On the perturbative expansion of exact bi-local correlators in JT gravity
PPrepared for submission to JHEP
On the perturbative expansion of exact bi-localcorrelators in JT gravity
Luca Griguolo, a Jacopo Papalini, a and Domenico Seminara b a Dipartimento SMFI, Universit`a di Parma and INFN Gruppo Collegato di Parma, Viale G.P.Usberti 7/A, 43100 Parma, Italy b Dipartimento di Fisica, Universit`a di Firenze and INFN Sezione di Firenze, via G. Sansone 1,50019 Sesto Fiorentino, Italy
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We study the perturbative series associated to bi-local correlators in Jackiw-Teitelboim (JT) gravity, for positive weight λ of the matter CFT operators. Starting fromthe known exact expression, derived by CFT and gauge theoretical methods, we reproducethe Schwarzian semiclassical expansion beyond leading order. The computation is donefor arbitrary temperature and finite boundary distances, in the case of disk and trumpettopologies. A formula presenting the perturbative result (for λ ∈ N /
2) at any given orderin terms of generalized Apostol-Bernoulli polynomials is also obtained. The limit of zerotemperature is then considered, obtaining a compact expression that allows to discuss theasymptotic behaviour of the perturbative series. Finally we highlight the possibility toexpress the exact result as particular combinations of Mordell integrals. a r X i v : . [ h e p - t h ] J a n ontents λ ∈ N λ values 144.2 The case of generic n 16 β → ∞
196 Mordell integrals and bi-local correlators for integer λ
217 Final comments and outlook 23Appendices 25A Evaluation of the Residue 25B Some useful expansion for generalized Apostol-Bernoulli Polynomials 27C Computing gaussian integrals of Hermite polynomials 28 – 1 –
Introduction
Understanding the quantum behavior of gravitational theories is one of the most fascinatingproblems in contemporary theoretical physics. Jackiw-Teitelboim (JT) gravity [1, 2] isprobably the simplest example where many questions concerning the nature of a quantumspace-time can be tackled and answered, being solvable as a quantum field theory butstill retaining quite non-trivial dynamics. It represents a particular example of AdS/CFTcorrespondence in which we can study bulk and boundary properties with high precision[3–6]. Recent investigations have shown that a detailed knowledge of the quantum theoryis compulsory to obtain an adequate understanding of quantum gravity’s fundamentalquestions. In particular the physical interpretation of different topological contributionsto the gravitational path-integral [7, 8] and the advances on the black hole informationparadox [8–10] in this setting heavily relied on explicit quantum results. In this sense, JTgravity has played an almost unique role, and it would be crucial to extend our controlto more complicated models, generalizing safely some important lessons learned there [11–15]. An appealing aspect of JT gravity is the existence of a particular class of n -pointfunctions that we can compute exactly, the so-called bi-local correlators [5]. They canbe viewed either as n -point vacuum expectation value for bi-local operators evaluatedat the boundary of the AdS space-time or as 2 n -point correlators of some 1D ’matterCFT’ at finite temperature coupled to the Schwarzian theory on the boundary [5]. Theirgeneral structure on disk and trumpet topologies has been studied by exploiting differenttechniques, and their explicit form can be systematically obtained for any n as an integralof momentum space amplitudes, using a simple set of diagrammatic rules. Originally thederivation relied on the precise equivalence between the 1D Schwarzian theory and a certainlarge central charge limit of 2D Virasoro CFT [16]. More recently, taking advantage of theSL(2 , R ) gauge theory formulation (see [17] for an exhaustive analysis of the subject), thecorrelation functions of bi-local operators have been computed as correlators of Wilsonlines anchored at two points on the boundary [18]. The time ordering is encoded into theintersection of the Wilson lines in the bulk, resulting in the appearance of momentum-dependent fusion coefficients and 6- j symbols inside the integrated amplitudes. AnchoredWilson lines also have a gravitational interpretation, representing the sum over all possibleworld-line paths for a particle moving between two fixed points on the boundary of theAdS patch [18, 19]. The computation of bi-local correlators have been later extended in thepresence of defects [20], and the inclusion of higher-genus corrections was also considered [8],with particular attention to their late time behavior and non-perturbative properties. Onthe other hand, correlation functions on the disk can also be studied through a perturbativeexpansion in the Schwarzian coupling constant [5, 21]. In this approach, one directlycomputes Feynman diagrams for boundary gravitons, i.e., the quantum mechanical degreesof freedom associated with the fluctuations of the wiggle AdS boundary. The semiclassicallimit and the first quantum correction to two-point and four-point functions were studied in[22]. Schwarzian perturbation theory has also found applications for higher-point functions[23], while higher loop corrections were analyzed in [24].Quite surprisingly, the consistency of the exact results obtained through CFT and gauge– 2 –heoretical techniques with the Schwarzian perturbative expressions has never been checkedor diskussed in details until recently [25]. More generally, the structure of the perturba-tive series and its convergence properties have been somehow overlooked despite certaininteresting pieces of information that could be directly extracted from it, as the relationwith the gravitational S-matrix or the trigger of the full quantum regime at large timewith respect to Schwarzian coupling constant. A particularly intriguing point concerns theconvergence itself of the perturbative series and the presence of non-perturbative contri-butions inside the exact expressions derived in [16, 18]. A first attempt to answer thisquestion has been taken in [25]: the exact two-point correlator on the disk for a bi-localoperator of conformal weight λ ∈ − N / κ , theSchwarzian coupling, and confronted successfully with the perturbative result beyond thesemiclassical regime. Moreover, exploiting the simplicity of the cases λ = − / λ = − λ the series is asymp-totic, implying the presence of non-perturbative contributions. Exactly for λ ∈ − N / e − κ inside these correlators, competing thereforewith the higher-genus corrections of order e − GN , derived by matrix-model techniques [7, 8],because κ is proportional to gravitational Newton constant G N .In this paper, we obtain some progress in these directions, performing explicit computationsand elucidating the analytical structure of the bi-local correlators in the case of generalpositive conformal weight and general temperature. We also slightly extend our analysisbeyond the disk topology by considering the trumpet configuration: this could be relevantin view of further studies on higher-genus topologies [25, 26] or for investigating one-point functions in the presence of defects [20]. Our first aim is to recover the Schwarzianperturbative result beyond the leading semiclassical order in the case of positive λ , on thedisk and trumpet topologies. We have obtained a perfect agreement by evaluating theexact expression through a saddle-point approximation: the computation heavily relies onthe relevant amplitudes’ analytical properties. In the general case, it reduces to an integralaround branch-cuts determined by the conformal weight of the operators involved. Theresult is obtained for finite boundary separations; it exhibits the correct time periodicityand, as expected in this case, the bi-local correlator is singular at coincident points. Theoutcome completes and generalizes the analysis of [25], performed for negative semi-integerweights λ and in the particular case of zero temperature, and strengths our trust in theanalytical approach. Actually, in the case of λ ∈ N / κ . Finally we have also examined the zero-temperature case,in order to recover the results of [25] in this limit: although being potentially singular, The semiclassical limit for the two-point and the four-point functions has been checked in [22] – 3 –s seen from the previous expansions, we have obtained a nice and compact expression interms of Bernoulli polynomials, consistent with the general result. We can draw from thislimit some conclusions on the convergence properties of the perturbative series, confirmingits asymptotic character for positive semi-integer weights. Moreover, the alternate signof the perturbative orders points towards a possible Borel summability for the full series.As a final observation, we point out that the exact expression for the bi-local correlatorcan also be written in terms of Mordell integrals [27], suggesting a link with the world ofMock-modular forms [28, 29].The paper’s structure is the following: in Section 2, we review the exact results [16, 18]and diskuss the perturbative computation of the bi-local correlator in JT gravity from theSchwarzian perspective. We present the explicit calculation on the trumpet to elucidatethe procedure, slightly generalizing the previous calculations. In Section 3, we perform thesaddle-point analysis of the exact expressions on the disk and the trumpet, successfullyrecovering the first subleading correction to the semiclassical result. Section 4 is devotedto the all-order expansion in the case λ ∈ N /
2. We give the explicit (although a littlecumbersome) general expression and examine in more detail the weights λ = 1 / λ = 1. The zero-temperature limit is instead the subject of Section 5. Subsequently, inSection 6, we illustrate how the bi-local correlators for integer 2 λ can be written in closedform in terms of Mordell integrals. In Section 7, we draw our conclusions and diskuss somepossible directions to extend the present analysis. A certain number of Appendices deepenthe technical aspects of the work, completing the paper. This section briefly summarizes how to write down the exact formula for the bi-local bound-ary correlators in JT gravity, starting from the BF-like picture presented in [18]. Here, wewill also introduce most of the conventions that we will use later.
Partition function
JT gravity was rephrased in [18] as a BF theory with gauge group SL(2 , R ), whose action is S BF ( φ, A ) = − i (cid:90) M Tr ( φF ) − κ (cid:90) ∂ M dτ Tr (cid:0) φ (cid:1) . (2.1)The gauge field A and the scalar φ belong to the adjoint representation of sl (2 , R ) algebra.The boundary potential is needed to recover the Schwarzian dynamics on ∂ M . We assumethat the manifold M has the topology of the disk. The coupling costant κ is related tothe gravitational coupling costant G N by κ = π G N φ r where φ r is the renormalized value ofdilaton on the boundary.The structure of the action (2.1) is reminiscent of that of 2D Yang-Mills. However thereare two main differences: the quadratic potential is localized on the boundary and thegauge group is non-compact. The partition function for YM theory on a disk can be Actually the true gauge group is a certain central extension of PSL(2 , R ) by R [18]. For anotherdiskussion of the gauge structure of JT gravity see [17] – 4 –onstructed by applying standard Hamiltonian quantization techniques and one gets thefollowing partition function: Z
2D YMdisk ( g, a ) = (cid:88) R dim R χ R ( g ) e − κac (R) , (2.2)where the sum extends over all possible representations R of the compact gauge group, g isthe holonomy of A around the boundary, a the total area of the disk, c (R) the quadraticCasimir. Finally χ R stands for the character basis χ R ( g ) = Tr R ( g ). Figure 1 . The blue line crossing the disk cor-responds to the boundary anchored Wilson line W λ (C τ τ ) , which splits the original surface intotwo regions that can be continuosly deformed intoa disk. Following the same logic that led to (2.2),we find that the disk partition function ofthe theory (2.1) must have the followingstructure Z BFdisk ( g, β ) = (cid:90) d R ρ (R) χ R ( g ) e − κβc (R) . (2.3)In the case of a non-compact gauge group,such as SL (2 , R ), the unitary irreduciblerepresentations span a continuous spec-trum, so we have replaced the sum withan integral, where each representationis weighted with the Plancherel measure ρ ( R ). Moreover the renormalized bound-ary length β takes the place of the area a .For the theory (2.1) one can argue thatonly the representations belonging to thePrincipal series contribute [18] and one is left with Z BFdisk ( β ) = (cid:90) + ∞ ds s sinh (2 πs ) e − κβs = (cid:18) πκβ (cid:19) e π βκ , (2.4)where we have chosen the trivial holonomy around the boundary. Boundary anchored Wilson lines.
