Exact expressions for $n$-point maximal $U(1)_Y$-violating integrated correlators in $SU(N)$ $\mathcal{N}=4$ SYM

The exact expressions for integrated maximal $U(1)_Y$ violating (MUV) $n$-point correlators in $SU(N)$ ${\mathcal N}=4$ supersymmetric Yang--Mills theory are determined. The analysis generalises previous results on the integrated correlator of four superconformal primaries and is based on supersymmetric localisation. The integrated correlators are functions of $N$ and $\tau=\theta/(2\pi)+4\pi i/g_{_{YM}}^2$, and are expressed as two-dimensional lattice sums that are modular forms with holomorphic and anti-holomorphic weights $(w,-w)$ where $w=n-4$. The correlators satisfy Laplace-difference equations that relate the $SU(N+1)$, $SU(N)$ and $SU(N-1)$ expressions and generalise the equations previously found in the $w=0$ case. The correlators can be expressed as infinite sums of Eisenstein modular forms of weight $(w,-w)$. For any fixed value of $N$ the perturbation expansion of this correlator is found to start at order $( g_{_{YM}}^2 N)^w$. The contributions of Yang--Mills instantons of charge $k>0$ are of the form $q^k\, f(g_{_{YM}})$, where $q=e^{2\pi i \tau}$ and $f(g_{_{YM}})= O(g_{_{YM}}^{-2w})$ when $g_{_{YM}}^2 \ll 1$. Anti-instanton contributions have charge $k<0$ and are of the form $\bar q^{|k|} \, \hat f(g_{_{YM}})$, where $\hat f(g_{_{YM}}) = O(g_{_{YM}}^{2w})$ when $g_{_{YM}}^2 \ll 1$. Properties of the large-$N$ expansion are in agreement with expectations based on the low energy expansion of flat-space type IIB superstring amplitudes. We also comment on the identification of $n$-point free-field MUV correlators with the integrands of $(n-4)$-loop perturbative contributions to the four-point correlator. In particular, we emphasise the important r\^ole of $SL(2, \mathbb{Z})$-covariance in the construction.

1. Properties of the large-N expansion are in agreement with expectations based on the low energy expansion of flat-space type IIB superstring amplitudes. We also comment on the identification of n-point free-field MUV correlators with the integrands of (n − 4)-loop perturbative contributions to the four-point correlator. In particular, we emphasise the important rôle of SL(2, Z)-covariance in the construction.

Overview and outline
In recent work [1,2] we conjectured an exact expression for an integrated four-point correlator of superconformal primaries of the stress tensor multiplet of N = 4 SU (N ) supersymmetric Yang-Mills theory, that is given by a two-dimensional lattice sum and manifests the SL(2, Z) modular symmetry of the theory. In this paper we will extend these results to n-point correlation functions that violate U (1) Y charge conservation maximally.

Overview
The standard correlators of operators in supersymmetric conformal field theory are position dependent and therefore in general break supersymmetry. However, integrating over the positions of the operators in a correlator with suitable measure leads to a supersymmetric integrated correlator. The form of certain integrated correlators in N = 4 supersymmetric SU (N ) Yang-Mills (SYM) theory can be determined by supersymmetric localisation in the manner described in [3] and briefly reviewed in appendix A. These are obtained by exploiting the fact that N = 4 SYM theory is a limit of N = 2 * SYM theory in which the hypermultiplet mass vanishes. The particular integrated correlator considered in the large-N expansion in [3] is the correlator of four superconformal primaries, O 2 (x 1 , Y 1 ) · · · O 2 (x 4 , Y 4 ) , integrated over their positions, x i , with a particular measure. 1 This is given by taking four derivatives of the logarithm of the partition function of N = 2 * SU (N ) SYM on S 4 , where ∆ τ = 4τ 2 2 ∂ τ ∂τ is the hyperbolic Laplacian. The partition function of N = 2 * SYM, Z N (m, τ,τ ), is precisely determined by supersymmetric localisation [4]. The parameter m is a mass parameter and in the limit in which it vanishes (m = 0), the hypermultiplet mass vanishes and N = 2 supersymmetry is extended to N = 4.
Our notation follows usual conventions where the complex Yang-Mills coupling constant is defined by with θ the topological theta angle and g Y M the Yang-Mills coupling constant. The large-N 't Hooft expansion (in which the 't Hooft coupling λ = g 2 Y M N is fixed) of G N (τ,τ ) was considered in some detail in [3,5]. The large-N expansion with fixed g 2 Y M was considered in [6], where the instanton contributions to the correlator play an essential rôle in implementing Montonen-Olive SL(2, Z) duality [7][8][9]. The considerations in [1,2] led to a reformulation of this correlator as a two-dimensional lattice sum, which makes the modular properties of G N (τ,τ ) manifest for all values of N and greatly simplifies and extends the analysis of the large-N expansion. These properties of the four-point correlator are also briefly summarised in appendix A.
A second example of an integrated correlator presented in [10] is obtained from four derivatives with respect to the masses, ∂ 4 m log Z N (m, τ,τ )| m=0 of N = 2 * SYM partition function. and is again an integral of O 2 (x 1 , Y 1 ) · · · O 2 (x 4 , Y 4 ) over x i , but with a different measure. Its large-N expansion in the 't Hooft and fixed g 2 Y M limits were discussed in [10] and [11], respectively. The exact results of these integrated correlators have been used to determine scattering amplitudes of type IIB superstring theory in AdS 5 × S 5 , after taking flat-space limit, which match precisely with known results [12][13][14][15][16].

