Exact properties of an integrated correlator in \mathcal{N}=4 SU(N) SYM
QQMUL-PH-21-09DCPT-21/03
Exact properties of an integrated correlator in N = 4 SU ( N ) SYM
Daniele Dorigoni ( τ ) , Michael B. Green ( λ )( N ) and Congkao Wen ( N ) ( τ ) Centre for Particle Theory & Department of Mathematical Sciences Durham University, Lower Mountjoy,Stockton Road, Durham DH1 3LE, UK( λ ) Department of Applied Mathematics and Theoretical PhysicsWilberforce Road, Cambridge CB3 0WA, UK( N ) School of Physics and Astronomy, Queen Mary University of London,London, E1 4NS, UK [email protected], [email protected], [email protected] Abstract
We present a novel expression for an integrated correlation function of four superconformal primariesin SU ( N ) N = 4 supersymmetric Yang–Mills ( N = 4 SYM) theory. This integrated correlator, which isbased on supersymmetric localisation, has been the subject of several recent developments. In this paperthe correlator is re-expressed as a sum over a two dimensional lattice that is valid for all N and all valuesof the complex Yang–Mills coupling τ = θ/ π + 4 πi/g Y M . In this form it is manifestly invariant under SL (2 , Z ) Montonen–Olive duality. Furthermore, it satisfies a remarkable Laplace-difference equation thatrelates the SU ( N ) correlator to the SU ( N + 1) and SU ( N −
1) correlators. For any fixed value of N the correlator can be expressed as an infinite series of non-holomorphic Eisenstein series, E ( s ; τ, ¯ τ ) with s ∈ Z , and rational coefficients that depend on the values of N and s . The perturbative expansion ofthe integrated correlator is an asymptotic but Borel summable series, in which the n -loop coefficient oforder ( g Y M /π ) n is a rational multiple of ζ (2 n + 1). The n = 1 and n = 2 terms agree precisely withresults determined directly by integrating the expressions in one-loop and two-loop perturbative N = 4SYM field theory. Likewise, the charge- k instanton contributions ( | k | = 1 , , . . . ) have an asymptotic,but Borel summable, series of perturbative corrections. The large- N expansion of the correlator withfixed τ is a series in powers of N − (cid:96) ( (cid:96) ∈ Z ) with coefficients that are rational sums of E ( s ; τ, ¯ τ ) with s ∈ Z + 1 /
2. This gives an all orders derivation of the form of the recently conjectured expansion. Wefurther consider the ’t Hooft topological expansion of large- N Yang–Mills theory in which λ = g Y M N isfixed. The coefficient of each order in the 1 /N expansion can be expanded as a series of powers of λ thatconverges for | λ | < π . For large λ this becomes an asymptotic series when expanded in powers of 1 / √ λ with coefficients that are again rational multiples of odd zeta values, in agreement with earlier resultsand providing new ones. We demonstrate that the large- λ series is not Borel summable, and determineits resurgent non-perturbative completion, which is O (exp( − √ λ )). a r X i v : . [ h e p - t h ] F e b ontents SU (2 ) theory 9 SU ( N ) theory 18 G N ( τ, ¯ τ ) . . . . . . . . . . . . . . . . . . . . . . . . 21 SU ( N ) integrated correlator 23 N . . . . . . . . . . . . . . . . . . 235.1.1 Comparison with perturbation theory using the Laplace-difference equation . . . . . . 245.1.2 Comparison with one-loop and two-loop perturbation theory . . . . . . . . . . . . . . 255.2 The large- N expansion with fixed ’t Hooft coupling . . . . . . . . . . . . . . . . . . . . . . . . 275.2.1 Resurgence of the strong coupling expansion . . . . . . . . . . . . . . . . . . . . . . . 295.3 The large- N expansion with fixed g Y M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.3.1 Large- N constraints from the Laplace-difference equation . . . . . . . . . . . . . . . . 37 D.1 Median resummation at leading order in the ’t Hooft expansion . . . . . . . . . . . . . . . . . 481
Overview and outline
The structure of N = 4 supersymmetric Yang–Mills ( N = 4 SYM) [1] has been the subject of intensestudy over a number of years. It is a highly nontrivial four-dimensional conformal field theory and manyfeatures of its correlation functions have been determined by making use of a variety of symmetries, such asintegrability and superconformal symmetry combined with crossing symmetry and causality. Furthermore,the holographic relationship between N = 4 SYM and type IIB superstring theory in AdS × S providesfurther constraints on the structure of these correlators.Of particular significance to this paper is the analysis of the integrated correlation function of four N = 4superconformal primaries that was formulated making use of supersymmetric localisation in [3], and furtherdeveloped in [4, 5, 6, 7]. This correlator was defined in terms of the partition function of N = 2 ∗ SYMtheory, which is a mass deformation of the superconformal N = 4 SU ( N ) SYM theory with mass parameter m . The suitably normalised N = 2 ∗ partition function, on a round S , Z N ( m, τ, ¯ τ ), was determined byPestun using supersymmetric localisation [8] and will be reviewed in section 2. Our notation follows usualconventions where the complex Yang–Mills coupling constant is defined by τ = τ + iτ := θ π + i πg Y M , (1.1)with θ the topological theta angle and g Y M the SU ( N ) Yang-Mills coupling constant.In [3] the integrated correlator of four primaries of the stress tensor supermultiplet of the N = 4 theorywas identified with the m → Z N that has the form G N ( τ, ¯ τ ) := 14 ∆ τ ∂ m log Z N ( m, τ, ¯ τ ) (cid:12)(cid:12)(cid:12)(cid:12) m =0 = (cid:90) (cid:89) i =1 dx i µ ( { x i } ) (cid:104)O ( x ) . . . O ( x ) (cid:105) , (1.2)where ∆ τ = 4 τ ∂ τ ∂ ¯ τ is the hyperbolic laplacian and O ( x i ) is a superconformal primary in the (cid:48) of SU (4)R symmetry. The first equality in (1.2) guarantees that G N ( τ, ¯ τ ) preserves half the supersymmetries andthis is the condition that determines the form of the integration measure, µ ( { x i } ).An integrated four-point function with a different integration measure was identified with ∂ m log Z N ( m, τ, ¯ τ )in [6] and was considered in more detail in [7]. A further generalisation of (1.2) considered in these refer-ences is based on the Pestun partition function on a squashed S with squashing parameter b (where theunsquashed S is recovered when b = 1). Including derivatives with respect to b in the limit b = 1 potentiallygenerates other integrated correlation functions. This paper gives more details of the results presented in letter format in [2]. The normalisation of the integrated correlator differs from that in [3] by a factor of c/ c = ( N − / Here and in much of the following we will suppress the SU (4) quantum numbers. The expression for µ ( { x i } ) is given in [3],where it is expressed in terms of the two independent cross ratios. This will be reviewed and made more precise in section 2. S tothe flat-space correlator on R , as discussed in [9, 3].In this paper we will re-express the integrated correlator G N ( τ, ¯ τ ) as a two-dimensional lattice sum thatmakes manifest many of its properties for all values of N and τ . Since this reformulation is based on a wealthof evidence concerning the structure of G N ( τ, ¯ τ ) in various limits, rather than being based on a mathematicalderivation we present this in the form of a conjecture rather than a theorem: Conjecture : The integrated correlation function (1.2) of four superconformal primary operators in thestress tensor multiplet of N = 4 SU ( N ) supersymmetric Yang–Mills theory is given by the lattice sum G N ( τ, ¯ τ ) = 12 (cid:88) ( m,n ) ∈ Z (cid:90) ∞ exp (cid:16) − tπ | m + nτ | τ (cid:17) B N ( t ) dt , (1.3) where B N ( t ) has the form B N ( t ) = Q N ( t )( t + 1) N +1 , (1.4) and where Q N ( t ) is a polynomial of degree N − that takes the form Q N ( t ) = − N ( N − − t ) N − (1 + t ) N +1 (cid:26) (3 + (8 N + 3 t − t ) P (1 , − N (cid:18) t − t (cid:19) + 11 + t (cid:0) t − N t − (cid:1) P (1 , − N (cid:18) t − t (cid:19)(cid:27) , (1.5) and P ( α,β ) N ( z ) is a Jacobi polynomial. In the case of SU (2) the polynomial pre-factor is given by Q ( t ) = 9 t − t + 9 t . A general propertyof B N ( t ) that will prove to be important is its inversion invariance B N ( t ) = 1 t B N (cid:18) t (cid:19) . (1.6)Equation (1.3) is manifestly invariant under the SL (2 , Z ) transformations τ → γ · τ = aτ + bcτ + d , γ = (cid:18) a bc d (cid:19) ∈ SL (2 , Z ) (1.7)which is in accord with the expectations of Montonen–Olive duality [10, 11, 12]. We will also show that thelattice sum (1.3) is convergent for τ in the upper half plane τ = Im τ > G N ( τ, ¯ τ ) satisfies the following corollary: Corollary : The localised integrated correlator satisfies a Laplace-difference equation of the form (∆ τ − G N ( τ, ¯ τ ) = N (cid:104) G N +1 ( τ, ¯ τ ) − G N ( τ, ¯ τ ) + G N − ( τ, ¯ τ ) (cid:105) − N (cid:104) G N +1 ( τ, ¯ τ ) − G N − ( τ, ¯ τ ) (cid:105) . (1.8)This is a very powerful equation that relates the dependence on τ and the dependence on N , therebyproviding powerful constraints on properties of the correlator, as will be discussed later in this paper.3 .1 Outline The outline of the paper is as follows. The construction of the localised correlator starting from Z N ( m, τ, ¯ τ )is reviewed in section 2.In order to analyse the perturbative and non-perturbative behaviour of the correlator in various regionsof the parameters N and τ it is useful to consider a Fourier expansion with respect to θ = 2 πτ , G N ( τ, ¯ τ ) := (cid:88) k ∈ Z G N,k ( τ, ¯ τ ) := (cid:88) k ∈ Z e πikτ F N, | k | ( τ ) . (1.9)This is a sum of contributions from sectors with Yang–Mills instanton charge k . The k = 0 term is thesector described by conventional N = 4 SYM perturbation theory and originates from a part of Z N ( m, τ, ¯ τ )corresponding to the one-loop contribution to the localised N = 2 ∗ partition function [8]. The k > k < Z N ( m, τ, ¯ τ ) described by the Nekrasov instanton partition function [13, 14].The example of SU (2) is considered in detail in section 3. We will determine the perturbative sector byanalysing the zero instanton part of Z N ( m, τ, ¯ τ ) and the definition in (1.2). We will show, in particular, thatthis sector can formally be expressed in terms of an infinite sum over s ∈ Z of zero modes of non-holomorphicEisenstein series, E ( s, τ, ¯ τ ). Each zero mode is the sum of two terms, proportional to τ s and τ − s . Since s > g Y M ) expansion (which is why we have stressed that this is a ‘formal’ expression). Our conventions regardingnon-holomorphic Eisenstein series, together with some of their properties, are described in appendix A. Wewill then see that the non-perturbative k -instanton sectors (with k (cid:54) = 0) determined from (1.2) have aform that combines beautifully with the k = 0 sector, to give the SL (2 , Z )-invariant expression (1.3) forthe SU (2) theory. We find that the integrated correlator can be expressed formally as an infinite series ofnon-holomorphic Eisenstein series with integer indices G ( τ, ¯ τ ) = 14 + 12 ∞ (cid:88) s =2 c (2) s E ( s ; τ, ¯ τ ) . (1.10)with c (2) s = ( − s s − − s ) Γ( s + 1) . (1.11)Finally, in section 3.3 we will use a standard integral representation of E ( s ; τ, ¯ τ ) to rewrite (1.10) in aconvergent form as the integral of a lattice sum, which is the conjectured form for the case N = 2 in (1.3).In section 4 we will present strong motivation for the form (1.3) of the integrated correlator for the theorywith gauge group SU ( N ) for general N . The procedure in this section begins with the evaluation of theone-instanton contributions for a large number of values of N ≥
2. The determination of these contributionsis based on the self-consistency of the perturbative evaluation of the matrix model of the Nekrasov instantonpartition function and the form of the large- N expansion (1.14) that was presented in [5]. This is describedin appendix B. This leads to expressions that reproduce the one-instanton contributions to (1.3) with specificexpressions for the polynomials Q N ( t ). The form of these polynomials generalises to arbitrary values of N
4n an obvious fashion. Furthermore, by considering a large number of examples, we verify that Q N ( t ) isindependent of the instanton number, k and that G N,k ( τ, ¯ τ ) is correctly reproduced by (1.3). As in the SU (2) case, the SU ( N ) correlator can be expressed as a formal sum of non-holomorphic Eisenstein serieswith integer indices, G N ( τ, ¯ τ ) = N ( N − ∞ (cid:88) s =2 c ( N ) s E ( s ; τ, ¯ τ ) . (1.12)where the coefficients c ( N ) s are rational numbers that depend on N and are generated by the expansion of B N ( t ) in the form B N ( t ) = ∞ (cid:88) s =2 c ( N ) s Γ( s ) t s − . (1.13)In section 4.1 we will show that G N ( τ, ¯ τ ) satisfies the Laplace-difference equation in the corollary (1.8).This is obtained by applying a Laplace operator to (1.3), which leads to a generalised Laplace equation thatdefines the relationship between G N ( τ, ¯ τ ) and G N +1 ( τ, ¯ τ ) and G N − ( τ, ¯ τ ). By inputting the SU (2) correlatorthis equation recursively determines correlators for all N .Section 5 discusses a number of properties of the expression (1.3). In section 5.1 we will show that theperturbative expansion of G N ( τ, ¯ τ ) for general values of N has the very simple form of a sum of powers( g Y M
