation through O ( y ) using twist fields would not have revealed the nonzeroresult! This speaks to the different strengths of the twist field method and the sewingexpansion that we have performed. We can compare (5.16) directly to I in [33]. The mutual information I n , cf. (2.7), has an overall factorof 1/2 for n = 3; including the anti-holomorphic part, as they do in [33], contributes an overall factor of 2,so the two factors cancel. O ( y ) follow from (5.5) and (5.6): F vac; (cid:96)> ( y ) = y · (cid:18) c + 125 c )5 c (5 c + 22) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c − (cid:96) + O ( y ) . (5.18) Consider the three-interval R´enyi entropy on the plane with n = 2. In this case, the replicamanifold R , is a genus-two manifold characterized by three moduli that parameterize thepositions of the intervals modulo conformal symmetry. Our results for the free energy for p , p (cid:28) x can therefore be regarded as (universal contributions to) R´enyientropies for the case of three disjoint intervals and n = 2.The period matrix of R , is known in terms of Lauricella functions [50]. To apply ourresults, one would first need to understand the relative spacings of intervals that correspondsto p , p (cid:28)
1, by using the map from Schottky space to the period matrix. We do not pursuethis geometric picture here. It is clear, however, that not all intervals need to be short,because x is allowed to be general. Thus, we have implicitly provided the first computationsof universal contributions to 2D CFT R´enyi entropies that do not require all intervals to beshort.One can also consider the case of one interval on the torus with n = 2. The replicamanifold has two moduli, namely, the temperature and interval length. Our methods canagain be applied to this case to derive universal contributions to the R´enyi entropy fromthe stress tensor sector. This has been done perturbatively in a high or low temperatureexpansion in [15, 51, 52] using different methods that cannot access terms at two-loop andbeyond in a large- c expansion, unlike the sewing method here. We note that p and p asa function of the moduli have been computed perturbatively in [15, 51, 52]. We leave theremaining explicit calculation for future work. An important predecessor of the present work is [28], where the relation between Z vac and3D gravity was first enunciated precisely. Yin tested this relation at genus two, focusing onthe separating degeneration limit of the Riemann surface, where Σ becomes the union of twotori. Before we probe this region of moduli space with our new results, let us briefly reviewthe work of [28]. Besides the case of n = 2 R´enyi entropy for two intervals (for which the replica manifold is a torus withcomplex structure τ given by a known function of the interval length [49]), this is the only replica manifoldthat spans its entire genus g = ( N − n −
1) moduli space. = ρ , Ω = σ , Ω = ν . (5.19)We also define the multiplicative periods q = e πiρ , s = e πiσ , v = 2 πiν . (5.20)The separating degeneration limit corresponds to the limit v → q, s ) fixed, where q and s parameterize the complex structure of the two tori.In the 1 /c expansion, [28] computed parts of F vac; 0 , F vac; 1 and F vac; 2 at genus two usinga variety of methods, all of which agree: • Demanding a match to the polar parts of extremal CFT partition functions at low val-ues of k = c/
24, which are fixed by invariance under the genus-two modular group Sp (4 , Z ).That this match should hold follows from the definition of extremal CFTs, theories thathave no non-trivial Virasoro primaries of dimension less than k + 1 above the vacuum. • For F vac; 1 , direct computation of (5.7). • Direct computation of Z vac written as a sum over bilinears of torus one-point functionsof Virasoro vacuum descendants. This is similar to what we do in the present work.Although it is not our focus here, F is given by a certain Liouville action whose originswe explain in Appendix D. In order to write the expressions for F vac; 1 and F vac; 2 , we mustdefine the holomorphic Eisenstein series, normalized asˆ E ρn = ∞ (cid:88) m =1 m n − q m − q m . (5.21)For n = 2 ,
4, these hatted versions relate to the usual Eisenstein series asˆ E ρ = 1 − E ( q )24 ≈ q + 3 q + 4 q + O ( q )ˆ E ρ = E ( q ) − ≈ q + 9 q + 28 q + O ( q ) . (5.22)The results of [28], which we denote F Yinvac , are as follows: in the separating degeneration47imit v → F Yinvac; 1 = − ∞ (cid:88) n =2 log[(1 − q n )(1 − s n )] + v (cid:18) q − q ˆ E σ + 2 s − s ˆ E ρ − E σ ˆ E ρ (cid:19) + v (cid:32) − (cid:18) q − q ˆ E σ + 2 s − s ˆ E ρ − E σ ˆ E ρ (cid:19) (5.23)+ qs (cid:16) − q + s ) + 45( q + s ) + 72 qs + 745( q s + qs ) + 3720 q s (cid:17) + O ( q , s ) (cid:33) + O ( v )and F Yinvac; 2 = 2 v (cid:18) q − q − ˆ E ρ (cid:19) (cid:18) s − s − ˆ E σ (cid:19) + 24 v (cid:18) q s (cid:18) q + s ) − qs (cid:19) + O ( q , s ) (cid:19) + O ( v ) . (5.24)Note that these are non-perturbative in q and s through O ( v ), and the leading term in F Yin1 is just the sum of one-loop free energies on two tori with periods q and s . (Note thatin order to recover the O ( v ) piece of F Yinvac; 1 , one relies on (4.24).) Expanding everythingthrough O ( q s ), we find F Yinvac; 1 = ( q + q + s + s ) − v (cid:16) qs ( q + s + 3 qs )(2 + 3 q + 3 s + 8 qs ) (cid:17) + v (cid:32) qs (cid:16) − q + s ) + 45( q + s ) + 72 qs + 745( q s + qs ) + 3624 q s (cid:17)(cid:33) + O ( v , q , s ) (5.25)and F Yinvac; 2 = 24 v (cid:18) q s (cid:18)