U (1) Y -violating correlators
Our aim here is to extend the preceding considerations to a class of n-point correlators of operators in the stress tensor supermultiplet that are modular forms with non-zero modular weights (w, −w), so they transform under SL(2, Z) by a U (1) Y transformation, where the U (1) Y charge is given by q U = 2w (see appendix B for a brief summary of some relevant SL(2, Z) properties). Here U (1) Y , which was called the 'bonus U (1) Y ' in [17], is the holographic image of the U (1) R-symmetry in type IIB supergravity and breaks to Z 4 when stringy corrections are turned on.
The U (1) Y charge of a correlation function of operators in the stress tensor supermultiplet is the sum of the charges of the individual operators in the correlator. Any of these superdescendent operators has the form δ nδn O 2 , where δ is a chiral supersymmetry transformation carrying U (1) Y charge +1/2 andδ is an anti-chiral supersymmetry transformation with U (1) Y charge −1/2. Since the superconformal primary, O 2 (x, Y ) has zero U (1) Y charge these descendants δ nδn O 2 possess a charge equal to (n −n)/2. Furthermore, the stress tensor supermultiplet is ultra-short so that n +n ≤ 4.
Super-descendent operators of particular significance in the following are the chiral and anti-chiral Lagrangian operators, O τ = δ 4 O 2 andŌτ =δ 4 O 2 , which carry U (1) Y charge +2 and −2, respectively. The N = 4 SYM Lagrangian can be expressed as the sum of two complex conjugate parts where the chiral and anti-chiral Lagrangians are defined by where F αβ ,Fαβ are the self-dual and anti self-dual Yang-Mills field strengths and ". . . " indicates terms involving fermion and scalar fields in the Yang-Mills supermultiplet. The pattern of U (1) Y symmetry breaking in the low energy expansion of type IIB superstring amplitudes was discussed in [18] and in the large-N expansion of N = 4 SYM in [17] (see also [19]). In either case, this symmetry is broken so that in general the U (1) Y charge is violated by the n-point correlator with n > 4. The magnitude of the modular weight of any such correlator has an upper bound given by |w| ≤ n − 4. This means that the maximum U (1) Y charge violation is given by (1.5) General features of such correlators were considered in [20] where particular emphasis was placed on those terms in the large-N expansion that correspond to the BPS protected terms in the low energy expansion of the holographic dual type IIB string theory studied in [21]. A characteristic feature of such MUV string amplitudes is that they do not possess massless poles in any channel. 2 This extended the analysis of the large-N expansion of the correlators in [3,5,6,10,11] to n-point MUV correlators.
There is a convenient harmonic superspace description which packages MUV correlators together [22,23]. In this approach the operators in the stress-tensor super-multiplet, T (Ψ), are functions of the superspace variables Ψ = (x, y, ρ,ρ), where the y-dependence determines the dependence on Y (the R-symmetry SU (4)) 3 and ρ,ρ are Grassmann coordinates in the (2, 1) 1 and (1, 2) −1 representations of SU (2) × SU (2) × U (1) ⊂ SU (4). 4 The n-point correlators of interest to us are the coefficients in the expansion of the correlator of n T 's in 2 "Next-to-MUV" (NMUV) n-point correlators were also defined in [21]. These are dual to type IIB string theory amplitudes that have a massless pole in one channel with a residue that is the product of a (n − 1)-point MUV amplitude and a three-point supergravity vertex. Furthermore, "Next-to-next-to-MUV" (NNMUV) n-point correlators were defined to be correlators that are dual to string amplitudes in which a massless pole either has a residue proportional to the product of a n 1 -point and a n 2 -point MUV amplitude (with n 1 + n 2 = n − 2 and n 1 , n 2 ≥ 4) or into the product of a three-point supergravity amplitude and a NMUV (n − 1)-point amplitude. 3 The coordinate y a a is related to the SO(6) null vector Y I by The U (1) factor is a subgroup of the SU (4) R-symmetry and should not be confused with the U (1) Y bonus symmetry, which is an automorphism of P SU (2, 2|4), and which is broken to Z 4 .
where the variables denoted by each label are j i = (x i , y i , k i ,¯ i ) and the superscript (w) indicates the modular weight of the correlator, which equals to w = n r=1 (k r −¯ r )/4. The fact that the stress-tensor multiplet is ultra-short implies that the sums are subject to the restrictions and furthermore, as explained in [23], supersymmetry and superconformal symmetry imply where (k r ,¯ r ) label the components of the stress tensor super-multiplet (for example, . For the MUV correlators, we have n r=1 k r − n r=1¯ r = 4n − 16, or equivalently w = n − 4. For much of the following we will restrict our considerations to MUV correlators of chiral operators, which have the form δ n O 2 (with n ≤ 4), in which case¯ r = 0 in (1.8). One example of such a correlator, which is particularly relevant in the following discussion, is which is the four-O 2 correlator with m insertions of O τ . Each insertion increases the modular weight w by 1, so that the total weight of the correlator is w = m. This correlator is related by superconformal symmetry to other MUV correlators with the same modular weight (or equivalently the same number of operators). For example, when m = 12 (1.9) is related to the product of sixteen fermionic operators of the form Λ( . This is the holographic dual of the sixteen-dilatino interaction in type IIB superstring theory. An important property of chiral MUV correlators is that they can be written in the form G (n−4) N (j 1 , j 2 , . . . , j n ) = I k 1 ,...,kn n (x 1 , . . . , x n ; y 1 , . . . , y n ) G and so the dependence on the operator content of the correlator (including the SU (4) quantum numbers) is contained in the pre-factor factor I k 1 ,...,kn n (x 1 , . . . , x n ; y 1 , . . . , y n ), which is fixed by the symmetries and independent of the coupling. This is the generalisation of the factor of I 4 (U, V ; Y ) in the case of the four-point function in (A.3). The remaining factor, G (n−4) N (x 1 , . . . , x n ; τ,τ ) is the "reduced correlation function" that has the same form for any MUV n-point correlator, and is the analogue of T N (U, V ) in (A.3). The explicit expression for I k 1 ,...,kn n (x 1 , . . . , x n ; y 1 , . . . , y n ) was determined in [23] and is reproduced in section 2 of [20] (where references to the original observations can be found). The fact that MUV correlators of a given modular weight are explicitly related by supersymmetry is the analogue of the property of MUV superamplitudes in type IIB string theory. There, the n-point amplitudes possess an overall prefactor of δ 16 where Q i is the sixteen-component supercharge acting on the i th particle. 6 The challenge is to determine properties of the reduced correlation function, G (n−4) N (x 1 , . . . , x n ; τ,τ ). General properties of G (n−4) N (x 1 , . . . , x n ; τ,τ ) and its large-N expansion at finite coupling τ were studied in detail in [20]. A key result is obtained by applying the SL(2, Z) covariant derivative, D w , to a correlator. This acts on the factor of e d 4 x L(x) in the definition of the expectation value (A.5), thereby inserting an integrated chiral lagrangian, d 4 x O τ (x). Care must be taken to include the contributions of the integrated contact terms arising from this insertion, which have the form , for each operator O wr (x r ) in the correlator with modular weight (w r , −w r ) (as discussed in [20,25,26]). The derivative also acts on the factor of τ 2 in the normalisation of each of the operators in the correlator. The net result is the recursion relation which expresses the content of a soft dilaton condition in the dual holographic superstring theory. Here w = n − 4 and the covariant derivative D w is defined as which acts on a modular form of weights (w,ŵ) and changes it to be a modular form with weights (w+1,ŵ−1). Thus, the application of D w to a correlator of weight (w, −w) results in the insertion of dx O τ (x), which shifts w to w + 1. Detailed properties of D w are discussed in appendix B. We are here interested in the integrated MUV correlators that generalise G N (τ,τ ) ≡ G Such insertions are obtained by applying multiple covariant derivatives D w to G N (τ,τ ). The resulting expression is a (w, −w) modular form given by 13) which is in accord with the soft dilaton properties of the holographically conjugate type IIB amplitudes [21,27], as argued in [20]. The leading terms in the large-N expansion of the MUV correlators that were studied in [20] have a holographic correspondence with the BPS protected terms in the low energy expansion of the MUV amplitudes in type IIB superstring studied in [21]. These are terms with dimension up to 14, i.e. up to the dimension of the d 6 R 4 interaction in the w = 0 sector. In the following we will generalise the lattice expression (A.8) to the expression that describes MUV correlators and determine their behaviour in various limits.