N/π ) L with coefficients that depend on N and are rational multiples of odd zeta values, ζ (2 L + 1).We will verify that the coefficients of the L = 1 and L = 2 terms are precisely the values that are obtainedin weakly-coupled Yang–Mills perturbation theory for any value of N . To verify this we need to integratethe known expressions for the standard Yang–Mills correlators over the positions of the operators with theappropriate measure. This integration will be described in appendix C by making use of exact results in theliterature for arbitrary L -loop ladder diagrams. The coefficients in the perturbative expansion also have aspecific dependence on powers of N , corresponding to non-planar contributions in Yang–Mills perturbationtheory. The first non-planar dependence starts at four loops, again in agreement with standard field theorycalculations in the literature.Section 5.2 discusses the large- N expansion of the integrated correlator in the ’t Hooft limit, which isan expansion of the form G N ( τ, ¯ τ ) ∼ (cid:80) ∞ g =0 N − g G ( g ) ( λ ), where λ = g Y M N and we denote with G ( g ) ( λ ) thegenus g contribution. We will demonstrate that for small values of λ the expansion of the leading term, G (0) ( λ ), in a power series in λ , converges for | λ | < π . This is interpreted as the perturbative sum of planarFeynman diagrams. We will show that the sum of this series is equal to the expression in [3], although thatwas derived from the power series in 1 /λ that is appropriate for the the strong-coupling limit. However,as we will show, the strong coupling series is not Borel summable. We will give a resurgence analysis thatdetermines the form of the non-perturbative completion, which behaves as e − √ λ . A brief description ofBorel summation and resurgence is given in appendix D. We will also determine G (1) ( λ ), the term of order N in the large- N expansion, as well as higher order terms, which can also be resummed and reproduce theresults in [4].In section 5.3 we will consider the 1 /N expansion with fixed g Y M . This is the expansion of the integratedcorrelator that was considered in [5], in which S-duality (invariance under SL (2 , Z )) is manifest. We will5how how the B N ( t ) recursion relation determines the expansion of the integrated correlator in powers of1 /N around the large- N limit. We will show that the integrated correlator has a large- N expansion withcoefficients that are sums of non-holomorphic Eisenstein series, but now with half-integer index (apart froma leading N term). The expansion takes the following form G N ( τ, ¯ τ ) ∼ N ∞ (cid:88) (cid:96) =0 N − (cid:96) (cid:96) + (cid:88) s = + mod ( (cid:96), d s(cid:96) E ( s ; τ, ¯ τ ) , (1.14)so that the coefficient of N − (cid:96) is the sum of a finite number of Eisenstein series E ( s ; τ, ¯ τ ) with index s starting at s = 3 / (cid:96) , or s = 5 / (cid:96) , up to s = 3 / (cid:96) . At fixed τ , this expansion is anasymptotic series in 1 /N . The first few terms in this expression coincide with the result found in [5], but theprocedure in this paper gives a simple recursive algorithm for determining the coefficients d s(cid:96) to arbitrarilyhigh values of (cid:96) and s , which follows from the large- N expansion of B N ( t ). These coefficients are againrational numbers. We do not have a closed formula for general d s(cid:96) , but it is straightforward to determine d (cid:96) + (cid:96) , which are the coefficients with maximum s . Furthermore, once d (cid:96) + (cid:96) is given, the Laplace-differenceequation (1.8) efficiently determines the rest. At the end of section 5.3 we will provide explicit examples ofa few sets of coefficients.The paper will end with a discussion in section 6 where we will comment on the interpretation andpossible extensions of these results. Our analysis will be based on the expression for the integrated correlation function of four superconformalprimaries that was formulated in [3]. This correlator was defined in terms of the partition function of N = 2 ∗ SYM theory, which is a mass deformation of the superconformal N = 4 SYM theory with mass parameter m . The (suitably normalised) N = 2 ∗ partition function on S , Z N ( m, τ, ¯ τ ), was determined by Pestunusing supersymmetric localisation in [8], where it was shown to have the form Z N ( m, τ, ¯ τ ) = (cid:90) d N a i e − π g Y M (cid:80) i a i (cid:89) i 6) totake care of the R-symmetry indices, and the operator O ( x, Y ) := O IJ (cid:48) Y I Y J . The quantity T N, free ( U, V ; Y i )represents the correlator in free theory, which is trivial. The factor I ( U, V ; Y i ) is fixed by the superconformalsymmetry [15, 16], and we follow the conventions of [3]. Its explicit expression will not be important for thefollowing discussion. Most of our focus will therefore be on the dynamical part of the correlator, T N ( U, V ).The integrated correlator G N ( τ, ¯ τ ) is the main quantity we will study in this paper. A second integratedcorrelator was considered in [6], and is obtained from the N = 2 ∗ partition function by applying fourderivatives with respective to mass, G (cid:48) N ( τ, ¯ τ ) := ∂ m log Z N ( m, τ, ¯ τ ) (cid:12)(cid:12) m =0 = I [ T N ( U, V )] = (cid:90) dU dV µ (cid:48) ( U, V ) T N ( U, V ) , (2.6)with a different integration measure µ (cid:48) ( U, V ), which is given in [6]. As mentioned in the introduction,integrated correlators can also be obtained by starting with the N = 2 ∗ partition function on a squashed Here we have used the simplified version of the integration measure as given in [6], and we have changed the overallnormalisation to agree with our normalisation conventions. with squashing parameter b (where b = 1 corresponds to the round S ) [6]. Many interesting and non-trivial relations are found among the integrated correlators obtained by acting on the partition function withderivatives with respect to m , τ , ¯ τ and b [7]. These relations have also been generalised to cases where morethan four derivatives act on the partition functions [17].Certain properties of the large- N expansion of the correlator (1.2) were determined in [3] starting with theexpression for Z N ( m, τ, ¯ τ ) in (2.1) in the ’t Hooft limit (fixed λ = g Y M N ), in which instantons are suppressedso Z instN = 1. On the other hand instantons play an essential rˆole in ensuring S-duality (Montonen–Oliveduality [10]). The regime in which S-duality is manifest is the large- N limit with fixed g Y M considered in[5] where the effects of Yang–Mills instantons were taken into account by virtue of the | ˆ Z instN | factor in(2.1). In this limit the integrated correlator has a large- N expansion of the form (1.14). The expansionwas conjectured in [5], based on a combination of analytic calculations at low orders in the 1 /N expansion,which led to a determination of the coefficients d s(cid:96) for the first few values of (cid:96) . We will see in section 5 howthe all-order expression arises by expanding (1.3), or by using the Laplace-difference equation (1.8) once theinitial data is given.There are many interesting features of this expansion. Firstly, it has a close holographic connection withthe low energy expansion of the four-graviton amplitude of type IIB superstring theory in an AdS × S background. For example, the leading term of order N has a coefficient proportional to E ( , τ, ¯ τ ) thatmatches the coefficient of R in the string theory amplitude (where R denotes the space-time Riemanncurvature). Furthermore, since the coefficient of N − (cid:96) is a sum of Eisenstein series with half-integer indices( s ∈ Z + ) it is obvious that the expression is invariant under SL (2 , Z ) S-duality. From (A.10) we see thatthis also means the perturbative expansion of these coefficients is in half-integer powers of y = 4 π /g Y M ,proportional to y s and y − s . By contrast, Yang–Mills perturbation theory for any finite value of N is definedby an expansion in integer powers of g Y M . This means that there must be a transition from the expansionin integer to half-integer powers of g Y M as N → ∞ .These properties will be manifest in the considerations of subsequent sections of this paper, which willfocus on the exact form of the integrated correlation function, 1.3, for all values of N and of the complexcoupling constant, τ = θ/ π + i π/g Y M . Before considering the structure and properties of the integrated correlator we would like to return to anissue mentioned in the introduction. It was noted in [3] that the expression ∆ τ ∂ m log Z N ( m, τ, ¯ τ ) | m =0 inthe first line of (1.2) does not simply describe the four-point correlator (cid:104) (cid:81) i =1 O ( x i ) (cid:105) in the second line inan obvious fashion. The N = 2 ∗ SYM Lagrangian includes a mass deformation of the form m K ( x ), where K ( x ) is the non-BPS Konishi operator. As a result, the quantity ∆ τ ∂ m log Z N ( m, τ, ¯ τ ) | m =0 seems to containan additional piece proportional to the integral of a three-point correlator (cid:104)K ( x ) O ( x ) O ( x ) (cid:105) , where the τ and ¯ τ derivatives bring down two O operators and ∂ m brings down K ( x ).However, as pointed out in [3], in the large- N and strong-coupling limit ( g Y M N (cid:29) N / . Therefore, the relation between the integrated correlator and Pestun’s partition8unction (1.2) is expected to be valid in this limit. However, we expect that such a non-BPS Konishicomponent in (1.2) is absent for all values of N .Indeed, recall that the arguments of [3] were based on analysing supersymmetric Ward identities thatrelate different components of the integrated -BPS super-stress tensor correlators. The analysis involvedintegration by parts that picked up certain boundary terms, which can be associated with OPEs of pairs ofexternal stress tensor operators. It was argued that these boundary terms cancel with integrated three-pointsuper-stress tensor correlators. But the OPE of two external O operators also includes non-BPS operators,such as K . Although these decouple at large- N , as described in [3], at finite N the boundary term mustcontain a component that cancels the non-BPS three-point correlator, (cid:104)K ( x ) O ( x ) O ( x ) (cid:105) , This wouldensure the BPS condition for the integrated four-point correlator.Another way to see that the Konishi operator should not be relevant is to recall that any operator thatcouples to m must have dimension exactly equal to two. The Konishi operator is not a BPS operator,hence in the interacting theory it does not have dimension exactly equal to two and cannot appear in the ∂ m log Z N ( m, τ, ¯ τ ). As we will discuss in section 5.1, the perturbative expansions that follow from our analysis are in precisenumerical agreement with known results for the correlator of four O ’s for all N at one and two loops, andproduce correct N dependence for higher loop terms. A contribution from a Konishi three-point correlatorwould spoil this consistency. This provides direct evidence that the relation between the integrated correlatorand the localised partition function is exact for any value of N and of g Y M . SU (2 ) theory In this section we will determine the expression for the correlator for arbitrary values of g Y M in the theorywith SU (2) gauge group. In the following subsections we will consider the contributions from the perturbativesector (that arises from ˆ Z pertN ) and from the sectors with instantons (that arise from | ˆ Z instN | ). As we willsee, rather surprisingly, the expressions for these contributions have a similar structure that demonstrateshow they combine into a SL (2 , Z )-invariant expression of the form (1.3). The sector with zero instanton number has Z instN = 1 and the perturbative contribution to the integratedcorrelator arises from log Z pertN and has the form G N, ( τ, ¯ τ ) = 14 ∆ τ ∂ m log Z pertN (cid:12)(cid:12) m =0 = 14 ∆ τ (cid:42) ∂ m (cid:89) i 1) = (cid:80) ∞ j =1 j − s , and erfc( z ) = (2 / √ π ) (cid:82) ∞ z e − t dt is the comple-mentary error function.Equivalently we can use the integral identity2 s − Γ(2 s ) (cid:90) ∞ dw w s − sinh ( w ) = ζ (2 s − , (3.8) With this convention there are no factors of π in this equation. 10o replace ζ (2 s − 1) in (3.6) and rewrite that expression via the modified Borel transform integral G , ( y ) = y (cid:90) ∞ e − t (6 t − t + 2 t )2 sinh ( √ yt ) dt . (3.9)We note that, after a suitable change of the integration variable, the above expression is identical to equation(3.33) of [4], which was obtained by a different method.Since the weak coupling expansion (3.6) is Borel summable as one can easily see from the lack of singular-ities along the t > G , ( y ). We will shortly see that this is a property shared by all instanton sectors. The integral representation (3.9) is well-defined for any value of y . We have also made a numericalcomparison of (3.9) with the definition (3.1) and indeed found complete agreement. It is therefore possibleto expand it in both the large- y and the small- y limits. The expansion at large- y (weak coupling) obviouslygives the asymptotic series (3.6). Furthermore, it is significant that this expression is a sum of terms of thesame form as the second term (i.e. the term proportional to ζ (2 s − y − s ) in the zero mode, F ( y ), of theEisenstein series, E ( s ; τ, ¯ τ ), which is defined in (A.10).However, the integral (3.9) can also be expanded in an asymptotic series at strong coupling (i.e. inpositive powers of y ). This gives the asymptotic series G , ( y ) ∼ 12 + 12 ∞ (cid:88) s =2 ( − s ( s − s − Γ( s + 1) 2 ζ (2 s ) π s y s , (3.10)which is a sum of terms proportional to ζ (2 s ) y s , so each term in this sum is proportional to the first termin F ( s, y ) in (A.10).We find rather strikingly, that the coefficients in the series (3.6) and (3.10) are such that the large- y andsmall- y expansions of the integral (3.9) can be combined to give an expression for zero instanton part of thecorrelator that has the form G , ( y ) ∼ 14 + 14 ∞ (cid:88) s =2 ( − s ( s − s − Γ( s + 1) F ( s, y ) , (3.11)where, again, F ( s, y ) is defined in (A.10) as the complete zero mode of the Eisenstein series E ( s ; τ, ¯ τ ). The fact that the zero mode of the SU (2) correlator has such a non-trivial expression in terms of a sum ofzero modes of Eisenstein series motivates a much more detailed conjectural expression for the exact correlatorthat includes the contributions of instantons. The suggestion is that the integrated correlator in the SU (2)theory can be expressed as the following sum of Eisenstein series G ( τ, ¯ τ ) = 14 + 14 ∞ (cid:88) s =2 ( − s ( s − s − Γ( s + 1) E ( s ; τ, ¯ τ ) . (3.12) The issue of Borel summability for N = 2 ∗ and other supersymmetric localizable theories has been discussed in [19, 20]. τ .The contributions to the correlator from sectors of non-zero instanton number are accounted for by thefactor of | ˆ Z instN ( m, τ, a ij ) | in (2.1), which is the square of the Nekrasov instanton partition function [13, 14].Such contributions were discussed in detail in [5]. More precisely, these contributions are given by ∞ (cid:88) k =1 ∂ m log Z instN,k = ∞ (cid:88) k =1 (cid:0) e ikθ + e − ikθ (cid:1) e − π kg Y M (cid:68) ∂ m ˆ Z instN,k ( m, a ij ) (cid:69) (cid:12)(cid:12)(cid:12)(cid:12) m =0 = ∞ (cid:88) k =1 e πikτ (cid:68) ∂ m ˆ Z instN,k ( m, a ij ) (cid:69) (cid:12)(cid:12)(cid:12)(cid:12) m =0 + c.c. (3.13)where ˆ Z instN,k is the k -instanton and anti- k -instanton contribution to the Nekrasov partition function and θ = 2 πτ = 2 π Im τ . The symbol c.c. represents the complex conjugate expression, which is a sum overant-instanton contributions. The one-instanton contribution The | k | = 1 contribution to the SU ( N ) partition function, G , ± ( τ, ¯ τ ) in (1.9), is determined by equations(3.7)–(3.9) of [5]. which state ∂ m ˆ Z instN, (cid:12)(cid:12)(cid:12)(cid:12) m =0 = − N (cid:88) l =1 (cid:89) j (cid:54) = l ( a lj + i ) a lj ( a lj + 2 i ) . (3.14)In the case of SU (2) there is a single integration variable since a = − a and the expression for G , ± ( τ, ¯ τ )is given by a straightforward one-dimensional integral, which gives (after applying ∆ τ and normalising as in(3.1)) G , ( τ, ¯ τ ) = e πiτ (cid:104) y − √ πe y y / (1 + 8 y )erfc (2 √ y ) (cid:105) , (3.15)and from (1.9) we know that G , − ( τ, ¯ τ ) is simply the complex conjugate of G , ( τ, ¯ τ ). The term in parenthesesis an exact expression analogous to (3.7) in the k = 0 case, that can be expanded in an asymptotic series atweak coupling G , ( τ, ¯ τ ) ∼ e πiτ (cid:20) − 38 + 932 y − y + 3151024 y + · · · (cid:21) . (3.16) The k -instanton contributions The form of the k -instanton contribution, G ,k ( τ, ¯ τ ), is more difficult to determine when | k | > 1, even in the SU (2) case. It involves an enumeration of contributions of Young diagrams as was discussed in detail [5]. Forgeneral N such diagrams are characterised by k boxes that can be sited at any of N locations. Although theenumeration of such diagrams is very complicated