Generating functions and large-charge expansion of integrated correlators in $\mathcal{N}=4$ supersymmetric Yang-Mills theory

We recently proved that, when integrating out spacetime dependence with a certain measure, four-point correlators $\langle \mathcal{O}_2\mathcal{O}_2\mathcal{O}^{(i)}_p \mathcal{O}^{(j)}_p \rangle$ in $SU(N)$ $\mathcal{N}=4$ super Yang-Mills are governed by a universal Laplace-difference equation. Here $\mathcal{O}^{(i)}_p$ is a superconformal primary with charge $p$ and degeneracy $i$. These observables, called integrated correlators, are modular functions of coupling $\tau$. The Laplace-difference equation relates integrated correlators of different charges recursively. In this paper, we introduce generating functions for the integrated correlators that sum over the charge. By utilising the Laplace-difference equation, we determine the generating functions given initial data. We show that the transseries of the integrated correlators in the large-$p$ (large-charge) expansion consists of three parts: 1) is independent of $\tau$ as a power series in $1/p$, plus an additional $\log(p)$ term if $i=j$; 2) is a power series in $1/p$, with coefficients given by a sum of the non-holomorphic Eisenstein series; 3) is a sum of exponentially decayed modular functions in the large-$p$ limit, which can be viewed as a generalisation of the non-holomorphic Eisenstein series. When $i=j$, there is an additional modular function that is independent of $p$ and is determined by the integrated correlator with $p=2$. The Laplace-difference equation was obtained with a reorganisation of the operators that means the large-charge limit is taken in a particular way here. From the $SL(2,\mathbb{Z})$-invariant results, we also determine the generalised 't Hooft genus expansion and associated large-$p$ non-perturbative corrections of the integrated correlators by introducing $\lambda = pg^2_{YM}$. The generating functions have subtle differences between even and odd $N$ with important consequences in resurgence.


Introduction
It was proposed in [1][2][3] and recently proved in [4] that the four-point correlation function of superconformal primary operators of the stress tensor multiplet in N = 4 supersymmetric Yang-Mills (SYM) theory, once the spacetime dependence has been integrated out with a certain measure, can be expressed in a lattice sum.
These observables are often called integrated correlators. 1 For example, in the case of the SU (N ) gauge group, the integrated correlator can be expressed as [1,2] C N (τ,τ ) = (m,n)∈Z 2 ∞ 0 e −tYm,n(τ,τ ) B N (t)dt , (1.1) where the coupling τ dependence only appears through the quantity Y m,n (τ,τ ), which is defined as , (1.2) and B N (t) is a rational function that encodes all the dynamics of the integrated correlator. In [4], the generating function that sums over the N dependence for the integrated correlator C N (τ,τ ) was introduced, and takes the following form, where B(z; t) = N B N (t)z N and its explicit expression is given in (3.21). We see that all the non-trivial information about the integrated correlator for any N is contained in a simple function B(z; t). Analogous expressions can be found in [4,11] for other classical gauge groups, built on earlier results [12]. These expressions make manifest the Montonen-Olive duality [13], as well as its generalisation, the Goddard-Nuyts-Olive (GNO) duality [14].
In this paper we generalise the lattice-sum representation (1.1) and the generating function (1.3) to integrated correlators that are associated with four-point functions of the form , where the operator O p is a superconformal primary of charge (or dimension) p. There are multiple operators which have the same dimension p (when p > 3), and the index i is to distinguish such degeneracy. When p = 2, it reduces to the integrated correlator C N (τ,τ ) that we discussed above. Because the operators O (i) p are the bottom component of supersymmetry transformation, they are not charged under the bonus U (1) Y symmetry [15,16]. This means that these more general integrated correlators are also SL(2, Z) modular invariant due to Montonen-Olive duality of N = 4 SYM. These integrated correlators were recently studied in [17], following the earlier work [5]. In [17], the SL(2, Z) spectral decomposition analysis was applied to these integrated correlators that generalises the earlier work of [18]. It was assumed in [17] that only the continuous spectrum (i.e. non-holomorphic Eisenstein series) contributes to these integrated correlators. 2 This is the same property that was used in [18] for the integrated correlator C N (τ,τ ). This proposal was verified by explicit perturbative and non-perturbative instanton computations [17]. However, as already pointed out in [3,18] for C N (τ,τ ), this particular form of the SL(2, Z) spectral decomposition (i.e. only non-holomorphic Eisenstein series contribute) is mathematically equivalent to the lattice-sum representation in (1.1). We will therefore assume that these more general integrated correlators associated with also have a lattice-sum representation, analogous to (1.1).
As we showed recently in [19], instead of simply organising the operators according to their charges (or equivalently dimensions), it is vital to reorganise operators into different towers and subtowers. We define the operators as O (τ,τ ) that we will consider are associated with the four-point correlation functions of the following form, (1.5) We have removed the explicit dependence on p ′ because of the relation 2p + M = 2p ′ + M ′ . This is because the four-point correlation functions are non-trivial only when the two higher-dimensional operators have the same dimension (see e.g. [20][21][22]  (τ,τ ), which in general is a highly non-trivial modular function of (τ,τ ), will be the main focus of the paper. In this paper, we will simply refer to C (1.7) Once again, all the non-trivial information of the integrated correlators is encoded in the rational functions B (M,M ′ |i,i ′ ) N,p (t). We will mostly be interested in the integrated correlators for a fixed N but arbitrary p.
Following the ideas of [4], it is very convenient to further introduce generating functions by summing over (t) w p . We will determine the generating functions, and use them to study the properties of the integrated correlators, especially in the large-charge limit. The Laplace-difference equation that is satisfied by the integrated correlators becomes a differential equation for B (M,M ′ |i,i ′ ) N (w; t).
We will determine the generating functions by solving the differential equation, and the generating functions will enable us to study the charge dependence of the integrated correlators in a straightforward manner.
We will be particularly interested in the large-charge (i.e. large-p) limit of the integrated correlators. Understanding the behaviour of quantum field theories at the large global charge limit has been of great interest [23][24][25][26][27][28] (see [29] for a recent review). Particularly related to our study, the large-charge limit has been studied in the context of extremal correlators in N = 2 supersymmetric theories, which can also be computed using supersymmetric localisation, see e.g. [30][31][32][33][34][35][36][37][38]. We find that, for the integrated correlators in N = 4 SYM that are considered in this paper, the large-p expansion takes a universal form, summarised in (4.32), and contains three parts. The first part is independent of τ . It behaves as log(p) plus a power series in 1/p when M = M ′ , i = i ′ (i.e. when two higher-dimensional operators are the same), otherwise we only have a power series in 1/p. The second part is a power series in 1/p, with coefficients given by a sum of the non-holomorphic Eisenstein series with half-integer indices. Finally, the third part is a sum of exponentially decayed modular functions in the large-p limit. These modular functions can be viewed as a generalisation of the non-holomorphic Eisenstein series. When M = M ′ , i = i ′ , there is also an additional p-independent modular function that is completely determined in terms of C N (τ,τ ). The second and third parts are remarkably similar to the large-N expansion of the integrated correlator C N (τ,τ ) in (1.1) that was recently obtained in [4], with N and p exchanged. However, it should be stressed that even though the large-p and large-N expansions look similar, there are interesting and subtle differences. In particular, when N is even, the number of non-holomorphic Eisenstein series that appear in the large-p expansion do not grow indefinitely as we consider higher orders in the expansion. This is very different from what has been seen in [2,4,8]. This fact also implies that in the generalised 't Hooft genus expansion, at a given order in 1/p, the large-λ expansion truncates. 4 These structures have interesting consequences in understanding the connections between the power series terms and exponentially decayed terms in the large-p limit from the point of view of resurgence.
The rest of the paper is organised as follows. Section 2 mainly summarises the results of [19]. We will begin by briefly reviewing the definition of the integrated correlators and their connection with the partition function of N = 2 * SYM on S 4 . We will particularly emphasise the importance of reorganising the operators in the localisation computation of the integrated correlators. With an important appropriate normalisation 3 To distinguish the resummation parameter z in summing over N as given in (1.1), we will use the parameter w for summing over the charge p. 4 In the large-p expansion, we will also consider the generalised 't Hooft genus expansion by introducing λ = p g 2 Y M .
factor, the integrated correlators obey a universal Laplace-difference equation, as proved in [19]. In section 3, we will introduce the lattice-sum representation and generating functions for the integrated correlators, following the ideas of [4]. We will determine the generating functions (in terms of some initial conditions) by utilising the Laplace-difference equation, which becomes a differential equation of the generating functions. The generating functions have interesting differences depending on whether N is even or odd. We will further determine the singularity structures (discontinuities and poles) of the generating functions, since they are most relevant in the study of the large-charge expansion of the integrated correlators. We will give several explicit examples to illustrate the general structures. In section 4, we will discuss how to obtain the large-charge expansion of the integrated correlators by utilising the generating functions derived in the previous section. By a simple contour deformation argument, it is easy to show that the p-dependence properties of integrated correlators are determined by the singularities of the generating functions. We will find that the large-charge expansion of the integrated correlators takes a universal form for any N , up to a subtle and interesting difference depending on whether N is even or odd. This difference is sharpened in the generalised 't Hooft limit, and makes connections with resurgence analysis. In section 5, we will give several non-trivial examples of the large-p expansion of the integrated correlators. We will conclude and discuss future research directions in section 6. The paper also includes three appendices. In appendix A, we will provide more examples of generating functions for the integrated correlators and their singularity properties. Appendix B briefly reviews the connections between the lattice-sum representaion and the SL(2, Z) spectral decomposition of integrated correlators. Appendix C concerns the large-N expansion of the integrated correlators with fixed charge p, generalising the results of [4].
Note added: While we were finalising this paper, [39] appeared on arXiv, which also studied some large-charge properties of the integrated correlators C (M,M ′ |i,i ′ ) N,p (τ,τ ) for the cases with M = M ′ = 0 (the corresponding operators are called maximal-trace operators in [39]). We found perfect agreement between our results where there are overlaps, up to an overall factor of 4 due to different normalisations.

