11-loop partition function in
AdS /C F T , , ∗ and Jie-qiang Wu † Department of Physics and State Key Laboratory of Nuclear Physics and Technology,Peking University, Beijing 100871, P.R. China Collaborative Innovation Center of Quantum Matter, 5 Yiheyuan Rd,Beijing 100871, P. R. China Center for High Energy Physics, Peking University, 5 Yiheyuan Rd,Beijing 100871, P. R. China
Abstract
The 1-loop partition function of the handlebody solutions in the AdS gravity have beenderived some years ago using the heat kernel techniques and the method of images. In thesemiclassical limit, such partition function should correspond to the order O ( c ) part in thepartition function of dual conformal field theory(CFT) on the boundary Riemann surface.The higher genus partition function could be computed by the multi-point functions inthe Riemann sphere via sewing prescription. In the large central charge limit, the CFTis effectively free in the sense that to the leading order of c the multi-point function isfurther simplified to be a summation over the products of two-point functions of single-particle states. Correspondingly in the bulk, the graviton is freely propagating withoutinteraction. Furthermore the product of the two-point functions may define the links, eachof which is in one-to-one correspondence with the conjugacy class of the Schottky groupof the Riemann surface. Moreover, the value of a link is determined by the multiplier ofthe element in the conjugacy class. This allows us to reproduce exactly the gravitational1-loop partition function. The proof can be generalized to the higher spin gravity and itsdual CFT. ∗ [email protected] † [email protected] a r X i v : . [ h e p - t h ] S e p Introduction
The AdS gravity, whose action includes a negative cosmological constant besides the Einstein-Hilbert term, provides a new angle to understand the AdS/CFT correspondence [1]. It lookstrivial as there is no local bulk degree of freedom, but it actually has global degrees of freedom.As shown in [2] , under appropriate boundary conditions the asymptotic symmetry of the AdS spacetime in the theory is generated by two copies of Virasoro algebra with the central charge c = 3 l G , (1.1)where l is the AdS radius and G is the three-dimensional gravitational coupling constant. Thissuggests that there are boundary degrees of freedom which may describe the physics in thebulk. The AdS /CFT correspondence states that the quantum gravity in AdS is dual toa two-dimensional(2D) CFT with the central charge (1.1). In this correspondence, the BTZblack hole [3, 4] is dual to the highly excited states in CFT, and its macroscopic Bekenstein-Hawking entropy could be counted by the degeneracy of the excited states and therefore hasa microscopic interpretation [5]. Moreover, the AdS gravity could be topological in nature.It was found in [6] that its action in the first order formulation could be written in forms ofthe Chern-Simons action. This raised the proposal that the AdS gravity could be equivalentto a Chern-Simons theory with a gauge group SL (2 , C ) [7, 8]. The Chern-Simons formulationof the AdS gravity has been generalized to include the higher spin fields [9, 10] and led to thecorrespondence between the higher spin AdS gravity and 2D CFT with W symmetry [11, 12].Even though a precise definition of AdS quantum gravity has not been well-established,its semiclassical limit is well-understood. In the AdS gravity, all classical solutions are locallymaximal symmetric and could be obtained as the quotients of the global AdS by the subgroupof the isometry group SL (2 , C ) [4]. Consequently the path integral of the AdS gravity could bewell defined in principle. In the semiclassical regime, the partition function should include thecontributions from all the saddle points. Among all semi-classical solutions, the handlebodysolutions have been best understood. This class of solutions could be obtained as the quotientsof the AdS spacetime by the Schottky group. At the asymptotic boundary, the configurationis a Riemann surface, which could be uniformized by the Schottky group. It was shown in [13]that the regularized semi-classical action of the handlebody solution could be described bya Liouville type action, the so-called Zograf-Takhtajan(ZT) action [14] . Furthermore, the1-loop partition function of the handlebody solutions has been conjectured to be [17]log Z | − (cid:88) γ ∞ (cid:88) m =2 log | − q mγ | , (1.2) For the study on the semiclassical action of other hyperbolic solutions see [15, 16]. γ is the primitive conjugacy class of the Schottky group and q − γ is the larger eigenvalueof γ . This relation has