13 + 12 q + 12 s + 34 qs (cid:19)(cid:19) + 24 v (cid:18) q s (cid:18) q + s ) − qs (cid:19)(cid:19) + O ( v , q , s ) . (5.26)We are now in a position to extend these results using our computations. As in the The semiclassical expansion of the free energy in [28] was performed in powers of 1 /k = 24 /c . We expandin 1 /c , and define F Yin according to the 1 /c , rather than the 1 /k , expansion. Thus, our F Yinvac; 2 equals 24times the S found in [28]. { p , p , x } (cid:55)→ { q, s, v } , express F vac inthese variables, and expand at large c .In [28], the following relations were established: p = q (cid:18) − v (2 ˆ E σ ) − v (cid:18)
2( ˆ E σ ) + 23 ˆ E ρ ˆ E σ −
16 ˆ E σ + 103 ˆ E ρ ˆ E σ (cid:19) + O ( v ) (cid:19) p = p ( σ ↔ ρ ) . (5.27)All we need now is to derive x ( q, s, v ). We do so by inverting one of the Schottky relationsin equation (2.17), e v = x + x ∞ (cid:88) n,m =1 p n p m n + m (cid:88) r = − n − m d ( n, m, r ) x r . (5.28)Note that for s = q = 0, we have x = e v . So we can write this as x = e v + O ( q, s, qv, sv ).The final result for x is rather appealing, x = e v − E σ ˆ E ρ ( v + v ) + O ( v qs ) . (5.29)We derive this in Appendix C.Plugging equations (5.27) and (5.29) into (4.12) enables us to extend the results of [28] intwo ways. First, we can now give the O ( v ) part of the one- and two-loop free energies (5.25)and (5.26), respectively, through O ( q s , q s ), not only through O ( q s ). Second, and moreimportantly, we can write down some of the leading terms as v → all loops.We find F vac; 1 = F Yinvac; 1 + v (cid:18) qs (cid:0) q (210 + 2764 s + 11865 s ) + s (210 + 2764 q + 11865 q ) (cid:1) + O ( q s ) (cid:19) + O ( v ) (5.30)and F vac; 2 = F Yinvac; 2 − v (cid:16) q s (cid:0) q + s ) + 283 qs ( q + s ) (cid:1) + O ( q s ) (cid:17) + O ( v ) . (5.31)To read off the free energy at three loops and beyond, we plug (5.29) into (5.4)–(5.6). Atthree-loop order, we find F vac; 3 = q s (cid:18) v + 8652875 v + O ( v ) (cid:19) + O ( q s v , q s v ) . (5.32)49o derive terms at higher orders in v for fixed q and s , we would need to expand x beyond O ( v ). Likewise, to derive terms at higher orders in q and s for fixed v , we would need toinclude more terms in the sewing expansion, like p p C , ( x ). Finally, the all-loop expansionis completed by the terms F vac; (cid:96)> = q s (cid:18) v + 136 v + O ( v ) (cid:19) · (cid:18) c + 125 c )5 c (22 + 5 c ) (cid:19) (cid:12)(cid:12)(cid:12) c − (cid:96) + O ( q s v , q s v ) . (5.33) We close with a discussion of some open questions, progressing from obvious directions forfuture work to the more speculative. Some directions for future work were mentioned in thetext. • In the realm of R´enyi entropy, performing the calculation suggested in section 5.2.2for three intervals would give a satisfying derivation away from a short-interval expansion. Inaddition, one can straightforwardly apply our results to the case of the n = 2 R´enyi entropyfor a single interval on the torus, at least in a high or low temperature expansion. Oneneed only perform the map between Schottky coordinates and the temperature and intervallength; this map has already been partially performed in [15, 51, 52]. No results have yetbeen derived for the torus case beyond one loop. • One can consider including local operator insertions on Σ. The sewing procedureremains a sum over sphere correlation functions, now with these extra operator insertions.The operators generate non-vacuum states in the CFT. Taking Σ to be a replica manifold,one can thus compute excited-state R´enyi entropies by the sewing procedure. Such entropieshave been computed in CFT using twist-field and holographic methods (e.g. [18, 19, 53–55]).As we have tried to demonstrate, the sewing construction is likely to provide a complementaryapproach that operates at finite c , so this seems like an especially worthwhile pursuit. Itwould be easy, for instance, to read off the O ( c ) terms from the above procedure: thesewould be predictions for bulk one-loop corrections to the Einstein action evaluated on the“punctured handlebody”. • It would be nice to prove that nowhere in the moduli space, except at the separat-ing degeneration point, does the genus-two partition function truncate in a 1 /c expansion(whereas our method could only access the regime of small p , p ). This seems highly likelyto be the case. Understanding the structure of the Schottky sum rules in Appendix C could This result should be contrasted with footnote 6 of [28]. • In our analytic bootstrap of section 3, we could equally have used Virasoro conformalblocks rather than global conformal blocks. In this case, the crossed blocks are related tothe original blocks by the fusion and braiding matrices. These are known in closed form [56],so our conclusions can also be phrased in terms of OPE coefficients of Virasoro primariesrather than quasi-primaries. The Virasoro approach is in principle more efficient, as it willfix the four-point function in terms of even fewer pieces of data, and it would be worthwhileto make this precise. An interesting demonstration of this fact comes by way of the W correlator (cid:104) W W W W (cid:105) , as computed in (3.41)–(3.42): up to the norm of W , the Virasoroapproach would fix (cid:104) W W W W (cid:105) without having to compute even a single OPE coefficient. • We briefly considered CFTs with higher-spin symmetry; it would be straightforwardto extend our computation of Z vac to higher orders for such theories. A more excitingprospect would be to compute