Integrated correlators and Laplace-difference equation
In this section, we will review the construction of [19], where the superconformal primary operators are reorganised into different towers and subtowers, which greatly simplifies the Gram-Schmidt procedure in the localisation computation of the integrated correlators. The integrated correlators organised in this particular way, with some crucial normalisation factor, were shown to obey a universal Laplace-difference equation that relates integrated correlators with different charges.

Review of integrated correlators
The operators O (i) p that appear in the four-point correlation functions that we will study are half-BPS superconformal primary operators. They are in the [0, p, 0] representation of the SU (4) R-symmetry, and have scaling dimension p, which is protected by supersymmetry. There are in general more than one distinguishable operator for a given charge (or dimension) p, and so the superscript i denotes such degeneracy. In terms of the fundamental scalar fields Φ I (with I = 1, · · · , 6), they take the following form: where p = p i , and each T p (x, Y ) is a single-trace operator, and Y I is a null SO(6) vector that conveniently sums over the R-symmetry index I. Note for a finite N , not all the T p1,··· ,pn (x, Y ) are independent. For example, T 4 = 1 2 T 2,2 when N = 2, 3. In fact, for N = 2 the only independent operator at dimension 2p is T 2,2,...,2 (with p copies of T 2 ); they are often called maximaltrace operators. We will consider a special class of correlators of the form . Because of the constraints from superconformal symmetry, the correlators can be decomposed as [40,41] where we separate the correlator into a free part, G (i,j) free , which may be simply computed by free-field Wick contractions, and a part that non-trivially depends on the dynamics of the theory. For the dynamic part, one can further factor out all the R-symmetry dependence, denoted as is completely fixed by the symmetry; we will follow the convention of [5], and the expression for I 4 (x, Y ) can be found in that reference. The function H (i,j) N,p (U, V ; τ,τ ) is our main focus, which is a function of the Yang-Mills coupling τ and the cross ratios, As argued in [5] (see also [17]), this particular type of correlator, once the spacetime dependence has been integrated out with a certain measure, can be computed by supersymmetric localisation. More precisely, we define the integrated correlators as [5] C (i,j) with U = 1 − 2r cos θ + r 2 and V = r 2 . Importantly, as shown in [5], the integrated correlators C p . The partition function, and thus the integrated correlators, can be computed by supersymmetric localisation [6,42]. The integrated correlators C (i,j) N,p (τ,τ ) are modular functions of (τ,τ ), and of course also have non-trivial dependence on the parameters N, p and i, j.