for a generic N = 2 ∗ theory, the diagrams that contribute12o ∂ m log ˆ Z instN,k (cid:12)(cid:12) m =0 are limited to diagrams in which all boxes are connected and form rectangles with p columns and q rows, where p q = | k | . We can associated a p × q matrix k a,b = a + b − k -instanton partition function in the SU ( N ) theory, we have [5] ∂ m ˆ Z instN,k (cid:12)(cid:12) m =0 = (cid:88) p,q> pq = k (cid:73) dz π p (cid:89) a =1 q (cid:89) b =1 N (cid:89) j =1 ( z − a j + i k a,b ) ( z − a j + i k a,b ) + 1 × (cid:20)(cid:18) p + 2 q (cid:19) + N (cid:88) j =1 if ( p, q )( z − a j + i ( p + q − z − a j + i ( q − z − a j + i ( p − , (3.17)where the integration contour z is a counter-clockwise contour surrounding the poles at z = a j + i (with j = 1 , . . . , N and (cid:80) j a j = 0). The function f ( p, q ) is f ( p, q ) = 2( q + p )( q − p ) pq . (3.18)The k -instanton contribution to the integrated correlator in the SU (2) theory involves evaluating G ,k ( τ, ¯ τ ) = 14 e πikτ ∆ τ ∂ m log Z inst ,k | m =0 , (3.19)where we have assumed k > 0, while for k < z contour integration, and setting a = − a , for SU (2), we find ∂ m log ˆ Z inst ,k takesthe following form, ∂ m ˆ Z inst ,k (cid:12)(cid:12) m =0 = − (cid:88) p,q> pq = k (cid:18) p + 2 q (cid:19) (cid:0) a + 4 (cid:0) p + 3 pq + 2 q (cid:1) a + 5 (cid:0) p q + pq (cid:1) + p + q (cid:1) (4 a + ( p + q ) ) . (3.20)The expectation value (cid:104) ∂ m ˆ Z inst ,k (cid:12)(cid:12) m =0 (cid:105) is a one-dimensional integral, which can be performed explicitly, andwe find (cid:104) ∂ m ˆ Z inst ,k (cid:12)(cid:12) m =0 (cid:105) = 2 (cid:88) p,q> pq = k (cid:20) − (cid:18) p + 1 q (cid:19) + 2( p + q ) y (cid:0) p − q ) y (cid:1) − y / √ πe y ( p + q ) (cid:16)(cid:0) p + q (cid:1) + (cid:0) p − q (cid:1) y (cid:17) erfc (( p + q ) √ y ) (cid:105) . (3.21)To obtain the integrated correlator, we need to apply a laplacian to this expression and compute ∆ τ (cid:104) ∂ m ˆ Z inst ,k (cid:105) , More precisely, Young diagrams that contribute also include diagrams that can can be transformed into rectangles by‘partial transposition’, as defined in section 3 of [5]. Here we have slightly changed the notation and the form of the expression used in (3.48) of [5]. It should also be stressedthat the expression for ∂ m log ˆ Z instN,k (cid:12)(cid:12) m =0 in (3.17) was conjectured in [5] by classifying the pattern of Young diagrams thatcontribute for a very large number of values of k . Although this gives us overwhelming confidence that the expression is correct,we do not have a mathematical proof of the statement. k -instanton contribution to the integrated correlator, G ,k ( τ, ¯ τ ) = e πikτ (cid:88) p,q> pq = k y ( p + q ) (cid:2) y (cid:0) p + 2 pq + 11 q (cid:1) ( p − q ) + 2 y ( p + q ) ( p − q ) (3.22)+9 p − pq + 9 q (cid:3) − √ π y / e y ( p + q ) (cid:104) y (cid:0) p − q (cid:1) + 24 y (cid:0) p + q (cid:1) (cid:0) p − q (cid:1) + 3 y (cid:0) p − p q + 9 q (cid:1) + 3 (cid:0) p + q (cid:1)(cid:3) erfc ( √ y ( p + q )) , where again we have assumed k > 0, while for k < p = 0 , q = j (or q = 0 , p = j ) so that k = 0, the above expressionreduces to half of the perturbative contribution given in (3.7). This is a non-trivial fact, which is not atall obvious when comparing the factors of ˆ Z pertN ( m, a ij ) and (cid:12)(cid:12)(cid:12) ˆ Z instN ( m, τ, a ij ) (cid:12)(cid:12)(cid:12) that enter into (cid:104) Z N (cid:105) . Thisproperty is crucial in ensuring that the correlator (1.3) is SL (2 , Z ) invariant.In order to understand whether non-perturbative corrections are present in the k -instanton sector we mayexpress (3.22) as a Borel-like integral. For this purpose we will make use of two useful integral representationsfor the complementary error function,erfc( z ) = 2 π e − z (cid:90) ∞ e − z t t + 1 dt , (3.23)erfc( z ) = 1 √ π e − z (cid:90) ∞ e − t √ t + z dt . (3.24)We will see that the first identity is the natural one when discussing the relation to non-holomorphic Eisen-stein series, while the second one is more directly related to a Borel-like resummation formula.After substituting (3.24) into (3.22) G ,k ( τ, ¯ τ ) can be expressed in the form G ,k ( τ, ¯ τ ) = e πikτ (cid:88) p,q> pq = k (cid:18) p q − p q + 3 pq )( p + q ) + 916 (cid:90) ∞ e − ( p + q ) ty P ( q, p, t )( p + q ) (1 + t ) / dt (cid:19) , (3.25)where P ( q, p, t ) is the polynomial P ( q, p, t ) =105( p − q ) − p − q ) ( p + q )(1 + t )+ 5(9 p − p q + 9 q )(1 + t ) − p + q ) ( p + q )(1 + t ) . (3.26)The integral in equation (3.25) can be interpreted as an inverse transform Borel transform. It is easier atthis stage to see that when p = 0 , q = j (or q = 0 , p = j ) the above expressions (3.25) and (3.26) reduce toprecisely half of the perturbative contribution given in (3.7).Either of the expressions (3.22) or (3.25) can be expanded in an asymptotic series in powers of 1 /y atweak coupling producing a Borel summable factorially divergent asymptotic series. Just as we saw in thepurely perturbative sector, from (3.25) we see that there are no singularities along the t > k -instanton sector. This is consistent with the absence of instanton/anti-instantoncontributions to G ,k ( τ, ¯ τ ). 14nterestingly, the series multiplying e πikτ in (3.22) also has a sensible expansion at strong coupling, y → 0, although this is not of particular interest since in this limit e πikτ = O (1) so all the instantoncontributions are of the same order. We will now verify that the conjectured SL (2 , Z )-invariant expression (3.12), which was based on the structureof the k = 0 sector, reproduces the k -instanton sector for all k . For this purpose it is very useful to rewrite(3.12) as a sum over a two-dimensional lattice in manner that makes its convergence manifest. We begin bywriting the non-holomorphic Eisenstein series as an integral of a double sum in the form (A.1) E ( s ; τ, ¯ τ ) = 1 π s (cid:88) ( m,n ) (cid:54) =(0 , τ s | nτ + m | s = (cid:88) ( m,n ) (cid:54) =(0 , (cid:90) ∞ e − tπY t s − Γ( s ) dt , (3.27)where Y := | m + nτ | τ . (3.28)Substituting this integral representation in (3.12) we obtain an expression that has the form of a Borelintegral, which we conjecture to be the integrated four-point correlator in SU (2) N = 4 SYM. In otherwords we conjecture that G ( τ, ¯ τ ) = 14 + 14 (cid:88) ( m,n ) (cid:54) =(0 , (cid:90) ∞ e − tπY ∞ (cid:88) s =2 ( − s (1 − s ) ( s − s + 1) t s − Γ( s ) dt = 14 + 12 (cid:88) ( m,n ) (cid:54) =(0 , (cid:90) ∞ exp (cid:16) − tπ | m + nτ | τ (cid:17) B ( t ) dt , (3.29)where B ( t ) = 9 t − t + 9 t ( t + 1) . (3.30)The t integral is manifestly finite for every value of m and n with τ in the upper half plane Im τ = τ > m, n ) ∈ Z it defines an analytic function of τ . Furthermore, for large Y the integral vanishesas Y − thus making the lattice sum over m and n convergent, so this representation of G ( τ, ¯ τ ) is well-definedfor all values of τ with τ > SL (2 , Z ) invariant and reproduces the sum ofEisenstein series in (3.12), which in turn reproduces the zero mode of the integrated correlator, G , ( τ, ¯ τ ).In order to check its validity we need to determine its non-zero Fourier modes. For this purpose we willseparate the ( m, n ) sum into two contributions:(i) G ( i )2 , ( τ ) is defined as the sum over all m with n = 0. The subscript 0 indicates that this term manifestlycontributes to the k = 0 sector (and is independent of τ );(ii) G ( ii )2 ( τ, ¯ τ ) is the sum over mode numbers n (cid:54) = 0, all m . This will contain all the k -instanton sectors,but it will also include a k = 0 part so it will also contribute to the perturbative sector.15ontribution (i) has the form G ( i )2 , ( τ ) = 14 + (cid:88) m> (cid:90) ∞ exp (cid:16) − tπ m τ (cid:17) B ( t ) dt . (3.31)This contribution takes the form of a Borel transform for the strong coupling limit τ → 0, which we knowfrom (3.10) contributes half of the zero-instanton sector.In order to analyse contribution (ii) we need to perform a Poisson sum over the index m , which leadsto G ( ii )2 ( τ, ¯ τ ) = 12 (cid:88) ˆ m ∈ Z ,n (cid:54) =0 e πikτ (cid:90) ∞ exp (cid:16) − π ˆ m τ t − tπn τ (cid:17)(cid:114) τ t B ( t ) dt , (3.32)where k = ˆ mn (and ˆ m is the integer that replaces m after the Poisson sum). Note that the ˆ m = 0 term inthe above equation contributes to the k = 0 sector and is given by G ( ii )2 , ( τ ) = (cid:88) n> (cid:90) ∞ exp (cid:16) − tπn τ (cid:17) B ( t ) dt , (3.33)which has the form of a Borel transform of the weak coupling ( τ → ∞ ) series and provides half of thecontribution displayed in (3.6).Combining terms that contribute to the purely perturbative part (the k = 0 sector) we have G , ( τ ) = G ( i )2 , ( τ ) + G ( ii )2 , ( τ ) (3.34)= 14 + (cid:88) m> (cid:90) ∞ exp (cid:16) − tπ m τ (cid:17) B ( t ) dt + (cid:88) n> (cid:90) ∞ exp (cid:16) − tπn τ (cid:17)(cid:114) τ t B ( t ) dt , which gives us precisely half of the strong coupling expansion (3.10) plus half of the weak coupling expansion(3.6).It is notable that B ( t ) satisfies the inversion property B ( t ) = 1 t B (cid:16) t (cid:17) . (3.35)This is important for ensuring the equality of the strong coupling expansion (3.10) and the weak couplingexpansion (3.6). In the next section we will give a general proof that the property (3.35) extends to thepolynomials B N ( t ) that arise for general N in our main result (1.3).The remaining terms in G ( ii )2 ( τ, ¯ τ ) arise from the sum over ˆ m (cid:54) = 0 in (3.32), which gives the sum of k -instanton sectors of the form G ,k ( τ, ¯ τ ) = 12 (cid:88) ˆ m (cid:54) =0 , n (cid:54) =0 ˆ mn = k e πikτ (cid:90) ∞ exp (cid:16) − π ˆ m τ t − tπn τ (cid:17)(cid:114) τ t B ( t ) dt . (3.36)This integral can be expanded as an infinite sum of K -Bessel functions using the integral representation(A.7) in appendix A with a = √ πy ˆ m and b = √ πyn , which reproduces the contribution of F k ( s ; y ) to (3.12). This analysis is very similar to the analysis of the Fourier modes of the non-holomorphic Eisenstein series, E ( s ; τ, ¯ τ ) displayedin appendix A. 16e will now check that (3.36) reproduces the expression for G ,k ( τ, ¯ τ ) in (3.25) that was derived from theintegrated correlator. To this end, we first notice that the exponent in the integral of (3.36) has a minimumfor t = | ˆ m/n | and its minimal value equals (cid:16) − π ˆ m τ t − tπn τ (cid:17)(cid:12)(cid:12)(cid:12) t = | ˆ m/n | = − π | ˆ mn | τ . (3.37)Hence we can rewrite (3.36) as G ,k ( τ, ¯ τ ) = 12 (cid:88) ˆ m (cid:54) =0 , n (cid:54) =0 ˆ mn = k e π ( −| k | τ + ikτ ) (cid:90) ∞ exp (cid:104) − (cid:16) | ˆ m |√ t − | n |√ t (cid:17) πτ (cid:105)(cid:114) τ t B ( t ) dt , (3.38)which makes manifest the exponential suppression factor e π ( −| k | τ + ikτ ) characteristic of the k -instantonand anti k -instanton sectors.In order to analyse the integral in (3.38) we will change integration variable by defining x = (cid:16) | ˆ m |√ t − | n |√ t (cid:17) , (3.39)and noting that the integration range 0 ≤ t ≤ ∞ is a double cover of the integration range 0 ≤ x ≤ ∞ .Following this change (3.38) becomes G ,k ( τ, ¯ τ ) = (cid:88) ˆ m (cid:54) =0 , n (cid:54) =0 ˆ mn = k e π ( −| k | τ + ikτ ) √ τ (cid:90) ∞ e − x πτ [( ˆ m + n ) + x ] 3( ˆ m + n ) × (cid:104) ˆ m ( ˆ m − n )(3 ˆ m − n ) n ( ˆ m + n ) + ( ˆ m + n ) (3 ˆ m − 39 ˆ m n + 76 ˆ m n − 39 ˆ mn + 3 n ) x − m − m n + 2 ˆ m n − mn + 2 n ) x + 3( ˆ m − ˆ mn + n ) x (cid:105) dx , (3.40)valid for k > k < 0. This is a gaussian-like integral that can be evaluatedand compared with the localization result (3.22). The two expressions are identical (with ( ˆ m, n ) ↔ ( p, q )and noting that ( ˆ m, n ) run over positive and negative integers while in (3.22) ( p, q ) are purely positive).Thus, we have verified that the expression (3.29) reproduces the integrated correlator derived fromlocalisation as conjectured earlier.In order to study the convergence properties of our conjectured expression for the integrated correlator(1.3), we have made numerical estimates of its dependence on the the complex coupling with τ in the upperhalf plane and with N = 2. Firstly we have checked that the sums that defines the zero mode (3.7) (orequivalently (3.9)) and the k -instanton sectors (3.22), both computed via localisation, agree within numericalerrors with the proposed lattice sum integral representation (1.3). More interestingly, figures 1a, 1b. illustratethe numerical evaluation of G ( τ, ¯ τ ) along one-dimensional cross-sections of the upper-half plane that passthrough the fixed points of SL (2 , Z ). The point τ = i , which is invariant under S · τ = − /τ , is a saddlepoint of G ( τ, ¯ τ ), while τ = e i π , which is invariant under T S with T · τ = τ + 1, is a maxima, finally the cusp τ = i ∞ is a global minimum (together with its images, which include τ = 0 = S · i ∞ ). These numericalresults are consistent with the expectation that the sum of the zero-mode (3.7) and the k -instanton sectors(3.22) produce the conjectured modular invariant lattice sum given by (3.29).17 Τ (cid:71) (a) G ( τ, ¯ τ ) along the imaginary axis τ = iτ . (cid:45) (cid:45) Τ (cid:71) (b) G ( τ, ¯ τ ) on the unit circle | τ | = 1 with τ ∈ [ − , ]. Figure 1: Numerical evaluation of G ( τ, ¯ τ ) in two one-dimensional subspaces of the τ plane. (a) τ takes valueson the imaginary axis showing a maximum at τ = i . The red line plots the purely perturbative expansion(3.6) truncated to fifth order. (b) τ takes values on the unit circle | τ | = 1. The integrated correlator hasmaxima at τ = exp( πi/ 3) and τ = exp(2 πi/ τ = i . SU ( N ) theory The preceding analysis of the SU (2) case will now be generalised to SU ( N ). This will lend very strongsupport for the conjecture that the integrated correlator is given by the expression (1.2) for all values of N and generates precisely-determined expressions for the rational functions B N ( t ) that enter into the integrand.Although we do not have a deductive mathematical derivation of (1.2) it embodies the structure of manyexamples that we have studied. For example, we have determined the form of the one-instanton contributionto the localised correlator for values of N up to N = 22 and to very high orders in the coupling, g Y M ,as is reviewed in appendix B. This determines the k = 1 contribution to (1.2) for these values of N withspecific expressions for B N ( t ). If we make the ansatz that B N ( t ) is independent of the instanton numberthe conjectured expression (1.2) leads to predictions for the k -instanton contributions to the integratedcorrelator. We have checked for many values of k and N and to high order in g Y M that these predictionsare in agreement with the expressions deduced from localisation and displayed in (3.22). Furthermore, aswe will see later, the expression (1.2) reproduces (and extends) the large- N expansion determined in [5]. Wehave also made a numerical comparison of the localised correlator with the lattice sum (1.2) for finite valuesof τ and a few values of N and k , which again demonstrates striking agreement.As a result we find that the lattice sum structure of the SU (2) case (3.29) extends to the general SU ( N )case, giving