the partition function on Σ in the presence of insertionsthat carry higher-spin charge. There is natural motivation for this from holography. Inparticular, while much work has been done to construct solutions of higher-spin gravity withnonzero higher-spin charge and solid-torus topology [57], there has been no work on buildingsolutions of higher-spin gravity of higher genus and with nonzero higher-spin fields turnedon. A subset of such “higher-spin handlebodies” would be saddle points of the Euclideanhigher-spin gravitational path integral with replica boundary conditions and nonzero higher-spin charge [48]; accordingly, their action would be expected to match CFT computations ofR´enyi entropy in states with higher-spin charge and/or chemical potentials. This calculationwould be analogous to the one peformed in [14] in the spin-2 case. Constructing such R´enyientropies via partition functions on replica manifolds endowed with higher-spin charge, ratherthan via twist fields [58] or Wilson lines [21], would be an interesting application of the replicatrick to the higher-spin setting. One might also try to make contact with the “spin-3 entropy”of [59]. • Our results can be used to test the idea that Liouville theory provides an effectivedescription of irrational CFTs with large central charge (see e.g. [25] for a recent refinementof this idea and references to earlier work). In particular, the 1 /c expansion of the genus-twopartition function can be checked against a diagrammatic calculation in Liouville theory. • Upon first glance, the relation between Z vac; 1 = ∞ (cid:88) h ,h =0 p h p h lim c →∞ C h ,h ( x ) (6.1)and the bulk graviton determinant (5.7) seems opaque. Nevertheless, these two quantities51re equal. Both formulae are written in terms of Schottky data, so it should be possible tofind a clean mapping between them. This would be a useful stepping stone to writing downa closed formula for the two-loop contribution to the bulk partition function, in analogy tothe determinant (5.7). In the sewing prescription, the two-loop result simply requires us tosum over the O (1 /c ) parts of C h ,h ( x ) instead of just the O ( c ) parts. Is there an equallysimple prescription in the bulk, and if so do the primitive elements of the Schottky groupplay a privileged role as they do at one loop? • Part of the motivation for the present work was the one-loop exactness of the pure-gravity partition function on the solid torus. The current understanding of this result relieson an elegant and simple argument about Virasoro representation theory, which can beunderstood holographically. It can be derived without recourse to CFT by computing theenergies of bulk excitations, or equivalently, by quantizing the phase space given by twocopies of diff S /SL (2 , R ) [2]. Still, it would be very satisfying to derive this result from amore direct perspective in the bulk. For example, while the solid torus partition function ofa pure higher-spin theory is also believed to be one-loop exact, we do not know the analogof diff S in that context; there should be a more direct argument one can make in the bulk.Understanding this exactness from the perspective of the bulk diagrammatic expansion couldprovide insights useful for higher genus.On the other hand, a perhaps cleverer approach would be to derive the partition functionfrom the SL (2 , R ) × SL (2 , R ) Chern-Simons formulation of 3D gravity. Einstein-Hilbertgravity and Chern-Simons theory are non-perturbatively inequivalent, but it is believed thatthe semiclassical expansion around a well-defined saddle point can be performed in eitherformulation. The Chern-Simons approach builds in the topological nature of 3D gravity,whereas the loop expansion of 3D gravity in the metric formulation is no simpler than it is inhigher dimensions, despite the absence of propagating bulk degrees of freedom. Presumably,such a computation would be manifestly one-loop exact, in analogy to similar truncations incompact Chern-Simons theory [60]. (More precisely, all higher-loop effects could be absorbedin a renormalization of the Newton constant.) A Chern-Simons approach would also havethe benefit of immediately generalizing to pure higher-spin gravity. The challenge to carryingthis out is that both the gauge group and the topology are non-compact. There has beenprogress in recent years in computing Chern-Simons partition functions for non-compactgauge groups (see e.g. [61]), but the requisite technology does not yet exist for the solidtorus. This technology would represent a significant advance in our understanding of 3Dgravity. 52 cknowledgments We thank Matthias Gaberdiel, Jared Kaplan, Christoph Keller, Albion Lawrence, DavidPoland, David Simmons-Duffin, Herman Verlinde, Roberto Volpato and Xi Yin for helpfuldiscussions. MH, AM, and EP wish to thank the Aspen Center for Physics for hospitalityduring this work, which was supported in part by National Science Foundation Grant No.PHYS-1066293. MH and IGZ were supported in part by the National Science Foundationunder CAREER Grant No. PHY10-53842. EP was supported in part by funding from theEuropean Research Council under the European Union’s Seventh Framework Programme(FP7/2007-2013), ERC Grant agreement STG 279943, “Strongly Coupled Systems”, andin part by the Department of Energy under Grant No. DE-FG02-91ER40671. IGZ wassupported in part by the Department of Energy under Grant No. DE-SC0009987. AM issupported by the National Science and Engineering Research Council of Canada.