The most interesting and natural observables in thetheory (2.1) are Wilson loops or, more generically, Wilson lines. Since any closed contouris homotopic to a point on the disk, Wilson loops are all trivial for this ( almost ) topologicaltheory. On the contrary, Wilson lines whose extrema are anchored to the disk’s boundaryhave a non-trivial expectation value. This kind of holonomies splits the original disk intotwo patches, which are homeomorphic to two disks (see fig. 1). Then we can use the cut andsewing techniques typical of topological theories to compute the expectation value of thisobservable [18]. Specifically we glue together two partition functions of the disk along thecommon boundary (the blu line C τ τ in fig. 1) where we have also inserted the observable Glue operatively means that we integrate over all possible holonomies h on the path C τ τ connectingthe boundary points τ and τ . – 5 –orresponding to the Wilson line, namely the character χ λ ( h ) with h = P exp (cid:82) C τ τ A and λ the representation of the Wilson line. We get (cid:104)W λ (C τ τ ) (cid:105) ∝ (cid:90) dh Z BFdisk ( g h, τ ) χ λ ( h ) Z BFdisk (cid:0) h − g , τ (cid:1) (2.5)In eq. (2.5) τ ≡ τ and τ = β − τ are the lengths of the two complementary red boundariesappearing in fig. 1. Next we substitute the expression for the partition functions of thetwo disks and we get (cid:104)W λ (C τ τ ) (cid:105) ∝ (cid:90) + ∞ ds s sinh (2 πs ) e − κτs (cid:90) + ∞ ds s sinh (2 πs ) e − κ ( β − τ ) s ×× (cid:90) dh χ s ( h ) χ λ ( h ) χ s ( h − ) (2.6)The last integral in (2.6) defines the so called fusion numbers N s s ,λ , namely the coefficientscounting how many times the irreducible representation s appears into the tensor productof the representations s and λ . For the group SL (2 , R ), the fusion number is given by a3-j symbol coefficient N s s ,λ = Γ [ λ ± is ± is ]Γ [2 λ ] (2.7)where Γ [ x ± y ± z ] ≡ Γ [ x + y + z ] Γ [ x + y − z ] Γ [ x − y + z ] Γ [ x − y − z ]. Thus the exactevaluation of the Wilson line (cid:104)W λ ( τ ) (cid:105) disk on the disk topology finally yields (cid:104)W λ ( τ ) (cid:105) disk = N d (cid:90) ∞ (cid:90) ∞ ds ds s s sinh (2 πs ) sinh (2 πs ) Γ [ λ ± is ± is ]Γ [2 λ ] e − κτs − κ ( β − τ ) s . (2.8)where N d is a normalization con stant proportional to the inverse of the partition func-tion: we choose it as N d ≡ κ λ π Z − = κ λ ( κβ ) π e − π βκ . The result (2.8) perfectly agreesfor instance with the computation performed in [16] via the conformal bootstrap in theSchwarzian theory. The first non-trivial topology beyond the disk is the trumpet , a two dimensional manifoldwith the topology of the cylinder. The first boundary is analogous to that of the disk,while the other is an asymptotic one whose geodesic length is related to the parameter b .The partition function of JT gravity on this genus-one manifold is given by the integral ofa slightly modified spectral density [20], namely Z trump . ( β ) = (cid:90) + ∞ ds cos (2 πbs ) e − κβs = (cid:18) πκβ (cid:19) e − b π βκ (2.9)Following the same logic that led to (2.8) in the case of the disk, we can write down theexpectation value of a boundary anchored Wilson line. Since this type of path will split From now on we set the total holonomy around the boundary to be trivial g g = 1 and so for semplicitywe choose g = g = 1. – 6 –he trumpet into two regions, homeomorphic to a disk and a trumpet, we can easily showthat (cid:104)W λ ( τ ) (cid:105) trump . = N t (cid:90) ∞ (cid:90) ∞ ds ds s sinh (2 πs ) cos (2 πbs ) Γ [ λ ± is ± is ]Γ [2 λ ] e − κτs − κ ( β − τ ) s . (2.10)with N t ≡ κ λ π Z − = κ λ (cid:16) κβπ (cid:17) e b π βκ . From the point of view of anchored Wilsonloops, this correlator describes bi-local lines not winding around the defects. It was observedin [20] that this observable could not arise from free matter in the bulk since it does notsatisfy the KMS condition (which is equivalent to periodicity around the boundary circle).Therefore they generalized the bi-local operator to satisfy the KMS condition, including anexplicit sum over integers in its definition. Computing the correlators with the improvedoperator is equivalent to sum over Wilson lines encircling the defect, with fixed anchoredpoints. Self-intersections naturally appear for non-trivial windings with the associated6 j -symbols, complicating the evaluation of the two-point function. In [18] it was suggested that the bulk Wilson line anchored to the points τ and τ of theboundary is dual to the bi-local correlator of conformal dimension λ : O ( τ ≡ τ − τ ) = (cid:20) t (cid:48) ( τ ) t (cid:48) ( τ )( t ( τ ) − t ( τ )) (cid:21) λ (2.11)computed in the Schwarzian theory, whose action is S Sch [ t ] = φ r π G N (cid:90) ∂ M dτ { t ( τ ) , τ } = 12 κ (cid:90) ∂ M dτ { t ( τ ) , τ } . (2.12)Here the fundamental field t ( τ ) plays the role of a boundary reparameterization mode,or boundary graviton. The expectation value (cid:104)O ( τ ) (cid:105) is found by inserting (2.11) insidethe path integral over the boundary mode t weighted by the Schwarzian action . We canuse this representation to compute these observables perturbatively. This result indirectlyprovides a check for the exact formulae (2.8) and (2.10). Below, we shall briefly describehow to do this perturbative analysis in the trumpet’s less trivial case. For the disk, we referto [5, 21]. As already remarked in the previous subsection, we do not take into accounthere the modified definition of the bi-local operator proposed in [20] to implement the KMScondition.The classical equations of motion for (2.12) are solved by a field t ( τ ) with a constantSchwarzian derivative. In the trumpet case, the classical saddle can be parameterized as t ( τ ) = e − ϑ ( τ ) ϑ ( τ ) = 2 πbβ ( τ + ε ( τ )) , (2.13) On the disk the Schwarzian path integral is (cid:104)O ( τ ) (cid:105) = (cid:90) D t SL(2 , R ) e − S Sch [ t ] O ( τ )where SL (2 , R ) are gauge redundancies of the Schwarzian action. When a hole is inserted, this breaks thegauge group to U(1). – 7 –here ε ( τ ) is a small fluctuation over the classical background . Plugging eq. (2.13) intoeq. (2.11), we find at zero order in ε (cid:104)O ( τ ) (cid:105) tr . tree = πbβ sinh (cid:16) πbβ τ (cid:17) λ , (2.14)which is the tree level amplitude for the correlator on the trumpet geometry. To computethe quantum correction to the tree-level result, we must determine the propagator for thefield ε . Expanding the action (2.12) to order (cid:15) around the saddle (2.13) we find S ε = − κ (cid:90) β d τ (cid:34) ε (cid:48)(cid:48) ( τ ) + (cid:18) πbβ (cid:19) ε (cid:48) ( τ ) (cid:35) = − β κ (cid:18) πβ (cid:19) (cid:88) n ∈ Z ε n ε − n n (cid:0) n + b (cid:1) , (2.15)where we have Fourier-expanded the fluctuation as ε ( τ ) = (cid:80) n ∈ Z ε n e πinτβ . We recognizethe presence of a zero mode ( n = 0) associated with the residual U(1) gauge redundancypresent in the trumpet geometry. The propagator can be found by inverting the quadraticaction and we get (cid:104) ε (0) ε ( τ ) (cid:105) = κβ π (cid:88) n (cid:54) =0 e πinτβ n ( n + b ) = (2.16)= βκ (cid:16) π b (cid:0) β − βτ + 6 τ (cid:1) + 3 β − πβ b csch( πb ) cosh (cid:16) πb ( β − τ ) β (cid:17)(cid:17) π b . where the sum over negative and positive integers has been computed in terms of elementaryfunctions by exploiting standard complex analysis techniques [21].To obtain the correction of order κ to this observable, we do not need to proceed further inexpanding the action. We have instead to expand the bi-local correlator (2.11) around thesaddle (2.13) up to order ε . The O ( ε ) has vanishing expectation value since the one-pointfunction is zero for the quadratic action (2.15). Normalizing with respect to the tree level(2.14), we get λ β (cid:26) b π (cid:18) λ coth πbτβ + 12 csch πbτβ (cid:19) ( ε ( τ ) − ε ( τ )) + β (cid:104) λ (cid:0) ε (cid:48) ( τ ) + ε (cid:48) ( τ ) (cid:1) −− ε (cid:48) ( τ ) − ε (cid:48) ( τ ) (cid:3) + 4 πbλβ coth (cid:18) πbτβ (cid:19) (cid:2) ( ε ( τ ) − ε ( τ )) (cid:0) ε (cid:48) ( τ ) + ε (cid:48) ( τ ) (cid:1)(cid:3)(cid:27) (2.17)We now substitute every appearance of ε − combination with their expectation value atthis order, i.e. with the propagator (2.16) (cid:104) ε (0) ε ( τ ) (cid:105) ≡ G ( τ ) or its derivatives. Introducing This parametrization can be justified by looking at the metric solution for the disk and the trumpet inRindler coordinates, which are respectevelyd s = d (cid:37) + sinh (cid:37) d τ d s = d σ + cosh σ d ϑ where the coordinate ϑ obeys the twisted periodicty ϑ ∼ ϑ + b . The relation between the τ and ϑ coordinatesat the boundary of the regular hyperbolic disk is cos τ = tanh ϑ and therefore this implies t = tan τ = e − ϑ . – 8 –he auxiliary combination ξ = τβ , the first perturbative term finally reads as (cid:104)O ( τ ) (cid:105) tr . (cid:104)O ( τ ) (cid:105) tr . tree = 1 + βκλ π b csch ( πbξ ) (cid:2) λ − − π b ( λ + 1)( ξ − ξ + (2.18)+ bπ ( λ (4 ξ − −
1) sinh(2 πbξ ) + (cid:0) − λ (cid:0) π b ( ξ − ξ + 1 (cid:1)(cid:1) cosh(2 πbξ ) (cid:3) + O ( κ ) . We will reproduce the above expression from the exact formula (2.10) in the next section.
