Exact Path Integral for 3D Quantum Gravity II
Masazumi Honda, Norihiro Iizuka, Akinori Tanaka, Seiji Terashima
aa r X i v : . [ h e p - t h ] O c t OU-HET-864, RIKEN-STAMP-14, WIS/06/15-Oct-DPPA, YITP-15-64
Exact Path Integral for 3D Quantum Gravity II
Masazumi Honda, ∗ Norihiro Iizuka, † Akinori Tanaka, ‡ and Seiji Terashima § Department of Particle Physics, Weizmann Institute of Science, Rehovot 7610001, Israel Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, JAPAN Interdisciplinary Theoretical Science Research Group, RIKEN, Wako 351-0198, JAPAN Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, JAPAN (Dated: September 11, 2018)Continuing the work arXiv:1504.05991, we discuss various aspects of three dimensional quantumgravity partition function in AdS in the semi-classical limit. The partition function is holomorphicand is the one which we obtained by using the localization technique of Chern-Simons theory inarXiv:1504.05991. We obtain a good expression for it in the summation form over Virasoro charactersfor the vacuum and primaries. A key ingredient for that is an interpretation of boundary localizedfermion. We also check that the coefficients in the summation form over Virasoro characters ofthe partition function are positive integers and satisfy the Cardy formula. These give physicalinterpretation that these coefficients represent the number of primary fields in the dual CFT in thelarge k limit. I. INTRODUCTION
Solving and writing down an explicit form of the par-tition function for quantum gravity is one of the mostimportant remaining problems in theoretical physics. In[1], the authors write down the explicit partition functionfor 3D quantum pure gravity in asymptotic AdS space-time. The partition function is the direct product of holo-morphic and anti-holomorphic function, and is obtainedby using the Chern-Simons formulation of 3D gravity,and its localization technique, for the holomorphic La-grangian L , L = ik π Z M Tr (cid:16) A d A + 23 A − λλ + 2 Dσ (cid:17) + t Z M Tr (cid:16) F µν F µν + 12 D µ σD µ σ + 12 ( D + σl ) + i λγ µ D µ λ + i λγ µ D µ ¯ λ + i λ [ σ, λ ] − l ¯ λλ (cid:17) , (1)and by summing over the geometries with Rademachersum regularization [2]. The need for Rademacher sum isdue to the fact that, for gravity path-integral, one needsto sum over all the geometries consistent with localiza-tion locus, F µν = 0, but F µν = 0 is nothing but theEinstein’s equation. The final expression for the holo-morphic partition function, Z hol ( q ), becomes Z hol ( q ) = R ( − k eff / ( q ) − R ( − k eff / ( q ) , (2)where q ≡ e πiτ with τ the complex moduli for theboundary torus, k eff = k + 2, k = ℓ/ G N with ℓ as an ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] AdS scale, and the function R ( m ) ( q ) is defined as R ( m ) ( q ) ≡ e πimτ + X c> , ( c,d )=1 (cid:16) e πim aτ + bcτ + d − r ( a, b, c, d ) (cid:17) = q m + (const) + ∞ X n =1 c ( m, n ) q n , (3)where r ( a, b, c, d ) = e πim ac is τ -independent quantity toregularize c , d sum, and the coefficients c ( m, n ) for thepositive powers of q are given in [1].For later purpose, we express c ( m, n ) in terms of mod-ified Bessel function I ( z ), I ( z ) = z ∞ X ν =0 (cid:16) z (cid:17) ν ν ! ( ν + 1)! , (4)as c ( m, n ) = 2 π r − mn ∞ X c =1 A c ( m, n ) c I (cid:16) π √− mnc (cid:17) , (5)where A c ( m, n ) is so-called Kloosterman sum, A c ( m, n ) = X ≤ d ≤ c ( c,d )=1 e πi ( m ac + n dc ) . (6)The summation in (6) is for d , given the c , where a isuniquely determined up to ( a, b ) ≈ ( a + c, b + d ) to satisfy ad − bc = 1 given c and d satisfying ( c, d ) GCD = 1.The (const) term in (3) is undetermined, because itdepends on the regularization scheme for the modu-lar sum, namely how to choose τ -independent quantity r ( a, b, c, d ), and r ( a, b, c, d ) = e πim ac for Rademachersum is just one possible regularization scheme. In di-rect gravity calculation, there is apparently no principleto choose the right scheme. In this paper, we set the(const) term in (3) to be zero for convenience, though weadmit that this is an open issue.One of the outstanding results in [1] is that for k eff =4 in which the central charge c Q is expected to be24, Z hol ( q ) becomes the J -function