been obtained by the direct computation in the gravity by using theheat kernel techniques and the method of images [18].On the dual CFT side, the semiclassical gravity corresponds to the large central chargelimit. Despite of many efforts (see for example [8]), the explicit construction of the CFT isnot clear yet. It is expected to have a sparse light spectrum in the large c limit [19] [20]. Tothe leading order of c in the partition function, the light spectrum dominate the contribution,while the heavy operators only contribute nonperturbatively as e − c . The vacuum moduleplays a special role. It is universal for all CFT, includes the stress tensor and its descendants.In the AdS /CFT correspondence, the stress tensor is dual to the massless graviton in thebulk. In fact, it was shown in [21] that the genus-1 partition function is 1-loop exact and thecontribution comes purely from the vacuum module.The recent study on the R´enyi and entanglement entropy revives the AdS /CFT corre-spondence. The multi-interval R´enyi entropy of a 2D CFT is determined by the partitionfunction on a higher genus Riemann surface resulted from the replica trick. For the CFTdual to the AdS gravity, the CFT partition function should be equal to the partition func-tion of corresponding gravitational configuration ending on the Riemann surface at the AdSboundary. In the large c limit, the leading contribution of the R´enyi entropy, which is domi-nated by the vacuum module [19], is equal to the ZT action [22]. This leads to the proof of theRyu-Takayanagi formula for the holographic entanglement entropy [23,24]. More interestingly,from the study on the R´enyi entropy of double intervals with a small cross-ratio and the singleinterval on a torus [25–34], it turns out that the holographic computation is even correct at1-loop level. Now the next-leading contribution to the R´enyi entropy from the vacuum moduleis dual to the 1-loop graviton partition function (1.2) of corresponding gravitational configura-tion. More generically, one may find the following picture. For a handlebody instanton whoseboundary is a higher genus Riemann surface, the partition function of the instanton includingthe quantum corrections should be exactly the same as the partition function of the Riemannsurface in the dual CFT in the large c limit. In this paper, we try to prove that the 1-looppartition function (1.2) for any handlebody solution agrees exactly with the next-leading c part of the partition function in CFT.To prove the 1-loop partition function (1.2) for any handlebody solution, we need to com-pute the partition function of a higher genus Riemann surface in CFT. Here we use the gluingprescription to compute the higher genus partition function [17, 35–37]. Every compact Rie-mann surface could be described by the Schottky uniformization. For a genus g Riemannsurface, the Schottky uniformization allows us to identify g pairs of nonintersecting circles in3he Riemann sphere. In CFT language, the identification is equivalent to cut open a handleand insert a complete set of states there. On the Riemann sphere, this means that one hasto insert pairs of the vertex operators at the fixed points in the pairwise circles. As a result,the partition function of a genus- g Riemann surface is the summation of 2 g -point functions onthe Riemann sphere. As there is a uniformization map from the Riemann surface to the Rie-mann sphere, the resulting conformal anomaly is proportional to the central charge, thereforethe linear c contribution in the partition function is captured purely by the ZT action. Thesub-leading contribution is encoded in the 2 g -point functions Z g | z = (cid:88) m ,m ,...m g (cid:104) L ¯ O (1) m O (1) m L ¯ O (2) m O (2) m ... L g ¯ O ( g ) m g O ( g ) m g (cid:105) , (1.3)where m , m , ...m g denote the summation of all of the states on the circles C , C , ...C g and C (cid:48) , C (cid:48) , ...C (cid:48) g , L i denotes the Schottky generator identifying C i and C (cid:48) i , and O ( i ) m i , L i ¯ O ( i ) m i arethe vertex operators corresponding to the same state but being inserted at the different fixedpoints of the generator L i . As we are interested in the next-leading contribution in the large c limit, the computation of 2 g -point functions is very much simplified.One essential fact is that the CFT in the large c limit becomes effectively free, whichmeans that the multi-point function is dominated by the product of two-point functions ofsingle-particle states. First of all, a general state in the vacuum module could be of the form (cid:89) m =2 ˆ L r m − m | >, (1.4)where ˆ L − m ’s are the normalized