A An OPE limit of C h ,h ( x ) The goal in this Appendix is to show that the x → C h ,h − h ( x ) is given, as in(4.25), by lim c →∞ lim x → C h ,h − h ( x ) ∼ O ( x − h ) + 1 c O ( x − h +2 ) + (cid:32) ∞ (cid:88) n =2 c n (cid:33) O ( x − h +4 ) (A.1)where we again have written h = h − h , and we restrict to h > h >
0. We are ignoring h -and h -dependent coefficients at each order, and displaying only the leading singular behaviorat each order in 1 /c .The upshot is that (A.1) follows from considering the t -channel OPE limit of the four-point functions that define the C h ,h ( x ). The expansion (A.1) follows from the c -scalingof the OPE coefficients and norms that appear in the conformal block decomposition. Theleading O ( c ) term in (A.1) arises from identity exchange; the leading O (1 /c ) term, from T exchange; and all higher order terms in 1 /c , from exchange of all other quasi-primaries inthe Virasoro identity representation.Let us give more detail. Recall that the C h ,h ( x ) are defined in terms of sums overfour-point functions of vertex operators, C h ,h ( x ) = (cid:88) φ i ,ψ i ∈H hi G − φ ,ψ G − φ ,ψ (cid:28) V out ( ψ , ∞ ) V out ( ψ , x ) V in ( φ , V in ( φ , (cid:29) (A.2)where the Hilbert subspaces H h i are spanned by operators of holomorphic dimensions h i .53ertex operators V out ( ψ, z ) and V in ( φ, z ) are just chiral CFT operators ψ ( z ) and φ ( z ),respectively, dressed with z -dependent factors. In the z → x → C h ,h ( x ) is simplythe t -channel limit of a weighted sum over four-point functions of CFT operators, includingdescendants.A four-point function of pairwise identical quasi-primary operators of dimensions h and h can be written in a global conformal block decomposition as (cid:104) ψ ( ∞ ) φ ( x ) φ (1) ψ (0) (cid:105) = x − h (cid:88) O C ψφ O N O x h O F ( h O , h O ; 2 h O ; x ) (A.3)where C ψφ O are OPE coefficients, and h = h + h . We are expanding this correlator inthe x → O ∈ { , T, Λ , . . . } .Indeed, for the purposes of establishing (A.1), consideration of the exchange of these threeoperators alone will be sufficient: that is, we associate the scaling in (A.1) with specific termsin (A.3). Let us write out the first three terms in (A.3) coming from the Virasoro identityblock, O ∈ { , T, Λ } : (cid:104) ψ ( ∞ ) φ ( x ) φ (1) ψ (0) (cid:105) = x − h (cid:18) C ψφ + 2 c C ψφT x F (2 ,
2; 4; x ) + 10 c (5 c + 22) C ψφ Λ x F (4 ,
4; 8; x ) + O ( x ) (cid:19) . (A.4)We have substituted the explicit operator norms.When ψ and φ are in the same global conformal family, fusion onto the identity is allowed,and C ψφ (cid:54) = 0 and independent of c . This yields a term of order O ( c ) and O ( x − h ). Fora given ( h , h ), there is always such a term in the definition of C h ,h ( x ), since the latterare defined as a sum over all correlators involving operators at levels ( h , h ). This is mostobvious when h = h , because C h ,h ( x ) will include correlators of four identical operators;but even when h (cid:54) = h , the definition of C h ,h ( x ) includes correlators of arbitrary derivativesof T , all of which are in the same global conformal family. For instance, C , ( x ) includes (cid:104) T ( ∞ ) ∂ T ( x ) ∂ T (1) T (0) (cid:105) , which permits fusion onto the identity when x →
0. There aregenerically no cancellations among terms in the sum (A.2). This accounts for the first termon the right-hand side of (A.1). This leading behavior was also observed in [40].The second term in (A.1), at O (1 /c ), comes from the second term in (A.4). Because C ψφT is c -independent and N T = c/