up to a constantshift which depends on the modular sum regularizationscheme. Therefore, this quantum gravity partition func-tion agrees with the extremal CFT partition function ofFrenkel, Lepowsky, and Meurman [3], predicted by Wit-ten in [4]. Note that in the case k eff = 4, G N ∼ ℓ , there-fore we are in full quantum gravity parameter regime.On the other hand, what happens when G N ≪ ℓ ? Inthis parameter regime, the semi-classical gravity descrip-tion is guaranteed. The purpose of this short letter is,based on the result in [1], to clarify the various physi-cal aspects of the partition function in the semi-classicallimit, where G N ≪ ℓ . II. BOUNDARY FERMION
As is seen from (2) and (3), the holomorphic partitionfunction Z hol ( q ) has a negative pole in the large k limit.Let us first comment on the physical origin of this minussign: we claim that this is due to the fact that our parti-tion function Z hol ( q ) contains additional fermion degreesof freedom, in addition to 3D gravity. However, we canextrapolate 3D pure gravity partition function at least inthe large k limit, which we will explain now.First of all, the computation for (2) is based on the lo-calization of N = 2 supersymmetric Chern-Simons the-ory [1], where we supersymmetrize the Chern-Simons for-mulation of the pure gravity by introducing auxiliarygauginos and scalars. The supersymmetrized action con-tains a “mass term” for the gaugino, k π ¯ λλ . Furthermore,for the localization we also added a supersymmetric exactterm, i.e., a super Yang-Mills term, which has a kineticterm for the gaugino, t ¯ λγ µ D µ λ . Note that we are forcedto introduce the kinetic term even for the auxiliary fieldfor localization. Furthermore in order to preserve somesupersymmetries at the AdS boundary, which is equiv-alent to the boundary torus, we imposed the followingboundary condition for the gauginos [6] λ (cid:12)(cid:12)(cid:12) bdry = e − i ( ϕ − t E ) γ ¯ λ (cid:12)(cid:12)(cid:12) bdry , (7)where e − i ( ϕ − t E ) is a phase factor depending on the toruscoordinates, t E and ϕ and γ = (cid:18) − (cid:19) . Since theboundary condition (7) is incompatible with the gaugino“mass term”, this leads to the conclusion that the massterm vanishes at the boundary and therefore there is aboundary localized fermion.If we look at the kinetic term and the mass term inthe Lagrangian for the auxiliary fermions λ and ¯ λ , onecan see that by using the doubling trick argument in [1],they are like L λ , ¯ λ ∼ t ψ (cid:16) ∂ x + kt sign( x ) (cid:17) ψ , (8) where auxiliary fermions ψ and ψ are some componentsof λ and ¯ λ (see [1] for detail) and x = 0 corresponds tothe boundary. Then it is easy to see that this admitsboundary localized fermion wave function as ψ ( x, ϕ, t E ) = e − kt | x | ψ boundary2 ( ϕ, t E ) , (9)and that ψ boundary2 is sharply peaked at x = 0 in the t → x = 0, are justified only in the classical limit, which is, inthe large k limit. This also implies that at finite k , thereis no reason that the auxiliary fermion admits bound-ary localized mode. In fact, the analysis of [1] stronglysuggests that there is no boundary localized fermion inthe case of k eff = 4, where central charge is expectedto be 24. This is because then Z hol itself becomes theconjectured J -function by Witten [4].Following these, we claim that the boundary fermioncontributes to the partition function as Z B-fermion ( q ) = ∞ Y n =1 (1 − q n ) . (10)Since this boundary localized fermion decouples at leastfrom the bulk gravity in the large k limit, and thereforeit should give a common contribution to the partitionfunction given by (10), independent of the bulk geometry.Then we claim that the quantity, Z hol ( q ) Z B-fermion ( q ) ≡ Z large k gravity ( q ) (11)represents physically “bulk pure gravity” partition func-tion, at least in the semi-classical limit at large k . Forlater use, let us write down our holomorphic partitionfunction (2) explicitly, Z hol ( q ) = (1 − q ) q − k eff4 + ∞ X ∆=1 c ( k eff )∆ q ∆ , (12) c ( k eff )∆ ≡ c (cid:18) − k eff , ∆ (cid:19) − c (cid:18) − k eff , ∆ (cid:19) . (13) III. LARGE k PURE GRAVITY PARTITIONFUNCTION