In the following our goal is to illustrate how the perturbative results for the bi-local corre-lator (cid:104)O ( λ ) ( τ ) (cid:105) disk β on the disk can be recovered from its exact integral representation (2.8).A simple-minded Taylor-expansion of the integrand would lead to divergent expressionssince the representation (2.8) is naturally suited to derive the large κ expansion for thetwo-point function.To obtain the perturbative series in κ we have to rearrange the κ dependence of (2.8) andwe start by using the following identity, first derived by Ramanujan: (cid:90) ∞−∞ d p sech a (cid:16) p (cid:17) e ipx = 2 a − Γ(2 a ) Γ ( a + ix ) Γ ( a − ix ) . (3.1)which holds for Re( a ) >
0. In this way we can express the 3-j symbol in Fourier space andwe get (cid:104)O ( λ ) ( τ ) (cid:105) disk β = N d Γ(2 λ )2 λ − (cid:90) ∞ (cid:90) ∞ d s d s s s sinh 2 πs sinh 2 πs e − κ ( β − τ ) s − κτs ×× (cid:90) ∞−∞ dp (cid:90) ∞−∞ dq e ip ( s + s )+ iq ( s − s ) sech λ p λ q s and s , we can extend the region of integration to theentire real line and subsequently perform the gaussian integration over s and s . We areleft with a double integral over p and q : (cid:104)O ( λ ) ( τ ) (cid:105) disk β = π N d Γ(2 λ )2 λ +2 κ τ / ( β − τ ) / (cid:90) ∞−∞ dpdq q − ( p − iπ ) cosh λ p cosh λ q e − ( p + q − iπ )24 κ ( β − τ ) − ( p − q − iπ )24 κτ (3.3)The original symmetry in the exchange τ ↔ β − τ in (2.8) is now realized by the changeof variables q ↔ − q . Next we perform the shift p → p + 2 πi by considering the contourdisplayed in fig. 2 in the complex p − plane. In the following we shall assume that 2 λ (cid:54)∈ N .Then the contour encircles the branch cut, due to (cosh p ) − λ , that has been chosen to runfrom p = πi to p = ∞ + πi . Moreover, we take 0 < λ < λ > The case 2 λ ∈ N , where the cut is replaced by a pole, will be diskussed in detail in sec. 4. This position of the cut is obtained by choosing the phase around the branch point between ( − π , π ]. – 9 – e pIm p π i π i I IIIIIV VIVIIVIII IV
Figure 2 . Contour in the complex p-plane used to perform the shift p → p + 2 πi . The contributions of the vertical edges (II, VI and VIII) of the contour vanish when weapproach infinity and thus the original integral (edge I) can be replaced by the two termscoming respectively from the horizontal edge (VII) and the diskontinuity around the cut (cid:104)O ( λ ) ( τ ) (cid:105) disk β = π N d e − πiλ Γ(2 λ )2 λ +2 κ τ / ( β − τ ) / (cid:90) ∞−∞ dpdq q − p cosh λ p cosh λ q e − ( p + q )24 κ ( β − τ ) − ( p − q )24 κτ −− πi N d e πiλ sin (2 πλ ) Γ(2 λ )2 λ +1 κ τ ( β − τ ) (cid:90) ∞−∞ dq (cid:90) ∞ dt q − ( t − iπ ) cosh λ q sinh λ t e − ( t + q − iπ )24 κ ( β − τ ) − ( t − q − iπ )24 κτ (3.4)The first integral in (3.4) vanishes because of the antisymmetry in the exchange p ↔ q . Inthe second one we can safely perform the following shift q → q + ( t − iπ )( β − τ ) β (3.5)since we do not encounter any branch cut or singularity of the integrand during this process(at least for generic values of β and τ ). This shift centers the integral at q = 0 and weobtain (cid:104)O ( λ ) ( τ ) (cid:105) disk β = − πi N d e πiλ sin (2 πλ ) Γ(2 λ )2 λ +1 κ β ξ (1 − ξ ) × (3.6) × (cid:90) ∞−∞ dq (cid:90) ∞ dt ( q + 2( t − πi )(1 − ξ ))( q − t − πi ) ξ )cosh λ q +( t − πi )(1 − ξ )2 sinh λ t e − q κβξ (1 − ξ ) − ( t − πi )2 βκ where we have found it convenient to introduce the auxiliary combination ξ ≡ τβ . Theform (3.6) of the integral representation is suited to identify the origin of the dominantcontributions in the limit κ →
0. A neighborhood around q = 0 dominates the integrationover q due to the integrand’s gaussian weight. For the same reason, one might assumethat the integration over t is also primarily controlled by a small interval around t = πi .On the other hand, since t spans the semi-infinite interval [0 , + ∞ ], we have to considera second candidate, namely the neighbourhood around t = 0 (see [30] for the general– 10 –heory). Comparing the two possibilities, we find that the integral (3.6) in the limit κ → e π κβ .The simplest way to construct systematically the asymptotic series in the limit κ → q (cid:55)→ √ κq t (cid:55)→ κt. (3.7)The different scaling of the variable t takes into account that the leading contribution comesfrom the lower extremum of the integral and not from a saddle-point. Using the explicitform of the normalization N d we get (cid:104)O ( λ ) ( τ ) (cid:105) disk β = − ie πiλ κ λ sin(2 πλ )Γ(2 λ )2 λ +2 π / β / (1 − ξ ) / ξ / × (3.8) × (cid:90) ∞−∞ dq (cid:90) ∞ dt ( √ κq + 2( κt − πi )(1 − ξ ))( √ κq − κt − πi ) ξ )cosh λ √ κq +( κt − πi )(1 − ξ )2 sinh λ κt e − q βξ (1 − ξ ) − κt β − πitβ The non-analytic factor e − π βκ present in N d cancels exactly against the constant term in thegaussian weight for t and we can Taylor-expand the integrand (3.8) around κ = 0, obtaininga series with both integer and semi-integer powers of κ . The latter is always proportionalto an odd power of q and vanish when the integral is performed. Moreover, the expansiongenerates integrals over t , which are divergent for real β . We can take care of this issue byrotating our path of integration in t of a small positive angle α before expanding. Oncewe have integrated over t , the final result does not depend on α. Alternatively, we couldassume that β has a small imaginary part and then analytically continue to real values.After integrating over t and q term by term, we find (cid:104)O ( λ ) ( τ ) (cid:105) disk β = π λ β λ sin λ ( πξ ) (cid:104) κβλ π sin ( πξ ) (2 π ( λ + 1) ξ − π ( λ + 1) ξ −− π (2 λ + 1)(2 ξ −
1) sin(2 πξ ) + (2 λ ( π ( ξ − ξ − −
1) cos(2 πξ ) + 2 λ + 1) + O ( κ ) (cid:105) (3.9)In this expansion, we recognize the classical term and the one-loop contribution obtainedby a direct diagrammatic computation in [21]. A systematic all order expansion can alsobe obtained by expanding the integrand in terms of generalized Apostol-Eulerian andBernoulli polynomials. However, the final expression is not particularly appealing, and wewill concentrate on the particular case 2 λ ∈ N . The structure of the bi-local operator on the trumpet (2.10) is quite similar to the case ofthe disk and if we use the symmetry of the integrand, we can rearrange it in the followingform: (cid:104)O ( λ ) ( τ ) (cid:105) tr .β = N t (cid:90) ∞−∞ (cid:90) ∞−∞ d s d s s e π ( s + ibs ) − κ ( β − τ ) s − κτs ×× Γ ( λ − is − is ) Γ ( λ + is + is ) Γ ( λ + is − is ) Γ ( λ − is + is )Γ(2 λ ) . (3.10)– 11 –s in the case of the disk, we can use the identity (3.1) to eliminate the Gamma function andperform the gaussian integration over s and s . We find this new integral representationfor the bi-local correlator (2.10): (cid:104)O ( λ ) ( τ ) (cid:105) tr .β = iπ N t Γ(2 λ )2 λ +1 κ √ τ ( β − τ ) (cid:90) ∞−∞ dpdq ( p + q − πi )cosh λ (cid:0) p (cid:1) cosh λ (cid:0) q (cid:1) e − (2 πb + p − q )24 κ ( β − τ ) − ( p + q − iπ )24 κτ . (3.11)Next we shift the variables of integration as follows p (cid:55)→ p − πb q (cid:55)→ q + πb and we get (cid:104)O ( λ ) ( τ ) (cid:105) tr .β = iπ N t Γ(2 λ )2 λ +1 κ √ τ ( β − τ ) (cid:90) ∞−∞ dpdq ( p + q − πi ) e − ( p − q )24 κ ( β − τ ) − ( p + q − πi )24 κτ cosh λ (cid:16) p − πb (cid:17) cosh λ (cid:16) q + πb (cid:17) . (3.12)Again we perform the shift p → p +2 πi by considering a contour similar to the one displayedin fig. 2. The only difference is the position of the branch cut that now runs p = πb + πi to p = ∞ + πi . As in the case of the disk we assume that 2 λ (cid:54)∈ N and 0 < λ <
1. We get (cid:104)O ( λ ) ( τ ) (cid:105) tr .β = iπe − πiλ N t Γ(2 λ )2 λ +1 κ √ τ ( β − τ ) (cid:90) ∞−∞ dpdq ( p + q ) e − ( p − q +2 πi )24 κ ( β − τ ) − ( p + q )24 κτ cosh λ (cid:16) p − πb (cid:17) cosh λ (cid:16) q + πb (cid:17) + (3.13)+ 2 π N t Γ(2 λ ) e πiλ sin (2 πλ )2 λ +1 κ √ τ ( β − τ ) (cid:90) ∞−∞ dq (cid:90) ∞ dt ( t + q − πi + πβ ) e − ( t − q + πb + πi )24 κ ( β − τ ) − ( t + q − πi + πb )24 κτ sinh λ (cid:0) t (cid:1) cosh λ (cid:16) q + πb (cid:17) . The first integral vanishes because it is odd under the transformations p (cid:55)→ − q and q (cid:55)→ − p .In the second integral we perform a shift in q to center the gaussian weight around q = 0and we obtain the analog of (3.6): (cid:104)O ( λ ) ( τ ) (cid:105) tr .β = π N t Γ(2 λ ) sin(2 πλ )16 λ β κ √ − ξξ / ×× (cid:90) ∞−∞ dq (cid:90) ∞ dt ξ ( πb + t ) + q sinh λ (cid:0) t (cid:1) sinh λ (cid:0) (2 ξ ( πb + t ) + q − t ) (cid:1) e − ( πb + t )2 βκ − q βκξ (1 − ξ ) (3.14)where we have again introduced the auxiliary combination ξ ≡ τβ . As in the previous case,the integration over q is again dominated by a neighbourhood around q = 0 due to thegaussian weight in the integrand. Since t spans the semi-infinite interval [0 , + ∞ ] and inthis interval the gaussian weight is monotonic (for b > t in the limit κ → t = 0.Next we scale the variables t and q as in (3.7) and expand the integrand around κ = 0.Performing the two integrations term by term we find (cid:104)O ( λ ) ( τ ) (cid:105) tr .β = π λ b λ β λ sinh λ ( πbξ ) (cid:20) βκλ π b sinh ( πbξ ) (cid:0) − π b ( λ + 1) ξ + 2 π b ( λ + 1) ξ ++ (cid:0) − λ (cid:0) π b ( ξ − ξ + 1 (cid:1)(cid:1) cosh(2 πbξ )+ πb ( λ (4 ξ − −