Organisation of operators and Laplace-difference equation
Due to the dimensionful radius of S 4 , operators with different dimensions will mix. In general, a dimension-p operator on S 4 could mix with operators of dimensions (p − 2), (p − 4), etc. This is resolved by a Gram-Schmidt procedure [5,43]. To efficiently perform the procedure, we will reorganise the operators into towers and subtowers, such that operators in different (sub)towers are orthogonal to each other. We will then only need to perform a Gram-Schmidt procedure within the (sub)towers. We therefore reorganise the operators in the following manner: O 0|M is defined as follows. We begin by considering a general operator T p1,p2,...,pn as defined in (2.1) with all p i > 2 (namely, T 2 is excluded). These operators are ordered according to their dimensions, which are labeled as B For a general mass m, the partition function Z N (τ, τ ′ A ; m) can be expressed as where the integration variables a i are constrained by N i=1 a i = 0, and Z 1−loop and Z inst give the perturbative and non-perturbative contributions, respectively. When m = 0, both Z 1−loop (a; m) and Z inst (τ, τ ′ , a; m) simply become 1, and we have (2.10) One may find more detailed localisation computation of the integrated correlators in [5,17,19]. Using (2.7) and the above expression for the partition function, it is straightforward to obtain O 0|M . For M < 6, there is no degeneracy, and we find They coincide with the so-called single-particle operators [44]. For M ≥ 6, in general there is non-trivial degeneracy. For example, when M = 6, we can choose B (1) 6 = T 3,3 and B (2) 6 = T 6 , with which we found (2.12) The operators O (i) p|M (for any p) with different M are denoted as different towers, and within a given M -th tower, i labels different subtowers [19]. One of the crucial properties of O (i) p|M is that they are mutually orthogonal on S 4 [43]. More precisely, they obey the following conditions, for any p 1 , p 2 . Precisely because of this property, we only need to perform the Gram-Schmidt procedure within a given (sub)tower, as was done in [19]. We will briefly review the results of [19] below.
As we commented earlier, the integrated correlators are related to the partition function of N = 2 * SYM on S 4 . Following [19], they can be expressed as, 15) with Z N (τ, τ ′ A ; m) being the partition function of N = 2 * SYM on S 4 as given in (2.9). Here ∂ τ ′ (i) µ|M (and ) is from 0 to p for M > 0 (or 0 to p ′ for M ′ > 0), and from 1 to p for M = 0 (or 1 to p ′ for M ′ = 0) 5 . We will now explain all the other ingredients as follows. The mixing coefficients v µ p|M for the operators O where the connected two-point function is defined as This uniquely defines the mixing coefficients v µ p|M as v µ p|M = p µ Another crucial ingredient of the construction is the normalisation factors in the denominator in (2.15), which are given as [19] where Θ(x) = 0 if x ≥ 0 and Θ(x) = x if x < 0, and (2.20) Without loss of generality, we will assume M ≥ M ′ and we simply have Θ(δ) = 0. Finally, the prefactor R in (2.14), which is independent of the coupling τ , is given by (when M ≥ M ′ ) determined by the two-point function, In the special case M = M ′ , i = i ′ , the denominator in (2.15) reduces to After setting up all the necessary ingredients, we are ready to present the Laplace-difference equation where the laplacian ∆ τ = 4τ 2 2 ∂ τ ∂τ and a and δ are given in (2.20). The "source term" C (0,0) N,1 (τ,τ ) denotes the integrated correlator associated with O 2 O 2 O 2 O 2 , which appeared in (1.1) as C N (τ,τ ) and has been studied in [1,2,4] (see also [7-10, 18, 45]). Correspondingly, B N (t), which appeared in its lattice-sum representation, will be denoted as B (0,0) N,1 (t) (note that we also omit i, i ′ indices since there is no degeneracy in this case). Furthermore, the prefactor R (0,0) N,1 is simply (N 2 − 1)/2, namely, The Laplace-difference equation provides a powerful recursion relation, which in principle determines C also satisfies a three-term recursion of a similar manner

Generating functions of integrated correlators
As anticipated in the introducation, we will assume the lattice-sum representation for the integrated correlators, is a rational function of t that contain all the dynamics of the integrated correlators. It has a Taylor expansion, and obeys several important properties, such as It should be stressed that assumption of the lattice-sum representation is mathematically equivalent to assume that the SL(2, Z) spectral decomposition of the integrated correlators only contains the continuous spectrum (i.e. the non-holomorphic Eisenstein series) [3] (see also appendix B for more details). This proposal (in the language of SL(2, Z) spectral decomposition) has been checked in [17] by explicit perturbative and non-perturbative instanton computations for some non-trivial examples. These checks provide highly nontrivial evidence for this representation.
We argue that the SL(2, Z)-invariant Laplace-difference equation (2.25), which recursively determines C (τ,τ ) has the lattice-sum representation, all the integrated correlators will enjoy the same representation. 6 When M = M ′ = 0, it was proved recently in [4] that indeed C (0,0) N,1 (τ,τ ) can be expressed in the lattice-sum form for any N , as in (1.1). Furthermore, using the results of [17], it was shown in [19] (see appendix A of the reference) that all initial conditions C Once the lattice-sum representation is given, following [4], it is natural to further introduce the generating functions for the integrated correlators by summing over the p-dependence of the integrated correlators, 7 The integrated correlators can then be expressed in terms of generating functions as a contour integral, where the contour C circles the origin w = 0 clockwise. It will prove to be convenient to introduce an intermediate quantity, and then B One can also introduce the generating function for H with its lattice-sum representation given by In the next section, we will determine the generating functions using the Laplace-difference equation.
To obtain the generating functions, we will first determine B (w), and we will see that the solutions to these equations are Hypergeometric functions, from which one can perform the summation over the index s. We will begin with the generating functions for M = M ′ , i = i ′ . By including some extra differential operators, we then generalise our method to obtain B where we have set δ = 0 in (2.25) and here a = (N 2 + 2M − 3)/2 since M = M ′ . Here we have used the fact that the laplacian ∆ τ , when acting on the lattice-sum representation, becomes t∂ 2 gives the first term in the differential equation (3.12). The solution to this differential equation is given by the standard Hypergeometric function. After taking into account the boundary conditions, we obtain the general formula for H   equivalently the lattice-sum representation), we only need to focus perturbative terms to determine these is a polynomial in s, and we find it has degree where ⌊y⌋ is the floor function.
To complete the final step of obtaining the generating functions for the integrated correlators, we also need to sum over the parameter s in B Firstly, for the last term in (3.14), which has a pole at w = 1, we have It is easy to see that B (M,M|i,i) N (w; t) pole is independent of M and i, and obeys the following relation, where B (0,0) N,1 (t) again is simply B N (t) that appears in the lattice-sum representation (1.1) (here we write it in our unified notation). The generating function for B (0,0) N,1 (t), by summing over the N -dependence, was determined recently in [4], and is given by The result (3.19) shows that the generating functions in general have a pole at w = 1, and from (3.20), the residue at this pole obeys the following relation, , when translated into the τ -plane, becomes ∆ τ . Therefore the contribution from the residue at w = 1 of the generating function cancels precisely the "source term" −4 C (w) depending on whether N is even or odd, due to the Hypergeometric function containing integer or half-integer arguments. In particular, the parameter a defined in (2.20) that appears in the solution (3.14) is an integer if N is odd and a half integer if N is even. This distinction leads to different structures of the final expression for the generating function B (M,M|i,i) N (w; t), and consequently the integrated correlators, especially in the large-charge expansion. We will now discuss each case separately.