Combining the equations (10), (11), (12) all together,we obtain Z large k gravity ( q )= q − k eff4 ∞ Y n =2 − q n + ∞ X ∆=1 c ( k eff )∆ q ∆ ∞ Y n =1 − q n . (14)Note that the first terms in (14) is the usual Verma mod-ule made from the vacuum | i , satisfying L n> | i = L | i = L − | i = 0 , (15)and its descendants Q n ≥ L N n − n | i . Therefore this is theVirasoro character for the vacuum Z vac ( q ), q − k eff4 ∞ Y n =2 − q n = Tr V | i q L − k eff4 ≡ Z vac ( q ) . (16)Similarly, the second term in (14) is the Verma modulemade from the primary state | ∆ i , satisfying L n> | ∆ i = 0 , L | ∆ i = (∆ + k eff | ∆ i , (17)and its descendants Q n ≥ L N n − n | ∆ i , with positive integer∆. Note that L − is included since | ∆ i is not vacuum.Therefore this is the Virasoro character for generic pri-mary Z primary ( q ), q ∆ ∞ Y n =1 − q n = Tr V | ∆ i q L − k eff4 ≡ Z primary ( q ) . (18)In the end we can write (14) as Z large k gravity ( q ) = Z vac ( q ) + ∞ X ∆=1 c ( k eff )∆ Z ∆primary ( q ) . (19)This is our main point. The holomorphic partition func-tion Z hol [ q ] given by (2) can be written as direct productof partition function from the boundary fermion (10) andone from bulk pure gravity (11) in the large k limit. Fur-thermore, the partition function from pure gravity con-tribution (11) can be written as the sums over conformalfamilies as (19).This strongly suggests that there is a dual 2D CFT inthe large k limit and the coefficients c ( k eff )∆ represent thenumber of primary operators labelled by ∆ in the dualCFT. Clearly for that nice CFT interpretation possible,all c ( k eff )∆ need to be positive integers. The fact that itis integer is easily confirmed step by step, following itsdefinition (13). We can also show that it is positive.However a few comments are worth making before we gointo that.The fact that in the large k limit, there is a huge gap oforder k eff / ≈ k/ ℓ/ G N for conformal dimen-sions between vacuum and primary operators are exactlythe half of the gap between AdS vacuum and BTZ blackhole [7]. Half is because there is a same amount of contri-bution from the anti-holomorphic part. This gap is alsothe reason why Witten conjecture in [4] that the dualCFT, if exists, should be an extremal CFT, where BTZblack holes correspond to primary states.Another point is that Z B-fermion in (10) is not modu-lar invariant, so is Z large k gravity ( q ) in (11). One might won-der if modular non-invariant quantity is physical or not.Note that in the large k limit, this modular invariancewill be lost. This is because in the semi-classical limit,only the dominant saddle point survives and the rest sad-dle points vanish exponentially. However these vanishing saddle points are important ingredients for modular in-variance property. Therefore we claim that only in thesemi-classical limit, k → ∞ , quantity Z large k gravity ( q ) (11)becomes physically meaningful since only in that limit,boundary fermion decouples and quantity Z B-fermion ( q )becomes meaningful. IV. POSITIVITY OF c ( k eff )∆ We now show the positivity for c ( k eff )∆ . Since c ( k eff )∆ in(19) is defined through (11) only in the large k limit,we focus on k eff = k + 2 → ∞ limit in (13), whichcorresponds to the limit m → −∞ in c ( m, n ) given by(5). Since asymptotic behaviour of the modified Besselfunction I ν ( z ) is given by the exponential growthlim z →∞ I ν ( z ) ∼ e z √ πz (cid:18) − ν − z + O ( z − ) (cid:19) , (20)in the summation over c expression for c ( m, n ) in (5), c =1 contribution dominates and the contributions c > c = 1 case, for any integervalues m and n = ∆, A c =1 ( m, ∆) = 1 (21)is easily seen from (6). Note that k eff / m to be negative integer.Therefore, the dominant contribution in the summationexpression for c ( m, n ) in (5) is given bylim m →−∞ c ( m, n ) = ( − m ) / / n / e π √− mn (cid:16) − π √− mn + O ( − mn ) − (cid:17) + O ( e π √− mn ) . (22)Then, from (13) it is clear thatlim k →∞ c ( k eff )∆ = 2 π ( k eff ) / ∆ / e π √ k eff ∆ (cid:16) O ( k − / eff ) (cid:17) > , (23)and this leads to, in the leading order,lim k →∞ log c ( k eff )∆ = 2 π p k eff ∆ = 2 π r c Q ∆6 , (24)where c Q is for central charge and we have used the re-lationship c Q = 6 k eff . The result (24) perfectly matcheswith the boundary CFT’s Cardy formula in the largecentral charge c Q limit.Since