Outline of paper
In section 2 we will consider features of integrated n-point MUV correlators for general values of N , which extend the results of the n = 4 case. For example, in section 2.1 we will show that the correlator G (w) N (τ,τ ) (w = n − 4) defined in (1.13) satisfies a Laplace-difference equation, which follows directly from the equation satisfied by the four-point correlator G N (τ,τ ). We will demonstrate in section 2.2 that a weight-w MUV correlator can be expressed as a two dimensional lattice sum, extending the analysis of the w = 0 case given in [1,2]. This lattice sum can also be expressed as an infinite sum of Eisenstein modular forms (which are defined and summarised in appendix B). The structure of the perturbative expansion of G (w) N (τ,τ ) in powers of λ = g 2 Y M N is determined to any desired order for any value of N . As in the w = 0 case the perturbation series contains non-planar contributions, which start at (4 − w)-loop order when w < 4 and at free theory if w ≥ 4.
Instanton and anti-instanton contributions are extracted from the exact expression for the correlator in section 2.3. Unlike in the w = 0 case, when w > 0 the systematics of the perturbation expansion around an instanton is different from that around an anti-instanton. This will be seen to be in accord with semi-classical arguments concerning the fermionic zero modes contained in the profile of the operators in the correlator in an instanton or anti-instanton background.
The large-N expansion of G (w) N (τ,τ ) is discussed in section 3 where we will determine both the fixed λ and fixed g 2 Y M expansions and demonstrate the similarities and differences from the w = 0 case. At small λ we find a convergent perturbative expansion for |λ| < π 2 , while for λ 1 perturbation theory produces an asymptotic, factorially growing, divergent series. This strong coupling series is not Borel summable and its non-perturbative completion, which behaves as O(λ w/2 e −2 √ λ ), is determined using resurgence techniques.
In section 4 we will briefly discuss the insertion of d 4 x O τ (x) in the non-integrated correlator O 2 (x 1 , Y 1 ) · · · O 2 (x 4 , Y 4 ) , and its application for constructing perturbative loop integrands [23,28]. We will argue that it is important to use the covariant derivatives (rather than the ordinary derivatives with respect to g Y M ) in this procedure of determining perturbative loop integrands of the 4-point correlator.
We end with a conclusion and discuss some future directions in section 5.