1) sinh(2 πbξ )+ 2 λ − (cid:1) + O ( κ ) (cid:21) (3.15)In this expansion we recognize the classical term and the one-loop contribution obtainedby a direct diagrammatic computation in subsec. 2.2.– 12 – e uIm u π i I IIIIIVI -v+ π i v+ π i IVVVIIVIII Γ Figure 3 . The red contour C used to perform the integration over u λ ∈ N In this section we focus our attention on a particular but very interesting case, namely2 λ ∈ N . For semi-integer values of λ , the cut present in fig. 2 is replaced by a poleof order 2 λ . For this reason it is convenient to start over our analysis from the integralrepresentation (3.3) and use as new variables of integration p = u + v q = u − v . (4.1)We get a nice and symmetric representation for the bi-local correlator on the disk: (cid:104)O ( λ ) ( τ ) (cid:105) disk β = − π N d Γ(2 λ )2 λ +3 κ τ / ( β − τ ) / (cid:90) ∞−∞ dudv ( u − πi )( v − πi )(cosh u + cosh v ) λ e − ( u − iπ )24 κ ( β − τ ) − ( v − iπ )24 κτ (4.2)Next we evaluate the integral over u in (4.2) using residues. Consider the closed red contour C depicted in fig. 3. Along this path the integral of the function f ( u, v ) = − π N d Γ(2 λ )2 λ +3 κ τ / ( β − τ ) / ( u − πi )( v − πi )(cosh u + cosh v ) λ e − ( u − iπ )24 κ ( β − τ ) − ( v − iπ )24 κτ (4.3)is identically zero as (4.3) defines a holomorphic function in the enclosed region. Sincethe contributions of the two vertical edges II and VIII vanish when they approach infinity,the integral along the entire real u − axis, i.e. the original integral, is equal to minus theintegral of f ( u, v ) along Γ (see fig. 3): (cid:104)O ( λ ) ( τ ) (cid:105) disk β = π N d Γ(2 λ )2 λ +3 κ τ / ( β − τ ) / (cid:90) ∞−∞ dv (cid:90) Γ d u ( u − πi )( v − πi ) (cid:0) cosh v + cosh u (cid:1) λ e − ( v − iπ )24 κτ − ( u − iπ )24 κ ( β − τ ) (4.4)The path Γ is composed by three straight segments (III,V and VII) and two semi-circum-ferences (IV and VI). The former three contributions either cancel or vanish because the– 13 –esulting integrand is an odd function under reflection with respect to the axis Im u . Insteadthe latter two (i.e. IV and VI) yield (cid:104)O ( λ ) ( τ ) (cid:105) disk β = − π i N d Γ(2 λ )2 λ +4 κ τ / ( β − τ ) / (cid:90) ∞−∞ dv e − ( v − iπ )24 κτ ( v − πi ) ×× Res u = v +2 πi ( u − πi ) e − ( u − iπ )24 κ ( β − τ ) (cid:0) cosh v + cosh u (cid:1) λ + Res u = − v +2 πi ( u − πi ) e − ( u − iπ )24 κ ( β − τ ) (cid:0) cosh v + cosh u (cid:1) λ = (4.5)= − π i N d Γ(2 λ )2 λ +3 κ τ / ( β − τ ) / (cid:90) ∞−∞ dv ( v − πi ) e − ( v − iπ )24 κτ Res u = v ue − u κ ( β − τ ) (cid:0) cosh v − cosh u (cid:1) λ , where we used the symmetry of the integrand to show that the two residues are equal. λ values The representation (4.5) is very efficient in reconstructing the perturbative series at allorders. To illustrate how we can recover the series for small κ , we first focus on the case λ = . Then the residue in (4.5) can be easily evaluated and is given byRes u = v u e − u κ ( β − τ ) cosh v − cosh u = − v csch v e − v κ ( β − τ ) . (4.6)Next we can recast the integral (4.5) as follows (cid:104)O ( ) ( τ ) (cid:105) disk β = iπ N d e π βκ λ +1 κ τ / ( β − τ ) / (cid:90) + ∞−∞ d v v ( v − iπ )sinh v e − β (cid:18) v − iπ ( β − τ ) β (cid:19) κτ ( β − τ ) . (4.7)It is convenient to center the gaussian weight in (4.7) v = 0 through the shift v → v + iπ ( β − τ ) β ; (cid:104)O ( ) ( τ ) (cid:105) disk β = − iκ − β π τ / ( β − τ ) / (cid:90) + ∞−∞ d v (cid:16) v + iπ ( β − τ ) β v + π τ ( β − τ ) β (cid:17) e − βv κτ ( β − τ ) sinh (cid:16) v − iπτβ (cid:17) (4.8)Now we replace 1 / sinh( · · · ) with its representation in terms of exponentialscsch (cid:18) v − πiτβ (cid:19) = 2 v e − πiτβ (cid:32) v e v e v − πiτβ − (cid:33) , (4.9)and recognize that the quantity between parenthesis is the generating functional of theso-called generalized Apostol-Bernoulli polynomials of degree one. The definition of thesepolynomials for general degree and some of their properties are briefly diskussed in app.A. Therefore we directly writecsch (cid:18) v − πiτβ (cid:19) = 2 e − πiτβ ∞ (cid:88) n =0 B (1) n (cid:18) , e − πiτβ (cid:19) v n − n ! . (4.10)– 14 –f we integrate in v term by term using the expansion (4.10), we encounter only powers of v averaged over a gaussian weight. We find convenient to treat separately the even and theodd powers in (4.10). Reordering the powers in κ produced by the gaussian integrations,we obtain the following perturbative series for the expectation value of the bi-local operatorwith λ = : (cid:104)O ( ) ( τ ) (cid:105) disk β = e − πiτβ π ∞ (cid:88) p =0 p κ p τ p ( β − τ ) p Γ [ p + ] (2 p )! β p (cid:20) ( β − τ ) τ ( β − τ ) B (1)2 p (cid:18) , e − πiτβ (cid:19) −− i (2 p +1) B (1)2 p +1 (cid:18) , e − πiτβ (cid:19) − ipβπτ ( β − τ ) B (1)2 p − (cid:18) , e − πiτβ (cid:19)(cid:21) . (4.11)The case λ = 1 is slightly more involved: the explicit form of the residue isRes u = v u e − u κ ( β − τ ) (cosh v − cosh u ) = − e − v κ ( β − τ ) sinh v (cid:18) v κ ( β − τ ) + v coth (cid:16) v (cid:17) − (cid:19) , (4.12)and we can perform again the previous analysis. This time the dependence on 1 / sinh( · · · ) isaccompanied by higher powers, namely 1 / sinh ( · · · ) and 1 / sinh ( · · · ). With computationssimilar to the case λ = 1 /
2, we can also obtain with little effort the all order expansion (cid:104)O (1) ( τ ) (cid:105) disk β = e − iπτβ π ∞ (cid:88) p =0 p κ p τ p ( β − τ ) p Γ ( p + ) (2 p )! β p (cid:104)(cid:16) p − p − β τ ( β − τ ) − τ ( β − τ ) (cid:17) B (2)2 p (cid:16) , e − πiτβ (cid:17) − iπ (2 p +1) β − τβτ ( β − τ ) B (2)2 p +1 (cid:16) , e − πiτβ (cid:17) − π β (2 p +1)(2 p +2) B (2)2 p +2 (cid:16) , e − πiτβ (cid:17) − ip π β ( β − τ ) τ ( β − τ ) B (2)2 p − (cid:16) , e − πiτβ (cid:17)(cid:105) (4.13)where also generalized Apostol-Bernoulli polynomials of degree 2 appears in the expansion.The same analysis can be easily carried out for the trumpet. Here the starting point is (cid:104)O ( λ ) ( τ ) (cid:105) tr .β = − π Γ(2 λ ) N λ +1 κ ( β − τ ) τ (cid:90) + ∞−∞ d v Res u = v u e − u κ ( β − τ ) (cid:0) cosh v − cosh u (cid:1) λ e − ( v +2 bπ )24 κτ (4.14)The case λ = 1 / λ = 1 are again obtained along the same lines diskussed above andone gets (cid:104)O ( ) ( τ ) (cid:105) tr .β = e − bπτβ π ∞ (cid:88) p =0 p κ p τ p ( β − τ ) p Γ [ p + ] (2 p !) β p (cid:18) τ B (1)2 p (cid:18) , e − bπτβ (cid:19) - bπ (2 p +1) β B (1)2 p +1 (cid:18) , e − bπτβ (cid:19)(cid:19) (4.15)– 15 –nd (cid:104)O (1) ( τ ) (cid:105) tr .β = e − bπτβ π ∞ (cid:88) p =0 p κ p τ p ( β − τ ) p Γ [ p + ] (2 p !) β p (cid:104)(cid:16) τ − β + bπττ ( β − τ )(2 p − (cid:17) B (2)2 p (cid:16) , e − bπτβ (cid:17) − bπβτ (2 p +1) B (2)2 p +1 (1 , e − πbτβ )+ b π β (2 p +1)(2 p +2) B (2)2 p +2 (cid:16) , e − bπτβ (cid:17) + pβτ ( β − τ )(2 p − B (2)2 p − (cid:16) , e − πbτβ (cid:17) + τ ( β − τ )(2 p − (cid:16) β B (3)2 p (cid:16) , e − πbτβ (cid:17) − bπτ (2 p +1) B (3)2 p +1 (cid:16) , e − πbτβ (cid:17)(cid:17)(cid:105) (4.16)The trumpet expressions become a little bit more cumbersome since we lose the symmetry τ → β − τ , as expected since we are working in the zero winding sector. As a consequence,we also notice the presence of generalized Apostol-Bernoulli polynomials of degree 3. Having trained with the simplest cases, we are ready now to perform the computation forgeneric semi-integer λ . In app. A we have shown that the residue in (4.5) can always becomputed in terms of generalized Apostol-Bernoulli polynomials. The structure of theanswer is Res u = v (cid:32) u e − αu (cid:0) cosh v − cosh u (cid:1) λ (cid:33) = e − αv f ( v ) (4.17)where α = κ ( β − τ ) and f ( v ) = e λv λ − (cid:88) (cid:96) =0 ( − λ − (cid:96) − α λ − (cid:96) − H λ − (cid:96) ( √ αv )(2 λ − (cid:96) − (cid:96) (cid:88) j =0 B (2 λ ) (cid:96) − j ( λ ) B (2 λ ) j +2 λ ( λ, e v )( (cid:96) − j )!