Virasoro generators, r m ’s are non-negative integers. In thelarge c limit, different states are orthogonal to each other, and all of the states constitutea complete set. Every state ˆ L − m | > is a one-particle state as it could be constructed as( L − ) m − | T > , where | T > = ˆ L − | > . From the state-operator correspondence, the corre-sponding vertex operator of ˆ L − m | > is of a form V m ∼ ∂ m − T so that it duals to a singlegraviton in the bulk. The particle number of a general state (1.4) is r = (cid:80) r m . For a particle- r state, the normalized vertex operator is just the normal ordered product of single-particleoperators ˆ O =: r (cid:89) j =1 V m j : . (1.5)Secondly, the leading contribution in the 2 g -point functions on the Riemann sphere is of order O ( c ), so that a 2 g -point function is dominated by the products of the two-point functionsbetween the single-particle operators. Moreover the products of two-point functions may definevarious links. By using SL (2 , C ) transformations and the reduced completeness condition, thevalue of a link is captured by the correlator of two single-particle vertex operators being related4y an element in the Schottky group. Consequently the value of the link is only determinedby the multiplier of the element. More interestingly, it turns out that every oriented link isactually in one-to-one correspondence with the conjugacy class of the Schottky group. Thispaves the way to prove the 1-loop partition function (1.2) for any handlebody solution bytaking into account all possible combination of the products of two-point functions in Z g .In the next section, after briefly reviewing the Schottky uniformization, we discuss how tocompute the partition function of a CFT on a higher genus Riemann surface. In section 3,we prove the relation (1.2) for any handlebody solution for pure AdS gravity. We discuss thestates in the vacuum module of the CFT in the large c limit and the corresponding vertexoperators. As a warm up, we reconsider the genus-1 partition function. Then we move to thehigher genus partition function. In section 4, we generalize our study to the CFT with W symmetry. We end with the conclusion and discussion. In 3D AdS gravity, all the classical solutions could be obtained as the quotient of the globalAdS spacetime by a subgroup of the isometry group SL (2 , C ). In this work, we focus on thehandlebody solutions whose asymptotic boundaries are compact Riemann surface. For thehandlebody solutions, the subgroup is actually a Schottky group. In general, for a handlebodysolution, the boundary Riemann surface is of higher genus.From AdS/CFT correspondence, the partition function of AdS quantum gravity shouldcorrespond to the partition function of higher genus Riemann surface in the dual CFT. In thelarge central charge limit, the semiclassical gravitational action is captured by the leading c terms in the CFT partition function. In the large c CFT, the leading contribution is determinedby the conformal anomaly and the Schottky uniformization. For the next-leading contribution,it could be read from the summation of the multi-point functions on the Riemann sphere viathe sewing prescription.In this section, we first give a brief review on Schottky uniformization, mainly basing onthe work [13]. Then we discuss how to compute a higher genus partition function using thesewing prescription.
Every compact Riemann surface can be described by a Schottky uniformization. For a genus- g Riemann surface M , it can be represented by the quotient M = Ω / Γ, where Ω is the fullcomplex plane plus the point of infinity with the fixed points of Γ being removed, and Γ isthe Schottky group freely generated by g loxodromic SL (2 , C ) elements L i . Ω is called the5egion of discontinuity of Γ. Moreover, it is convenient to introduce the fundamental region todescribe the Schottky group. A fundamental region D is a subset of Ω, such that the interiorpoints in D are not Γ equivalent to each other. One may choose 2 g non-intersecting circles C , C , ...C g and C (cid:48) , C (cid:48) , ...C (cid:48) g in the Riemann sphere such that all circles lie to the exterior ofeach other. The loxodromic element L i ( L − i ) maps C i to C (cid:48) i such that the outer(inner) part of C i is mapped to the inner(outer) part of C (cid:48) i . Then the fundamental region is the part of theRiemann sphere exterior to all the circles, and its quotient is a compact Riemann surface ofgenus g . Each element L i is an SL (2 , C ) matrix , and it is represented by the action L i ( z ) − a i L i ( z ) − r i = p i z − a i z − r i . (2.1)where a i and r i are respectively