Even N
We begin with the case of even N , for which a is a half-integer. From (3.14), the remaining part of Focusing first on the sum of the Hypergeometric function, we define which for the even N case (again in this case the parameter a is a half integer) is given by where we have defined is a polynomial in w and t, which can be written as another way, the function f In this particular case, f 2 (w, t) = 0 according to (3.27), since g 2 (w, t) = 0. We now take into account the prefactors (−1) s and (1 − w) s−1 , giving which can be written as Finally, we take into account the prefactor q is a polynomial in s, it gets promoted to the differential operator q (M,M|i,i) N ;S acting on the function F (M) N (w, t(w − 1)). Putting all the terms together, the generating function is given by is given in (3.15). This equation is valid for both even and odd N .
For the even N case, we can use the expression of F (3.30), and the simple structure of q (3.36) The general structure (3.33) can be understood from (3.32) and using the fact that q is an order-x differential operator in t acting on F (3.33) arises from the first term given in (3.30), specifically when the derivatives act on the term in front of the G(w, t(w − 1)).
The pre-factor q (M,M|i,i) N ;S has a factor of (2S − 1), which leads to an extra factor of (w − 1) when acting 30) (and the contribution from the first term when the derivatives act on G(w, t(w − 1)), which becomes a rational function) leads to the second term in B (M,M|i,i) N (w; t). This term naively contains some higher-order pole 1/(w − 1) a−1/2 due to t → t(w − 1). However this is canceled out by the numerator f (M) N (w, t(w − 1)), which behaves as (w − 1) a−1/2 , as can be seen from the definition (3.27). Finally, the last term B is given in (3.19), which obeys the relation (3.20).
As we will see, by a simple application of the residue theorem, what is important for understanding the charge dependence of integrated correlators is the singularity structure of the generating functions on the w-plane. As shown in (3.33) explicitly, the generating functions have a pole at w = 1, which is determined by (3.20). The generating functions in general also have branch cuts along (1, w 1 ) (again w 1 = (t + 1) 2 /(t − 1) 2 ), and the discontinuities can be read off from (3.33), and take the following general form, where we have also used that G(w, t(w − 1)) √ w − 1 has branch cuts along (1, w 1 ) with discontinuity given by It is of importance to note that the parameter b, as defined in (3.34), is an integer when N is even. Therefore, in this case, (t + 1) 2b (1 − w/w 1 ) b is simply a polynomial in t and w. This will not be the case for odd N , and we will discuss the implications of this later. The discontinuity can also be found directly from F (M) N (w, t(w − 1)). As (a − 1/2) is an integer, the discontinuity in (3.30) arises solely from the first term, and so discF (M) will not affect the discontinuity on the w-plane, we find Then the general structure of the discontinuity given in (3.37) follows using the fact that q is an order-x differential operator in t with a factor of (2S − 1), with the operation S given in (3.31).

Odd N
We now consider the case of odd N . In this case, the parameter a (as given in (2.20)) is an integer, and so the Hypergeometric function in (3.23) has integer arguments. Using the Pfaff transformation and the definition of Jacobi polynomials (3.24) in terms of Jacobi polynomials, We can then apply the generating function of the Jacobi polynomials . As before, S should be understood as a differential operator, whereas its inverse from the factor (S) −1 a should be understood as an integral on any function f (t) it acts on, which is the inverse of (S + k).
Evaluating (3.45), we find, similarly to (3.25), where, similarly to the even N case . To obtain the total generating function, we rescale t → t(w − 1), and obtain Finally, from (3.32), we find the generating functions take the following general form, 10 , with the parameters a and b are given in (2.20) and (3.34), respectively. The function P in w, just as in the even N case, and the polynomial Q They are very similar to the polynomials that appeared in the even N case, and also satisfy similar symmetries, and The generating function again has a pole at w = 1 due to the last term B Similarly to the even-N case, this discontinuity can be obtained directly from (3.49). The discontinuity of and so the discontinuity of B We see that, very interestingly, even though the generating functions for even N and odd N take different forms, they have identical singularity structures. However, when N is odd, the parameter b as given in (3.34) is a half-integer, and so unlike the even N cases, the combination (t + 1) is not a polynomial of t and w anymore. In particular, it has branch cuts along (w 1 , ∞). This will lead to an important difference in the large-p expansion of the integrated correlators for even and odd N .
The solution to this differential equation is again given in terms of Hypergeometric function, and takes the following form, (3.58) We therefore see that the generating functions in general can be written as It is easy to see that the role of K , which again contains all the initial data, is given by It is therefore clear that q is a polynomial in s, and its degree is which generalises (3.17) for δ = 0.
We now perform the summation over the index s. As there is a factor of (s , which can be seen from the expansion of the Hypergeometric function, we can start the sum at s = δ + 1.
We therefore find where S is the differential operator defined in (3.31). To compute the summation of the first term in (3.64), we first find the generating function for the Hypergeometric function, We can shift s so that the sum starts from s = 1, as in the M = M ′ case, so we have (3.67) Using the identity As a + δ = (N 2 + 2M − 3)/2, this is exactly the same Hypergeometric function that appeared in (3.24), and so taking the sum inside the differential, we find N (w, t) is given in (3.25) for even N and (3.47) for odd N . Therefore, finally, the generating function is given by   (1, w 1 ). We will now compute the corresponding discontinuity. From (3.25) and (3.47), and using (3.38), we find that the discontinuity of B To see the form of the discontinuity, we can perform the derivatives ∂ 2δ w explicitly, and so disc B can then be expressed as where h(w, t) is a polynomial in w and t, explicitly given by (3.74) A factor of (w − 1) a−δ−1/2 has been factored out in (3.73), and h(w, t(w − 1)) has an extra factor of (w − 1) δ .
Furthermore, the prefactor q (w) is an order-x (as given in (3.63)) differential operator in t, and so disc B and P In writing down the general expression, we have used the fact that an extra factor of (w − 1) comes from the factor of (2S − 1)   Going through the procedure we outlined in the previous section, we find the generating function is given by,