Exact properties of MUV correlators
We will now consider properties of the MUV correlators that are obtained from (1.13) using the exact expression for G

The Laplace-difference equation
It is straightforward to determine the Laplace equation satisfied by G (w) N (τ,τ ) given the Laplace difference equation for the w = 0 case in (A. 16), which was derived in [1,2] and we rewrite here for convenience: As described in appendix B, the hyperbolic Laplacian ∆ τ acting on G N (τ,τ ) can be identified with the SL(2) Casimir operator Ω (0,0) , defined in (B.7), when restricted to the space of modular functions, i.e. modular forms M (w,ŵ) with holomorphic and anti-holomorphic weights (w,ŵ) = (0, 0) .
From equation (1.13), we know that G (w) N is obtained by repeated applications of the covariant derivative to G (0) N . Furthermore, the covariant derivative changes the modular weights according to D w : M (w,ŵ) → M (w+1,ŵ−1) , and since the Casimir operator Ω commutes with D w , using (B.7) it follows that where Ω w,−w denotes the restriction of the Casimir operator to the vector space of modular forms M (w,−w) with weights (w, −w). The second line follows from the Laplace-difference equation (2.1) satisfied by G N (τ,τ ), and the fact that Ω 0,0 = ∆ τ . Given the explicit forms of Ω w,−w in (B.10) and (B.11) (2.2) can be expressed in either of two ways: Just as in the w = 0 case described in [1,2], this equation determines G (w) It is easy to see that in the perturbative sector, where there is no dependence on τ 1 , the operator on the left-hand side of (2.3) reduces to which is identical to the differential operator of the w = 0 case. The same is true for the operator on the left-hand side of (2.4). In other words, the perturbative part of the Laplace-difference equation is not sensitive to the value of w. This does not mean that the perturbative part of G . In the following section, we will explicitly discuss the perturbative expansion of G

Yang-Mills perturbation theory
The expression for G (w) N (τ,τ ) can be obtained by substituting the expression for G N (τ,τ ) in terms of non-holomorphic Eisenstein series (A.14) into (1.13), giving where δ w,0 denotes the Kronecker delta and ∇ = 2iτ 2 2 ∂ τ is the Cauchy-Riemann derivative discussed in appendix B. Using the relation τ −w is the Eisenstein modular forms that is discussed in appendix B (and defined by (B.25)), one can further express the integrated correlator as Now, using the Lattice sum expression of E (w) (s; τ,τ ) given in (B.30) together with (A.15), we find that the weight-w integrated correlator can be further expressed as This lattice sum representation is a well-defined analytic modular form for all values of τ with τ 2 > 0. If the rational function B N (t) is expanded around the origin, as in (A.15), and if we restrict our attention to a single monomial term of the form t s−1 we obtain the same integrand as that of E (w) (s; τ,τ ) in (B.30). In other words, (2.8) can formally be expanded as an infinite sum of E (w) (s; τ,τ ) modular forms with rational coefficients.

The relationship between weak and strong coupling
The perturbative terms in the small-g 2 Y M limit can be extracted by proceeding as in the w = 0 case considered in [1,2]. Recall that the perturbative terms come from the zero mode of non-holomorphic Eisenstein series as in (A.14), which is given by the sum of two pieces shown in (A.18). The sum of the τ 1−s N,0 (τ 2 ) and both parts of the zero mode sum contribute equally.
We will now see how this extends to MUV correlators starting from the expression (2.8) and following closely the procedure used to analyse the modes of E (w) (s; τ,τ ) in appendix B. When w > 0 the term (m, n) = (0, 0) is absent since it is killed by the α derivative in (2.8). The Fourier expansion of (2.8) is again obtained by performing a Poisson resummation in m (and later changing n → −n), resulting in where the integers k =mn labelling these modes are interpreted as instanton numbers. As in the analysis of the w = 0 case the perturbative terms (the k = 0 terms) arise from two classes of terms: (i) the terms with n = 0 with a sum over allm; (ii) the terms withm = 0 with a sum over all n.
The n = 0 case: In this case we can rewrite the contribution as (2.10) In this case we can rewrite the contribution as Although it appears that the (m, n) = (0, 0) term has been double counted, it is fairly simple to show that this actually vanishes thanks to (A.12).
If we redefine the variable α in (2.11) by setting α = −tα so that d/dα = −t −1 d/dα, and then change variable from t to 1/t, (2.11) becomes , which extends the result previously found when w = 0.