the attracting and repelling fixed points, 0 < | p i | < L i are √ p i and (cid:113) p − i . Therefore each generator L i is completely characterized by the fixed points a i , r i and the multiplier p i . Among 3 g complex parameters a i , b i and p i , i = 1 , · · · , g , one can fix three of them by using Mobiustransformation. The Schottky group satisfying the above conditions are called a normalizedand marked Schottky group. The remaining 3 g − g .One may define the map γ a i ,r i ( z ) = r i z + a i z + 1 (2.2)such that γ a i ,r i (0) = a i , γ a i ,r i ( ∞ ) = r i . It maps the standard unit circle centered at the originto the circle C i . With γ p ( z ) ≡ pz , a Schottky generator L i in (2.1) is just L i = γ a i ,r i γ p i γ − a i ,r i . (2.3)Actually every Schottky generator could be constructed in this way.The uniformization map from the region of discontinuity to the Riemann surface could bedetermined by the help of a second order differential equation ψ (cid:48)(cid:48) ( u ) + 12 R S ( u ) ψ ( u ) = 0 , (2.4)where R S ( u ) is the projective connection on a marked Riemann surface and ψ could be takenas a multi-valued differential on M of order − /
2. The ratio of two independent solutions ψ and ψ of the Fuchsian equation (2.4) z = ψ ( u ) ψ ( u ) (2.5)gives the map. By imposing appropriate monodromy condition on the cycles of the funda-mental group of the Riemann surface, one can find the generators of the Schottky group. For6 general higher genus Riemann surface, this is a very difficult problem. However, for theRiemann surface resulted from the replica trick in computing the R´enyi entropy, the problemhas been solved explicitly in a perturbative way in the double-interval case [22, 26] and singleinterval on a torus case [26] [33]. We would like to compute the partition function of a large c CFT on a higher genus CFT. Itturns out that the leading c contribution is captured by the Zograf-Takhtajan(ZT) action. Onany compact Riemann surface of genus greater than 1, there is a so-called Poincar´e metric,which is a unique complete metric of constant negative curvature − d ˆ s = dtd ¯ t (Im( t )) . (2.6)Such a metric is related to the flat metric on the complex plane by a conformal transformation d ˆ s = e φ s ( z, ¯ z ) dzd ¯ z (2.7)where φ s is a real field on the Riemann sphere. The constant curvature condition requires thatthe field satisfy the Liouville equation ∂ z ∂ ¯ z φ s = 12 e φ s . (2.8)This equation is the Euler-Lagrange equation of the ZT action defined on the fundamentalregion in the Schottky uniformization [14] S ZT [ φ s ] = − c π (cid:90) (cid:90) D i dz ∧ d ¯ z (cid:18) ∂ z φ s ∂ ¯ z φ s + 12 e φ s (cid:19) + boundary terms . (2.9)This action is a Liouville action with boundary terms. The action evaluated on the solution ofthe Liouville equation gives exactly the regulated AdS gravitational action of the correspond-ing gravitational configuration. Actually this relation helps us to fix the overall factor in theabove action. Moreover the ZT action captures the conformal anomaly and depends only onthe choice of the metric in a fixed conformal class [13]. Under the conformal transformation,the partition function on the Riemann surface is related to the one on the Riemann sphere via Z | u = e − S ZT Z | z . (2.10)It is remarkable that the ZT action captures the whole leading contribution in the partitionfunction in the large c limit.The partition function on a higher genus Riemann surface can be computed using gluingprescription, following Segal’s approach to conformal field theory [35]. As nicely reviewed in7he appendix C of [36], the partition function is defined to be the summation of 2 g -pointfunctions on the Riemann sphere Z g = (cid:88) φ i ,ψ i ∈H g (cid:89) i =1 G − φ i ψ i (cid:104) g (cid:89) i =1 φ i [ C i ] ψ i [ C (cid:48) i ] (cid:105) D , (2.11)where D is the fundamental region with boundary ∂D = ∪ i ( C i ∪ C (cid:48) i ). Here φ i , ψ i are thestates in the Hilbert space H , and φ i [ C i ] denote the states associated with the boundarycircle C i . The circle C i could be related to the standard circle around the origin by a Mobiustransformation (2.2): C i = γ a i ,r i C . To simplify the notation, we will write γ a i ,r i = γ i . Due tothe state-operator correspondence, the states on the circle C i is created by the vertex operatorat γ i (0). More precisely, there is a correspondence φ i [ C i ] → V ( U ( γ i ) p L i φ i , a i ) , (2.12)where the operator U is U ( γ ) = γ (cid:48) (0) L e L γ (cid:48)(cid:48) (0)2 γ (cid:48) (0) . (2.13)Moreover, the state on the circle C (cid:48) i corresponds to the vertex operator ψ i [ C (cid:48) i ] → V ( U ( γ i ˆ γ ) ψ i , r i ) (2.14)where ˆ γ ≡ /z maps the origin to the infinity. In (2.11), G φψ is the metric on the space of thestates G φψ = lim z →∞ (cid:104) V ( z L e zL ψ, z ) V ( φ, (cid:105) . (2.15)With the vertex operators, the partition function (2.11) is changed to the summation over2 g -point functions of the vertex operators inserted at 2 g fixed points Z g = (cid:88) φ i ,ψ i ∈H g (cid:89) i =1 G − φ i ψ i (cid:104) g (cid:89) i =1 V ( U ( γ i ) p L i φ i , a i ) V ( U ( γ i ˆ γ ) ψ i , r i ) (cid:105) , (2.16)The relation (2.11) could be understood in the following way: one can insert a completeset of states in the Hilbert space at each pair of the circles C i and C (cid:48) i , which are related bythe Schottky generator L i , and compute all the possible 2 g -point functions of correspondingvertex operators on the Riemann sphere. One may apply this relation to compute the partitionfunction of any CFT on a higher genus Riemann surface. The computation could be simplifiedif one can choose a complete set of orthogonal state basis, in which case the metric on thespace of the states becomes trivial (cid:104) ¯ O m (cid:48) | O m (cid:105) = lim z →∞ (cid:104) V ( z L e zL O m (cid:48) , z ) V ( O m , (cid:105) = δ mm (cid:48) . (2.17)8et us reconsider the genus-1 partition function in a CFT. In this case, the partitionfunction is decomposed into two-point functions Z = (cid:88) φ,ψ ∈H G − φψ (cid:104) V ( U ( γ ) p L φ, a ) V ( U ( γ ˆ γ ) ψ, r ) (cid:105) . (2.18)As the two-point function is conformal invariant, we may apply a SL (2 , C ) transformation γ − to the two-point functions and get Z = lim z →∞ (cid:88) φ,ψ ∈H G − φψ (cid:104) V ( p L φ, V ( U (ˆ γ ) ψ, z ) (cid:105) = Tr H ( p L ) , (2.19)which is the standard result for the thermal partition function of a CFT. In the computation,the transformation brings the circles C and C (cid:48) to the boundary circles of the annulus aroundthe origin with the radius being p and the unit respectively. Now the Schottky generator issimply the diagonal matrix, and p is the modular parameter of the torus formed from theannulus by identifying two boundary circles.More generally we may consider the two-point function of the vertex operators inserted atthe fixed points in two circles which are related by an element L of the Schottky group. Asevery such element could be put in the form of (2.1), the two-point function is simply (cid:104) L ¯ V V (cid:105) = lim z →∞ (cid:104) V ( U (ˆ γ ) φ, z ) V ( p L φ, (cid:105) = p h , (2.20)where V is the vertex operator corresponding to the state φ with conformal weight h , and p is the multiplier of the element L . Here we use the notation that the operator V denote theoperator inserted at the fixed point of one circle and L ¯ V is the one in the other circle. Bothoperators correspond to the same state, with V generating the ket state and L ¯ V generatingthe bra state.In the following, as every Schottky generator is characterized by the fixed points and themultiplier, there is no need to write them explicitly. Formally, the partition function could bewritten as Z g | z = (cid:88) m ,m ,...m g (cid:104) L ¯ O (1) m O (1) m L ¯ O (2) m O (2) m ... L g ¯ O ( g ) m g O ( g ) m g (cid:105) , (2.21)where m , m , ...m g denote the summation of all of the states on the circles C , C , ...C g and C (cid:48) , C (cid:48) , ...C (cid:48) g . gravity The partition function on a higher genus Riemann surface (2.11) could be decomposed into asummation of 2 g -point correlation functions on Riemann sphere. This is workable for any CFT.9t certainly depends on the the spectrum and the OPE of the CFT. Here we are interested inthe large central charge limit of the CFT dual to the AdS quantum gravity. In this case, thedual CFT has a sparse light spectrum [19, 20], and only the vacuum module contributes to thepartition function perturbatively, and other heavy modules give non-perturbative contributionas O ( e − c ). Therefore we focus on the large central charge limit of the vacuum module. Itturns out that the theory becomes essentially free, and the interaction is suppressed in thelimit [38,39]. After a detailed study of the states in the vacuum module, we compute the genus-1 partition function as a warm up and reproduce the thermal partition function computed inother ways. Next we turn to the computation of the partition function on a higher genusRiemann surface, and find the perfect agreement with (1.2) as well. The vacuum module can be generated by the Virasoro generators acting on the vacuum | (cid:105) .The holomorphic sector of the Virasoro algebra is[ L m , L n ] = ( m − n ) L m + n + c