(2 λ + j )! . (4.18)In (4.18) H n ( x ) stands for the usual Hermite polynomials, while B nk ( x, y ) and B nk ( x ) arerespectively generalized Apostol-Bernoulli of degree n and generalized Bernoulli polyno-mials. Their definition and some of their properties are diskussed in app. A. Then theremaining integral in v takes the following form: (cid:104)O ( λ ) ( τ ) (cid:105) disk β = − iπ Γ(2 λ ) N d λ +2 κ τ / ( β − τ ) / (cid:90) + ∞−∞ d v ( v − iπ ) f ( v ) e − ( v − iπ )24 κτ − v κ ( β − τ ) . (4.19)Next we perform the shift v (cid:55)→ v + πi ( β − τ ) β to move the gaussian center around v = 0. Wefind (cid:104)O ( λ ) ( τ ) (cid:105) disk β = − iπ Γ(2 λ ) κ λ − β λ +3 π τ ( β − τ ) (cid:90) + ∞−∞ d v (cid:18) v − πiτβ (cid:19) f (cid:18) v + 2 πi ( β − τ ) β (cid:19) e − βv κτ ( β − τ ) . (4.20)Using the result (B.6), we can immediately expand f (cid:16) v + πi ( β − τ ) β (cid:17) in powers of vf (cid:18) v + 2 πi ( β − τ ) β (cid:19) = (4.21)= ( − λ ∞ (cid:88) m =0 v m m ! λ − (cid:88) (cid:96) =0 ( − λ − (cid:96) − α λ − (cid:96) − H λ − (cid:96) (cid:16) √ α (cid:16) v + πi ( β − τ ) β (cid:17)(cid:17) (2 λ − (cid:96) − c ( λ ) (cid:96),m ( β, τ ) , – 16 –here c ( λ ) (cid:96),m ( β, τ ) denotes the following combination c ( λ ) (cid:96),m ( β, τ ) = e − πiλτβ (cid:96) (cid:88) j =0 ( j + m )!( (cid:96) − j )!(2 λ + j + m )! j ! B (2 λ ) (cid:96) − j ( λ ) B (2 λ ) j +2 λ + m (cid:16) λ ; e − πiτβ (cid:17) . (4.22)Plugging the expansion (4.21) into the integral (4.20), we get a series representation forour correlator (cid:104)O ( λ ) ( τ ) (cid:105) disk β = − iπ ( − λ Γ(2 λ ) κ λ − β λ +3 π τ ( β − τ ) ∞ (cid:88) m =0 m ! λ − (cid:88) (cid:96) =0 ( − λ − (cid:96) − (2 λ − (cid:96) − c ( λ ) (cid:96),m ( β, τ ) × (4.23) × α λ − (cid:96) − (cid:90) + ∞−∞ d v v m H λ − (cid:96) (cid:18) √ α (cid:18) v + 2 πi ( β − τ ) β (cid:19)(cid:19) (cid:18) v − πiτβ (cid:19) e − βv κτ ( β − τ ) . To single out the dependence on κ , we scale our variable of integration as follows v → (cid:113) κτ ( β − τ ) β and we get (cid:104)O ( λ ) ( τ ) (cid:105) disk β = − iπ ( − λ Γ(2 λ ) κ λ − β λ +3 π τ ( β − τ ) ∞ (cid:88) m =0 m +1 ( κτ ( β − τ )) m +12 β m +12 m ! λ − (cid:88) (cid:96) =0 ( − λ − (cid:96) − (2 λ − (cid:96) − c ( λ ) (cid:96),m ( β, τ ) ×× (cid:18) κ ( β − τ ) (cid:19) λ − (cid:96) − (cid:90) + ∞−∞ d v v m H λ − (cid:96) (cid:18)(cid:114) τβ v + iπ √ β − τβ √ κ (cid:19)(cid:32) (cid:115) κτ ( β − τ ) β v − πiτβ (cid:33) e − v . (4.24)Next we exploit a simple rule holding for Hermite polynomials with shifted argument H λ − (cid:96) (cid:18)(cid:114) τβ v + iπ √ β − τβ √ κ (cid:19) = λ − (cid:96) (cid:88) k =0 (cid:18) λ − (cid:96)k (cid:19) H k (cid:18)(cid:114) τβ v (cid:19) (cid:18) πi √ β − τβ √ κ (cid:19) λ − (cid:96) − k , (4.25)to rearrange our correlator in the form (cid:104)O ( λ ) ( τ ) (cid:105) disk β = − iπ ( − λ Γ(2 λ ) κ λ − β λ +3 π τ ( β − τ ) ∞ (cid:88) m =0 m +1 ( κτ ( β − τ )) m +12 β m +12 m ! λ − (cid:88) (cid:96) =0 ( − λ − (cid:96) − (2 λ − (cid:96) − c ( λ ) (cid:96),m ( β, τ ) ×× (cid:18) κ ( β − τ ) (cid:19) λ − (cid:96) − λ − (cid:96) (cid:88) k =0 (cid:18) λ − (cid:96)k (cid:19)(cid:18) πi √ β − τβ √ κ (cid:19) λ − (cid:96) − k (cid:32)(cid:115) κτ ( β − τ ) β P k,m +1 − πiτβ P k,m (cid:33) , (4.26)where P k,m = (cid:90) ∞−∞ dv v m H k (cid:18)(cid:114) τβ v (cid:19) e − v . (4.27)This integral yields a polynomial of order k in (cid:113) τβ and its explicit expression in terms ofthe associated Legendre function is given in (C.6). Obviously P k,m is different from zeroonly when m + k is an even number. We shall use this selection rule to rearrange the twocontributions proportional to P k,m +1 [(A)] and P k,m [(B)] respectively. For the former the– 17 –election rule is m + k + 1 = 2 p with p = 1 , · · · , ∞ . We can use this result to replace thesum over m with a sum over p ( A ) = − iπ ( − λ Γ(2 λ )2 λ π ∞ (cid:88) p =1 κ p + (cid:96) (2 p − k − (cid:18) κτ ( β − τ ) β (cid:19) p + (cid:96) − λ × (4.28) × λ − (cid:88) (cid:96) =0 min(2 λ − (cid:96), p − (cid:88) k =0 ( − λ − (cid:96) − (2 λ − (cid:96) − c ( λ ) (cid:96), p − k − ( β, τ ) (cid:18) λ − (cid:96)k (cid:19)(cid:18) πi √ τ ( β − τ ) β (cid:19) λ − (cid:96) − k (cid:18) τβ (cid:19) λ − (cid:96) − P k, p − k . Next we introduce a new index n ≡ k + (cid:96) , which simply counts the power of the couplingconstant( A ) = ( − πi ) λ +1 Γ(2 λ )(2 β ) λ π ∞ (cid:88) n =1 (cid:18) κτ ( β − τ ) β (cid:19) n min( n, λ − (cid:88) (cid:96) =0 ( − λ − (cid:96) − (2 λ − (cid:96) − × (4.29) × min(2 λ − (cid:96), n − (cid:96) − (cid:88) k =0 c ( λ ) (cid:96), n − (cid:96) − k − ( β, τ )(2 n − (cid:96) − k − (cid:18) λ − (cid:96)k (cid:19)(cid:18) β πiτ ( β − τ ) (cid:19) (cid:96) + k (cid:18) τβ (cid:19) k − P k, n − (cid:96) − k . The same analysis is done for the latter contribution, taking into account the constraint m + k = 2 p with p = 0 , , · · · , ∞ :( B ) = (cid:16) πiτβ (cid:17) iπ ( − λ Γ(2 λ )2 λ π ∞ (cid:88) p =0 κ p + (cid:96) (2 p − k )! (cid:18) κτ ( β − τ ) β (cid:19) p + (cid:96) − λ × (4.30) × λ − (cid:88) (cid:96) =0 min(2 p, λ − (cid:96) ) (cid:88) k =0 ( − λ − (cid:96) − (2 λ − (cid:96) − c ( λ ) (cid:96), p − k ( β, τ ) (cid:18) λ − (cid:96)k (cid:19)(cid:18) πi √ τ ( β − τ ) β (cid:19) λ − (cid:96) − k (cid:18) τβ (cid:19) λ − (cid:96) − P k, p − k . Setting again n = p + (cid:96) , we get( B ) = − (cid:16) πiτβ (cid:17) ( − πi ) λ +1 Γ(2 λ )(2 β ) λ π ∞ (cid:88) n =0 (cid:18) κτ ( β − τ ) β (cid:19) n × (4.31) × min( n, λ − (cid:88) (cid:96) =0 min(2 n − (cid:96), λ − (cid:96) ) (cid:88) k =0 ( − λ − (cid:96) − c ( λ ) (cid:96), n − (cid:96) − k ( β, τ )(2 n − (cid:96) − k )!(2 λ − (cid:96) − (cid:18) λ − (cid:96)k (cid:19)(cid:18) β πiτ ( β − τ ) (cid:19) (cid:96) + k (cid:18) τβ (cid:19) k − P k, n − (cid:96) − k . Combining the two contributions, we obtain the final expansion (cid:104)O ( λ ) ( τ ) (cid:105) disk β = π λ β λ sin λ πτβ + ( − πi ) λ +1 Γ(2 λ )(2 β ) λ π ∞ (cid:88) n =1 (cid:18) κτ ( β − τ ) β (cid:19) n min( n, λ − (cid:88) (cid:96) =0 ( − λ − (cid:96) − (2 λ − (cid:96) − ×× (cid:34) min(2 n − (cid:96) − , λ − (cid:96) ) (cid:88) k =0 c ( λ ) (cid:96), n − (cid:96) − k − ( β, τ )(2 n − (cid:96) − k − (cid:18) λ − (cid:96)k (cid:19)(cid:18) β πiτ ( β − τ ) (cid:19) (cid:96) + k (cid:18) τβ (cid:19) k − P k, n − (cid:96) − k −− πi min(2 n − (cid:96), λ − (cid:96) ) (cid:88) k =0 c ( λ ) (cid:96), n − (cid:96) − k ( β, τ )(2 n − (cid:96) − k )! (cid:18) λ − (cid:96)k (cid:19)(cid:18) β πiτ ( β − τ ) (cid:19) (cid:96) + k (cid:18) τβ (cid:19) k P k, n − (cid:96) − k (cid:35) . (4.32)We make a couple of observations on the above expression: first of all, we notice that atsufficiently large order in κ the non-trigonometric dependence on τ cannot grow arbitrarily,– 18 –eing a polynomial bounded by the weight of the bi-local operators itself. Moreover weexpect the original symmetry τ → β − τ to be preserved by the expansion: looking at thestructure of the coefficients it is not manifest but we checked its presence till the order κ . Making explicit this symmetry should probably simplify the final formula. A secondremark concerns the trigonometric dependence of the generic perturbative term and itssingularity properties as τ →