Examples of generating functions
(3.80) For this particular N = 2 case, the coth −1 components of G(w, t(w − 1)) in F 2 (w, (w − 1)t) actually cancel out, and the generating function only contains poles but no branch cuts in the w-plane.

Large-charge expansion
As we commented in the introduction, it has been shown in the literature that the charge of some global symmetry provides a very useful expansion parameter for performing the analytical study of physical observables in QFT. In particular, it allows us to access the strong coupling regime, and can often be described by some effective field construction (see e.g. [23][24][25][26][27]). In this section, we will utilise the generating functions of integrated correlators we obtained previously to study the large-charge properties of the integrated correlators. It will provide explicit analytical results of the integrated correlators in the large-charge expansion. As we will show, all the integrated correlators behave universally in the large-charge expansion (up to some subtle difference between even and odd N ), and we will further find that the large-charge expansion of the integrated correlators take a rather similar form to that of the large-N expansion. In particular, the same types of modular functions appear in both cases.
Let us begin with the general expression for the integrated correlators. From (3.4), we have (τ,τ ) into several different parts. Firstly, we will separate out the (m, n) = (0, 0) case in the lattice-sum, which contributes to the τ -independent part of the integrated correlators since Y 0,0 (τ,τ ) = 0. In fact, in the large-p expansion, there is additional τ -independent contribution, as we show in (B.9). We will denote all the τ -independent contributions as C Below we will discuss each part separately; we will start with the discontinuity part and consider the remaining parts afterwards.

Discontinuity part
As we will see, the most interesting part of the large-charge expansion of integrated correlators arises from the discontinuity part, C (M,M ′ |i,i ′ ) N,p (τ,τ ) disc , which can be written as The deformed contour C ′ encircles the cut, together with the contour at infinity, C ∞ , which gives a vanishing contribution. with where δ N = 1 for odd N and δ N = 0 for even N . The factor ±iǫ in the integrations given in (4.5) is to choose the integration contour slightly above or below the real axis to avoid the cut in the odd-N case. Recall that the exponent b of (1 − w/w 1 ) b in the discontinuity disc B branch cuts along (w 1 , ∞), but for the even-N case it is a polynomial. For this reason we need to deform the integration slightly by ±iǫ for the cases with odd N .
For odd N , the factor ±iǫ in the integration has no effect in the case of C (w; t) has no branch cuts along (w 1 , ∞), and so we can simply integrate along the real axis. Therefore, there is no such ambiguity. As we will see, this is also in agreement with resurgence.
To perform the computation explicitly, we make a suitable change of variables. For the perturbative part, are all non-zero; for even N , however, the coefficients vanish when m is too large. More precisely, we have, This is because (1 + t) 2b (1 − w/w 1 ) b is a polynomial for even N , which truncates in the variable t (and w).
Therefore, according to the definition of E(s; τ,τ ) in (4.7), the index s cannot be too big if the expansion in t truncates.
For computing the non-perturbative part C and (t) does not contain exponentially behaved terms in the large-p limit, which is in the "action" S p (t), more precisely in e −Sp(t) . In the integration region, S p (t) > 0, therefore in the large-p limit C  To obtain the large-p expansion, we substitute t = t * + α back into (4.10) and perform Gaussian integrals over α. We find the result of this analysis takes the following form, where D p (s; τ,τ ) is a generalisation of E(s; τ,τ ) with an exponentially decayed piece, When the parameter p = 0, it reduces to E(s; τ,τ ). As we explained earlier, there may be an overall ±i ambiguity in (4.14) depending on whether N is odd or even. Furthermore, v 1 , v 2 are some simple rational numbers (such as 1/4, 1/2) depending on the parameters (M, M ′ |i, i ′ ) and N , and the coefficients d are rational numbers (again, up to some overall factors of π, √ 2, etc., depending on particular cases). We note that exactly the same modular function D p (s; τ,τ ) appeared in the large-N expansion of C

Remaining parts
We will now consider the additional contributions to the integrated correlators that are beyond what has been considered in subsection 4.1. They are relatively simpler to compute. The remaining parts can be separated into two parts: one is independent of the charge p, which arises from the pole at w = 1 of the integrated correlators, as given in (4.16); the other is independent of the coupling τ , as explained in appendix B (more precisely it is given in(B.9)).

The p-independent term from w = 1 pole
We begin by considering the contribution from the pole at w = 1, which is given by As we commented earlier, this contribution is associated with the source term C  One can in fact write down the generating function by summing over N . Using the result (3.21) and appropriate boundary conditions, we find, where we have excluded (m, n) = (0, 0) here in the lattice sum, as it will be considered in the next subsection.

The τ -independent term
We now consider the τ -independent part of the integrated correlators. As shown in (B.9) in appendix B, it consists of two contributions in the large-p expansion, which we quote below Since it is independent of τ , the Laplace-difference equation (  (t), we can evaluate the first contribution in (4.21) straightforwardly. We find  We now consider the second term in (4.21), which we find to be where ρ is independent of p. When M = M ′ , i = i ′ , we find ρ = N (N − 1)/2 and the above expression  (4. 29) In summary, in the large-p expansion, the τ -independent part of the integrated correlators is given by (4.21), with each contribution defined in (4.23) and (4.27).