m ( m − δ m + n , (3.1)which has a non-homogenous term of order c . The anti-holomorphic sector has the samestructure. In the following discussion, we focus on the holographic sector. As the vacuumis invariant under SL (2 , C ) conformal symmetry, so is annihilated by the generators L ± , L .The vacuum module are built on the states ...L r n − n ...L r − L r − | (cid:105) , (3.2)where only finite number of r i ’s are non-zero, and their conformal dimensions are h = ∞ (cid:88) j =2 jr j , (3.3)in which there is only finite number of non-zero terms in the summation. A general state inthe module is the linear combination of these states. We note that the states in (3.2) are notorthogonal to each other.In the large c limit, the states in the vacuum module could be re-organized more nicely.Under this limit, we can renormalize the operatorsˆ L m = | cm ( m − | L m for | m | ≥ . (3.4)10he commutation relations for the renormalized operators are[ ˆ L m , ˆ L n ] = δ m + n + O ( 1 c )[ L , ˆ L m ] = m ˆ L m [ L , ˆ L m ] = − sgn ( m ) | m − | | m + 2 | ˆ L m +1 [ L − , ˆ L m ] = − sgn ( m ) | m + 1 | | m − | ˆ L m − . (3.5)In these relations, we have absorbed all of the large c factors into the normalizations of thegenerators. From the relations, we can read two remarkable facts if we only care about theleading c effects. The first is that the operators ˆ L m and ˆ L − m for a fixed m constitute a pair ofcreation and annihilation operators such that they may build a subspace of the Hilbert spacelike ˆ L r m − m | >, with m ∈ N, m ≥ r m ∈ N . (3.6)Note that the states in different subspace are orthogonal to each other. The other fact is thatthe state ˆ L − m | > could be constructed by acting L − repeatedly for m − L − | > = | T > ˆ L − m | > ∼ ( L − ) m − ˆ L − | > = ( L − ) m − | T > . (3.7)A general state in the vacuum module could be of the form ∞ (cid:89) m =2 ˆ L r m − m | (cid:105) , (3.8)with only finite number of r m ’s being non-zero. Now different states are orthogonal to eachother to order c . The normalization for the state is (cid:104) | ∞ (cid:89) m =2 ˆ L r m m ∞ (cid:89) m =2 ˆ L r m − m | (cid:105) = ∞ (cid:89) m =2 r m ! + O ( 1 c ) . (3.9)We may define the “particle number” for such a state to be r = (cid:80) r m . The physical reasonbehind this definition is that each single-particle state ˆ L − m | > corresponds to a graviton.By contour integral the corresponding operator for the state (3.2) is O r ,r ,...r n ... =: ... ( 1( n − ∂ ( n − T ( z )) r n ... ( ∂T ( z )) r T ( z ) r : , (3.10)which is a product of the stress tensors and their partial derivatives. It is clear that the“particle number” of this operator is the number of the stress tensors r = ∞ (cid:88) j =2 r j . The two-point function of O r ,r ,...r n ... is of order c r in the large c limit, which means the operator shouldbe normalized with c r/ . 11n the following discussion, the single-particle state is of particular importance. For asingle-particle state ˆ L − m | > , its corresponding vertex operator is of the following forms atthe origin and the infinity respectively V m = ( 12 cm ( m −
1) ) m − ∂ m − T ( z ) | z =0 , ¯ V m = ( 12 cm ( m −
1) ) m − − z ∂ z ) m − ( z T ( z )) | z →∞ for m ≥ . (3.11)At the origin, the normalized vertex operator for the particle- r state (3.2) readsˆ O =: ( r (cid:89) j =1 V m j ) : (3.12)In other words, the vertex operator of a multi-particle state is just the normal ordered productof the vertex operators for the single-particle states. The important point is that this fact iseven true for the states on the circle not around the origin. Under a conformal transformation,the form of the operator get complicated due to the existence of the partial derivatives. Ac-cording to (2.12), under a conformal transformation γ i (2.2), the vertex operator at the originis changed to the one at the fixed point a i V ( φ i , → V ( U ( γ i ) φ i , a i ) , (3.13)which could be of a complicated form if φ i is a multi-particle state. Taking φ i = L − m L − m · · · L − m r | >, m i ≥ , (3.14)we find that U ( γ i ) L − m L − m · · · L − m r | > = U ( γ i ) L − m U − ( γ i ) U ( γ i ) L − m U − ( γ i ) · · · U ( γ i ) L − m r | > (3.15)Actually the operators U ( γ i ) L − m and U ( γ i ) L − m U − ( γ i ) differs only the terms proportionalto L and L ± U ( γ i ) L − m = U ( γ i ) L − m U − ( γ i ) + terms involving L and L ± . (3.16)As the states induced by the terms involving L and L ± are subdominant in the large centralcharge limit, we may just take U ( γ i ) L − m L − m · · · L − m r | > ∼ ( U ( γ i ) L − m )( U ( γ i ) L − m ) · · · ( U ( γ i ) L − m r ) | > . (3.17)In terms of the