0. The trigonometric dependence is completely encoded intothe coefficients c ( λ ) (cid:96),m ( β, τ ): we expect the presence of negative powers of sin( τ /β ) , generatinga singular behaviour at small τ . This fact is also evident from the singularity appearing inthis limit for the generalized Apostol-Bernoulli polynomials. β → ∞ It is now interesting to concentrate on the zero temperature limit of the bi-local correla-tor to check explicitly the agreement with [25]. The structure of our integrals simplifiessignificantly as β → ∞ : moreover we observe that both the disk and the trumpet sharethe same behaviour in this regime, since we expected that as the total boundary lengthdiverges (while keeping b fixed) the presence of an hole in the interior becomes negligible.However this limit cannot be directly extracted from the final result of subsec. 4.2 sincethe limit of generalized Apostol-Bernoulli polynomial B ( (cid:96) ) n ( x, µ ) is discontinuous when µ approaches one. Therefore it is convenient to go back to eq. (4.2) and take the limit β → ∞ at this level. The limit of the integrand and of its normalization is smooth and we get (cid:104)O ( λ ) ( τ ) (cid:105) β →∞ = κ λ Γ(2 λ )2 λ +4 π κ τ (cid:90) ∞−∞ dudv ( u − πi )( v − πi )(cosh u + cosh v ) λ e − ( v − iπ )24 κτ (5.1)The integral over u can be now evaluated in closed form. The linear term in u vanishessince it is odd, while the contribution proportional to 2 πi yields (cid:104)O ( λ ) ( τ ) (cid:105) β →∞ = 8 πiκ λ Γ(2 λ )2 λ +4 π κ τ (cid:90) ∞−∞ dv ( v − πi )sinh λ v e − ( v − iπ )24 κτ Q λ − (cid:16) coth v (cid:17) , (5.2)where we have used that Q n (coth v )sinh n +1 v = 14 (cid:90) + ∞−∞ d u (cid:0) cosh v + cosh u (cid:1) n +1 . (5.3)In eq. (5.3) Q n ( z ) stands for the so-called Legendre function of the second kind. Nextwe eliminate the dependence on the linear factor ( v − πi ) by integrating by parts and wewrite (cid:104)O ( λ ) ( τ ) (cid:105) β →∞ = 16 πiκ λ Γ(2 λ ) κτ λ +4 π κ τ (cid:90) ∞−∞ dv e − ( v − iπ )24 κτ ddv (cid:32) Q λ − (cid:0) coth v (cid:1) sinh λ v (cid:33) , (5.4)Exploiting the recurrence relation for the Legendre function of the second kind and itsderivatives for integer indices, it is straightforward to show thatdd v (cid:34) Q λ − (cid:0) coth v (cid:1) sinh λ v (cid:35) = − λ Q λ (cid:0) coth v (cid:1) sinh λ v (5.5)– 19 –hus (cid:104)O ( λ ) ( τ ) (cid:105) β →∞ = − πiλκ λ Γ(2 λ ) κτ λ +4 π κ τ (cid:90) ∞−∞ dv e − ( v − iπ )24 κτ Q λ (cid:0) coth v (cid:1) sinh λ v . (5.6)Although the structure of the integrand suggests the possible presence of a singularity at v = 0, it is not difficult to check this singularity is only apparent. In fact a careful analysisof the integrand shows that it is completely regular at v = 0. When 2 λ is an integer Q λ can be expressed in terms of the Legendre polynomials. Specifically the following identityholds Q λ (cid:0) coth v (cid:1) = 12 P λ (cid:0) coth v (cid:1) v − W λ − (cid:0) coth v (cid:1) (5.7)with W λ − (cid:0) coth v (cid:1) = λ (cid:88) k =1 k P k − (cid:0) coth v (cid:1) P λ − k (cid:0) coth v (cid:1) (5.8)Therefore the Legendre function of the second kind is not periodic under the shift v → v + 2 iπ , but we have Q λ (cid:0) coth v (cid:1) → Q λ (cid:0) coth v (cid:1) + iπ P λ (cid:0) coth v (cid:1) . (5.9)If we perform this shift in our integral we find (cid:104)O ( λ ) ( τ ) (cid:105) β →∞ = − πi ( − λ λκ λ Γ(2 λ ) κτ λ +4 π κ τ (cid:90) ∞−∞ dv e − v κτ (cid:2) Q λ (cid:0) coth v (cid:1) + πi P λ (cid:0) coth v (cid:1)(cid:3) sinh λ v . (5.10)The combination sinh − λ (cid:0) v (cid:1) Q λ (cid:0) coth v (cid:1) is an odd function and so its contribution to theintegral (5.10) identically vanishes. So we are left with the term proportional to P λ only,i.e. (cid:104)O ( λ ) ( τ ) (cid:105) β →∞ = 16 π ( − λ λκ λ Γ(2 λ ) κτ λ +4 π κ τ (cid:90) ∞−∞ dv e − v κτ P λ (cid:0) coth v (cid:1) sinh λ v . (5.11)A remark is now in order. The final integral is singular at v = 0. If we perform, as weshould, the translation v → v + 2 iπ as a change of path in the complex v − plane, we haveto deform a little bit the contour to avoid precisely v = 0 since the integrand possessesa pole there. This small deformation provides us the prescription on how to regularizethe singularity (it is PV-like prescription). In the following, we neglect this issue andregularize this singularity using an analytic regularization, which is more straightforwardand produces the same result.Next we use the following representation of the Legendre polynomials P n ( x ) P n ( x ) = 12 n n (cid:88) k =0 (cid:18) nk (cid:19) ( x − n (cid:18) x + 1 x − (cid:19) k (5.12)For x = coth v this representation simplifies P λ (cid:16) coth v (cid:17) = (cid:18) e v − (cid:19) λ λ (cid:88) k =0 (cid:18) λk (cid:19) e kv (5.13)– 20 –nd our integral becomes (cid:104)O ( λ ) ( τ ) (cid:105) β →∞ = 16 π ( − λ λκ λ Γ(2 λ ) κτ λ +4 π κ τ λ (cid:88) k =0 (cid:0) λk (cid:1) (cid:90) ∞−∞ dv e − v κτ (cid:18) e v − (cid:19) λ e v ( k + λ ) . (5.14)We can now expand part of the integrand in terms of generalized Bernoulli polynomials (cid:18) e v − (cid:19) λ e v ( k + λ ) = ∞ (cid:88) n =0 B (4 λ ) n ( k + λ ) n ! v n − λ . (5.15)This expansion explicitly exhibits the aforementioned poles present in the integrand. Equa-tion (5.15) is a Laurent series which contains negative powers up to − λ . Then we haveto compute (cid:104)O ( λ ) ( τ ) (cid:105) β →∞ = 16 π ( − λ λκ λ Γ(2 λ ) κτ λ +4 π κ τ λ (cid:88) k =0 (cid:0) λk (cid:1) ∞ (cid:88) n =0 B (4 λ ) n ( k + λ ) n ! (cid:90) ∞−∞ d v v n − λ e − v κτ . (5.16)Since 4 λ is even, the integral is different from zero only for even n . If we set n = 2 p with p ∈ N , the gaussian integral can be now easily performed and we always get (cid:90) ∞−∞ d v v p − λ ) e − v κτ = (2 √ κτ ) p − λ )+1 Γ (cid:20) p − λ + 12 (cid:21) , (5.17)where we have defined the integral for negative powers of v by analytic continuation. Bysubstituting it in (5.16) we find (cid:104)O ( λ ) ( τ ) (cid:105) β →∞ = (2 λ )!( − λ λ π τ λ ∞ (cid:88) p =0 (4 κτ ) p (2 p )! λ (cid:88) k =0 (cid:0) λk (cid:1) B (4 λ )2 p ( k + λ )Γ (cid:20) p − λ + 12 (cid:21) . (5.18)To better understand the structure of this perturbative expansion it is convenient to sep-arate positive and negative powers of τ . Recalling the value of the Gamma function forseminteger values of the argument, we immediately find (cid:104)O ( λ ) ( τ ) (cid:105) β →∞ = (2 λ )! τ λ λ − (cid:88) p =0 ( κτ ) p (2 p )! ( − p (2 λ − p )!(4 λ − p )! λ (cid:88) k =0 (cid:0) λk (cid:1) B (4 λ )2 p ( k + λ ) ++ ( − λ λ ∞ (cid:88) r =2 λ (2 κτ ) r (2 r )! (2 r − λ − λ (cid:88) k =0 (cid:0) λk (cid:1) B (4 λ )2 r ( k + λ ) (cid:41) (5.19)One can easily check that for λ = and λ = 1 this result exactly reproduces the expressionsgiven by [25], where the first perturbative orders are presented. λ As anticipated in the introduction, in this section our goal is to show that the bi-localcorrelator can be in general expressed as a combination of Mordell integrals. Concretely,– 21 –et us go back to the case 2 λ ∈ N and to be more specific we focus on λ = 1 / (cid:104)O ( ) ( τ ) (cid:105) disk β = − π iκ − β e iπτβ τ / ( β − τ ) / (cid:90) + ∞−∞ d v v a ) + i ( β − τ ) β v ( b ) + τ ( β − τ ) β ( c ) e − π βv κτ ( β − τ ) + πv e πv − e πiτβ , (6.1)where we have also scaled the integration variable by 2 π . We recognize three different con-tributions: the third one can be immediately identified with the so-called Mordell integral[27], which appears in number theory and in the theory of Mock-theta functions [28]. Thegeneral form of the Mordell integral is M ( x, θ, ω ) = (cid:90) ∞−∞ dt e πiωt − πxt e πt − e πiθ = e − πi ( θ ω +2 θx +2 θ ) F [( x + θ ) /ω, − /ω ] + iωF [ x + θω, ω ] ωθ ( x + θω, ω ) . (6.2)The function F ( x, ω ) admits a q − expansion of the form F [ x, ω ] = − i (cid:88) m ∈ Z ( − m q ( m +1 / e πi ( m +1 / x q m +1 , (6.3)where q = e πiω . The denominator is one of the usual Jacobi theta function and its q − expansion is θ ( x, ω ) = − i (cid:88) m ∈ Z ( − m q ( m +1 / e πi ( m +1 / x . (6.4)In our case, we have x = − , θ = τβ and ω = πiβκτ ( β − τ ) . The other two contributions, (a)and (b), are proportional to the second and first derivatives of the Mordell integral withrespect to x . Then the complete result for the bi-local correlator at λ = 1 / (cid:104)O ( ) ( τ ) (cid:105) disk β = − π iκ − β e iπτβ τ / ( β − τ ) / (cid:18) π ∂ x M (cid:18) − , τβ , πiβκτ ( β − τ ) (cid:19) −− i ( β − τ )2 πβ ∂ x M (cid:18) − , τβ , πiβκτ ( β − τ ) (cid:19) + τ ( β − τ ) β M (cid:18) − , τβ , πiβκτ ( β − τ ) (cid:19)(cid:19) (6.5)The structure of the bi-local correlator for 2 λ generic integer is not so different. In fact,by carefully inspecting (4.20) we can easily verify that it is given by a sum of integrals ofthe following form (cid:90) ∞−∞ dt t n e πiωt − πxt ( e πt − e πiθ ) m , (6.6)where m and n are integers. However any integral of this kind can be evaluated in termsof the original Mordell integral: (cid:90) ∞−∞ dt t n e πiωt − πxt ( e πt − e πiθ ) m = ( − n ( m − (cid:18) π ∂ x (cid:19) n (cid:18) e − πiθ πi ∂ θ (cid:19) m − M ( x, θ, ω ) (6.7)In other words, the correlators are completely controlled by this kind of functions.– 22 – Final comments and outlook
In this work, we have considered JT bi-local correlators of operators with positive weight λ , on the disk and the trumpet topologies. The perturbative series associated to thesecorrelation functions is harder to obtain than in the parent case λ ∈ − N /
2, recently studiedin [25] in the zero temperature limit. We have been able nevertheless to distill someaspects of the κ expansion of the two-point function, checking the agreement of the exactnon-perturbative expression with the Schwarzian perturbation theory for any value of β .In the particular case of λ ∈ N /