2, this term contributes at O (1 /c ) compared to the For instance, when ψ = φ is a quasi-primary, C φφT = h N φ . C h ,h ( x ) alwaysincludes such terms. This accounts for the second term on the right-hand side of (A.1).The final terms in (A.1), at O (1 /c ) and beyond, come from exchange of the level-fourquasi-primary Λ in (A.4). Because its inverse norm has an infinite expansion in 1 /c , thiswill contribute a term O ( x − h +4 ) to all orders in a 1 /c expansion, thus accounting for theremaining terms in (A.1). B More details on the sewing construction
B.1 Operators and norms
In this section, we list the operators and their norms at the first six levels of the Virasorovacuum representation. To ensure that we have not missed any, it is useful to expand theholomorphic Virasoro vacuum character, χ vac : χ vac = Tr vac ( q L − c/ )= q − c/ ∞ (cid:89) n =2 − q n ≈ q − c/ (1 + q + q + 2 q + 2 q + 4 q . . . ) . (B.1)One can branch χ vac into global SL (2 , R ) characters, thereby counting the number of quasi-primary fields. The resulting generating function, call it χ qp , is χ qp = ( q c/ χ vac − − q ) ≈ q + q + 2 q + . . . . (B.2)In terms of the degeneracy d ( h ) of all level h operators, the degeneracy of level h quasi-primaries is d ( h ) − d ( h −
1) (for h > O = O (0) to denoteoperators. • Level 2: there is one quasi-primary operator, the stress-energy tensor, T = L − | (cid:105) ,with norm N T = c/ • Level 3: there is one secondary operator, O = ∂T = L − | (cid:105) , with norm N O = 2 c . • Level 4: there are two operators, O (1)4 = Λ = (cid:18) L − − L − (cid:19) | (cid:105) , O (2)4 = L − | (cid:105) . (B.3)55he operator O (1)4 is the commonly studied quasi-primary often denoted Λ, and O (2)4 is secondary. Their norms are N Λ = c (cid:18) c + 225 (cid:19) , N O (2)4 = 5 c . (B.4) • Level 5: there are two operators , O (1)5 = L − (cid:18) L − − L − (cid:19) | (cid:105) , O (2)5 = L − | (cid:105) , (B.5)where both of them are secondary. Their norms are N O (1)5 = 4 c (cid:18) c + 225 (cid:19) , N O (2)5 = 10 c . (B.6) • Level 6: there are four operators (all acting on | (cid:105) ), O (1)6 = − L − − L − L − + 59 L − L − , (B.7) O (2)6 = − (60 c + 78)(70 c + 29) L − − c + 67)(70 c + 29) L − L − + 93(70 c + 29) L − L − + L − L − L − , O (3)6 = L − L − (cid:0) L − L − − L − (cid:1) , O (4)6 = L − , where O (1)6 and O (2)6 are quasi-primary, and O (3)6 and O (4)6 are secondary. Their normsare N O (1)6 = 463 c (70 c + 29) , N O (2)6 = 34 c (2 c −
1) (5 c + 22) (7 c + 68)(70 c + 29) , N O (3)6 = 72 c (cid:18) c + 225 (cid:19) , N O (4)6 = 352 c . The secondary operators L − n | (cid:105) , n > ∂ ( n − T / ( n − n ( n − c/ B.2 Four-point functions of vertex operators
In this section we provide more details on the transformation properties of the vertex opera-tors which were used in computation of the four-point functions C h ,h ( x ) in subsection 4.2.The final expressions for C h ,h ( x ) in terms of x are reported in the main text and are not56epeated here.We will need the explicit expressions for the vertex operators at infinity. For h = 2 , V out ( T, ∞ ) = lim z →∞ z T ( z ) V out ( O , ∞ ) = lim z →∞ (cid:16) − z ∂T ( z ) − z T ( z ) (cid:17) . (B.8)For h = 4, we have V out (Λ , ∞ ) = lim z →∞ z Λ( z ) V out ( O (2)4 , ∞ ) = lim z →∞ (cid:16) z ∂ T ( z ) + 5 z ∂T ( z ) + 10 z T ( z ) (cid:17) . (B.9)For h = 5, we have V out ( O (1)5 , ∞ ) = lim z →∞ (cid:16) − z − z L (cid:17) O (1)5 ( z ) V out ( O (2)5 , ∞ ) = lim z →∞ (cid:16) − z ∂ T ( z ) − z ∂ T ( z ) − z ∂T ( z ) − z T ( z ) (cid:17) . (B.10)Finally, for h = 6, we have quasi-primary vertex operators V out ( O ( i )6 , ∞ ) = lim z →∞ z O ( i )6 ( z ) , i = { , } , (B.11)and secondary vertex operators V out ( O (3)6 , ∞ ) = lim z →∞ (cid:16) z + z L + z L (cid:17) O (3)6 ( z ) ,V out ( O (4)6 , ∞ ) = lim z →∞ (cid:16) z ∂ T ( z ) + 76 z ∂ T ( z ) + 212 z ∂ T ( z )+ 35 z ∂T ( z ) + 35 z T ( z ) (cid:17) . (B.12)With these in hand, we start with C h, ( x ). Using the definitions V out ( T, x ) = ( x − T ( x ) V in ( T,
1) = ( x − T (1) V in ( O ,