Some features of the Yang-Mills perturbation expansion
Making use of (2.6) and the identity in (B.15) as well as the definition of the holomorphic Eisenstein series G k (τ ) in (B.16) it is easy to see that for all s ≤ w Consequently the τ 1−s 2 term in the zero mode of the Eisenstein series with s ≤ w does not contribute to the perturbative expansion. Therefore the first contribution comes from (2.14) Therefore we conclude that the perturbation expansion of G In the w = 0 case (the four-point correlator) the leading term is of order τ 0 2 , which is the free-field contribution that arose in (A.3). However, following [3] this term cancels out of the supersymmetric localization calculation and the interacting part, which is described by T N (U, V ), begins with the one-loop contribution of order τ −1 2 . If, however, we were to explicitly integrate the free-field contribution in (A.3) (appropriately normalised) with the measure (A.6), this would produce a divergent τ 0 2 coefficient, which can be interpreted as a rational multiple of ζ(1). This is formally consistent with uniform trascendentality as we will shortly see in (2.15).
On the other hand, for MUV correlators with w > 0, we need to include the free-field contributions, which are of order τ −w 2 . Indeed, as will be explained in section 4, the free part of a n-point MUV correlator can also be interpreted as the (n − 4)-loop correction to the four-point correlator, which provides an efficient method for constructing perturbative loop integrands [23].
Using (2.10) and/or (2.11) it is straightforward to determine the perturbative expansion of G N (τ,τ ) to any order and for any value of N . The following expressions for the perturbative expansion of correlators in the SU (2) theory with different weights (including w = 0, 2, 4) illustrate the general structure, It is of interest to exhibit the N -dependence of the SU (N ) Yang-Mills perturbation expansion for generic N , which takes the form where a = g 2 Y M N/(4π 2 ) = N/(πτ 2 ) and arbitrary N ≥ 2. Since the Pochhammer symbol (−n) w , with n ∈ N, vanishes when n < w perturbation theory starts at order a w ∼ τ −w 2 for any N . As anticipated earlier for w = 0 the a 0 term, which would correspond to a divergent ζ(1) free-field theory contribution, does not appear. So we see that although in the case of the w = 0 correlator, non-planar terms enter the perturbative expansion at four loops, when w > 0 non-planar corrections start earlier. For example, for w = 2 the first non-planar correction enters at three loops, When w ≥ 4 the first non-planar correction enters at leading order (tree-level). 7 For example, for w = 4:

Instanton and anti-instanton contributions
We will now study the non-zero Fourier modes of G (w) N (τ,τ ), by using its expression in terms of non-holomorphic Eisenstein modular forms given in (2.6). The k-instanton contribution to a single non-holomorphic Eisenstein modular form E (w) (s; τ,τ ) (the k th positive Fourier mode with k > 0), behaves as e 2πikτ τ w 2 as τ 2 → ∞, while the k anti-instanton contribution (the k th Fourier mode with k < 0) vanishes for s ≤ w and behaves as e 2πikτ τ −w 2 when s > w. We will see that the MUV integrated correlator (2.6) has the same general properties.
It follows from (A.15) that the k th Fourier mode for the weight-w integrated correlator can be expressed as where k =mn and again if we expand the rational function B N (t) around the origin we find that the integrand can be written as an infinite sum of integrands for E (w) (s; τ,τ ) in (B.34). 7 Note the leading order term arises from Wick contractions of the free theory.
Given the definitions of B N (t) in (A.9)-(A.10), this integral can be explicitly evaluated for fixed mode number k and fixed number of colours N . Alternatively, these non-zero modes of the weight-w integrated correlator can be determined by applying the Cauchy-Riemann derivative (B.17) to the w = 0 integrated correlator, G N (τ,τ ).
To illustrate the generic features of the instanton terms we will now present some simple explicit examples. These are the k = ±1 (charge-one instanton and charge-minus one antiinstanton) sectors of the weight-2 and weight-4 correlators in the SU (2) theory. In each case we will present the exact expression together with the first few terms in its perturbative expansion around the y = πτ 2 → ∞ limit, G (2.23) The general structure of these contributions is in accord with expectations from the analysis of semi-classical instanton contributions to MUV correlators in special cases treated in the literature, see, for example, [29][30][31]. These references were all restricted to the holographically related leading low-energy expansion of superstring amplitude, or to leading orders in the 1/N expansion of N = 4 SYM correlators, and only considered the semi-classical approximation. Our present results go far beyond the semi-classical approximation and apply to any value of N ≥ 2, but nevertheless some general features are explained by the leading order calculations.
For example, the fact that the leading power of g 2 Y M ∼ τ −1 2 in the instanton background is of order τ w 2 is a direct reflection of the presence of 16 superconformal zero modes. The counting of powers of τ 2 to leading order in 1/τ 2 is as follows. The instanton profile of each operator insertion involves the product of (2∆ − 4w) fermionic zero modes (where ∆ is the dimension of the operator), each contributing a power τ −1/4 2 , in addition to the power of τ 2 in the normalisation of each operator. The leading order instanton contribution to the n-point correlator necessarily absorbs all 16 superconformal fermion zero modes and is therefore of order τ n−16×1/4 2 = τ w 2 as τ 2 → ∞, as in G 2,1 and G 2,1 exhibited above. More explicitly, the instanton profile of the operator O 2 (x) (∆ = 2, w = 0) has four fermionic zero modes, while O τ (x) (∆ = 4, w = 2) has no fermionic zero modes, and so G (w) N,k (τ,τ ) behaves as The contributions to the profiles of operators in an anti-instanton background acquire more powers of τ −1 2 from two distinct sources.
(i) Firstly, they involve more fermionic modes, which are quasi-zero modes -these are classical zero modes in the ADHM construction, that arise with non-zero coefficients in the moduli space action when interactions are taken into account (see, for example, [30]). The contributions of such modes to various correlators is discussed in [31].
(ii) Secondly (as is also discussed in detail in [31]), there are perturbative corrections to the anti-instanton contribution that arise by Wick contractions of fields inside the operators in the correlators. Such contractions appear as propagators joining operators, where the propagator in an instanton background is rather complicated [31], but has the same power of τ −1 2 as the propagator in a trivial background.
Instead of considering the correlator (2.24) in a k-anti-instanton background with k > 0 we may consider the complex conjugate correlator in a k-instanton background with k > 0, Since the operatorŌτ is related to O τ by the action of eight supercharges, its profile contains the product of eight fermionic zero modes, which may be a mixture of true zero modes and quasi-zero modes. In general the evaluation of the semi-classical contribution to the correlator involves the sum of both types of contributions, (i) and (ii), described above, together with the contribution of the 16 true superconformal zero modes. For illustrative purposes we can consider the contribution in which all the fundamental fields inŌτ are contracted by propagators in the instanton background and the 16 true fermionic zero modes are soaked up by the four O 2 operators. In this contribution the counting of the powers of τ 2 has the form where the factor τ −2w 2 arises from 2w propagators contracting the fields inŌ τ 's 8 . The factor of τ w 2 arises from the normalisation factorŌτ ∼ τ 2 . This counting extends to all possible terms involving both integration over quasi-zero modes in the profiles of the operators as in item (i) above, and propagator contractions as in item (ii) above. All terms contribute the same net power of τ −w 2 . 9 By taking the complex conjugate of equation (2.26) we then deduce that the correlator (2.24) in a k-anti-instanton background, k > 0, behaves as e −2πikτ τ −w 2 . We have thus seen that the leading behaviour of G (w) N,−k with k > 0 at large τ 2 is of order τ −w 2 , which in the cases G 2,−1 and G 2,−1 is in accord with the τ 2 -dependence in (2.21) and (2.23).