it is defined only in the large k limit through thedefinition (11), the expression (19) is meaningful onlyin the large k limit. However one can also easily checkthat c ( k eff )∆ is positive and also integer by using the defini-tion (13) in a straightforward way numerically for finite k case, where our coefficients with analytic method agreewith numerics. But we take other approach here: wewrite R ( m ) ( q ) in terms of polynomial of J -function as [4]in order to know the c ( m, n ) order by order, where J isgiven in terms of Klein’s j -invariant as J = j − R ( m ) ( q ) and positive powers of J -function are modular invariant and have no pole otherthan q = 0. For example, R ( − ( q ) = J ( q )= 1 q + 196884 | {z } c ( − , q + 21493760 | {z } c ( − , q + . . . ,R ( − ( q ) = J ( q ) − q + 42987520 | {z } c ( − , q + 40491909396 | {z } c ( − , q + . . . ,R ( − ( q ) = J ( q ) − J ( q ) − q + 2592899910 | {z } c ( − , q + 12756069900288 | {z } c ( − , q + . . . , (25)and so on. As one can see, the coefficient c ( m, n ) growingvery fast with respect to − m with fixed n . Therefore,this yields positivity for c ( k eff )∆ . One can observe, forexample, c (4)∆=1 = 196884 , c (4)∆=2 = 21493760 ,c (8)∆=1 = 42790636 , c (8)∆=2 = 40470415636 ,c (12)∆=1 = 2549912390 , c (12)∆=2 = 12715577990892 , (26)from the above numerics. We observed that positivityof c ( k eff )∆ > k eff / k region, we can conduct step bystep check for the positivity and integer nature, and aswe have already shown in (23), in the large k parameterregion, it is positive. V. SUMMARY AND DISCUSSION
In this short paper, we focused and analyzed the var-ious physical aspects of our previous results [1] in thelarge k limit. Large k corresponds to, through the rela-tion ℓ/G N ∝ k , the semi-classical limit of quantum grav-ity. We obtain an plausible representation of our parti-tion function (14) just by dividing the original our holo-morphic partition function by the contribution comingfrom the boundary-localized fermion. This suggests thatour full quantum gravity partition function (2) containscontributions from both bulk gravity and the boundaryfermion, and in the dual CFT, they couple in the finite k ,but only in the large k limit, they decouple. The gravitypartition function, obtained by dividing the boundaryfermion contribution as (11), is written as summationover Virasoro characters for the vacuum and primariesas (19). We have shown that the coefficients c ( k eff )∆ in(19) are positive definite integers and satisfy the Cardy formula in the large k limit. These facts give a consis-tency check to interpret (19) as a dual 2D CFT partitionfunction.In this paper, we claim that in nonperturbative for-mulation of 3D quantum “pure” gravity, involving addi-tional fermion degrees of freedom is unavoidable. Onemight wonder why we cannot obtain non-perturbativepartition function for just pure gravity, without any ad-ditional degrees of freedom. Let us discuss this point indetail now. Our claim is that, in order to conduct themetric path integral exactly at the nonperturbative level,it is better to use the localization technique and for that,“ t regularization” [11]. Otherwise, one has to rely on theperturbative analysis, unless one can solve it exactly.About exact solvability, since Chern-Simons theory istopological, it could be that one can solve it exactly, with-out relying on perturbation. In [8], purely bosonic Chern-Simons theory is discussed in detail. Therefore by usingthe results of [8], one might be able to obtain Z ( c,d ) in ourprevious paper [1] without relying localization. However,even if one could, it gives at most the non-perturbativeresults for the Chern-Simons theory with fixed topologyonly. It does not give the non-perturbative results forquantum gravity, i.e., the justification for the emergenceof modular/Rademacher sum.About perturbative analysis, it is pointed out in severalliteratures [5, 9] that gravity path integral is “one-loopexact”. However there is no direct calculation to confirmthis solely in gravity side. Many literatures assume someproperties motivated by dual CFT. Furthermore, even ifthe one-loop exactness is true, it is at most perturbativelevel. On the other hand, by t regularization