G N ( τ, ¯ τ ) = N ( N − (cid:88) ( m,n ) (cid:54) =(0 , (cid:90) ∞ exp (cid:16) − tπ | m + nτ | τ (cid:17) B N ( t ) dt . (4.1)The function B N ( t ) is a rational function of t , which is the generalisation of B ( t ) and takes the form B N ( t ) = Q N ( t )(1 + t ) N +1 , (4.2)18here Q N ( t ) is a polynomial in t of order 2 N + 1 defined in (1.5). Although it is not obvious from thisexpression that Q N ( t ) is a polynomial, this can be seen by using various identities that express Jacobipolynomials in terms of hypergeometric functions.We may write (1.5) in the form Q N ( t ) = t N − (cid:88) i =0 a ( N ) i t i , with a ( N ) i = a ( N )2 N − − i . (4.3)The symmetry property of the coefficients a ( N ) i implies that B N ( t ) obeys the same inversion symmetry as B ( t ) B N ( t ) = 1 t B N (cid:16) t (cid:17) . (4.4)From the explicit expression (4.2) for B N ( t ) we see that, in addition to this inversion symmetry it satisfiesthe interesting integral identities: (cid:90) ∞ B N ( t ) dt = N ( N − , (cid:90) ∞ B N ( t ) 1 √ t dt = 0 . (4.5)Using the first identity the integrated correlator, G N ( τ, ¯ τ ) in (4.1), may be expressed as G N ( τ, ¯ τ ) = 12 (cid:88) ( m,n ) ∈ Z (cid:90) ∞ exp (cid:16) − tπ | m + nτ | y (cid:17) B N ( t ) dt , (4.6)where the summation over m, n now includes the case m = n = 0. This is the result quoted in (1.3).We can now decompose G N ( τ, ¯ τ ) into its perturbative ( k = 0) and instantonic ( k (cid:54) = 0) components,following the discussion of the SU (2) case in (3.34) and (3.36). This again involves splitting the sum over( m, n ) into a sum over m with n = 0, which defines G ( i ) N, ( τ ) and a sum over m and n (cid:54) = 0. A Poisson sumconverts the sum over m into a sum over ˆ m and the product k = ˆ mn is identified with the instanton number.This includes a sector with ˆ m = 0, which has k = 0.Therefore, the complete zero mode sector again has the form G N, ( τ, ¯ τ ) = G N, ( τ ) = G ( i ) N, ( τ ) + G ( ii ) N, ( τ ) , (4.7)where G ( i ) N, ( τ ) = N ( N − (cid:88) m> (cid:90) ∞ exp (cid:16) − tπ m τ (cid:17) B N ( t ) dt , (4.8) G ( ii ) N, ( τ ) = (cid:88) n> (cid:90) ∞ exp (cid:16) − tπn τ (cid:17)(cid:114) τ t B N ( t ) dt . (4.9)Once again the k = 0 sector is the sum of two terms that are related by Weyl reflection. The proof of thisstatement relies crucially on the inversion property (4.4) together with the identities (4.5). In fact using thefirst identity in (4.5) we have G ( i ) N, ( τ ) = 12 (cid:88) m ∈ Z (cid:90) ∞ exp (cid:16) − tπ m τ (cid:17) B N ( t ) dt , (4.10)19o that after Poisson summation it becomes G ( i ) N, ( τ ) = 12 (cid:88) n ∈ Z (cid:90) ∞ exp (cid:16) − πn τ t (cid:17)(cid:114) τ t B N ( t ) dt . (4.11)Using the second identity in (4.5) we see that the n = 0 term vanishes and after the change of variables t → /t we arrive at G ( i ) N, ( τ ) = (cid:88) n> (cid:90) ∞ exp (cid:16) − tπn τ (cid:17) √ τ t B N (cid:16) t (cid:17) dtt = G ( ii ) N, ( τ ) , (4.12)where in the last step we made use of the inversion property (4.4).The k -instanton sectors (with k (cid:54) = 0) have the form G N,k ( τ, ¯ τ ) = 12 (cid:88) ˆ m (cid:54) =0 , n (cid:54) =0 ˆ mn = k e πikτ (cid:90) ∞ exp (cid:16) − π ˆ m τ t − tπn τ (cid:17)(cid:114) τ t B N ( t ) dt . (4.13)Just as in the SU (2) case we can rewrite G N,k ( τ, ¯ τ ) to make manifest the characteristic exponential suppres-sion factor of the k -instanton sector, G N,k ( τ, ¯ τ ) = 12 (cid:88) ˆ m (cid:54) =0 , n (cid:54) =0 ˆ mn = k e π ( −| k | τ + ikτ ) (cid:90) ∞ exp (cid:104) − (cid:16) | ˆ m |√ t − | n |√ t (cid:17) πτ (cid:105)(cid:114) τ t B N ( t ) dt . (4.14)Furthermore, it is straightforward to see that the lattice sum expression for G N ( τ, ¯ τ ) (4.6) can be expressedas a sum of non-holomorphic Eisenstein series, E ( s ; τ, ¯ τ ), with integer s as shown in (1.12), G N ( τ, ¯ τ ) = N ( N − ∞ (cid:88) s =2 c ( N ) s E ( s ; τ, ¯ τ ) , (4.15)where the coefficients c ( N ) s are defined from B N ( t ) via the expansion B N ( t ) = ∞ (cid:88) s =2 c ( N ) s t s − Γ( s ) . (4.16)The following are a few examples of these coefficients, c ( N )2 = 32 N (cid:0) N − (cid:1) , c ( N )3 = − N (cid:0) N − (cid:1) , c ( N )4 = 1472 N (cid:0) N − (cid:1) ,c ( N )5 = − N (cid:0) N − (cid:1) (cid:0) N + 2 (cid:1) , c ( N )6 = 1815 N (cid:0) N − (cid:1) (cid:0) N + 1 (cid:1) , (4.17) c ( N )7 = − N (cid:0) N − (cid:1) (cid:0) N + 25 N + 4 (cid:1) , c ( N )8 = 5252 N (cid:0) N − (cid:1) (cid:0) N + 605 N + 332 (cid:1) . Note the expansion (4.15) can be re-expressed by making use of the functional equation (A.12) so thatit becomes G N ( τ, ¯ τ ) = N ( N − ∞ (cid:88) s =2 c ( N ) s Γ(1 − s )Γ( s ) E (1 − s ; τ, ¯ τ ) . (4.18)20he coefficients c ( N ) s are constrained by the condition (4.4). More explicitly, using the integral representation(3.27) and requiring that the analytic continuation s → − s leads to the same expression for the correlatorrequires the function B N ( t ) to satisfy B N ( t ) = ∞ (cid:88) s =2 c ( N ) s Γ(1 − s )Γ( s ) t − s Γ(1 − s ) = 1 t B N (cid:16) t (cid:17) . (4.19)In section 5 we will further analyse (1.2) (equivalently (4.1)) and show that it has the properties one wouldexpect for the integrated four-point correlator. G N ( τ, ¯ τ ) We will now demonstrate that G N ( τ, ¯ τ ) satisfies (1.8). This is a Laplace equation in the variable τ andsimultaneously a difference equation in N – hence the terminology ‘Laplace-difference’ equation. In order toapply the Laplace operator to G ( τ, ¯ τ ) we first note the relation∆ τ e − tπY ( τ, ¯ τ ) = e − tπY ( τ, ¯ τ ) (cid:2) ( πtY ( τ, ¯ τ )) − πtY ( τ, ¯ τ ) (cid:3) = t ∂ t (cid:16) t e − tπY ( τ, ¯ τ ) (cid:17) , (4.20)where Y ( τ, ¯ τ ) = | m + nτ | τ . (4.21)It therefore follows from (1.3), after integration by parts, that∆ τ G N ( τ, ¯ τ ) = 12 (cid:88) ( m,n ) ∈ Z (cid:90) ∞ e − tπ | m + nτ | τ t d dt ( t B N ( t )) dt . (4.22)We now substitute the expression (1.4) for B N ( t ) where Q N ( t ) is defined in (1.5) by a sum of Jacobipolynomials.In order to proceed it is important to note that Jacobi polynomials satisfy the following three-termrecursion relation2( n + α − n + β − n + α + β ) P ( α,β ) n − ( z ) + 2 n ( n + α + β )(2 n + α + β − P ( α,β ) n ( z )= (2 n + α + β − (cid:104) (2 n + α + β )(2 n + α + β − z + α − β (cid:105) P ( α,β ) n − ( z ) . (4.23)Together with (1.4) and (1.5), this implies that B N ( t ) (or alternatively Q N ( t )) also satisfies a thee-termrecursion relation that takes the form p ( N, t ) B N − ( t ) + q ( N, t ) B N ( t ) + r ( N, t ) B N +1 ( t ) = 0 , (4.24)where the coefficients are the functions p ( N, t ) = 16( t + 1)( t − t − ( t − ( t − t − t + 2 t + 3) N − ( t − (9 t + 10 t + 9) N ,q ( N, t ) = − t + 1) t + 6( t + 1)( t + 3)(3 t + 1) N − t (46 t − t + 46) N , (4.25) r ( N, t ) = 16( t + 1)( t + 1) t − ( t + 1) ( t + 3)(3 t + 1)(3 t − t + 3) N + ( t + 1) (9 t − t + 9) N . N expansion of thecorrelator, which determines the coefficients d s(cid:96) in (1.14).Using the recursion relation (4.24) together with the property( z − ddz P ( α,β ) n ( z ) = n P ( α,β ) n ( z ) − ( α + n ) P ( α,β +1) n − ( z ) , (4.26)we find that t d dt ( t B N ( t )) = N ( N − B N +1 ( t ) − N − B N ( t ) + N ( N + 1) B N − ( t ) . (4.27)Substituting (4.27) in (4.22) gives the Laplace-difference equation,(∆ τ − G N ( τ, ¯ τ ) = N (cid:104) G N +1 ( τ, ¯ τ ) − G N ( τ, ¯ τ ) + G N − ( τ, ¯ τ ) (cid:105) − N (cid:104) G N +1 ( τ, ¯ τ ) − G N − ( τ, ¯ τ ) (cid:105) , (4.28)which is the equation in the corollary (1.8). This equation is of great significance since it determines manyof the properties of G N ( τ, ¯ τ ). We will make use of it in the following ways. • Given the expression for G ( τ, ¯ τ ) as input, together with G ( τ, ¯ τ ) = 0, (1.8) determines G N ( τ, ¯ τ )recursively. This powerful result demonstrates that the integrated four-point correlators for any SU ( N )group are precisely determined by the N = 2 case. • We will shortly use (1.8) to show that the terms in the perturbative expansion in powers of a = g N/ (4 π ) are independent of N up to order a , apart from an overall factor of ( N − N = 4 SYM perturbation theory. At higher orders the coefficients have adependence on N that is characteristic of non-planar contributions to perturbative SYM. The detailedexpressions will be determined in the next section. • The Laplace-difference equation determines a recursion relation for the coefficients c ( N ) s in the expan-sion (4.15). This follows upon substituting (4.15) into (1.8) and making use of the Laplace eigenvalueequation (A.13) satisfied by each E ( s ; τ, ¯ τ ). Taking these Eisenstein series to be linearly independentresults is a three-term recursion relation for the coefficients c ( N ) s , N ( N − c ( N +1) s − (cid:16) N − 1) + s ( s − (cid:17) c ( N ) s + N ( N + 1) c ( N − s = 0 . (4.29)It is easy to check that the coefficients (4.17) solve this recursion relation for all values of N , subjectto the initial condition (1.11) for c (2) s . Similarly, with the input of the leading term (the N s +1 term),one may solve the recursion relation (4.29) order by order in 1 /N and for all values of s . The first fewof the resulting coefficients are c ( N ) s = ( − s s − ( s − s − (cid:0) s + (cid:1) √ π Γ( s + 2) N s +1 + ( − s s − (1 − s ) ( s − s − (cid:0) s − (cid:1) √ π Γ( s − N s − + ( − s s − (1 − s ) ( s − s − s + 30)Γ (cid:0) s − (cid:1) √ π Γ( s − N s − + (4.30)+ ( − s s − (1 − s ) ( s − (cid:0) s − s + 2749 s − s + 1680 (cid:1) Γ (cid:0) s − (cid:1) √ π Γ( s − N s − + · · · . In section 5.3, we will use the Laplace-difference equation to show that the large- N expansion has theform (1.14), which is a series of half-integer powers of 1 /N with coefficients proportional to Eisensteinseries with 1 / d s(cid:96) will again be determined recursively.The Laplace-difference equation (1.8) can also be written in terms of the shift-operator D N defined by( D N ) α f ( N ) = f ( N + α ), and takes the form(∆ τ − G N = (cid:2) N ( D N − D − N ) − N ( D N − D − N ) (cid:3) G N . (4.31)Since the function G N is differentiable in N we can substitute the representation D N = exp( ∂ N ) and expand(4.31) as (∆ τ − G N = ∞ (cid:88) k =1 (cid:2) N ( N − 1) + ( − k N ( N + 1) (cid:3) ∂ kN k ! G N . (4.32)From this expression we can easily see that when N → ∞ the Laplace-difference equation (1.8) becomes aLaplace equation in both τ and N , taking the form(∆ τ − G N ( τ, ¯ τ ) = N →∞ ( N ∂ N − N ∂ N ) G N ( τ, ¯ τ ) . (4.33)In writing this equation we have assumed that G N ( τ, ¯ τ ) is a power series in 1 /N and non-leading terms in1 /N have been suppressed. This expression will be important when we discuss the large- N , fixed τ expansionin section 5.3. For the moment we note that in the large- N limit, if we focus on a single power-like term ofthe form G N ( τ, ¯ τ ) ∼ N − s F s ( τ, ¯ τ ), the Laplace-difference equation (1.8) reduced to (4.33) becomes[∆ τ − s ( s − F s ( τ, ¯ τ ) = 0 , (4.34)which is the Laplace eigenvalue equation (A.13) satisfied by the Eisenstein series E ( s ; τ, ¯ τ ). SU ( N ) integrated correlator Since we believe that (1.3) should reproduce the integrated correlation function (1.2) for all N and for all g Y M it is of interest to verify that it matches known properties of the correlator in various limits. In thissection we will consider the expansions around three such limits, namely(i) small g Y M and finite N ;(ii) large- N with fixed ’t Hooft coupling λ = N g Y M ;(iii) large- N with fixed g Y M . N In this subsection, we will study the perturbative expansion of the integrated correlator. It is straightforwardto obtain the small coupling expansion from the general result expressed in terms of a sum of Eisensteinseries (4.15). The perturbative expansion in the k = 0 sector is an expansion in powers of 1 /τ , which is23btained by summing the τ − s terms in the zero modes of the Eisenstein series in (A.10) using the expression G N, ( τ ) = 2 G ( i ) N, ( τ ) = 2 G ( ii ) N, ( τ ) exhibited in (4.12) (we are here recalling that the Borel sum of the τ s terms is identical to the sum of the τ − s terms). The first few terms of this expansion are as follows G N, ( τ ) = ( N − (cid:34) ζ (3) a − ζ (5) a ζ (7) a − ζ (9) (cid:0) N − (cid:1) a 32+ 114345 ζ (11) (cid:0) N − (cid:1) a − ζ (13) (cid:0) N − + N − (cid:1) a ζ (15) (cid:0) N − + N − (cid:1) a O ( a ) (cid:35) , (5.1)where a = g Y M N/ (4 π ) = N/ ( πτ ) with arbitrary N ≥ 2. When N = 2, this reduces to (3.4), theresult we obtained earlier for the case of SU (2). Here we have only listed the first few orders in a butit is straightforward to compute the expansion to arbitrary orders. In fact, we can write down a generalexpression for the first few orders in 1 /N . We will come back to this later.In obtaining this expression, we have used the coefficients c ( N ) s listed in (4.17), and the definition of thezero mode of the Eisenstein series. Since the sum of the τ s terms in the zero modes of the Eisenstein seriesin (A.10) contributes the same as the sum of the τ − s terms we can simply keep the terms proportional to τ − s and double the result.The expression (5.1) has a remarkably simple structure – the coefficients of each power of a = g Y M N/ (4 π )is a rational ( N -dependent) multiple of an odd zeta value. By contrast, the perturbative expansion of theunintegrated correlation function is a sum of highly non-trivial transcendental functions of the positioncoordinates, and only the first orders in perturbation theory (i.e. up to three loops) have so far beenevaluated [21].Among many interesting features, the expression (5.1) shows clearly that the contribution of non-planarFeynman diagrams first enter at order a . This is consistent with the known result [18] that the correlatordoes not receive any non-planar corrections at the first three loops. We further note that the next-ordernon-planar correction only starts to enter at six loops (and not at five loops). We have checked that thispattern continues at higher orders, where higher order non-planar effects (higher powers of N − ) only enterevery two loops. The same pattern was observed up to seven loops in the computation of the Sudakovform factor that involves the same BPS operators we are considering here [22]. Finally, it would also beinteresting to make connections between our results and those obtained in [4], where the zero-instantonresults are expressed in terms of the Laguerre polynomials. An alternative way of arriving at the perturbative expansion for N > G , ( τ ) in the SU (2) case (3.6) and determine the expressions for higher values of N iteratively bymaking use of the Laplace-difference equation (1.8). For example, substituting G , ( τ ) in (1.8) leads to the24xpression for the N = 3 case G , ( y ) ∼ ∞ (cid:88) s =2 ( − s (2 s − − s + s )Γ(2 s + 1) ζ (2 s − s Γ( s − y − s . (5.2)In this manner the corollary (1.8) determines the correlator for any desired value of N .Similarly, other features of the perturbative expansion (5.1) can be deduced directly from (1.8). Thus, ifwe input a general perturbation expansion into G N, ( τ ) ∼ ∞ (cid:88) s =2 α s ( N ) τ − s (5.3)into (1.8), we obtain equations relating the coefficients of each power τ − s . This leads to an infinite systemof three-term recursion relations of the form N ( N − α s ( N + 1) − [2( N − 1) + s ( s − α s ( N ) + N ( N + 1) α s ( N − 1) = 0 , (5.4)subject to the initial condition α s (1) = 0 for all s ≥ 2, which is reminiscent of (4.29). Although we cannotprovide the general solution for arbitrary s , it is simple to determine an interesting pattern, as follows. The N -dependence of the coefficients of first three orders in perturbation theory, i.e. at order τ − s = ( g Y M / π ) s − with s = 2 , , 4, takes the extremely simple form α s ( N ) = N − − N s s α s (2) , s = 2 , , , (5.5)which is consistent with the form exhibited in (5.1). From the fourth order onward the N -dependence of thecoefficients gets non-planar corrections, which have the structure, α ( N ) = N − − N (cid:16) N − (cid:17)(cid:16) − (cid:17) α (2) , α ( N ) = N − − N (cid:16) N − (cid:17)(cid:16) − (cid:17) α (2) ,α ( N ) = N − − N (cid:16) N − + N − (cid:17)(cid:16) − + − (cid:17) α (2) , · · · (5.6)If we substitute the values of the SU (2) perturbative coefficients, given in (3.4) and (3.6), for α s (2) then theseries (5.3) reproduces (5.1). We would now like to compare the numerical coefficients of the perturbation expansion of the integratedcorrelator (5.1) with explicit loop calculations of the four-point correlator in the literature. In order to makethis comparison we will need to integrate the perturbative expression for the unintegrated correlator overthe positions of the operators as given in (2.3) (where the superscript L indicates the order in perturbationtheory), G ( L ) N, ( τ, ¯ τ ) = I (cid:104) T ( L ) N ( U, V ) (cid:105) = − π (cid:90) ∞ dr (cid:90) π dθ r sin ( θ ) U T ( L ) N ( U, V ) , (5.7)25ith U = 1 + r − r cos( θ ) and V = r .We will make use of the fact that the one-loop and two-loop contributions are L = 1 and L = 2 examplesof ladder diagrams, so we can proceed by making use of the known expression, Φ ( L ) ( U, V ), associated with L -loop ladder diagrams. This expression was determined in [23], for arbitrary L and has the formΦ ( L ) ( U, V ) = − z − ¯ z f ( L ) (cid:18) z − z , ¯ z − ¯ z (cid:19) , (5.8)where f ( L ) ( z, ¯ z ) = L (cid:88) r =0 ( − r (2 L − r )! r !( L − r )! L ! log r ( z ¯ z ) (Li L − r ( z ) − Li L − r (¯ z )) , (5.9)and z, ¯ z are related to the cross ratios U, V by z ¯ z = U and (1 − z )(1 − ¯ z ) = V .The one-loop contribution to the unintegrated correlator is given by the expression [24, 25] T (1) N ( U, V ) = − ( N − a UV Φ (1) ( U, V ) . (5.10)As shown in the appendix C, the effect of the integration in (5.7) basically turns a L -loop four-point integralinto a ( L + 1)-loop two-point integral. Such two-point integrals for ladder diagrams were evaluated in [26],leading to I (cid:20) UV Φ ( L ) ( U, V ) (cid:21) = − (cid:18) L + 2 L + 1 (cid:19) ζ (2 L + 1) . (5.11)It turns out, as also shown in appendix C, that the integration of the product of two ladder diagrams,Φ ( L ) ( U, V )Φ ( L ) ( U, V ) with measure defined by (5.7), behaves as a single ladder diagram Φ ( L + L ) ( U, V ).In other words, I (cid:20) UV Φ ( L ) ( U, V )Φ ( L ) ( U, V ) (cid:21) = − (cid:18) L + 2 L + 1 (cid:19) ζ (2 L + 1) , with L = L + L . (5.12)We have also checked these identities numerically to high precision for various values of L .Using (5.11) and applying it to the L = 1, case gives, I (cid:104) T (1) N ( U, V ) (cid:105) = ( N − 1) 3 ζ (3) a , (5.13)which agrees precisely with localisation computation, namely the leading-order term in (5.1).The two loop contribution to the correlator is given by [27, 28] T (2) N ( U, V ) = ( N − a UV (cid:20) Φ ( U, V ) + 14 (1 + U + V ) (cid:16) Φ (1) ( U, V ) (cid:17) (cid:21) , (5.14)where Φ ( U, V ) = Φ (2) ( U, V ) + 1 V Φ (2) (1 /V, U/V ) + 1 U Φ (2) (1 /U, V /U ) , (5.15) The relative normalisation between one-loop and two-loop results are consistent with the OPE analysis. See for instance(4.14) of [29]. (2) defined in (5.8). Using (5.11), we have I (cid:20) UV Φ ( U, V ) (cid:21) = − ζ (5) , (5.16)where we have used the fact that each term in Φ ( U, V ) gives the same result since they are related bypermuting external legs, and are indistinguishable after integration. Similarly, using (5.12), we find the termproportional to (cid:0) Φ (1) ( U, V ) (cid:1) in (5.14) gives an identical contribution. Combining these contributions givesthe total two-loop result I [ T (2) N ( U, V )] = − ( N − a 16 120 ζ (5) × 54 = − ( N − 1) 75 ζ (5) a , (5.17)which again agrees with the localisation computation of the integrated correlator as shown in (5.1).So we see that the simple expressions for the coefficients in the perturbative expansion of the integratedcorrelator correctly correspond to the explicit evaluation of these terms using standard field theory techniques,although these are immensely more complicated. It would be of obvious interest to compute the integratedcorrelator to higher orders making use of known results for the unintegrated correlator. For example, theplanar limit integrand has been constructed up to ten loops [30, 31], and the first non-planar contribution(i.e. at four loops) was obtained in [18]. As shown in (5.11) and (5.12), a particular contribution to thecorrelator at L loops does lead to a result that is proportional to ζ (2 L + 1) as expected from the localisationcomputation in (5.1).Of course, beyond two loops ladder diagrams are only part of the story and so we do not expect to matchthe exact coefficients of ζ (2 L + 1) without including non-ladder contributions. As we argue in appendixC, for any L the integrated L -loop correlator is simply an ( L + 1)-loop two-point function. It is plausiblethat this will allow us to compute the integrated correlator beyond the orders presented here. For instance,at three loops, there is a contribution arising from the so-called tennis court diagram, the integration overthis diagram again leads to a 4-loop two-point function whose result is also known [32]. We will leave asystematic study of these integrals as a future direction. N expansion with fixed ’t Hooft coupling In this section, we will study the large- N expansion of the integrated correlator with fixed ’t Hooft cou-pling. In this limit the instanton contributions are exponentially suppressed and so the correlator only hasperturbative contributions from the zero mode sector, G N ( τ, ¯ τ ) = N →∞ G N, ( λ ) = G ( i ) N, ( λ ) + G ( ii ) N, ( λ ) . (5.18)We will define the genus expansion of the integrated correlator in the ’t Hooft limit by G N ( τ, ¯ τ ) ∼ ∞ (cid:88) g =0 N − g G ( g ) ( λ ) , (5.19)where G ( g ) ( λ ) is the genus g contribution to the ’t Hooft large- N expansion with λ = g Y M N = 4 πN/τ , andfor fixed value of λ we have an asymptotic series in 1 /N . As anticipated, it is possible to write down allorder expressions for the perturbative expansion of G ( g ) ( λ ) at small λ or at large λ for any value of g .27he computation of the perturbative (small- λ ) expansion of G ( g ) ( λ ) follows the same path that led to(5.1) in the previous section. We will use (4.15) with the large- N expansion of the coefficients c ( N ) s givenin (4.30). The leading term in this expansion is proportional to N s +1 , which multiplies the zero mode of E ( s ; τ, ¯ τ ) for each value of s in (4.15) . The perturbative terms proportional to τ − s = (4 πN/λ ) − s thereforecontribute to the leading coefficient proportional to N (the g = 0 term in (5.19)). As before, the sum of theterms proportional to τ s = (4 πN/λ ) s resum in a manner that doubles the coefficient of N . The net resultfor the coefficient of the leading power of N is G (0) ( λ ) = ∞ (cid:88) n =1 − n +1 ζ (2 n + 1)Γ (cid:0) n + (cid:1) π n +1 Γ( n )Γ( n + 3) λ n . (5.20)It is easy to see that the series is in fact convergent with a finite radius | λ | < π . After using the integralidentity (3.8) or alternatively performing a modified Borel resummation (D.11), it is straightforward toevaluate the sum and obtain G (0) ( λ ) = λ (cid:90) ∞ dw w F (cid:16) ; 2 , (cid:12)(cid:12)(cid:12) − w λπ (cid:17) π sinh ( w ) . (5.21)The above integral representation is the analytic continuation of the perturbation expansion given in (5.20),which is well-defined for λ beyond the convergence radius, i.e. for | λ | ≥ π . The result is in fact identical tothe expression given in [3] obtained (see equation (3.48) of [3] for p = 2) after using the relation, λ π w F (cid:18) 52 ; 2 , (cid:12)(cid:12)(cid:12) − w λπ (cid:19) = J (cid:16) w √ λ/π (cid:17) − J (cid:16) w √ λ/π (cid:17) , (5.22)where J α is the Bessel function. We stress however, that the derivation in [3] was in the context of thelarge- λ limit. As we will discuss in the next sub-section, the large- λ expansion is not Borel summable andits form needs to be supplemented by exponentially small (resurgent) contributions.A similar analysis can be performed for the higher order terms using (4.30). As an example we willconsider the g = 1 term in (5.19). Using the N s − term in (4.30) and (4.15), we find G (1) ( λ ) = ∞ (cid:88) n =1 ( − n ( n − n + 1) ζ (2 n + 1)Γ (cid:0) n − (cid:1) Γ (cid:0) n + (cid:1) π n +1 Γ( n ) λ n . (5.23)This is again a convergent series with a finite radius of convergence | λ | < π . Using the integral identity(3.8), a straightforward resummation of the series leads to G (1) ( λ ) = λ (cid:90) ∞ dw w F (cid:16) , 2; 1 , , (cid:12)(cid:12)(cid:12) − w λπ (cid:17) − F (cid:16) ; 1 , (cid:12)(cid:12)(cid:12) − w λπ (cid:17) − J (cid:16) w √ λπ (cid:17) π sinh ( w ) . (5.24)After using identities relating Hypergeometric functions and Bessel functions, we see the above result isequivalent to that obtained in [4] (notably, the first equation (cid:101) F in (3.19) of that reference). In similarfashion, the analysis reproduces all the higher-order terms given in equation (3.19) of the reference [4], andalso leads to new higher order term, although we will not list the these results here.28 .2.1 Resurgence of the strong coupling expansion We have seen that the integral representations of G ( g ) ( λ ) based on the small- λ expansion are well-definedfor λ beyond its radius of convergence at every order in the 1 /N expansion, We may therefore explore theirconvergence and resurgence properties in the large- λ domain. The large- λ expansion of these terms wasstudied in [3, 4] by using the Mellin-Barnes representation of the product of Bessel functions J a ( x ) J b ( x ) = 12 πi (cid:90) c + i ∞ c − i ∞ ds Γ( − s )Γ(2 s + a + b + 1)( x/ a + b +2 s Γ( s + a + 1)Γ( s + b + 1)Γ( s + a + b + 1) . (5.25)It leads to the large- λ expansion of the leading term of order N in the 1 /N expansion of the integratedcorrelator, G (0) ( λ ) ∼ 14 + ∞ (cid:88) n =1 b (0) n x − n − , (5.26)where x = √ λ , and the coefficients of the asymptotic series are given by b (0) n = Γ (cid:0) n − (cid:1) Γ (cid:0) n + (cid:1) Γ(2 n + 1) ζ (2 n + 1)2 n − π Γ( n ) . (5.27)Similarly for the sub-leading term, we have G (1) ( λ ) ∼ − x 16 + ∞ (cid:88) n =1 b (1) n x − n − (5.28)where b (1) n is given by b (1) n = − n (2 n + 11)Γ (cid:0) n + (cid:1) Γ (cid:0) n + (cid:1) ζ (2 n + 1)24 π Γ( n + 2) . (5.29)From the expressions of b (0) n and b (1) n we see that, unlike the weak coupling expansion, the strong couplingexpansion is factorially divergent and so we will analyse it using Borel summation (more details are given inappendix D). As in [33] we need to consider the modified Borel transformation, B : ∞ (cid:88) n =1 b (0) n x − n − → ∞ (cid:88) n =1 π b (0) n ζ (2 n + 1)Γ(2 n + 2) (2 w ) n +1 := ˆ φ ( w ) . (5.30)For the b (0) n given in (5.27), we find,ˆ φ ( w ) = − πw F (cid:18) − , 32 ; 1 (cid:12)(cid:12)(cid:12) w (cid:19) . (5.31)Then the directional Borel resummation of (5.26) leads to S θ G (0) ( x ) = 14 + xπ (cid:90) e iθ ∞ dw ( xw ) ˆ φ ( w ) , (5.32)which defines an analytic function for x > θ ∈ ( − π/ , π/ G (0) ( λ ) it defines a functionthat is neither unique nor is it real for x > θ . The reason is that the integral (5.32) isnot well defined along the real axis since ˆ φ ( w ) has a cut along [1 , ∞ ). This is a signal that the asymptoticseries is not Borel summable and standard resurgence arguments suggest that the large- λ expansion requiresthe addition of exponentially small non-perturbative terms. These terms are encoded by the discontinuity,related to what is usually called Stokes automorphism factor, which is given by:( S + − S − ) G (0) ( x ) = ∆ G (0) ( x ) = xπ (cid:90) ∞ dw 14 sinh ( xw ) Disc ˆ φ ( w ) , (5.33)where S ± define the lateral Borel resummations S ± G (0) ( x ) = lim θ → ± S θ G (0) ( x ) . (5.34)The discontinuity of the Borel transform is given byDisc ˆ φ ( w ) = ˆ φ ( w + i − ˆ φ ( w − i 0) = 16 πiw F (cid:18) − , 32 ; 1 (cid:12)(cid:12)(cid:12) − w (cid:19) , (5.35)valid for w > 1, where we have used the discontinuity of F given byDisc F ( a, b ; c | z ) = 2 πi Γ( c )Γ( a )Γ( b )Γ( c − a − b + 1) z − c ( z − c − a − b × F (1 − a, − b ; c − a − b + 1 | − z ) , (5.36)valid for z > G (0) ( x ) from (5.33) we first shift the integration variable w → w + 1 and then expandsinh( xw ) in the denominator, giving∆ G (0) ( x ) = 16 i x ∞ (cid:88) n =1 ne − nx (cid:90) ∞ e − nxw ( w + 1) F (cid:18) − , 32 ; 1 (cid:12)(cid:12)(cid:12) − w ( w + 2) (cid:19) dw . (5.37)This expression is a sum of ‘instantonic’ terms that are non-perturbative in 1 /x = 1 / √ λ , with coefficients e − x = e − √ λ . The integral can be performed by expanding the integrand in powers of w to obtain∆ G (0) ( x ) = 16 i x ∞ (cid:88) n =1 ne − nx (cid:104) (2 nx ) − + 92 (2 nx ) − + 1178 (2 nx ) − + 48916 (2 nx ) − + O ( x − ) (cid:105) . (5.38)This is a perturbative expansion around each ‘instantonic’ term in powers of 1 /x .The sum over n can be performed straightforwardly for each power of x , which leads to an expression forthe non-perturbative sector of the large λ expansion as a sum of polylogs (recall x = √ λ ),∆ G (0) ( λ ) = i ∞ (cid:88) (cid:96) =1 a (cid:96) (2 √ λ ) − (cid:96) L i (cid:96) − ( e − √ λ ) (5.39)= i (cid:104) ( e − √ λ ) + 18Li ( e − √ λ ) λ / + 117Li ( e − √ λ )4 λ + 489Li ( e − √ λ )16 λ / + · · · (cid:105) . a (cid:96) are rational numbers, which are determined by the following recursion relation,0 = ( (cid:96) − (cid:96) − (cid:96) + 1) (cid:0) (cid:96) − (cid:96) − (cid:1) a (cid:96) + 3 (cid:0) (cid:96) − (cid:96) − (cid:96) + 35 (cid:96) + 15 (cid:1) a (cid:96) +1 + 2( (cid:96) + 1) (cid:0) (cid:96) − (cid:96) − (cid:1) a (cid:96) +2 (5.40)with initial conditions a = 8 , a = 36.The above expression for ∆ G (0) ( x ) is input into the median resummation of the asymptotic formal powerseries (5.26), which is defined by S med G (0) ( x ) = S ± G (0) ( x ) ∓ 12 ∆ G (0) ( x )= 14 + xπ (cid:90) ∞ dw ( xw ) Re ˆ φ ( w ) , (5.41)which is manifestly real for x > 0, as expected from (5.26). Following a discussion very similar to thatin [33] we can show that this median resummation is identical to (5.21). In other words, we deduce thatthe leading N term (5.21), when expanded at strong coupling λ → ∞ , does include an infinite sum overnon-perturbative