vertex operators, we have the operator at the fixed point a i being of the form V ( U ( γ i ) φ i , a i ) =: r (cid:89) j V ( U ( γ i ) L − m r | >, a i ) : , (3.18)12p to a normalization. In other words, the vertex operator at a i could still be written as thenormal ordered product of the vertex operators corresponding to single-particle states.In the large c limit, the states constructed above are not only normalized and orthogonalto each other, but also constitute a complete set. Therefore, we may insert such a completeset of states at the pairwise circles in the Riemann sphere to compute the partition function.In other words, we have the relation I = | (cid:105)(cid:104) | + ∞ (cid:88) m =2 ˆ L − m | (cid:105)(cid:104) | ˆ L m + 12! ∞ (cid:88) m =2 ∞ (cid:88) m =2 ˆ L − m ˆ L − m | (cid:105)(cid:104) | ˆ L m ˆ L m + ... = ∞ (cid:88) r =0 r ! (cid:88) { m j } ( r (cid:89) j =1 ˆ L − m j ) | (cid:105)(cid:104) | ( r (cid:89) j =1 ˆ L m j ) + O ( 1 c ) , (3.19)where the summation over m j is from 2 to the infinity, and r is the “particle number” for theinserting state. Here we list the states with the fewest particle numbers in the above relation r = 0 | (cid:105)(cid:104) | r = 1 ∞ (cid:88) m =2 ˆ L − m | (cid:105)(cid:104) | ˆ L m r = 2 (cid:88) ≤ m
2, but also the modules generated by the W primariesand their descendants, in the large c limit. In the module generated by a W primary, thelowest-weight state is generated by the primary field. For example, in the CFT with W symmetry, the lowest-weight state in the W module is the one generated by the W − | > ,and the other states could be obtained by acting L − repeatedly on W − | > . This is verysimilar to the construction in the vacuum module, where the lowest weight state is generatedby L − . Correspondingly, the holomorphic vertex operators in the W module are of the forms W ( z ) , ∂W , ∂ W · · · . (4.5)For the CFT with other W symmetries, the discussion is similar. One has to keep in mindthat in the large c limit, the modules generated by different primaries are decoupled. Thecompleteness condition for the CFT with W symmetry now changes to I = | (cid:105)(cid:104) | + ∞ (cid:88) m =2 ˆ L − m | (cid:105)(cid:104) | ˆ L m + 12! ∞ (cid:88) m =2 ∞ (cid:88) m =2 ˆ L − m ˆ L − m | (cid:105)(cid:104) | ˆ L m ˆ L m + ... + ∞ (cid:88) n =3 ˆ W − n | (cid:105)(cid:104) | ˆ W n + 12! ∞ (cid:88) n =3 ∞ (cid:88) n =3 ˆ W − n ˆ W − n | (cid:105)(cid:104) | ˆ W n ˆ W n + ... + 12! ∞ (cid:88) m =2 ∞ (cid:88) n =3 ˆ L − m ˆ W − n | (cid:105)(cid:104) | ˆ W n ˆ L m + ... = ∞ (cid:88) r =0 r ! (cid:88) { m j }{ n k } ( r (cid:89) j =1 ˆ L − m j )( r − r (cid:89) k =1 ˆ W − n k ) | (cid:105)(cid:104) | ( r (cid:89) j =1 ˆ L m j )( r − r (cid:89) k =1 ˆ W n k ) + · · · + O ( 1 c ) , where the summation over m j is from 2 to the infinity, the summation over n k is from 3 tothe infinity, and r , r − r are the “particle numbers” for the inserting states coming from thevacuum module and W module respectively. This completeness condition can be transformedinto the one in terms of the vertex operators. 26s in the pure gravity case, if we are interested in the two-point functions, we still have thecompleteness relation (3.25) but now the single particle states should include the ones fromthe W primaries. Moreover, the states in different Verma modules are orthogonal to eachother so that the two-point function of the vertex operators coming from different modules arevanishing. As a result, one may consider the contributions of different modules to the partitionfunction (2.11) separately and finally multiply them together. Taking the W module as anexample, we find that the single-particle states in it contribute to the genus-1 partition function Z (1) W = ∞ (cid:88) n =3 (cid:104) L ¯ W n ( r ) W n ( a ) (cid:105) = Tr H ,W q L = ∞ (cid:88) n =3 q n (4.6)where H ,W means the Hilbert space of the single-particle states in the W module. For themulti-particle states, there are states like ( r (cid:89) j =1 ˆ L − m j )( r − r (cid:89) k =1 ˆ W − n k ) | (cid:105) . A multi-point functionof these states on the Riemann sphere is factorized into the product of two-point functions,each of them being between the operators from the same module. With the completenesscondition, only the operators from the same module can form link. Consequently, the finalpartition function is Z g = Z ( vacuum ) g Z W g (4.7)where Z ( vacuum ) g = (cid:89) γ (cid:32) ∞ (cid:89) m =2 − q mγ (cid:33) , Z W g = (cid:89) γ (cid:32) ∞ (cid:89) n =3 − q nγ (cid:33) (4.8)Therefore, the partition function Z g is exactly the same as the 1-loop partition function (4.1)for the higher spin