for local-ization, we succeeded in conducting the exact path inte-gral for the metric nonperturbatively, and as a result, weobtain full partition function Z hol ( q ). Note that in ourcalculation, we do not assume the existence of dual CFT,nor one-loop exact, nor Virasoro algebra. In this paper,we propose how to extrapolate pure gravity contributionat least in the large k semi-classical limit, and as a resultof our calculation, we obtain the Virasoro characters.Furthermore our localization calculation gives verynaturally the reason for the emergence of modu-lar/Rademacher sum, very important non-perturbativeeffects: this is because in localization calculation, onlythe localization locus F µν = 0 contributes in the path-integral but we have to sum over all of the field configu-rations satisfying F µν = 0. Since F µν = 0 is nothing butthe Einstein’s equation written in terms of the holomor-phic gauge field A µ , summing over all the field configura-tions satisfying F µν = 0 exactly corresponds to summingover all of the complex solutions of the Einstein’s equa-tion. Complex is due to the holomorphic property. Thiscorresponds to summation over ‘cosets of SL (2 , Z )’, i.e., the summation over c and d for the Rademacher sum,because c and d characterise how to embed all of thecomplex solutions of the Einstein’s equation into solidtorus, and therefore, characterise all the complex valuedsaddle points. Thus, in our localization calculation, wedo not have to impose summing over ‘cosets of SL (2 , Z )’ by hand , rather it arises naturally as localization locusfrom the exact path-integral [12]. Furthermore, this givesthe explanation why geometries like singular ones do notcontribute to the path integral. This is simply becausethey are not the localization locus, satisfying F µν = 0.Note that even though the partition function is writ-ten as the direct product of holomorphic and anti-holomorphic one, and therefore there are complex val-ued saddle points parametrised by c and d , in the semi-classical limit the dominant saddle point gives the realvalued saddle point, see § in more great detail throughthese generalization. We will come back to these issuesin near future. Acknowledgments
AT would also like to thank Osaka University for itshospitality where part of this work was done. ST wouldalso like to thank M. Shigemori for discussion. The workof NI was supported in part by JSPS KAKENHI GrantNumber 25800143. The work of AT was supported inpart by the RIKEN iTHES Project. [1] N. Iizuka, A. Tanaka and S. Terashima, “Exact Path In-tegral for 3D Quantum Gravity,” arXiv:1504.05991 [hep-th].[2] H. Rademacher, “The Fourier Coefficients and the Func-tional Equation of the Absolute Modular Invariant j ( τ ),”Am. J. Math. , 237 (1939).[3] I. B. Frenkel, J. Lepowsky and A. Meurman, “A NaturalRepresentation of the Fischer-Griess Monster With theModular Function J As Character,” Proc. Natl. Acad.Sci. USA (1984) 3256-3260.[4] E. Witten, “Three-Dimensional Gravity Revisited,”arXiv:0706.3359 [hep-th].[5] A. Maloney and E. Witten, “Quantum Gravity PartitionFunctions in Three Dimensions,” JHEP , 029 (2010)[arXiv:0712.0155 [hep-th]].[6] S. Sugishita and S. Terashima, “Exact Results in Su-persymmetric Field Theories on Manifolds with Bound-aries,” JHEP , 021 (2013) [arXiv:1308.1973 [hep-th]].[7] M. Banados, C. Teitelboim and J. Zanelli, “The Blackhole in three-dimensional space-time,” Phys. Rev. Lett. , 1849 (1992) [hep-th/9204099].[8] S. Elitzur, G. W. Moore, A. Schwimmer and N. Seiberg,“Remarks on the Canonical Quantization of the Chern-Simons-Witten Theory,” Nucl. Phys. B , 108 (1989).[9] S. Giombi, A. Maloney and X. Yin, “One-loop Parti-tion Functions of 3D Gravity,” JHEP , 007 (2008)[arXiv:0804.1773 [hep-th]].[10] M. Honda, N. Iizuka, A. Tanaka and S. Terashima, toappear.[11] Here we mean ‘ t regularization’ as deforming the originaltheory by addition of t L SY M term as (1).[12] Precisely speaking, for this to work, we have to mod-ify the meaning of Chern-Simons theory path integral tomore appropriate one for gravity: we have to sum over allof the solutions of F µν = 0 whose boundary are relatedby cosets of SL (2 , Z ). Summing over different boundaryconditions related by cosets of SL (2 , Z, Z