corrections of the form e − n √ λ . Since the proof is extremely close to the analysis of [33]we have relegated the details to appendix D.1.We have also analysed the higher order 1 /N terms, and similar conclusions are found that the small- λ expansion is always convergent with a finite radius | λ | < π , whereas the large- λ expansion is alwaysasymptotic and not Borel summable, which requires exponential terms, whose leading term goes as e − √ λ .For example we can analyse the N term G (1) ( λ ) presented in (5.24) expanded at strong coupling (5.28).Similar to (5.41) we now have S med G (1) ( x ) = S ± G (1) ( x ) ∓ 12 ∆ G (1) ( x )= − x 16 + xπ (cid:90) ∞ dw ( xw ) Re ˆ ψ ( w ) , (5.42)where the modified Borel transform computed from (5.28) is given byˆ ψ ( w ) = − π w (cid:104) (10 w + 81 w + 36) F (cid:16) , 52 ; 1 (cid:12)(cid:12)(cid:12) w (cid:17) + 9(3 w + w − F (cid:16) , 52 ; 1 (cid:12)(cid:12)(cid:12) w (cid:17)(cid:105) . (5.43)The difference between the two lateral resummations can be computed once more using the knowndiscontinuity for the hypergeometric function∆ G (1) ( x ) = ( S + − S − ) G (1) ( x ) (5.44)= − ix (cid:90) ∞ dw w sinh ( xw ) (cid:104) w + 81 w + 36) F (cid:16) , 52 ; 4 (cid:12)(cid:12)(cid:12) − w (cid:17) + 27(3 w + w − F (cid:16) , 52 ; 5 (cid:12)(cid:12)(cid:12) − w (cid:17)(cid:105) . Proceeding as above we first shift the integration variable w → w + 1, then expand the sinh function andfinally integrate order by order in powers of w to arrive at∆ G (1) ( λ ) = − i (cid:104) ( e − √ λ )2 − ( e − √ λ )2 λ / + 3897Li ( e − √ λ )2 λ − ( e − √ λ )2 λ / + · · · (cid:105) , (5.45) The same behaviour was found in [34] in the computation of the exact slope for AdS/CFT. λ = L /α (cid:48) (where α (cid:48) is the square of the string length scale and L is the AdS length scale) wesee that the string theory translation of a e − √ λ term is a term proportional to e − L /α (cid:48) . This indicates acontribution that should arise from world-sheet instanton, which would be the effect of a string world-sheetpinned to the four operators in the correlator on the AdS × S boundary and stretching into the interior.It is also worth mentioning that similar behaviour has been found for other properties of N = 4 SYM. Forinstance, it is well known that the cusp anomalous dimension has also a convergent small λ expansion withthe same convergence radius, while in the large- λ expansion [35] it also requires a completion with similar,but slightly different, exponentially small terms of order λ / e −√ λ/ with a considerably more complicatedresurgence structure than in the present case [36, 37]. Similarly, the “anomalous dimension” associated withthe six-point MHV amplitude in N = 4 SYM studied in [38], behaves as e −√ λ . It would be of interestto understand the origin of the interesting similarities and differences of all these exponentially suppressedterms.We also note that, for fixed ’t Hooft coupling λ , the genus expansion (5.19) is expected to be an asymptoticseries in 1 /N . It would be interesting to understand the full non-perturbative definition of the genusexpansion, which should also contain terms exponentially suppressed in N . Once the complete large- N transseries is known it would be of interest to retrace our steps and re-derive the finite N results from theBorel-Ecalle resummation of the non-perturbative large- N genus expansion as discussed in [39].Finally, as shown in (4.30), the large- N expansion of the coefficients c N ( s ) is determined by the Laplace-difference equation once the term with the leading power of N is given. We also found in this section thatthe leading power of N multiplies G (0) ( λ ) given in (5.21). We therefore conclude that the higher genus terms G ( g ) ( λ ) must also be determined in terms of G (0) ( λ ) through the Laplace-difference equation. Furthermore,we will see in the next section that the large- N expansion with fixed g Y M is also determined by the Laplace-difference equation once G (0) ( λ ) is given. N expansion with fixed g Y M In this section, we will study the large- N expansion with fixed g Y M . This is the limit where Yang–Millsinstanton contributions are not suppressed, which is crucial for the study of the SL (2 , Z ) duality of N = 4SYM (see [5, 7, 40] for recent results related to this paper). The method we utilised in the previous section isclearly also applicable for this limit. Indeed, one can obtain the zero-instanton terms by simply re-expressing λ as g Y M N/ π = N/τ in the large- λ expansions of G ( g ) ( λ ), which reproduces the known large- N resultsof [5]. In fact this was precisely how the perturbative terms were obtained in [5], bur we are now able to32ompute terms to much higher order. Here we list the results of the first few orders in the 1 /N expansion, G N, ( τ ) ∼ N − N (cid:32) π / τ / − τ / ζ (3)4 π / (cid:33) + 1 N (cid:32) π / τ / + 45 τ / ζ (5)128 π / (cid:33) + 1 N (cid:32) π / τ / − π / τ / − τ / ζ (3)4096 π / + 4725 τ / ζ (7)16384 π / (cid:33) + 1 N (cid:32) τ / ζ (9)131072 π / − τ / ζ (5)32768 π / − π / τ / + 3 π / τ / (cid:33) + O ( N − ) . (5.46)The instanton contributions in this limit can, in principle, be obtained in the same way by using (4.15),and again reproduce the large- N results of [5]. However, performing the infinite sum over s in (4.15) becomesvery tedious for a general instanton number. We will take a much more efficient route to determining thelarge- N instanton sector. This also reveals interesting structure in the integrated correlator. The idea isthat, instead of performing the large- N expansion of (4.15), we will expand the integrand B N ( t ) of theintegrated correlator (1.3). We will see that although this leads to well-defined convergent t integrals in thesectors with non-zero instanton number, k (cid:54) = 0, such an expansion leads to ill-defined divergent t integralsin the zero instanton ( k = 0) sector, However, this problem can be circumvented since the k = 0 sector isalready taken care of by (5.46).In order to study the large- N expansion of B N ( t ), which was defined by (1.4) and (1.5), we will use thefollowing integral representation of Jacobi polynomials(1 − x ) α (1 + x ) β P ( α,β ) n ( x ) = 12 πi (cid:73) γ (cid:104) z − z − x ) (cid:105) n (1 − z ) α (1 + z ) β z − x dz , (5.47)where the closed contour γ circles around the pole z = x but not the (possibly singular) points z = ± B N ( t ) = N ( N − πi (cid:73) γ exp[ − N S ( t, z )] g ( t, z, N ) dz , (5.48)with S ( t, z ) = log (cid:16) t + 1) + z ( t − − z )( t − (cid:17) , (5.49) g ( t, z, N ) = (1 − z ) 8 t [ t + 1 + z ( t − N − t − [ t − z ( t + 1)] t ( t − z + 1) ( t + 1 + z ( t − , (5.50)and the closed contour γ circles around the pole z = (1 + t ) / (1 − t ) but not the points z = ± 1. Thisexpression exhibits the exponential N dependence in a form that is suitable for a saddle point analysis inthe limit N → ∞ .There are two saddle points, corresponding to the solutions of ∂S ( t, z ) ∂z = 0 , (5.51)33nd given by z = 1 + t − t , z = 1 − t t , (5.52)for which the exponent, S ( t, z ) takes the values S ( t ) = S ( t, z ) = 0 , S ( t ) = S ( t, z ) = log (cid:104) ( t + 1) ( t − (cid:105) . (5.53)We note that for t > S ( t ) > t > 0. This solution thereforecorresponds to a contribution that is exponentially small at large- N prior to integration over t . Therefore, forthe time-being we will assume that this second saddle point is negligible in the large- N limit and commenton this point later on.Expanding S ( t, z ) around the first saddle point to quadratic order gives S ( t, z + δ ) = − ( t − t δ + O ( δ ) , (5.54)so we see that the contour of integration can be replaced by the steepest descent contour δ ∈ ( − i ∞ , i ∞ ).The N -dependence is extracted by the rescaling δ → ˜ δ/ √ N , leading to B N ( t ) = N ( N − πi (cid:90) + i ∞− i ∞ exp (cid:104) ( t − t ˜ δ + O (cid:16) ˜ δ √ N (cid:17)(cid:105) g (cid:16) t, z + ˜ δ √ N , N (cid:17) d ˜ δ √ N , (5.55)Expanding both S and g in (5.48) to the appropriate order in ˜ δ , or equivalently in N , gives the large- N expansion which starts with the terms B N ( t ) ∼ − N √ π ( t / + t − / ) + 1 N √ π ( t / + t − / )+ 1 N √ π [105( t / + t − / ) − t / + t − / )] + O ( N − ) . (5.56)However, calculating higher order terms in the large- N expansion in this manner is complicated and itis far more efficient to use the recursion relation (4.24) using the initial term B N ( t ) ∼ − N √ π ( t / + t − / ) + O ( N − ) , (5.57)as input to determine higher order corrections. In this procedure we start by writing the large- N expansionof B N ( t ) as B N ( t ) = ∞ (cid:88) (cid:96) =0 N − (cid:96) p (cid:96) ( t ) , (5.58)from which it follows that B N ± ( t ) = ∞ (cid:88) (cid:96) =0 N − (cid:96) (cid:96) (cid:88) k =0 (cid:18) − (cid:96) + kk (cid:19) ( ± k p (cid:96) − k ( t ) . (5.59)34ubstituting this into (4.24) and imposing that the recursion relation is satisfied order by order in 1 /N wearrive at the infinite system of linear equations O ( N − / ) : 5( t + 1) p ( t ) + 8 t (1 + t ) p ( t ) = 0 ,O ( N − / ) : (45 t + 118 t + 45) p ( t ) + 240( t + t ) p ( t ) − t p ( t ) = 0 ,O ( N − / ) : 4 t (13 t + 11 t + 13) p ( t ) + ( t + 1)(63 t + 274 t + 63) p ( t ) − t (12 t + 19 t + 12) p ( t ) + 768( t + t ) p ( t ) = 0 ,O ( N − / ) : · · · , (5.60)where we have imposed p − ( t ) = p − ( t ) = 0.If we now substitute the initial condition (5.57) for the leading term p ( t ) = − √ π ( t / + t − / ) , (5.61)we can easily generate as many terms as we want in the large- N expansion using the infinite system ofequations (5.60). The first few of these are the following p ( t ) = 1564 √ π ( t / + t − / ) ,p ( t ) = 3154096 √ π ( t / + t − / ) − √ π ( t / + t − / ) ,p ( t ) = 94516384 √ π ( t / + t − / ) − √ π ( t / + t − / ) ,p ( t ) = 2598754194304 √ π ( t / + t − / ) − √ π ( t / + t − / ) + 45992097152 √ π ( t / + t − / ) , (5.62)which one can check are indeed consistent with the saddle point result (5.56) but are much easier to generateto higher orders.Although we have not determined a closed formula for p (cid:96) ( t ), it is easy to determine analytic expressionsfor the coefficients of the highest (and lowest) powers, as well as the sub-leading ones. These in turndetermine the coefficients of the highest index Eisenstein series, and the next to highest index Eisensteinseries, contributing at any order in the 1 /N expansion. In particular, using the recursion relation (4.24) wefind by induction that p (cid:96) ( t ) = ( (cid:96) + 1)Γ (cid:16) (cid:96) − (cid:17) Γ (cid:16) (cid:96) + (cid:17) (cid:96) +2 π / Γ( (cid:96) + 1) (cid:16) t (cid:96) + + t − − (cid:96) (cid:17) − ( (cid:96) − (2 (cid:96) + 9)Γ (cid:16) (cid:96) + (cid:17) (cid:96) +3 π / Γ( (cid:96) + 1) θ ( (cid:96) − (cid:16) t (cid:96) − + t − (cid:96) (cid:17) + ( (cid:96) − (20 (cid:96) + 48 (cid:96) − (cid:0) (cid:96) − (cid:1) Γ (cid:0) (cid:96) + (cid:1) 45 2 (cid:96) +5 π / Γ( (cid:96) ) θ ( (cid:96) − t (cid:96) − + t − (cid:96) ) − ( (cid:96) − (cid:0) (cid:96) − (cid:96) − (cid:96) + 90912 (cid:96) − (cid:1) Γ (cid:0) (cid:96) − (cid:1) Γ (cid:0) (cid:96) + (cid:1) (cid:96) +7 π / Γ( (cid:96) − θ ( (cid:96) − t (cid:96) − + t − (cid:96) )+ · · · (5.63)35here (cid:96) ≥ 0, and θ ( x ) is the unit step function. It is straightforward to determine higher order terms usingthe recursion relation (4.24). Furthermore, the above results also show clearly the general structure of thesecoefficients.We would now like to use the large- N expansion of the integrand B N ( t ) to obtain the large- N expansionfor the integrated correlator (1.3) with finite τ . We begin with the equivalent expression (4.13) for the k -instanton contribution. Using the large- N expansion of B N ( t ) given in (5.58) and exchanging the sum andthe integral, we have G N,k ( τ, ¯ τ ) ∼ ∞ (cid:88) (cid:96) =0 N − (cid:96) (cid:88) ˆ m (cid:54) =0 ,n (cid:54) =0 ˆ mn = k e πikτ (cid:90) ∞ exp (cid:16) − π ˆ m τ t − tπn τ (cid:17)(cid:114) τ t p (cid:96) ( t ) dt . (5.64)Substituting the expressions for p (cid:96) ( t ) given in (5.62) into this equation gives rise to t -integrals that areconvergent for all ˆ m, n (cid:54) = 0 (i.e. for k (cid:54) = 0). The t -integral is again the standard integral representation ofthe K -Bessel function. In particular, the terms in the sum in (5.64) take the following form, (cid:88) ˆ m (cid:54) =0 ,n (cid:54) =0 ˆ mn = k dt √ t (cid:0) t − s + t s − (cid:1) exp (cid:16) − π ˆ m τ t − tπn τ (cid:17) = (cid:88) ˆ m (cid:54) =0 ,n (cid:54) =0 ˆ mn = k (cid:0) n ˆ m (cid:1) s − (cid:16) ˆ m (cid:0) ˆ mn (cid:1) s − + n (cid:17) K s − (2 ˆ mnπτ ) √ ˆ mn = 4 k s − σ − s ( k ) K s − (2 kπτ ) , (5.65)where we assumed k > K -Besselfunctions that form the non-zero modes of the non-holomorphic Eisenstein series, as given in (A.11).As noted earlier, the t -integral for the zero-instanton part (which comes from the ( m, n = 0) sector andthe ( ˆ m = 0 , n ) sector) is divergent at each order in the 1 /N expansion. These divergences, which arise fromthe boundaries of the integration region at t = 0 or at t = ∞ , are absent in the k -nstanton sectors whereboth ˆ m and n are non-vanishing. The occurrence of such singular behaviour is a consequence of expandingthe integrand prior to performing the t integral. It is easy to see that a simple regularisation p (cid:96) ( t ) → p (cid:96) ( t ) t r , (5.66)would allow us to perform the t integral, with the regulator parameter set to r = 0 at the end. Althoughthis does lead to the correct expression for all the coupling dependent terms in (5.46) this procedure missesthe coupling independent constant term (i.e. the N / t since S ( t ) = 4 t + O ( t ) the t -integral is dominated by the boundary of integration t ∼ t → ∞ endpoint). When t → t integral is singular, in a similar manner to (5.62). However, these issues with the zero-mode sector arenot relevant for our discussion since we have already determined the expression for G N, ( τ ) in (5.46), whichdescribes the zero-instanton sector.This problem does not arise in the k (cid:54) = 0 instanton sector. When both ˆ m and n are non-vanishing theexponential exp( − π ˆ m τ /t − tπn τ − N S ( t )) has a saddle-point near t ∼ / √ N and the contribution to36he k (cid:54) = 0 sector coming from the saddle z in (5.52) is well-defined and exponentially suppressed in N .Clearly this second saddle-point is important in understanding non-perturbative, exponentially suppressedcorrections to the large- N expansion of the integrated correlator at fixed τ . However, we have not studiedthe possible presence of such large- N non-perturbative effects.Combining the zero mode sector, (5.46), with the sum over the non-zero modes arising from (5.64), weobtain G N ( τ, ¯ τ ) ∼ N − N E ( ; τ, ¯ τ ) + 452 N E ( ; τ, ¯ τ ) (5.67)+ 3 N (cid:104) E ( ; τ, ¯ τ ) − E ( ; τ, ¯ τ ) (cid:105) + 225 N (cid:104) E ( ; τ, ¯ τ ) − E ( ; τ, ¯ τ ) (cid:105) + 63 N (cid:104) E ( ; τ, ¯ τ ) − E ( ; τ, ¯ τ ) + 732 E ( ; τ, ¯ τ ) (cid:105) + 945 N (cid:104) E ( ; τ, ¯ τ ) − E ( ; τ, ¯ τ ) + 16392 E ( ; τ, ¯ τ ) (cid:105) + 33 N (cid:104) E ( ; τ, ¯ τ ) − E ( ; τ, ¯ τ ) + 614864252 E ( ; τ, ¯ τ ) − E ( ; τ, ¯ τ ) (cid:105) + O ( N − ) , which reproduces precisely the results of [5] as well as new higher order terms.We also note that, although we do not have a closed-form expression for thew general rational coefficients d s(cid:96) multiplying the Eisenstein function E ( s ; τ, ¯ τ ) in (1.14), we can determine the expression for the coefficientsof highest index s = + (cid:96) , as well as lower index ones, such as s = (cid:96) − , s = (cid:96) − , s = (cid:96) − . These aredetermined in a closed form by (5.63) at any given order N − (cid:96) and are given by d (cid:96) + (cid:96) = ( (cid:96) + 1)Γ (cid:16) (cid:96) − (cid:17) Γ (cid:16) (cid:96) + (cid:17) Γ (cid:16) (cid:96) + (cid:17) (cid:96) +2 π / Γ( (cid:96) + 1) , (5.68) d (cid:96) − (cid:96) = − ( (cid:96) − (2 (cid:96) + 9)Γ (cid:16) (cid:96) − (cid:17) Γ (cid:16) (cid:96) + (cid:17) (cid:96) +3 π / Γ( (cid:96) + 1) θ ( (cid:96) − , (5.69) d (cid:96) − (cid:96) = ( (cid:96) − (20 (cid:96) + 48 (cid:96) − (cid:0) (cid:96) − (cid:1) Γ (cid:0) (cid:96) − (cid:1) Γ (cid:0) (cid:96) + (cid:1) 45 2 (cid:96) +5 π / Γ( (cid:96) ) θ ( (cid:96) − , (5.70) d (cid:96) − (cid:96) = − ( (cid:96) − (cid:0) (cid:96) − (cid:96) − (cid:96) + 90912 (cid:96) − (cid:1) Γ (cid:0) (cid:96) − (cid:1) Γ (cid:0) (cid:96) − (cid:1) Γ (cid:0) (cid:96) + (cid:1) (cid:96) +7 π / Γ( (cid:96) − θ ( (cid:96) − . (5.71) N constraints from the Laplace-difference equation We shall now study the extent to which the coefficients of the large-N expansion (5.67), are determined bythe Laplace-difference equation (1.8). Had we not performed the preceding saddle point analysis, we couldhave started with an ansatz for the large- N expansion of the form: G N ( τ, ¯ τ ) ∼ N ˜ f ( τ, ¯ τ ) + ∞ (cid:88) (cid:96) =0 N − (cid:96) f (cid:96) ( τ, ¯ τ ) , (5.72)37or some unknown modular functions ˜ f and f (cid:96) . The fact that this should be an expansion in half-integerpowers of N , is implied by (5.46). After substituting this ansatz into (1.8) the condition that the large- N expansion of the equation is satisfied order by order in 1 /N leads to an infinite series of equations.Thus, at leading order N we simply find O ( N ) : ∆ τ ˜ f ( τ, ¯ τ ) = 0 , (5.73)which, together with SL (2 , Z ) invariance, requires ˜ f ( τ, ¯ τ ) = α , a constant. The next two orders N and N − ( (cid:96) = 0 , 1) lead to two homogenous Laplace eigenvalue equations, O ( N ) : (cid:16) ∆ τ − (cid:17) f ( τ, ¯ τ ) = 0 ,O ( N − ) : (cid:16) ∆ τ − (cid:17) f ( τ, ¯ τ ) = 0 . (5.74)The SL (2 , Z )-invariant solutions to these equations that are power behaved at weak coupling ( τ → ∞ ) arenon-holomorphic Eisenstein series with half-integer index (as described in appendix A), f ( τ, ¯ τ ) = β E ( ; τ, ¯ τ ) , f ( τ, ¯ τ ) = β E ( ; τ, ¯ τ ) , (5.75)with undetermined constants β and β .This result could have been anticipated given power series ansatz (5.72) and our previous discussionregarding the large- N expansion (4.33) of the Laplace-difference equation (1.8).With (cid:96) > (cid:96) = 2 we have O ( N − ) : (cid:16) ∆ τ − (cid:17) f ( τ, ¯ τ ) = − f ( τ, ¯ τ ) , (5.76)which has the solution f ( τ, ¯ τ ) = β E ( ; τ, ¯ τ ) + 13512 β E ( ; τ, ¯ τ ) , (5.77)where β is a new arbitrary constant associated with the homogeneous Laplace eigenvalue equation thatenters with (cid:96) = 2. The equation for (cid:96) = 3 is O ( N − ) : (cid:16) ∆ τ − (cid:17) f ( τ, ¯ τ ) = 7564 f ( τ, ¯ τ ) , (5.78)which has the solution f ( τ, ¯ τ ) = β E ( ; τ, ¯ τ ) − β E ( ; τ, ¯ τ ) , (5.79)Continuing in this fashion, we see that the (cid:96) = 4 equation is O ( N − ) : (cid:16) ∆ τ − (cid:17) f ( τ, ¯ τ ) = 59564 f ( τ, ¯ τ ) − f ( τ, ¯ τ ) , (5.80)that has solution that is a sum of E ( ; τ, ¯ τ ), E ( ; τ, ¯ τ ) and E ( ; τ, ¯ τ ). This solution again has one newundetermined constant, β , which is the coefficient of E ( ; τ, ¯ τ ). Likewise, the (cid:96) = 5 equation is O ( N − ) : (cid:16) ∆ τ − (cid:17) f ( τ, ¯ τ ) = 199564 f ( τ, ¯ τ ) + 483512 f ( τ, ¯ τ ) , (5.81)38hich has a solution that is a sum of E ( ; τ, ¯ τ ), E ( ; τ, ¯ τ ) and E ( ; τ, ¯ τ ), with an undetermined constant β . It is easy to see that this leads to the following general pattern. For any (cid:96) the coefficient of N − (cid:96) is thesum of Eisenstein series with coefficients determined by the values of the constants, β (cid:96) . There are two towersof solutions in which (cid:96) = 0 , , . . . and (cid:96) = 1 , , . . . . In other words the complete solution can be organised asa sum of the form G N ( τ, ¯ τ ) = ∞ (cid:88) (cid:96) =0 [ E ( + 2 (cid:96) ; τ, ¯ τ ) F (cid:96) ( N ) + E ( + 2 (cid:96) ; τ, ¯ τ ) F (cid:96) ( N )] , (5.82)where F (cid:96) ( N ) = N − (cid:96) (cid:16) β (cid:96) + ∞ (cid:88) k =1 a (cid:96) +2 k N − k (cid:17) , (5.83)and the coefficients a k are determined, as described above, by the Laplace-difference equation in terms ofthe undetermined constants β (cid:96) . Once the values of β (cid:96) are specified the values of all the a k are determinedand the complete set of constants reduces to the coefficients in (5.67) that were previously obtained from d s(cid:96) .The undetermined constants β (cid:96) are the coefficients of the Eisenstein series with highest index at a givenorder in the 1 /N expansion. These are precisely the coefficients d (cid:96) + (cid:96) in the previous terminology. Asmentioned earlier, these coefficients can be obtained from the large- λ expansion of G (0) ( λ ) (5.26), which hasthe form N G (0) ( λ ) ∼ N (cid:34) 14 + ∞ (cid:88) (cid:96) =1 b (0) (cid:96) λ − (cid:96) − (cid:35) = N ∞ (cid:88) (cid:96) =0 N − (cid:96) b (0) (cid:96) +1 (cid:16) τ π (cid:17) (cid:96) + , (5.84)where the coefficients b (0) (cid:96) +1 are given in (5.27). We can identify this series with the zero mode of the expansionof G N ( τ, ¯ τ ) in a series of half-integer index Eisenstein series shown in (1.14). Using the expression for thezero mode of E ( s ; τ, ¯ τ ) shown in (A.10) we find the value for the unknown coefficients d (cid:96) + (cid:96) = b (0) (cid:96) +1 (cid:96) +4 ζ (2 (cid:96) + 3) , (5.85)which indeed can be shown to agree with (5.68) using b (0) (cid:96) +1 given in (5.27).We have therefore seen that the coefficients, β (cid:96) , in the large- N expansion that are not determined by theLaplace-difference equation are given by the coefficients in the large- λ expansion of G (0) ( λ ). This paper has provided strong evidence for our main conjecture that the lattice sum (1.3) describes theintegrated correlation function of four superconformal primaries in N = 4 SU ( N ) SYM that was introducedin [3]. In that reference the integrated correlator was defined by (1.2) in terms of Pestun’s localised partitionfunction. Previous analysis explored the large- N expansion of the correlator, both in the perturbative ’tHooft regime [3, 6], and in the non-perturbative regime in which Yang–Mills instantons contribute [5, 7].39owever, the detailed dependence of the correlator on N and τ has proved difficult to extract from thedefinition (1.2). Furthermore, up to now the finite- N regime has been largely ignored.Although we have not produced mathematically rigorous arguments, we have produced compelling evi-dence for our main conjecture that G N ( τ, ¯ τ ) can be expressed as a sum over a two dimensional lattice given in(1.3). This representation of the correlator, which is defined for arbitrary values of N and of the Yang–Millscoupling, g Y M , is manifestly invariant under SL (2 , Z ), which is a manifestation of Montonen–Olive duality[10].Most notably, as a corollary to the main conjecture we showed that the lattice sum satisfies a Laplace-difference equation (1.8), which relates the Laplace operator, ∆ τ acting on G N ( τ, ¯ τ ) to a linear combinationof G N +1 ( τ, ¯ τ ) and G N − ( τ, ¯ τ ). This rather unusual equation leads to many of the properties of the correlatorthat we have discussed.In particular, we have shown that this reproduces the expected approximations to the correlator in variouslimits. These are:(i) The small- g Y M perturbative expansion at finite N . This has a remarkably simple structure withcoefficients that are rational multiples of odd zeta values. This striking observation is reminiscent ofthe fact that the coefficients of the leading terms in the large- N ’t Hooft limit are proportional to oddzeta values. We identified the contribution of these perturbative terms with the sum of contributions ofzero modes of non-holomorphic Eisenstein series with integer indices of the form (1.12). The coefficients c ( N ) s are rational numbers with values that depend on N and s .We have compared these results with one-loop and two-loop contributions to four-point correlatorsevaluated by standard quantum field theory methods. These expressions are complicated functionsof the positions of the operators. However, we found, upon integration over the positions of the fouroperators, these complicated expressions reduce to precisely the terms deduced from the localisedcorrelator.(ii) The large- N limit at fixed ’t Hooft coupling (fixed λ = g Y M N ). In this limit we reproduce the 1 /λ expansion which is encapsulated in equation (3.48) of [3]. This is the strong-coupling limit of the planardiagram contribution to the correlator. Furthermore, we noted that expanding the same expression atweak coupling reproduces the planar diagram contribution to perturbative Yang–Mills. Whereas thesmall- λ expansion is convergent with radius of convergence | λ | < π , the strong coupling expansion isdivergent and is not Borel summable making necessary the addition of an instantonic contribution oforder e − √ λ . This has the form of a non-perturbative world-sheet instanton contribution in the dualstring theory, which would be of interest to understand in more detail.(iii) The large- N limit at fixed g Y M , in which Yang–Mills instantons play an essential rˆole. This reproducesthe expansion that was suggested in [5] that is a sum of Eisenstein series of half-integral index of theform (1.14). Our expression generates the coefficients d s(cid:96) very efficiently to any given order in 1 /N .The terms in this series are manifestly SL (2 , Z ) invariant since they are proportional to Eisensteinseries. 40ur arguments have been based on discovering patterns satisfied by series expansions of the integratedcorrelator in the various limits described above. It would be gratifying to determine a more direct mathe-matical argument that systematically leads from the expression for the localised partition function to thelattice sum (1.3). Such an argument could explain the form of the rather cumbersome expression for Q N ( t )given in (1.5), which surely has an elegant geometrical origin.As mentioned in the introduction, in addition to the correlator defined by (1.2) an independent integratedcorrelator may be defined by the quantity ∂ m log Z N , as was discussed in [6] and [7]. The large- N structureof this correlator was found to be related to generalisations of Eisenstein series. It would be interesting todiscover whether this correlator can also be expressed as a lattice sum.There are other obvious directions in which these results could be extended. For example, it shouldbe possible to give a similar analysis of N = 4 SYM with other classical (or non-classical) gauge groups.The extension to correlators of other operators in the stress tensor multiplet also seems feasible. It wouldsimilarly be of interest to generalise this construction to n -point correlators that violate the bonus U (1) Y symmetry maximally, that were discussed in [40]. Beyond that, the extension of these ideas to correlatorsof more general BPS operators appears to be considerably more challenging.Finally, it would be of interest to see how these results may make contact with the low energy expansionof superstring scattering amplitudes in AdS × S , extending the results of [3]-[7]. Acknowledgements We would like to thank Shai Chester, Lance Dixon, Paul Heslop, Axel Kleinschmidt, Silviu Pufu, Yifan Wang,and Gang Yang for useful conversations and comments. DD would like to thank the Albert Einstein Institutefor the hospitality and support during the writing of this paper. MBG has been partially supported by STFCconsolidated grant ST/L000385/1. CW is supported by a Royal Society University Research Fellowship No.UF160350. A Non-holomorphic Eisenstein series A non-holomorphic Eisenstein series can be defined by a lattice sum that has a Fourier expansion of theform E ( s ; τ, ¯ τ ) = 1 π s (cid:88) ( m,n ) (cid:54) = (0 , τ s | m + nτ | s = (cid:88) k ∈ Z F k ( s ; τ ) e πikτ , (A.1)where the non-zero Fourier modes have the characteristic instantonic exponential behaviour F k ∼ e − π | k | τ in the large- τ (weak coupling) limit.We will now review the derivation of the detailed structure of the Fourier modes. The first step is toseparate the double sum over ( m, n ) (cid:54) = (0 , 0) into two sets of terms:(i) The sum of terms ( m, 0) with m (cid:54) = 0. We will denote this by F ( i )0 ( s ; τ ) := 2 π s ∞ (cid:88) m =1 τ s m s = 2 ζ (2 s ) π s τ s . (A.2)41ii) The sum of terms ( m, n ) with −∞ ≤ m ≤ ∞ and n (cid:54) = 0. In order to sum over these values it is usefulto introduce an integral representation, which takes the form F ( ii ) ( s ; τ, ¯ τ ) := (cid:88) m ∈ Z , n (cid:54) =0 s ) (cid:90) ∞ e − tπY t s − dt = (cid:88) m ∈ Z , n (cid:54) =0 s ) (cid:90) ∞ e − tπ ( m + nτ n τ τ t s − dt , (A.3)where Y := | m + nτ | τ . (A.4)This may be re-expressed by performing a Poisson summation over the integer m . Recall that a Poisson sumcan be expressed as the relation (cid:88) m ∈ Z f ( m ) = (cid:88) ˆ m ∈ Z ˜ f ( ˆ m ) , (A.5)where ˜ f ( ˆ m ) is the Fourier transform of f ( m ) (i.e. ˜ f ( ˆ m ) = (cid:82) ∞−∞ dme πim ˆ m f ( m )). This transforms (A.3) into F ( ii ) ( s ; τ, ¯ τ ) = √ τ Γ( s ) (cid:88) ˆ m ∈ Z , n (cid:54) =0 e πi ˆ mnτ (cid:90) ∞ e − πτ (cid:16) ˆ m t + n t (cid:17) t s − dt = F ( ii )0 ( s ; τ ) + ∞ (cid:88) k = −∞ k (cid:54) =0 e πikτ √ τ Γ( s ) | k | s − σ − s ( | k | ) K s − (2 π | k | τ ) . (A.6)where we have set k = ˆ mn and separated the term with ˆ m = 0 (i.e. k = 0) and used the integralrepresentation for a K -Bessel function for the k (cid:54) = 0 terms, (cid:90) ∞ e − a t − b t t ν − dt = 2 (cid:16) ab (cid:17) ν K ν (2 ab ) . (A.7)The divisor sum is defined by σ ν ( k ) = (cid:80) n | k n ν for k > 0. The asymptotic expansion of the Bessel functionfor large τ = 4 π/g Y M is given by K ν (2 π | k | τ ) ∼ (cid:114) π π | k | τ e − π | k | τ (cid:18) ν − π | k | τ + . . . (cid:19) . (A.8)This behaviour is consistent with the contribution of charge- k BPS instantons (when k > 0) and anti-instantons (when k < 0) to (A.6).The k = 0 term in (A.6) is given by F ( ii )0 ( s ; τ ) = 2 √ π Γ( s − ) ζ (2 s − π s Γ( s ) τ − s , (A.9)and so the total zero Fourier mode is given by F ( s ; τ ) = F ( i )0 ( s ; τ ) + F ( ii )0 ( s ; τ )= 2 ζ (2 s ) π s τ s + 2 √ π Γ( s − ) ζ (2 s − π s Γ( s ) τ − s . (A.10)42rom (A.6) we see that the k -th Fourier mode ( k (cid:54) = 0) is proportional to a K -Bessel function, F k ( s ; τ ) = 4Γ( s ) | k | s − σ − s ( | k | ) √ τ K s − (2 π | k | τ ) , k (cid:54) = 0 . (A.11)The extra factor of 2 in this equation comes from the two choices of signs for ˆ m and n It is noteworthy that an Eisenstein series satisfies the functional relationΓ( s ) E ( s ; τ, ¯ τ ) = Γ(1 − s ) E (1 − s ; τ, ¯ τ ) , (A.12)and it is the unique modular invariant solution to the Laplace equation (cid:16) ∆ τ − s ( s − (cid:17) E ( s ; τ, ¯ τ ) = 0 . (A.13) B Computing the one-instanton contribution The one-instanton contribution is obtained from the expectation value of ∂ m ˆ Z instN, (cid:12)(cid:12)(cid:12)(cid:12) m =0 given in (3.14), whichwe quote here ∂ m ˆ Z instN, (cid:12)(cid:12)(cid:12)(cid:12) m =0 = N (cid:88) l =1 (cid:89) j (cid:54) = l ( a lj + i ) a lj ( a lj + 2 i ) . (B.1)So we are interested in the following quantity, G N, ( τ, ¯ τ ) = − Z (0) N (cid:90) d N − a e − π g Y M (cid:80) Ni =1 a i (cid:89) i 1) + α τ + α N τ + α N ( N − τ + α N (cid:0) N − N + 58 (cid:1) τ + α N (cid:0) N − N + 293 N − (cid:1) τ + · · · (cid:35) , (B.3)where the coefficient a i is given by α i = ( − i +1 i − π i + Γ (cid:0) N − i + (cid:1) Γ( N − . (B.4)However, it becomes computationally difficult to expand the integrand to higher order. In the following wewill use a more efficient approach to generate such higher order terms,From (B.3), we observe that G N, ( τ, ¯ τ ) has a simple structure, which allows us to write a general expressionfor the one-instanton contribution to the integrated correlator in the form, G N, ( τ, ¯ τ ) = e πiτ (cid:34) − α N − 1) + α τ + N ∞ (cid:88) i =2 (cid:32) i − (cid:88) k =0 β i,k N k (cid:33) α i +1 τ i (cid:35) , (B.5)43here α i is given in (B.4) and β i,k is to be determined below.The procedure we will adopt is to use the conjecture in [5] for the large- N expansion of the integratedcorrelator, which expresses it in terms of non-holomorphic Eisenstein series as in (1.14). According to thisconjecture the one-instanton contribution has the form G N, ( τ, ¯ τ ) = e πiτ ∞ (cid:88) (cid:96) = − N − − (cid:96) (cid:98) (cid:96) (cid:99) (cid:88) s = − γ (cid:96),s √ τ K (cid:96) − s (2 πτ ) , (B.6)with γ (cid:96), (cid:98) (cid:96) (cid:99) = 0 if m is even and γ (cid:96),s is in general unknown.The equality of the expressions for the one-instanton contribution, (B.5) and (B.6), imposes non-trivialconstraints on both coefficients β i,k and γ (cid:96),s . After inputting some initial data we are able to solve for β i,k and c (cid:96),s recursively by comparing the two expressions order by order in the 1 /N and 1 /τ expansion.Importantly, this procedure often leads to over-constrained equations for β i,k and γ (cid:96),s . The existence of asolution to these equations gives consistency checks on the ansatz in (B.5). Explicitly, we obtain the resultsfor β i,k up to i = 25 and for γ (cid:96),s up to (cid:96) = 27. These results provide us a large amount of data, which isused in the main text to strongly constrain the structure of G N,k ( τ, ¯ τ ) with arbitrary N and k . C Evaluation of the integrated correlator in perturbation theory Recall that the integral that is relevant for computing the integrated correlator is I [ T N ( U, V )] = − π (cid:90) ∞ dr (cid:90) π dθ r sin ( θ ) U UV T (cid:48) N ( U, V ) , (C.1)here we have written T N ( U, V ) = UV T (cid:48) N ( U, V ). Using the relation U = 1 + r − r cos( θ ) and V = r , wehave I [ T N ( U, V )] = − π (cid:90) ∞ dr r (cid:90) π dθ sin ( θ ) T (cid:48) N ( U, V ) r (1 + r − r cos( θ )) . (C.2)This integral can be viewed as an integration over a four-dimensional vector P µ V with ( P V ) = r . Thefactors in the denominator can be viewed as two propagators, with an external unit momentum P µ with P = 1 by noting that (1 + r − r cos( θ )) = ( P V − P ) . Therefore, I [ T N ( U, V )] = − π (cid:90) d P V T (cid:48) N ( U, V ) P V ( P V − P ) , (C.3)where (cid:82) d P V = 4 π (cid:82) ∞ dr r (cid:82) π dθ sin ( θ ).This expression manifestly has the form of a Feynman integral for a two-point function with externalmomentum P , where P = 1. So we see that upon integration over U and V , as in (C.1), the L -loopFeynman diagram contributions to the four-point correlator inside T (cid:48) N ( U, V ) are transformed into ( L + 1)-loop Feynmann integrals with the external momentum being the unit vector P µ .Let us apply this observation to the ladder diagram shown in Fig. 2. The expression for the ladderdiagram is known to any number of loops [23] and is given in (5.8). The resulting integrated correlatorrelated to the ladder diagram contribution is simply another ladder diagram but now with two external legs44igure 2: The ladder diagram. Figure 3: The ladder diagram of the inte-grated correlator.and one extra loop, as shown in Fig. 3. The expression for the two-point ladder diagram is also well-known[26]. Using the known result we find, I (cid:20) UV Φ ( L ) ( U, V ) (cid:21) = − (cid:18) L + 2 L + 1 (cid:19) ζ (2 L + 1) . (C.4)We have further verified the above result numerically up to L = 15.We will now show that the following relation holds I (cid:20) UV Φ ( L ) ( U, V )Φ ( L ) ( U, V ) (cid:21) = I (cid:20) UV Φ ( L + L ) ( U, V ) (cid:21) , (C.5)that is used in the main text as part of the two-loop calculation, with L = L = 1. First, we note Φ ( L ) ( U, V )obeys a differential recursion relation, P V ( P V − P ) (cid:50) P V Φ ( L +1) ( U, V ) = Φ ( L ) ( U, V ) , (C.6)where Φ (0) ( U, V ) = 1 and (cid:50) P V = ∂ P µV ∂ P V,µ . Therefore we have I (cid:20) UV Φ ( L ) ( U, V )Φ ( L ) ( U, V ) (cid:21) = I (cid:20) UV Φ ( L ) ( U, V ) P V ( P V − P ) (cid:50) P V Φ ( L +1) ( U, V ) (cid:21) = − π (cid:90) d P V Φ ( L ) ( U, V ) (cid:50) P V Φ ( L +1) ( U, V ) , (C.7)where we have used (C.3) to obtain the final expression.To proceed, we perform integration by parts and use the recursion relation (C.6), from which we find I (cid:20) UV Φ ( L ) ( U, V )Φ ( L ) ( U, V ) (cid:21) = − π (cid:90) d P V P V ( P V − P ) Φ ( L − ( U, V )Φ ( L +1) ( U, V ) . (C.8)Using the relation (C.3) once again leads to I (cid:20) UV Φ ( L ) ( U, V )Φ ( L ) ( U, V ) (cid:21) = I (cid:20) UV Φ ( L − ( U, V )Φ ( L +1) ( U, V ) (cid:21) . (C.9)Finally, applying this relation repeatedly, we arrive at (C.5).45 Borel summation and median resummation In this appendix we will present some basic ideas concerning the Borel transform and resummation offactorially growing asymptotic power series. These ideas are contained in recent more detailed reviews[41, 42].Our starting point is a “strong coupling”, i.e. x → ∞ , asymptotic formal power series of the form F ( x ) = ∞ (cid:88) n =0 a n x − n − , (D.1)whose perturbative coefficients grow factorially, i.e. a n ∼ α R n n ! (D.2)for some α, R ∈ R \ { } .The standard Borel transform of (D.1) is given by B ( t ) = (cid:88) n ≥ a n n ! t n , (D.3)which has finite radius of convergence, thus defining a germ of an analytic function at the origin t = 0.The well-known integral (cid:90) ∞ e − t t n dt = n ! , (D.4)leads to the definition of a possible resummation, i.e. a possible analytic continuation, of the originalasymptotic series (D.1), using the steps, (cid:90) e − iθ ∞ e − tx B ( t ) dt = (cid:90) ∞ e − t B (cid:18) tx (cid:19) dtx = (cid:90) ∞ e − t (cid:88) n ≥ a n n ! (cid:18) tx (cid:19) n dtx ∼ ∞ (cid:88) n =0 a n x − n − , (D.5)where θ = arg x and in the last step we have simply commuted the series with the integral.Note that for generic θ ∈ [0 , π ] the directional Borel resummation S θ F ( x ) = (cid:90) e iθ ∞ e − tx B ( t ) dt , (D.6)will define an analytic function in the wedge Re ( e iθ x ) > x -plane with exactly the sameasymptotic expansion (D.1).In the course of this paper we have encountered several formal asymptotic power series of the form F ( x ) = ∞ (cid:88) n =1 c n ζ ( n + 1) x − n − , (D.7)with c n growing factorially. In general the standard Borel transform (D.3) cannot be written in closed formbecause of the presence of a Riemann zeta value, however we can simply replace ζ ( n + 1) by its Dirichlet46eries ζ ( n + 1) = (cid:80) k ≥ k − n − and obtain the resummation S θ F ( x ) = (cid:88) k ≥ (cid:90) e iθ ∞ e − tkx B ( t ) dt = (cid:88) k ≥ (cid:90) ∞ e − tkx (cid:16) (cid:88) n ≥ c n t n n ! (cid:17) dt . (D.8)Furthermore, when dealing with perturbative coefficients of the form c n ζ ( n + 1) one can also use amodified Borel transform, see e.g. [43, 44], using the modified integral kernel (cid:90) ∞ t n +1 ( t/ dt = ζ ( n + 1)( n + 1)! , (D.9)valid for n ≥ B ( t ) = (cid:88) n ≥ c n ζ ( n + 1) ζ ( n + 1)( n + 1)! t n +1 = (cid:88) n ≥ c n ( n + 1)! t n +1 , (D.10)and modified Borel resummation˜ S θ F ( x ) = x (cid:90) e iθ ∞ ˜ B ( t )4 sinh ( tx/ dt = x (cid:90) e iθ ∞ (cid:88) n ≥ c n t n +1 ( n + 1)! dt ( tx/ ∼ ∞ (cid:88) n =1 c n ζ ( n + 1) x − n − , (D.11)where again in the last step we have commuted the sum with the integral and computed the modified integralkernel as above.Note that in (D.11) we can substitute the expansion14 sinh ( tx/ 2) = (cid:88) k ≥ k e − tkx , (D.12)which is valid for Re ( tx ) > 0, and integrate by parts to arrive at the infinite sum of standard Borel transforms(D.8). The modified Borel resummation (D.11) is generically simpler to analyse compared to its standardcounterpart (D.8) but they capture exactly the same amount of information.Usually we say that the asymptotic expansion (D.1) is Borel summable if we can perform the directionalBorel resummation, of the form (D.6), (D.8) or (D.11), along the positive real axis, and we can then extendthe domain of analyticity by varying θ . In general however as we vary θ we will reach a singular direction,called Stokes direction , for the Borel transform B ( t ) and we say that (D.1) is not Borel summable along thatdirection; in many physically interesting cases θ = 0 is a Stokes direction.Whenever B ( t ) has a branch-cut starting at t = t (cid:63) in the direction θ (cid:63) = arg t (cid:63) , the lateral resummations(D.6) on the two sides of the Stokes direction, i.e. for θ > θ (cid:63) and θ < θ (cid:63) , define two different analyticcontinuations of the same asymptotic expansion (D.1).This is usually called an ambiguity in the resummation, but it can be quantified by defining lateral resummations across a singular direction θ = θ (cid:63) given by the limit of (D.6) from the two sides: S ± F ( x ) = lim θ → θ ± (cid:63) S θ F ( x ) . (D.13)47he difference between the two lateral resummations, related to what is usually called the Stokes auto-morphism , is given by( S θ + (cid:63) − S θ − (cid:63) ) F ( x ) = ∆ θ (cid:63) F ( x ) = (cid:90) γ e − tx Disc θ (cid:63) B ( t ) dt ∼ πie − t (cid:63) x (cid:88) n ≥ ˜ c n x − n − , (D.14)where Disc θ (cid:63) B ( t ) denotes the discontinuity of the Borel transform across the Stokes direction θ (cid:63) and thecontour of integration γ is a Hankel contour coming from ∞ below the cut, circling around the branch-point t = t (cid:63) , and going back to ∞ above the cut. Since this quantity is non-perturbative in nature, it shouldbe clear that the location of the singularities of B ( t ) plays a crucial role in understanding non-perturbativecorrections encoded in the asymptotic series (D.1), which are necessary for the definition of a unique analyticcontinuation of the physical quantity whose perturbative expansion is (D.1).The non-perturbative completion of (D.1), i.e. its trans-series expansion, is usually very difficult tocompute, however in many cases median resummation [45] S med F ( x ) = S ± F ( x ) ∓ 12 ∆ F ( x ) (D.15)gives the correct non-perturbative definition of the physical quantity associated with (D.1), i.e. the appro-priate, unambiguous, analytic continuation which is also real for real coupling x . D.1 Median resummation at leading order in the ’t Hooft expansion In this appendix we want to show that the median resummation S med G (0) ( x ) = 14 + xπ (cid:90) ∞ dw ( xw ) Re ˆ φ ( w ) , (D.16)with ˆ φ ( w ) = − πw F (cid:18) − , 32 ; 1 (cid:12)(cid:12)(cid:12) w (cid:19) , (D.17)obtained from the strong coupling asymptotic expansion (5.26), matches identically the exact result of [3]and [4]: G (0) ( λ ) = (cid:90) ∞ dw w J ( xw/π ) − J ( xw/π ) sinh w . (D.18)The reasoning very closely resembles the analysis carried out in [33].We want to show that median resummation (D.16) can actually be written as a contour integral. Firstof all we rewrite the real part of the Borel transform in the alternative form − πw Re F (cid:18) − , 32 ; 1 (cid:12)(cid:12)(cid:12) w (cid:19) = π Re F (cid:18) , 32 ; 3 (cid:12)(cid:12)(cid:12) w − (cid:19) , valid for w > 0, which one can easily prove using the Mellin-Barnes representation for the hypergeometricfunction.We can then rewrite our median resummation as S med G (0) ( x ) = 14 + x (cid:90) ∞−∞ dw sinh ( xw ) Re F (cid:18) , 32 ; 3 (cid:12)(cid:12)(cid:12) w − (cid:19) , (D.19)48*****... . . − ∞−∞ i ∞ inπx C C C C Figure 4: Integration contour γ for the integral I .using the fact that the integrand is an even function of w .At this point let us consider the following contour integral I = x (cid:90) γ dw sinh ( xw ) F (cid:18) , 32 ; 3 (cid:12)(cid:12)(cid:12) w − (cid:19) , (D.20)where the contour γ is presented in figure 4.Note that the integrand of (D.20) has a branch cut on the interval w ∈ [ − , 1] and the integration contour C runs just above this cut. Furthermore, since the integrand is symmetric with respect to x → − x theintegration just above the cut is equivalent to the integration along the real line of the real part of theintegrand. The segment C of the contour of integration circles the origin and picks out the residue at theorigin of (D.20). However it is simple to check that this residue vanishes. From these arguments we are leadto the conclusion that x (cid:90) C ∪ C dw sinh ( xw ) F (cid:18) , 32 ; 3 (cid:12)(cid:12)(cid:12) w − (cid:19) = x (cid:90) ∞−∞ dw sinh ( xw ) Re F (cid:18) , 32 ; 3 (cid:12)(cid:12)(cid:12) w − (cid:19) . (D.21)The contribution coming from C , which is the residue at infinity, gives x (cid:90) C dw sinh ( xw ) F (cid:18) , 32 ; 3 (cid:12)(cid:12)(cid:12) w − (cid:19) dw = x R →∞ (cid:104) Rx ) x (cid:105) = 14 . (D.22)Combining these expressions we arrive at the result that the median resummation is given by the contourintegral: I = x (cid:90) γ dw sinh ( xw ) F (cid:18) , 32 ; 3 (cid:12)(cid:12)(cid:12) w − (cid:19) = S med G (0) ( x ) . (D.23)We can now close the contour along the imaginary axis, picking up all the residues at w = inπx , i.e. I = x ∞ (cid:88) n =1 res (cid:104) F (cid:16) , ; 3 (cid:12)(cid:12)(cid:12) w − (cid:17) sinh ( xw ) (cid:105) w = inπx , which are easily evaluated by Cauchy integration, leading to I = ∞ (cid:88) n =1 x π n F (cid:18) , 52 ; 4 (cid:12)(cid:12)(cid:12) − x π n (cid:19) . (D.24)49e can now take the result (D.18) of [3] and [4], and after changing the variable of integration, w → w (cid:48) = wx/π , we can expand the sinh ( πw (cid:48) /x ) in the denominator, arriving at G (0) ( λ ) = ∞ (cid:88) n =1 π nx (cid:90) ∞ dw (cid:48) e − nπ w (cid:48) x w (cid:48) (cid:104) J ( w (cid:48) ) − J ( w (cid:48) ) (cid:105) . 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