AdS gravity. In this paper, we discussed the 1-loop partition function in the AdS /CFT correspondence.We focused on the handlebody solutions in the AdS gravity. These solutions end on theasymptotic boundary as compact Riemann surfaces, which could be described by Schottkyuniformization. The 1-loop partition function (1.2) of these solutions have been computed byusing the heat kernel techniques and the method of images in [18]. But the direct computationin the dual CFT has so far been missing. We filled this gap and proved the result (1.2) in thelarge central charge limit of the CFT in this work.We used the sewing technique to compute the CFT partition function on the Riemannsurface. In the large c limit, the leading contribution, which is linear in c and correspondsto the semiclassical action of the gravitational configuration, is captured by the ZT action.The sub-leading contribution to the partition function is encoded by the 2 g -point functions27n the Riemann sphere. These multi-point functions are at most of order c under the large c limit. Actually it is relatively easy to read the leading order terms in these 2 g -point functions.It turns out that at leading order every 2 g -point function could be reduced to the productsof the two-point functions of single-particle operators. The products of two-point functionsmay define the links. Every oriented link is one-to-one related to a conjugacy class in theSchottky group. The value of each link could be reduced to one two-point function, whosevalue is determined by the multiplier of the conjugacy element in the Schottky group. Byconsidering all possible ways to contract the operators and form the links, the result (1.2)has been reproduced. We generalized the study to the higher spin AdS gravity and foundagreement as well.The proof presented in this work relies on the essential fact that the dual CFT in the large c limit is effectively free. As the two-point function of single-particle states dominates, thecontribution from three-point function is suppressed. As a result, the multi-point functions onthe Riemann sphere is simplified. In the bulk side, the dominance of two-point function of thesingle-particle operator is reflected in the fact that the massless graviton is freely propagatingand the interaction among gravitons can be ignored. Certainly, this should be the case sincethe 1-loop gravitational partition function is only given by the functional determinant of thefree massless graviton.It would be interesting to study the higher loop partition function in the AdS gravityfrom the multi-point functions on the Riemann sphere. The recent study in [37] shows thatthe higher order 1 /c terms, corresponding to the higher loop corrections, are not vanishing forhigher genus Riemann surface. It would be great to develop a systematic way to compute suchterms in the large c limit. However, this problem is rather difficult as several approximationswe relied on have to be reconsidered carefully. First of all, the orthogonality of the differentstates in the vacuum module does not hold at the order 1 /c . Secondly, the vertex operators atthe fixed points could not be written as the normal ordered product of single-particle operators.Moreover, besides two-point function, the three-point function of single-particle state shouldbe taken into account. On the bulk side, this means that we have to consider the interactionof the gravitons.In this work, we mainly discussed the pure AdS gravity and its higher spin generalization.It is easy to see that the 1-loop partition function in the chiral gravity [40] can be proved.In this case, we only needs to consider the holomorphic sector of the CFT, then the result isimplied in our discussion. For the topologically massive gravity [41] at critical point, therecould be other consistent asymptotic boundary condition to allow the logarithmic modes so28hat the dual CFT is a logarithmic CFT . It would be interesting to check if we can reproducethe 1-loop partition function in this case and its higher spin generalization [43–47]. Acknowledgments
We would like to thank Jiang Long for valuable discussions, thank A. Maloney, I. Zadehfor usual conversations and correspondence. BC would like to thank the participants of theWorkshop on “Holography for black holes and cosmology” (ULB, Brussels), especially E.Perlmutter and T. Hartman, for stimulating discussion, which initialized this project. Thework was in part supported by NSFC Grant No. 11275010, No. 11335012 and No. 11325522.BC thanks Harvard University for hospitality during the final stage of this work.
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