Large-N expansion
We will now study large-N expansion of the integrated correlators. We begin by considering the standard 't Hooft limit, in which λ = g 2 Y M N is fixed and Yang-Mills instantons are suppressed by factors of e −cN/λ for some finite value of c. In this limit G (w) N (τ,τ ) is expressed as a power series in 1/N , which has the standard interpretation as a genus expansion of the form, where g denotes the genus. The expressions for G (w,g) (λ) and their small-λ and large-λ expansions are discussed in sections 3.1.1 and 3.1.2 for the cases with g = 0, 1.
In order to exhibit the SL(2, Z) covariance of the correlator it is necessary to consider the large-N limit with fixed g 2 Y M (sometimes called the "very strong coupling limit"), in which Yang-Mills instantons play an essential rôle. This will be the subject of section 3.2.

Large-N and fixed-λ
Applying the relation (1.13) to the perturbative contributions, while ignoring the instanton contribution (which is equivalent to ignoring the τ 1 dependence), the covariant derivative reduces to leading to the relation, Transforming from τ 2 to the 't Hooft coupling λ = 4πN/τ 2 and substituting in (3.1) implies, We may now consider the series expansions of G (w,g) (λ) at small λ or large λ. Note that if we apply the differential operator (3.4) to a general function F (λ) that has a small-λ expansion of the form F (λ) = n a n λ n the result is a new Taylor series with coefficients given by The quantity Γ(−n+w)/Γ(−n) is the Pochhammer symbol (−n) w that vanishes when n < w, while for n ≥ w it can be replaced by the regular expression (−1) w (n + 1 − w) w . Similarly, the action of the differential operator (3.4) on the large-λ expansion, which has the form F (λ) = n b n λ −n−1/2 , gives rise to an expansion of the form So the coefficient for any value of g is determined in terms of the corresponding coefficient in the w = 0 case. The factor Γ(n + 1/2 + w)/Γ(n + 1/2) = (n + 1/2) w is non-vanishing so, in contrast to the small-λ expansion, the coefficients in the large-λ expansion do not automatically vanish for any value of n.