Summary
Putting everything together, we find that the large-p expansion of all the integrated correlators can be summarised in the following form, 13 N,1 (τ,τ ) that was recently obtained in [4]. However, as we have emphasised, there are important differences. In particular, when N is even, the number of non-holomorphic Eisenstein series that appear in the large-charge expansion do not grow indefinitely as we consider higher orders in the 1/p expansion. More precisely, This is very different from what has been seen in the large-N expansion of integrated correlators [2,4,8].
Furthermore, when considering the generalised 't Hooft genus expansion by introducing λ = p g 2 Y M , the zero mode of the Eisenstein series give power series terms in the 1/ √ λ expansion, whereas the functions D p (s; τ,τ ) lead to exponentially decayed terms e −2n √ λ . The property (4.33) implies that in the 't Hooft expansion, at a given order of 1/p, the power series in the 1/ √ λ expansion also truncates. Often the power series terms and exponentially decayed terms are related by resurgence [2,4,18,45]. The truncation therefore has interesting consequences in understanding the connection between power series terms and exponentially decayed terms from the point of view of resurgence. Indeed this phenomenon, that certain physical observables have truncating perturbative expansions but still possess non-perturbative objects, has appeared in the literature of resurgence, under the name of "Cheshire cat" resurgence [47][48][49][50][51][52][53][54]. The general idea is that one may introduce some deformation parameter such that the truncated perturbative series becomes an asymptotic series with such a deformation parameter. One can then perform more standard resurgence analysis, and turn off the deformation parameter at the end of the calculation to obtain the non-perturbative terms. In our case, presumably the rank of gauge group N naturally plays the role of such a deformation parameter.
Finally, we would like to stress that, because it is completed with the exponentially decayed terms, the large-p transseries (4.32) should be understood as the full result of C (M,M ′ |i,i ′ ) N,p (τ ;τ ). The Borel resummation 13 We have expanded the integrated correlators C of the transseries provides a well-defined analytic continuation for all values of p, and should reproduce the starting-point expression in terms of generating functions. This is analogous to the large-N expansion of C (0,0) N,1 (τ,τ ) that was recently obtained in [4]. In the next section, we will consider specific examples and compute explicitly the coefficients.

SU(2) gauge group
We begin by considering the simplest case, the integrated correlators with gauge group SU (2). As we commented earlier, for this special case, the only independent operators are (T 2 ) p , or equivalently O p|0 , which have dimension 2p. So for this particular gauge group, the only relevant integrated correlators are C (0,0) 2,p (τ,τ ). The generating functions of the integrated correlators can be expressed as (3.80). The integrated correlator is then given by The generating functions with SU (2) gauge group are special in that they do not have any branch cuts. From (3.80), we find We now consider the large-p expansion of the integrated correlator following the general discussion in the previous section. First of all, the first term in (5.3) arises from the B (0,0) 2 (w; t) pole , which is independent of p and leads to The τ -independent part is given in (4.30) by setting N = 2 and M = M ′ = 0, We will now focus on the remaining part of the large-p expansion of the integrated correlators, namely the second line of (4.32), which is given by Firstly, we note that C (0,0),NP 2,p (τ,τ ) obeys the following Laplace-difference equation, where the "source term", −4C (0,0) 2,1 (τ,τ ) in (2.25), cancels out because of the relation (4.20). The superscript 'NP' is to indicate that this term is only exponentially decayed in the large-p limit; this is consistent with (4.33), as for this special case there are no 1/p power series terms with a sum of non-holomorphic Eisenstein series as the coefficients. We will study the large-p behaviour of C (0,0),NP 2,p (τ,τ ) by a saddle-point analysis as we outlined in the previous section. We write C (0,0),NP 2,p (τ,τ ) as, with S p (t) given in (4.12). The result of the saddle-point analysis can be written as, Not surprisingly, all the d 2;r,2r , which plays the role of the initial condition for the recursion relation. We find that d Finally, it is instructive to consider the generalised 't Hooft limit by introducing λ = p g 2 Y M in the region 1 ≪ λ ≪ p. We begin by considering the small-λ perturbative expansion, for which we find It is easy to see that the perturbative expansion is convergent with a finite radius |λ| < π 2 . This is identical to the small-λ perturbative expansion in the large-N expansion of integrated correlators with p = 2 [2].
However, we find the large-λ expansion is trivial. This can be seen easily from the fact that the coefficients of λ s−1 in (5.12) have vanishing residues for negative half-integer s. This peculiar property of the large-λ expansion is also related to the fact that the large-p expansion of the integrated correlators with finite τ only contains exponentially decayed terms, as we discussed earlier. The large-λ expansion of the integrated correlators can be obtained from the zero mode of the non-holomorphic Eisenstein series, and for this particular case there is no 1/p power series with non-holomorphic Eisenstein series.
In the 't Hooft limit, the zero-mode part (i.e. the zero-instanton sector) of C (0),NP 2,p (τ,τ ) leads to exponentially decayed terms in the large-λ limit,   (1.2). 14 From the zero mode of D p (s; τ,τ ), one may also obtain terms that behave as e −2m √λ forλ = 4πp/λ in a different region where 1 ≪λ ≪ p. As emphasised in [4], all these types of exponentially decayed terms in terms of the 't Hooft coupling are contained in our SL(2, Z)-invariant expression, as given in (5.9).
We can also consider the finite λ orλ region, generalising the results of [4,45]. In these regions, C (0,0),NP 2,p (λ) behaves as where the function A(x) is defined as A(x) = 4 x √ x 2 + 1 + sinh −1 (x) . All these behaviours are very similar to what have been found in large-N expansion of C (0,0) N,1 (τ,τ ) in the 't Hooft limit (for which λ = N g 2 Y M ), so we will not consider all the regions in detail.
In the large-N expansion of C (0,0) N,1 (τ,τ ), besides the exponentially decayed terms e −2n √ λ , at each order in 1/N expansion, C (0,0) N,1 (τ,τ ) also contains power series terms 1/ √ λ. In fact, the exponentially decayed terms may be considered as the non-perturbative completion for the power series terms through a resurgence procedure [2,18,45], because of the fact that the power series terms are not Borel summable. In comparison, for the case we are considering, there are no power series terms at all, therefore applying the standard resurgence procedure to obtain the above exponentially decayed terms given in (5.13) may not be straightforward. As we will show later for higher-rank gauge groups SU (N ), there will be both power series terms and exponentially decayed terms. However when N is even, the power series terms are not an infinite series as we commented on earlier, and once again relating the power series terms and exponentially decayed terms through resurgence procedure is not as straightforward. The resurgence analysis may be done through the so-called "Cheshire cat" resurgence as we commented earlier. We will leave such analysis for future work.