2, we derived an all-order formula for the perturbativecontributions, that becomes particularly handful in the limit of infinite β . We have alsoshown that the exact expression for our bi-local correlators is closely related to the Mordellintegral, a basic constituent in the theory of Mock-modular forms [29].There are some lessons that we can draw from our computations and some directions thatcould be worth to study further. A feature that we may explore from the knowledgeof the entire perturbative series is, for example, the nature of possible non-perturbativecontributions to the full answer. For instance, let us consider the coefficient c r in the case β → ∞ . We can easily read it from (5.19): c r = (2 λ )! (2 r − λ − λ r τ r − λ (2 r )! (cid:34)(cid:32) λ (cid:88) k =0 (cid:0) λk (cid:1) B (4 λ )2 r ( k + λ ) (cid:33) (cid:35) (7.1)To understand its behaviour for large value of r we need to know the behaviour of thegeneralized Bernoulli polynomials in that limit. This aspect was diskussed in detail in [31],where it was found that the dominant contribution is B m r ( z ) (cid:39) − (2 r )! (cid:20) β m e πiz (2 πi ) r + β m − e − πiz ( − πi ) r (cid:21) (7.2)where β mk ( n, z ) (cid:39) ( − m − n m − ( m − . At leading order these coefficients are independent of k and (7.2) collapses to B m r ( z ) (cid:39) (2 r )! ( − m + r (2 r ) m − π ) r ( m − πz ) . (7.3)If we choose m = 4 λ the coefficient c r for r → ∞ takes the form c r (cid:39) (2 λ )! (2 r − λ − λ r τ r − λ (2 r )! (2 r )! ( − λ + r (2 r ) λ − π ) r (4 λ − − λ λ (cid:88) k =0 (cid:18) λk (cid:19) == (2 λ )! (2 r − λ − λ r τ r − λ (2 r )! (2 r )! ( − λ + r (2 r ) λ − π ) r (4 λ − − λ λ !(2 λ !) , (7.4)where we have performed the sum over the square of the binomial coefficients. We can noweasily complete our large r − expansion with the help of the Stirling formula. After sometedious algebra we find c r = 4 λ ( − λ τ r − λ e λ √ π (2 λ )! ( − r r λ − r ! π r (7.5)– 23 –he coefficient grows as a power times r ! and its global sign alternates with the parityof r . Thus the perturbative series appears to be Borel-summable: in fact the leadingpole appearing in the Borel-transform is located on the negative axis and thus one couldargue that non-perturbative instanton-like configurations should not play any role here.On the one hand, there is no guarantee that the Borel resummation of a Borel summableseries reconstructs the non-perturbative answer. There are sufficient conditions for this tobe the case, which typically require strong analyticity conditions on the underlying non-perturbative function. On the other hand, in most of the examples of Borel summableseries in quantum theories, Borel resummation does reconstruct the correct answer (see[32] for a lucid diskussion of these topics). The actual determination of non-perturbativeconfigurations, if any, remains therefore an important issue for future investigations, as wellas to understand their possible physical meaning with respect to the boundary gravitonsappearing in Schwarzian perturbation theory.The obvious extension of the present work would consist in studying the perturbativeseries associated to general four-point correlators of bi-local operators. While the exactform on the disk and the trumpet is well known [16, 18], much less has been learned onits perturbative incarnation, due to the appearing in the out-of-time-ordered case of avery complicated vertex function inside the integrals. The relevant 6- j symbols involvedthere can be expressed through Wilson function and, in principle, one could try to performan expansion using the analytical structure of the full amplitude. The success of suchcomputation would certainly improve our understanding of the properties of the associatedgravitational S-matrix.Another generalization of our investigations would concern the perturbative aspects of two-point functions in presence of defects [20]. The trumpet correlators studied here are justa particular example within this class, being associated to a bi-local operator with theinsertion of a hyperbolic defect in the bulk and computed without taking into account thewinding sectors [20]. It could be interesting to extend our analysis to the winding caseand to consider elliptic and parabolic defects too. The fate and the physics of bi-localcorrelators in presence of multiple defects [12] or for deformed JT gravity [11] could be alsoexplored. It would be nice also to understand the character of perturbative contributionsto bi-local correlators from boundary fluctuations in higher-genus geometry [33].Finally we point out that the correlators studied here were obtained in [26] from boundarycorrelators of minimal Liouville string, exploiting a particular double-scaling limit. Itwould be interesting to see if the Mordell structure, underlying the exact form of thebi-local correlator on the disk, could be understood from a Liouville perspective. Acknowledgements:
We thank Marisa Bonini and Itamar Yaakov for several discussionon different aspects of this paper. This work has been supported in part by Italian Ministerodell’Istruzione, Universit`a e Ricerca (MIUR), and Istituto Nazionale di Fisica Nucleare(INFN) through the “Gauge and String Theory” (GAST) research project.– 24 – ppendicesA Evaluation of the Residue
Most of our results can be expressed in terms of the so-called generalized Apostol-Bernoullipolynomials B ( (cid:96) ) n ( x ; µ ). If (cid:96) ∈ N , they are defined through the generating function: (cid:18) tµe t − (cid:19) (cid:96) e xt = ∞ (cid:88) n = (cid:96) B ( (cid:96) ) n ( x ; µ ) t n n ! . (A.1)This definition implies that B ( (cid:96) ) n ( x ; µ ) = 0 for n = 0 , . . . , (cid:96) −
1. The generalized Apostol-Bernoulli numbers B ( (cid:96) ) n ( µ ) are then given by B ( (cid:96) ) n ( µ ) ≡ B ( (cid:96) ) n (0; µ ) . (A.2)The familiar Bernoulli polynomials are recovered when we set (cid:96) = 1 and µ = 1. The explicitform of this polynomials can be obtained as follows. First we consider the combination (cid:16) tµe t − (cid:17) (cid:96) and write its formal expansion in power of e t − (cid:18) tµe t − (cid:19) (cid:96) = t (cid:96) ( µ − (cid:96) (cid:18) µ − µe t − (cid:19) (cid:96) = t (cid:96) ( µ − (cid:96) (cid:18) µµ − e t − (cid:19) − (cid:96) == t (cid:96) ∞ (cid:88) k =0 (cid:18) k + (cid:96) − k (cid:19) ( − µ ) k ( µ − k + (cid:96) ( e t − k (A.3)Next we use that ( e t − k = k ! ∞ (cid:88) r = k S ( r, k ) t r r ! (A.4)where S ( r, k ) denotes the Stirling numbers of the second kind. Thus (cid:18) tµe t − (cid:19) (cid:96) = ∞ (cid:88) k =0 ∞ (cid:88) r = k (cid:18) k + (cid:96) − k (cid:19) k !( − µ ) k ( µ − k + (cid:96) S ( r, k ) t r + (cid:96) r ! == ∞ (cid:88) r =0 t r + (cid:96) r ! r (cid:88) k =0 (cid:18) k + (cid:96) − k (cid:19) k !( − µ ) k ( µ − k + (cid:96) S ( r, k ) == ∞ (cid:88) r =0 t r + (cid:96) ( r + l )! (cid:96) ! r (cid:88) k =0 (cid:18) r + (cid:96)r (cid:19)(cid:18) k + (cid:96) − k (cid:19) k !( − µ ) k ( µ − k + (cid:96) S ( r, k ) (A.5)From eq. (A.5) we can immediately extract a representation for the generalized Apostol-Bernoulli numbers B ( (cid:96) ) n ( µ ) by setting r = n − (cid:96) . B ( (cid:96) ) n ( µ ) = (cid:96) ! n − (cid:96) (cid:88) k =0 (cid:18) nn − (cid:96) (cid:19)(cid:18) k + (cid:96) − k (cid:19) k !( − µ ) k ( µ − k + (cid:96) S ( n − (cid:96), k ) == (cid:96) ! n − (cid:96) (cid:88) k =0 (cid:18) n(cid:96) (cid:19)(cid:18) k + (cid:96) − k (cid:19) k !( − µ ) k ( µ − k + (cid:96) S ( n − (cid:96), k ) . (A.6)– 25 –iven the generalized Apostol-Bernoulli numbers B ( (cid:96) ) n ( µ ) , it is easy to write down theexpansion for the polynomial B ( (cid:96) ) n ( x ; µ ) = n (cid:88) k =0 (cid:18) nk (cid:19) B ( (cid:96) ) n − k ( µ ) x k . (A.7)The case µ = 1 are simply known as generalized Bernoulli polynomials and we shall denotethem as B ( (cid:96) ) n ( x ). Obviously we can also introduce the generalized Bernoulli numbers,B ( (cid:96) ) n ≡ B ( (cid:96) ) n (0). These polynomials are not simply obtained by taking the limit for µ → λ is an integer can be expressed in terms of these generalized quantities.We start by observing that1 (cid:0) cosh v − cosh u (cid:1) n = − ( − n − e nv (cid:18) y (cid:19) n (cid:0) y (cid:1) n e ny (cid:16) e y − (cid:17) n (A) (cid:0) y (cid:1) n e ny (cid:16) e v + y − (cid:17) n (B) == − ( − n − e nv (cid:18) y (cid:19) n ∞ (cid:88) (cid:96) =0 (cid:16) y (cid:17) (cid:96) (cid:96) (cid:88) j =0 B ( n ) (cid:96) − j (cid:0) n (cid:1) B ( n ) j + n (cid:0) n ; e v (cid:1) ( (cid:96) − j )!