0) = O (0) (B.13)57hat follow from section 4.1, C h, ( x ) takes the general form C h, ( x ) = N − T ( x − d ( h ) (cid:88) i =1 N − O ( i ) h (cid:28) V out ( O ( i ) h , ∞ ) T ( x ) T (1) O ( i ) h (0) (cid:29) . (B.14)We next consider the functions C h, ( x ). For h = 3 we have C , ( x ) = N − O (cid:28) V out ( O , ∞ ) V out ( O , x ) V in ( O , V in ( O , (cid:29) , (B.15)where V out ( O , ∞ ) is given in (B.8) and V out ( O , x ) = − ( x − ∂T ( x ) − x − T ( x ) ,V in ( O ,
1) = ( x − ∂ T (1) − x − T (1) . (B.16)The expressions for C , ( x ) and C , ( x ) can then be easily obtained using (B.16) and thevertex operators given above. C Schottky parameters in the separating degenerationlimit
Here, we provide details of the map between Schottky space and the period matrix in theseparating degeneration limit considered in section 5.3: { p , p , x } (cid:55)→ { q, s, v } . (C.1)The final result, perturbative in v but non-perturbative in q and s , is p = q (cid:18) − v (2 ˆ E σ ) − v (cid:18)
2( ˆ E σ ) + 23 ˆ E ρ ˆ E σ −
16 ˆ E σ + 103 ˆ E ρ ˆ E σ (cid:19) + O ( v ) (cid:19) p = p ( σ ↔ ρ ) x = e v − E σ ˆ E ρ ( v + v ) + O ( v qs ) . (C.2)The hatted Eisenstein series were defined in (5.21). The first two relations were derivedin [28]; here, we will derive the last one.The computation is an exercise in series solutions of algebraic equations. We make aseries ansatz for x , x = ∞ (cid:88) j =0 x j ( q, s ) v j , (C.3)58lug this and the expressions for p and p in (C.2) into the perturbative Schottky relation e v = x + x ∞ (cid:88) n,m =1 p n p m n + m (cid:88) r = − n − m d ( n, m, r ) x r , (C.4)and solve order-by-order for x j ( q, s ).An immediate question that may occur to the reader is how we are able to obtain a result(C.2) that is non-perturbative in q and s , despite only having access to d ( n, m, r ) to finiteorder in ( n, m ). The answer is that we were able to infer various sum rules obeyed by the d ( n, m, r ) that we believe to hold for all ( n, m ): n + m (cid:88) r = − n − m d ( n, m, r ) = 0 (C.5a) n + m (cid:88) r = − n − m d ( n, m, r ) r = 0 (C.5b) n + m (cid:88) r = − n − m d ( n, m, r ) r = 0 (C.5c) ∞ (cid:88) n,m =1 q n s m n + m (cid:88) r = − n − m d ( n, m, r ) r = 24 ˆ E ρ ˆ E σ (C.5d) n + m (cid:88) r = − n − m d ( n, m, r ) r = 0 (C.5e)where ˆ E ρ was defined in (5.21). We have also found a set of sum rules obeyed by the c ( n, m, | r | ) that appear in the other two Schottky relations (2.17): n + m (cid:88) r = − n − m c ( n, m, | r | ) = 0 , ( n, m ) (cid:54) = (0 , n + m (cid:88) r = − n − m c ( n, m, | r | ) r = 0 , n (cid:54) = 0 (C.6) ∞ (cid:88) m =1 s m m (cid:88) r = − m c (0 , m, | r | ) r = 4 ˆ E s . We have checked all of these identities through m = n = 7 using the tables of [40]. Actually, we have proven (C.6), as well as (C.5a)-(C.5c). Proof of (C.6) follows from compar-ing a series solution for p and p using (2.17) to the known solution (C.2), and demanding We are grateful to the authors of [40] for sharing the relevant Mathematica notebooks. O ( v ). Proof of (C.5a)-(C.5c) follows from demanding that all threeperturbative relations in (2.17) yield the same result for { p , p , x } : hence, having proven(C.6), we can use these to derive sum rules obeyed by d ( n, m, r ). Proof of (C.5d) and (C.5e)is undoubtedly possible using similar methods.With these sum rules in hand, we proceed to invert (C.4). At O ( v ), we must solve1 = x ( q, s ) + x ( q, s ) ∞ (cid:88) n,m =1 q n s m n + m (cid:88) r = − n − m d ( n, m, r ) x ( q, s ) r . (C.7)But (C.5a) implies that x ( q, s ) = 1 solves (C.7). At O ( v ), we must solve1 = x ( q, s ) + x ( q, s ) ∞ (cid:88) n,m =1 q n s m n + m (cid:88) r = − n − m d ( n, m, r ) r . (C.8)This time, (C.5b) implies that the second term vanishes, leaving x ( q, s ) = 1. The analysisat O ( v ) is nearly identical, and (C.5c) implies x ( q, s ) = 1 / O ( v ), the first non-trivial sum appears in the series expansion:13! = x ( q, s ) + 13! ∞ (cid:88) n,m =1 q n s m n + m (cid:88) r = − n − m d ( n, m, r ) r . (C.9)Plugging in (C.5d) leads to x ( q, s ) = 13! − E ρ ˆ E σ . (C.10)Finally, at O ( v ), we must solve14! = x ( q, s ) + 13! ∞ (cid:88) n,m =1 q n s m (cid:88) r d ( n, m, r ) r + 14! ∞ (cid:88) n,m =1 q n s m (cid:88) r d ( n, m, r ) r . (C.11)The final sum rule (C.5e) eliminates the last term, and (C.10) leaves us with x ( q, s ) = 14! − E ρ ˆ E σ . (C.12)Putting this all together, we find the advertised result in (C.2).We believe that the above sum rules may have interesting applications in other studiesof genus-two Riemann surfaces. It would be interesting to understand, for instance, why thesum (C.5d) factorizes. A systematic exploration of all sum rules obeyed by these coefficientswould be worthwhile. It seems likely that higher order sum rules may be expressible in termsof holomorphic Eisenstein series (5.21). 60 The order- c part of the free energy and the sewingconstruction In subsections 2.2 and 4.1, we reviewed the sewing construction, which expresses the partitionfunction of an arbitrary CFT on a genus- g Riemann surface in terms of 2 g -point functionson the sphere, as illustrated in figure 4. However, the formulas in those subsections, such as(4.4), only give the order- c and higher (in 1 /c ) terms in the free energy F = − ln Z ; theymiss the order- c term. (This term depends on the full metric on the Riemann surface, notjust its complex structure; in other words it depends on the choice of representative of theWeyl class.) This is adequate for the purposes of this paper, since our main interest is inthe higher-order terms in 1 /c . However, for completeness, in this appendix we will explainhow to obtain the order- c term within the context of the sewing construction.As an illustrative example, consider an arbitrary CFT on a flat torus with modularparameter τ . The partition function is well-known to be Z ( τ ) = ( p ¯ p ) − c/ (cid:88) i p h i ¯ p ˜ h i = ( p ¯ p ) − c/ (cid:88) h, ˜ h d ( h, ˜ h ) p h ¯ p ˜ h , (D.1)where p := e πiτ and d ( h, ˜ h ) is the multiplicity of operators of weights h, ˜ h . To calculate Z ( τ ) using (4.4) (more precisely, its generalization including the antiholomorphic sector),we need to compute the coefficient C h, ˜ h . Applying the definitions (4.5) and (4.20), we findsimply C h, ˜ h = d ( h, ˜ h ). Hence (4.4) would give Z ( τ ) = (cid:80) h, ˜ h d ( h, ˜ h ) p h ¯ p ˜ h ; thus, we are missingthe factor ( p ¯ p ) − c/ .We proceed with a brief recap of the sewing construction. For convenience we willassume that the metric on our Riemann surface M is smooth. We now cut it along a circleand glue in two disks, which we call D and D , to obtain a new manifold M (which maybe connected or disconnected). We choose the metric on these disks in such a way that themetric on the new surface is still smooth. This implies that D and D can be glued togetherto make a sphere with a smooth metric, which we’ll call S . We consider coordinates z , on D , which, when extended to S , obey z = 1 /z .The path integral on M can be computed by inserting a complete set of states on thecircle where it has been cut. By the state-operator mapping, this is equivalent to insertinga complete set of operators on D , at z , = 0, with an appropriate inverse metric G ij (to We will closely follow the discussion of the sewing construction in sections 9.3 and 9.4 of [62]. However,that reference considered CFTs with vanishing total central charge (in the context of string theory), so theissue we are focusing on here did not arise there. When comparing to that reference, note also that thecorrelators there are unnormalized, whereas (as throughout this paper) ours are normalized.