M = M ′ = 0 with SU(3) gauge group
We now consider the large-charge expansion of the integrated correlator C (0,0) 3,p (τ,τ ). As described in section 4.2, the p-independent and τ -independent contributions are relatively simple, so we will focus on the more interesting part from the branch cuts (3.91), where the perturbative and non-perturbative terms are given by (4.5), which we quote again below:  For the non-perturbative part C (0,0),NP 3,p (τ,τ ), we make the change of variables w = w 1 e µ p to give, as in (4.10), with S p (t) defined in (4.12), and, as in (4.11), with which is only polynomial in p. Performing the analysis described in the above leads to the final result, which is given by It is straightforward to verify that the results (5.18) and (5.22) obey the Laplace-difference equation (2.25) with the "source term" −4C (0,0) 3,1 (τ,τ ) on the RHS removed. It is important to note the ±i ambiguity in the final expression, which is due to the branch cuts of the disc B (M,M ′ |i,i ′ ) N (w; t) along (w 1 , ∞) for any odd N , as we explained earlier.
As before, from the SL(2, Z)-invariant results, we can extract the expression in the 't Hooft limit. In particular, from (5.18), we find that the perturbative terms become 24) and the non-perturbative terms lead to (5.25) which can be expressed in terms of polylogarithm's, where the argument x = e −2 √ λ . Importantly, unlike the even-N cases, C (0,0),P 3,p (λ) contains infinite numbers of terms at each order in 1/p expansion. This property can also be seen from small-λ expansion, which is given by We see the coefficients of λ s−1 have non-trivial residues at s = −1/2 − n for all n = 0, 1, . . .. Using the contour integral representation described in appendix B, one can precisely rederive (5.24) from (5.28) by computing the residues at s = −1/2 − n. As we commented earlier, C (0,0),NP 3,p (λ) can be understood as the non-perturbative completion of C (0,0),P 3,p (λ) by the process of resurgence. In particular, the large-λ expansion of the integrated correlator at each order in 1/p expansion is asymptotic and not Borel summable, and C (0,0),NP 3,p (λ) gives the non-perturbative completion. This analysis was performed recently in [39]; we will not repeat the analysis here. Once again, in our analysis, all these results in the 't Hooft limit are simply the zero Fourier mode of the complete SL(2, Z) invariant expression.
The computation of the large-p expansion for N = 4, M = M ′ = 0 is very similar to those we described in the previous subsections. We will only list the results in this section. Let us begin with the power series terms. We once again find that they are expressed in terms of non-holomorphic Eisenstein series. However, as we already emphasised in the general expression (4.32), due to (4.33), it turns out not all the Eisenstein series appear in the 1/p expansion of the integrated correlators. In fact, we find it terminates at E(13/2; τ,τ) for the integrated correlators C (0,0),P 4,p (τ,τ ). Explicitly, using the result of the discontinuity given in (3.88), we find that the power series terms in the large-p expansion take the following form, We now move to the non-perturbative terms. For the same reason we discussed above (i.e. the factor (t + 1) 2b (1 − w/w 1 ) b is polynomial when N is even), the disc B (0,0) 4 (w; t) does not have branch cuts on the w-plane, therefore there is no ambiguity in the choice of contour as in the SU (3) case (or more general odd N cases). We find the non-perturbative terms are given by Once again, the complete finite-τ results allow us to obtain the 't Hooft limit of the integrated correlators expressed in terms of λ. We find It is notable that at each order in 1/p expansion the large-λ expansion terminates at O(λ −13/2 ). This is related to the fact that in (5.29), a (0,0) 4;r,m = 0 for m > 6, namely Eisenstein series with indices larger than 13/2 do not appear. This interesting property can also be seen from the small-λ expansion 90090 (−1) s s 2 − s + 4 s 2 − s + 18 ζ(2s − 1)Γ s + 1 2 π 2s−3/2 (2s + 1)(2s + 3)(2s + 5)(2s + 7)(2s + 9)(2s + 11)Γ(s) λ s−1 (5.34) π 2s−3/2 (2s + 1) 2 (2s + 3)(2s + 5)(2s + 7)(2s + 9)(2s + 11)Γ(s − 1) We see that the coefficients of λ s−1 at each order in 1/p have non-trivial residues at s = −1/2, . . . , −11/2, and these residues precisely lead to (5.33).
Even though there are only finite number of power series terms, our results show that at each order in the 1/p expansion, there are associated exponentially decayed terms. They can be obtained from (5.31) and are given by

Operators in other identical towers
We now consider integrated correlators involving operators in other identical towers, i.e. M = M ′ > 0. The computation is very similar. We begin with the operators with M = M ′ = 3, for which the discontinuity is given in (3.94). Because this tower of operators involve T 3 , we must have N ≥ 3 for the gauge group SU (N ). Let us take the simplest N = 3 case as an example; the computation and the results for higher N are very similar. The perturbative terms in this case are given by In the 't Hooft limit and expressed in terms of λ = p g 2 Y M , we find and from the non-perturbative term, we have,  We have also analysed the integrated correlators with higher ranks of gauge group, once again the results are all given in the general structure (4.32). We will move on to consider the integrated correlators C (4,4),P N,p (τ,τ ). For this case, we require N ≥ 4, and here we will consider the case with SU (4) gauge group. The perturbative terms are given by Importantly, we also find that the coefficients of the perturbative terms obey the condition that all a (4,4) 4;r,m = 0 for m > 10. Once again, this implies that the Eisenstein series E(s; τ,τ ) with s > 21/2 do not appear in the expansion. In terms of 't Hooft coupling λ, this means that there are only a finite number of terms in the large-λ expansion for a given order in 1/p. In the 't Hooft limit, we find

Operators in different towers
We now consider integrated correlators involving operators in different towers, i.e. M = M ′ . The analysis is very similar. We begin with the case M = 4, M ′ = 0, with the simplest case of SU (4). Using the result of (3.97) for the discontinuity, we find that the perturbative terms are given by, The non-perturbative terms are given by