( n + j )! , (A.8)where we have introduced y = u − v to keep a compact notation. We have expandedthe factor (A) in terms of generalized Bernoulli polynomials, while the remaining factor(B) has been expressed as series whose coefficients are the generalized Apostol-Bernoullipolynomials for λ = e v . If we use the property B ( n ) k ( n − x, µ ) = ( − k µ n B ( n ) k ( x, µ − ) (A.9)we find thatB ( n ) k (cid:16) n (cid:17) = ( − k B ( n ) k (cid:16) n (cid:17) e nv B ( n ) k (cid:16) n , e v (cid:17) = ( − k e − nv B ( n ) k (cid:16) n , e − v (cid:17) . (A.10)The first identity implies that B ( n ) k (cid:0) n (cid:1) vanishes for odd k . Next we observe that thecombination u exp (cid:0) − αu (cid:1) can be written as ue − αu = 12 ∞ (cid:88) n =0 ( − n α n − n ! H n +1 ( √ αv ) e − αv (cid:16) y (cid:17) n (A.11)where H n ( u ) stand for the usual Hermite polynomials. Thus we find the following Laurentexpansion for the function f ( u ) = ue − αu ( cosh v − cosh u ) n : f ( u ) = − ( − n − (cid:18) y (cid:19) n e − αv + n v ∞ (cid:88) p =0 (cid:16) y (cid:17) p e − αv ×× p (cid:88) (cid:96) =0 ( − p − (cid:96) α p − (cid:96) − ( p − (cid:96) )! H p − (cid:96) +1 ( √ αv ) (cid:96) (cid:88) j =0 B ( n ) (cid:96) − j (cid:0) n (cid:1) B ( n ) j + n (cid:0) n ; e v (cid:1) ( (cid:96) − j )!( n + j )! . (A.12)– 26 –t is a trivial exercise to extract the relevant residue form (A.12). We getRes[ f ( u )] u = v = e − αv + n v n − (cid:88) (cid:96) =0 ( − n − (cid:96) − α n − (cid:96) − H n − (cid:96) ( √ αv )( n − (cid:96) − (cid:96) (cid:88) j =0 B ( n ) (cid:96) − j (cid:0) n (cid:1) B ( n ) j + n (cid:0) n ; e v (cid:1) ( (cid:96) − j )!( n + j )! . (A.13) B Some useful expansion for generalized Apostol-Bernoulli Polynomials
In ref [34, 35] they provide the following expansion for the generalized Apostol-Bernoullipolynomials in terms of the generalized Bernoulli polynomials (i.e. µ = 1): B ( n ) j ( x, µ ) = e − x log µ ∞ (cid:88) k =0 (cid:18) j + k − nk (cid:19)(cid:18) j + kk (cid:19) − B ( n ) k + j ( x ) (log µ ) k k ! (B.1)This expansion suggests that it is possible to expand the generalized Apostol-Bernoullipolynomials at a given µ = µ µ in terms of the same polynomials at µ = µ . In fact,exploiting the properties of logarithms B ( n ) j ( x, µ µ ) = e − x (log µ +log µ ) ∞ (cid:88) k =0 (cid:18) j + k − nk (cid:19)(cid:18) j + kk (cid:19) − B ( n ) k + j ( x ) (log µ + log µ ) k k ! == e − x (log µ +log µ ) ∞ (cid:88) k =0 k (cid:88) (cid:96) =0 (cid:18) j + k − nk (cid:19)(cid:18) j + kk (cid:19) − B ( n ) k + j ( x ) 1 k ! (cid:18) k(cid:96) (cid:19) (log µ ) (cid:96) (log µ ) k − (cid:96) (B.2)We can disentangle the two sums by setting k = (cid:96) + m . Then the two sums becomesindependent:= e − x (log µ +log µ ) ∞ (cid:88) m =0 ∞ (cid:88) (cid:96) =0 (cid:18) j + m + (cid:96) − nm + (cid:96) (cid:19)(cid:18) j + m + (cid:96)m + (cid:96) (cid:19) − B ( n ) m + (cid:96) + j ( x ) (log µ ) (cid:96) (cid:96) ! (log µ ) m m ! . (B.3)Let use rearrange the binomials coefficient as follows and perform the sum over (cid:96) := e − x (log µ +log µ ) ∞ (cid:88) m =0 ∞ (cid:88) (cid:96) =0 (cid:0) j + m − nm (cid:1)(cid:0) j + (cid:96) + m − n(cid:96) (cid:1)(cid:0) j + mm (cid:1)(cid:0) j + l + m(cid:96) (cid:1) B ( n ) m + (cid:96) + j ( x ) (log µ ) (cid:96) (cid:96) ! (log µ ) m m ! == e − x log µ ∞ (cid:88) m =0 (cid:0) j + m − nm (cid:1)(cid:0) j + mm (cid:1) B ( n ) m + j ( x, µ ) (log µ ) m m ! . (B.4)Therefore we have shown B ( n ) j ( x, µ µ ) = e − x log µ ∞ (cid:88) m =0 (cid:0) j + m − nm (cid:1)(cid:0) j + mm (cid:1) B ( n ) m + j ( x, µ ) (log µ ) m m ! . (B.5)If we apply this result to our specific case we get e nv B ( n ) j + n (cid:16) n e v − πiτβ (cid:17) = ∞ (cid:88) m =0 (cid:0) j + mm (cid:1)(cid:0) j + m + nm (cid:1) B ( n ) m + j (cid:16) n e − πiτβ (cid:17) v m m ! . (B.6)– 27 – Computing gaussian integrals of Hermite polynomials
We consider the following integral I k = (cid:90) ∞−∞ dv H k ( av ) e − v + xv . (C.1)To find its expression for general k we construct the following generating functional G ( t ) = ∞ (cid:88) k =0 t k k ! I k = (cid:90) ∞−∞ dv e − v + xv +2 atv − t = √ πe ( a − ) t + atx + x == √ πe − ( √ − a t ) +2 a √ − a ( √ − a t ) x + x = √ πe x ∞ (cid:88) k =0 t k (1 − a ) k k ! H k (cid:18) ax √ − a (cid:19) , (C.2)where we used ∞ (cid:88) n =0 t n n ! H n ( av ) = e atv − t . (C.3)Thus I k = √ π (1 − a ) k H k (cid:18) ax √ − a (cid:19) e x . (C.4)The integral P km defined in (4.27) can be computed by taking the m th derivative withrespect to x of I k and then setting x = 0P km = √ π ∂ mx I k | x =0 = m (cid:88) j =0 (cid:18) mj (cid:19) (1 − a ) k ∂ jx H k (cid:18) ax √ − a (cid:19) ∂ m − jx e x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x =0 == √ π m (cid:88) j =0 (cid:18) mj (cid:19) k ! a j (cid:0) − a (cid:1) k − j H k − j (cid:16) ax √ − a (cid:17) ( k − j )! e x (cid:18) − i (cid:19) m − j H m − j (cid:18) ix (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x =0 == π m (cid:88) j =0 (cid:18) mj (cid:19) ( − i ) m − j Γ( k + 1) a j (cid:0) − a (cid:1) k − j k − j Γ( k − j + 1)Γ (cid:0) ( j − k + 1) (cid:1) Γ (cid:0) ( j − m + 1) (cid:1) . (C.5)This sum can be easily evaluated in terms of hypergeometric functions once we extendthe range of j to infinity. In fact the generic term, once written in terms of Γ − function,vanishes for j ≥ m + 1. We findP km = π / k ( − i ) m (cid:0) − a (cid:1) k F (cid:16) − k , − m ; ; a a − (cid:17) Γ (cid:0) − k (cid:1) Γ (cid:0) − m (cid:1) + 2 ia F (cid:16) − k , − m ; ; a a − (cid:17) √ − a Γ (cid:0) − k (cid:1) Γ (cid:0) − m (cid:1) ==( − i ) m k − m − π (cid:0) − a (cid:1) k − m − P ( k + m +1) ( k − m − (cid:18) − ia √ − a (cid:19) , (C.6)where P µν ( x ) is the Associated Legendre Function.– 28 – eferences [1] R. Jackiw, “Lower Dimensional Gravity” , Nucl. Phys. B 252, 343 (1985) .[2] C. Teitelboim, “Gravitation and Hamiltonian Structure in Two Space-Time Dimensions” , Phys. Lett. B 126, 41 (1983) .[3] A. Almheiri and J. Polchinski, “Models of AdS backreaction and holography” , JHEP 1511, 014 (2015) , arxiv:1402.6334 .[4] K. Jensen, “Chaos in AdS Holography” , Phys. Rev. Lett. 117, 111601 (2016) , arxiv:1605.06098 .[5] J. Maldacena, D. Stanford and Z. Yang, “Conformal symmetry and its breaking in twodimensional Nearly Anti-de-Sitter space” , PTEP 2016, 12C104 (2016) , arxiv:1606.01857 .[6] J. Engels¨oy, T. G. Mertens and H. Verlinde, “An investigation of AdS backreaction andholography” , JHEP 1607, 139 (2016) , arxiv:1606.03438 .[7] P. Saad, S. H. Shenker and D. Stanford, “JT gravity as a matrix integral” , arxiv:1903.11115 .[8] P. Saad, “Late Time Correlation Functions, Baby Universes, and ETH in JT Gravity” , arxiv:1910.10311 .[9] A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, “Replica Wormholesand the Entropy of Hawking Radiation” , JHEP 2005, 013 (2020) , arxiv:1911.12333 .[10] G. Penington, S. H. Shenker, D. Stanford and Z. Yang, “Replica wormholes and the blackhole interior” , arxiv:1911.11977 .[11] E. Witten, “Matrix Models and Deformations of JT Gravity” , arxiv:2006.13414 .[12] H. Maxfield and G. J. Turiaci, “The path integral of 3D gravity near extremality; or, JTgravity with defects as a matrix integral” , arxiv:2006.11317 .[13] A. Maloney and E. Witten, “Averaging over Narain moduli space” , JHEP 2010, 187 (2020) , arxiv:2006.04855 .[14] N. Afkhami-Jeddi, H. Cohn, T. Hartman and A. Tajdini, “Free partition functions and anaveraged holographic duality” , arxiv:2006.04839 .[15] J. Cotler and K. Jensen, “AdS gravity and random CFT” , arxiv:2006.08648 .[16] T. G. Mertens, G. J. Turiaci and H. L. Verlinde, “Solving the Schwarzian via the ConformalBootstrap” , JHEP 1708, 136 (2017) , arxiv:1705.08408 .[17] F. Ferrari, “Gauge Theory Formulation of Hyperbolic Gravity” , arxiv:2011.02108 .[18] L. V. Iliesiu, S. S. Pufu, H. Verlinde and Y. Wang, “An exact quantization ofJackiw-Teitelboim gravity” , JHEP 1911, 091 (2019) , arxiv:1905.02726 .[19] A. Blommaert, T. G. Mertens and H. Verschelde, “The Schwarzian Theory - A Wilson LinePerspective” , JHEP 1812, 022 (2018) , arxiv:1806.07765 .[20] T. G. Mertens and G. J. Turiaci, “Defects in Jackiw-Teitelboim Quantum Gravity” , JHEP 1908, 127 (2019) , arxiv:1904.05228 .[21] G. S´arosi, “AdS holography and the SYK model” , PoS Modave2017, 001 (2018) , arxiv:1711.08482 . – 29 –
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