61e determined below) on the space of operators: Z ( M ) = Z ( M ) (cid:88) i,j G ij (cid:104)O ( z ) i O ( z ) j (cid:105) M , (D.2)where the superscripts denote that O i is inserted at the origin of the z and O j at the originof the z coordinate system. More generally, we can start with arbitrary operators O a · · · on M : Z ( M ) (cid:104)O a · · ·(cid:105) M = Z ( M ) (cid:88) i,j G ij (cid:104)O a · · · O ( z ) i O ( z ) j (cid:105) M . (D.3)To fix the inverse metric G ij , we consider the case where M happens to include a patch D (cid:48) that is diffeomorphic to D , with an operator O k inserted at z (cid:48) = 0. Cutting M alongthe boundary of D (cid:48) and gluing in D and D yields M = M ∪ S , where the S is covered bycoordinates z (cid:48) and z = 1 /z (cid:48) . Equation (D.3) then becomes: Z ( M ) (cid:104)O a · · · O ( z (cid:48) ) k (cid:105) M = Z ( M ) (cid:88) i,j G ij (cid:104)O a · · · O ( z (cid:48) ) k O ( z ) i O ( z ) j (cid:105) M = Z ( M ) Z ( S ) (cid:88) i,j G ij (cid:104)O a · · · O ( z ) i (cid:105) M (cid:104)O ( z ) j O ( z (cid:48) ) k (cid:105) S . (D.4)For this to hold for arbitrary O k and arbitrary insertions O a · · · , it must be that G ij = Z ( S ) (cid:104)O ( z ) i O ( z ) j (cid:105) S = Z ( S ) G ij , (D.5)where G ij is the Zamolodchikov metric.Now that we have fixed G ij , the partition function (D.2) becomes Z ( M ) = Z ( M ) Z ( S ) (cid:88) i,j G ij (cid:104)O ( z ) i O ( z ) j (cid:105) M . (D.6)It is often useful to add a parameter p to the sewing construction, so that the coordinateidentification is z z = p . (Even though p can be absorbed in a coordinate transformationon M , it is useful to fix the coordinate system on M and use p to vary the modulus of M .)This can be done by replacing z by z (cid:48) in the formulas above, and defining z = pz (cid:48) . Wehave O ( z (cid:48) ) j = p h j ¯ p ˜ h j O ( z ) j , so Z ( M ) = Z ( M ) Z ( S ) (cid:88) i,j p h j ¯ p ˜ h j G ij (cid:104)O ( z ) i O ( z ) j (cid:105) M . (D.7)Cutting M along g non-contractible cycles, where g is its genus, reduces it to a sphere.62his yields the formula (4.4), except with a product of sphere partition functions Z ( S ) · · · Z ( S g )in the denominator. The free energy on any sphere is proportional to c , so these factors con-tribute such a term to F ( M ). (The coordinate transformation from local coordinates z , inthe vicinity of each operator insertion to the single coordinate z covering the plane leads tothe definition of the operators V out , V in explained in subsection 4.1.)To illustrate the application of (D.7), let us return to the example of the flat torus. Set β = Im τ , and let the horizontal cycle have circumference 2 π ; thus the total area is 4 π β . Wewill cut it along the horizontal cycle. For D and D we use unit round hemispheres. Thus S is a round unit sphere, while M is a cylinder of circumference 2 π and length 2 πβ with roundendcaps. In the next paragraph we will compute the ratio Z ( M ) /Z ( S ) using the Liouvilleaction, finding e πcβ/ = ( p ¯ p ) − c/ , precisely the prefactor appearing in the expression (D.1)for the torus partition function.In order to compute Z ( M ) /Z ( S ), we will compute the change in the partition function Z ( M ) under a small change in β , and then integrate the result up from β = 0 (notingthat M | β =0 = S ). Under a Weyl transformation, ds = e ω d ˆ s , the partition function getstransformed by the Liouville action: Z = e S L ˆ Z , S L = c π (cid:90) (cid:112) ˆ g (cid:16) ˆ g ab ∂ a ω∂ b ω + ˆ Rω (cid:17) . (D.8)We will let d ˆ s be the metric with cylinder length 2 πβ , and ds with cylinder length 2 π ( β + δβ ). Hence the Weyl transformation relating them is close to the identity, with ω of order δβ , and we can work to first order in ω . Since the cylinders have the same circumference, ω vanishes on the cylinder. ω can also be taken to vanish on, say, the bottom endcap, while onthe top endcap it transforms the hemisphere into a hemisphere attached to a thin cylinderof height 2 πδβ . On this endcap, ˆ R = 2, so (cid:90) (cid:112) ˆ g ˆ Rω = (cid:90) (cid:112) ˆ g ω = (cid:90) √ g − (cid:90) (cid:112) ˆ g = 4 π δβ . (D.9)Hence S L = πc δβ . (D.10)Integrating from β = 0, we find ln Z ( M ) = ln Z ( S ) + πc β , (D.11)as promised. 63 eferences [1] E. Witten, “Three-Dimensional Gravity Revisited,” .[2] A. Maloney and E. Witten, “Quantum Gravity Partition Functions in ThreeDimensions,” JHEP (2010) 029, .[3] A. Castro, M. R. Gaberdiel, T. Hartman, A. Maloney, and R. Volpato, “The GravityDual of the Ising Model,”
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