Small-λ expansion and resummation
We will now apply the above general discussion to concrete examples to obtain explicit results for G (w,g) (λ). To illustrate the structure of these expressions, in the following we will present the results for the first two genera, g = 0 and g = 1. Let us consider the w = 0 case discussed in [2] and define G (g) (λ) := G (0,g) (λ). At leading order in the large-N expansion (i.e. g = 0), we have which converges for |λ| < π 2 , and can be resummed leading to Applying the relation (3.5) to (3.7), the resulting series is again convergent for |λ| < π 2 , and after again performing the resummation, the result is where 1F2 is the regularised hypergeometric function, defined by with 1 F 2 the usual generalised hypergeometric function. It is easy to see that G (w,0) (λ) w=0 = G (0) (λ). The g = 1 contribution to the w = 0 correlator considered in [2] takes the form which converges for |λ| < π 2 and can be resummed to . (3.12) For non-zero w, we find that this result generalises to which reduces to (3.12) in the w = 0 limit, noticing that 1F2 Higher-genus terms can be obtained in a similar fashion, and the results have analogous structures to those of G (w,0) (λ) and G (w,1) (λ).

Large-λ expansion and resurgence
In this section, we will consider properties of the integrated MUV correlators in the large-λ limit. Using (3.8) we can straightforwardly obtain the series expansion, for G (w,0) (λ) and G (w,1) (λ) from the known results in the w = 0 case [2]. Equivalently, one may perform the large-λ expansion directly using the integral expressions given in (3.9) and (3.13) and the Mellin-Barnes representations of hypergeometric functions. Either way, we find the factorially growing expansion for the g = 0 coefficient in (3.1), ,(3.14) and similarly, for the g = 1 term We see that the large-λ expansion is asymptotic and not Borel summable for any value of w and at each order in 1/N . We will now see that a resurgence analysis, following closely the techniques of [2,32], leads to the non-perturbative completion of these asymptotic series. As in [2] this involves understanding the singularities of the asymptotic series after Borel summation. One may also obtain the same result by acting on the w = 0 expression of [2], with appropriate differential operators as discussed earlier.
We will now consider the convergence properties of the large-λ expansion that defines G (w,0) (λ) in (3.14). We start by defining a modified Borel transformation [32] B : which, when applied to the asymptotic series (3.14), produces the modified Borel transform Using the key identity 2 2s−2 Γ(2s) we can then provide an analytic continuation of the formal expansion (3.14) in terms of the directional Borel resummation which defines an analytic function for √ λ > 0 when θ ∈ (−π/2, π/2). Although (3.19) provides an analytic continuation for G (w,0) (λ) it is neither unique nor is it real for √ λ positive for any value of integration direction θ. This is due to the presence of the branchcut in the Borel transformφ (w,0) (x) along [1, ∞]. As anticipated, G (w,0) (λ) is non-Borel summable and standard resurgence arguments suggest that we are missing exponentially small non-perturbative terms. These terms are encoded in the discontinuity of the Borel transform and can be determined in terms of the so-called Stokes automorphism, which gives The discontinuity of the Borel transform can easily be computed for generic w using the known discontinuity for the hypergeometric function arriving at (3.21) with 2F1 again denoting a regularised hypergeometric function. Following the discussion in [2], we can evaluate (3.20) using (3.21), which results in the non-perturbative completion ∆G (w,0) (λ). Alternatively, we can apply (3.4) to the non-pertubative completion ∆G (0) (λ) derived in [2]. Either method results in It is easy to see that ∆G (w,0) (λ) behaves as O(λ w/2 e −2 √ λ ). For instance, for w = 1, 2, the non-perturbative completions take the following forms, Similarly, for the g = 1 coefficient in the large-N expansion, the non-perturbative term is given by (3.24) which again behaves as O(λ w/2 e −2 √ λ ). Let us also take w = 1, 2 as examples,

(3.25)
Using an argument that closely follows appendix D of [2], it is easy to prove that the median resummation gives a real expression when √ λ > 0, and the analytic continuation is unambiguous and coincides with the small-λ analytic continuation in (3.9)- (3.13). This demonstrates the importance of the non-perturbative completion ∆G (w,g) (λ).
As in [2], making use of the AdS/CFT dictionary we can translate these non-perturbative terms into string language, where they should arise from world-sheet instantons. Presumably these would come from a string world-sheet pinned to the n operators in the correlator on the AdS 5 × S 5 boundary and stretching into the interior. However, such a semi-classical picture of these configurations is presently missing. It is also worth mentioning that similar exponentially suppressed terms have been found [33] in the large-λ expansion of the cusp anomalous dimension in N = 4 SYM. The strong coupling expansion of this physical quantity, requires a non-perturbative completion with similar, but slightly different, exponentially suppressed terms of order λ 1/4 e − √ λ/2 . The cusp anomaly resurgence structure is considerably more complicated than in the present case and was discussed in [34,35]. Finally, we notice that the "anomalous dimension" associated with the six-point MHV amplitude in N = 4 SYM studied in [36], behaves as e − √ λ . It would be of interest to understand the semi-classical origin of the interesting similarities and differences of all these exponentially suppressed terms.