Conclusion and discussion
In this paper we proposed a lattice-sum representation for integrated correlators that are associated with fourpoint correlation functions in N = 4 SYM of the form p is a charge-p (or dimensionp) half-BPS operator. This generalised the earlier proposal for the simplest integrated correlator with p = 2 [1,2]. The formulation makes manifest SL(2, Z) invariance of the correlators, and allows us to introduce generating functions for the integrated correlators which sum over the dependence on charge p and/or gauge group rank N . As we reviewed in this paper, we recently proved [19] that all the integrated correlators for any N are governed by a universal Laplace-difference equation that relates integrated correlators of operators with different charges. The Laplace-difference equation is a recursion relation that completely determines all the integrated correlators, once the initial conditions are given. In this paper, we utilised the Laplacedifference equation to explicitly determine the generating functions for the integrated correlators, in terms of initial data. We found that the generating functions, and especially their singularity structures, which are what is relevant for understanding the charge dependence of the integrated correlators, take a universal form for all the integrated correlators. Using the generating functions (and the singularity structures), we determined the transseries for the integrated correlators in the large-charge limit. The Laplace-difference equation of [19] is satisfied only after reorganising the operators in a particular way (as we reviewed in subsection 2.2), so accordingly we have taken the large-charge limit in a specific manner.
In particular, for a fixed N , the large-charge expansion of integrated correlators universally contains three parts. The first part is independent of the coupling τ and behaves as a power series in 1/p as well as a log(p) term when i = j (i.e. when two higher-dimensional operators are identical). The second part is a power series in 1/p, where the coefficient of each term is an SL(2, Z) invariant function given by a sum of finite numbers of non-holomorphic Eisenstein series with half-integer indices and rational pre-factors. The third part is exponentially decayed in the large-p limit. It can be organised as a "power series" in 1/p, and the coefficient of each term is a sum of the new SL(2, Z) invariant functions D p (s; τ,τ ), which behave as e −c(τ,τ) √ p . Finally, when i = j, there is an additional modular function of τ that is independent of p and is determined in terms of the integrated correlator with p = 2. These results (especially the second and third parts) are remarkably similar to those found recently in [4] for the integrated correlator with p = 2 in the large-N expansion, by exchanging p ↔ N . Furthermore, in appendix C we also studied the large-N expansion of the integrated correlator with a fixed p = 3, and once again a similar structure is found.
There are however some interesting differences between the large-N expansion of integrated correlators with a fixed p and the large-p expansion with a fixed N . In particular, the precise form of the large-p expansion of integrated correlators depends on whether N is even or odd. In the case of odd N , the number of Eisenstein series for a given order in the large-charge expansion grows as the order of the 1/p expansion increases. This is the same as the large-N expansion of the integrated correlators [2,4]. In terms of the generalised 't Hooft coupling λ = p g 2 Y M , the power series terms in 1/p become asymptotic series in 1/ √ λ, which are not Borel summable. As has been seen in the large-N expansion [2,18,45], through resurgence this is tightly related to the exponentially decayed terms D p (s; τ,τ ), which behave as e −2n √ λ in the 't Hooft coupling. For even N , on the other hand, the number of Eisenstein series that appear in the large-p expansion does not grow indefinitely. In terms of 't Hooft coupling, this implies at a given order in 1/p expansion, the 1/ √ λ expansion terminates at a finite order, and there is in principle no ambiguity of defining such polynomials of 1/ √ λ. Nevertheless, we find that there are still always exponentially decayed terms e −2n √ λ for even N . This imposes interesting questions of relating these perturbative terms and exponentially decayed terms through the resurgence analysis, as in the case of odd N . As we commented earlier, this phenomenon is not new and has appeared in the literature of resurgence, see e.g. [47][48][49][50][51][52][53][54]. The general procedure of understanding such observables from the viewpoint of resurgence is to introduce some deformation parameters so that the finite perturbative series depends on the parameters and becomes asymptotic. This is often called "Cheshire cat" resurgence. For the integrated correlators, a natural parameter here is clearly N . The power series in the large-λ expansion truncates when we tune the parameter N to be even. We will leave the systematic study of the resurgence analysis for the integrated correlators, especially for even N , for future work.
As we commented earlier, we have studied the large-charge expansion of the integrated correlators C (M,M ′ |i,i ′ ) N,p (τ,τ ) by considering the large-p limit with fixed M, M ′ ; this allows us to utilise the Laplacedifference equation. It will be very interesting to understand the large-M (and large-M ′ ) behaviour of the integrated correlators with a fixed p (even for p = 0). Another research direction is to study the large-N limit along with the large-charge limit. It will be of particular interest to understand this double scaling limit, and its connections with the dual string theory on AdS 5 ×S 5 along the line of [55]. The double scaling limit has been considered for C (M,M ′ |i,i ′ ) N,p (τ,τ ) in the recent work [39], but mostly only for the special cases of M = M ′ = 0. The Laplace-difference equation and the concept of generating functions should provide powerful pathways to extend the analysis of [39].
B Lattice-sum representation and SL(2, Z) spectral decomposition The lattice-sum representation (1.7) of integrated correlators implies that they can be formally expressed as an formal infinite sum of non-holomorphic Eisenstein series, [1,2]  . In general, a SL(2, Z)-invariant quantity can be decomposed into a continuous spectrum in terms of non-holomorphic Eisenstein series and a discrete spectrum in terms cusp forms. It is remarkable that SL(2, Z) spectral decomposition of these integrated correlators only contain the non-holomorphic Eisenstein series. where σ ν (k) = n|k n ν is the divisor sum, and K ν (x) the Bessel function. The first line in the above expression is the zero mode of the non-holomorphic Eisenstein series, which gives perturbative contributions; whereas the second line accounts for non-zero modes, which lead to non-perturbative instanton contributions. with the contour closing from the right in the large-p limit. Besides the 1/τ 2 perturbative terms, as ζ(2s − 1) has a singularity at s = 1, we see that the residue at s = 1 leads to an additional τ -independent term, and 15 The overall factor of 2 on the second line is due to the fact that the resummation of the first term in (B.5) gives the same result as the second term in the large-τ 2 expansion [2]. This is related to the symmetry property,ĉ From point of view of generating functions, the second τ -independent term arises because when performing large-p expansion, the generating functions may develop 1/t singularity and the residue at t = 0 leads to Y m,n (τ,τ )-independent (i.e. τ -independent) terms. And that the contribution at t = 0 is essentially equivalent to s → 0 limit.