Large-N with fixed-g 2 Y M
To study the non-perturbative instanton effects, which are vital for understanding the SL(2, Z) symmetry, we will consider the large-N limit with g 2 Y M fixed. In this "very strong limit", when w = 0, as worked in [6] and furthered extended in [2], the integrated correlator is expanded in terms of non-holomorphic Eisenstein series with half-integer indices, A few comments are in order. Firstly, the leading large-N term (i.e. the N 2 term) in G (w) N (τ,τ ) now disappears due to the action of derivatives. This is consistent with the fact that the N 2 term is associated with the supergravity amplitudes, which cannot violate the U (1) Y bonus symmetry (or correspondingly the U (1) R-symmetry of type IIB supergravity). Secondly, the results are in accord with the α -expansion of the MUV superamplitudes in type IIB superstring [20,21]. In particular, the non-holomorphic modular Eisenstein series E (w) ( 3 2 ; τ,τ ) and E (w) ( 5 2 ; τ,τ ) are associated with the higher-derivative terms R 4 Z w and d 4 R 4 Z w (here Z is the dilaton), respectively. Finally, the result may also be obtained directly from the lattice-sum representation (2.8). This is done by expanding the integrand, especially B N (t), order by order in 1/N , as in [2].

Perturbative loop integrands
Apart from the interpretation of (1.13) in terms of SL(2, Z)-covariant MUV correlators, this equation also leads to a construction of the w-loop contribution to the four-point correlator in a manner reminiscent of [23]. To demonstrate this we will consider w insertions of Y 4 ) . This defines a partially integrated (4 + w)-point MUV correlator, which has a form given by (1.13), but without the integration over x 1 , . . . , x 4 . The perturbative expansion may be extracted by ignoring the dependence on τ 1 . In other words, by replacing the covariant derivative D w (defined in (B.2)) by 1 2 (τ 2 ∂ τ 2 + w), as given in (3.2). With this change the partially integrated correlator based on (1.13) reduces to, This equation can be used to determine the perturbative contributions to any MUV correlator, starting from the lowest order contribution to the four-point correlator. This follows by considering the perturbative expansion of where L denotes the number of loops in the perturbative expansion, and so the L-loop contribution to the correlator is written as As discussed in section 2.2 the product of SL(2, Z) covariant derivatives annihilates the perturbative terms up to order τ −w 2 . Indeed, substituting (4.2) into (4.1), leads to It is easy to check that the lowest-order contribution to the correlator on the right-hand side of this equation is also of order τ −w 2 , as follows from Wick contractions of the free theory and the fact that each O τ (x) is proportional to τ −1 2 . More generally, at L loops in perturbation theory, the correlator on the right-hand side behaves as τ −w−L 2 . This is obviously consistent with the left-hand side of the equation.
We can then redefine L = + w, with ≥ 0, and by matching the powers of τ − −w 2 on both sides of (4.3) we obtain The = 0 case is of particular interest, in that case the right-hand side of this equation is a partially integrated free MUV correlator. The integrand is a rational function of x 2 ij with known analytical properties. It is then identified with the integrand of the w-loop contribution to the four-point correlator O 2 (x 1 , Y 1 ) · · · O 2 (x 4 , Y 4 ) . The = 0 combinatorial factor 1/Γ(w + 1) accounts for the symmetry factor of the loop integrand. By further taking out the overall factor I w+4 as in (1.10), the integrand in fact enjoys a S w+4 permutation symmetry. All these facts lead to an efficient construction of the correlator to high orders using graph theory [23]. The idea has been utilised for constructing the loop integrand to a high number of loops [23,28,[37][38][39] in the planar limit and beyond. An important point is that it is crucial to use the covariant derivative in (4.1), rather than the ordinary derivative τ 2 ∂ τ 2 , which was the suggested prescription made in [23]. The problem is that a product of ordinary derivatives does not annihilate any low-order terms in the 1/τ 2 expansion and therefore (τ 2 ∂ τ 2 ) w O 2 (x 1 ) · · · O 2 (x 4 ) cannot be identified with the left-hand side of (4.1), whose lowest order is O(τ −w 2 ) as was discussed earlier. In fact, operating with τ 2 ∂ τ 2 = 1 2 [(τ 2 ∂ τ 2 + w) + (τ 2 ∂ τ 2 − w)] on the correlator inserts the sum of d 4 x O τ (x) and d 4 x Oτ (x). Therefore (τ 2 ∂ τ 2 ) w O 2 (x 1 , Y 1 ) · · · O 2 (x 4 , Y 4 ) is related to a linear combination of correlators with insertions of O τ (x) and Oτ (x).

Conclusion and Discussion
In this paper, we have extended the results of [1,2], which concerned properties of an integrated four-point correlator in N = 4 SU (N ) SYM, to general n-point integrated MUV correlators. In the earlier papers, which were based on the application of supersymmetric localisation techniques [4] to the integrated four-point function described in [3], the integrated correlator G N (τ,τ ) was recast as a two-dimensional lattice sum, which made its modular properties manifest and from which it was simple to analyse its dependence on N and the coupling constant, τ . Similarly, the n-point MUV integrated correlators, G (n−4) N (τ,τ ) are non-holomorphic modular forms of weight (n − 4, 4 − n), which can again be expressed as two-dimensional lattice sums. They satisfy a Laplace difference equation that is a simple generalisation of the equation satisfied in the n = 4 case discussed in [1,2]. The dependence of G (n−4) N (τ,τ ) on N and τ is straightforward to analyse and we presented expansions at weak and strong coupling in the finite-N and large-N limits. Various systematic features of the dependence of these expansions on the modular weight w were demonstrated in sections 2 and 3.
The results of section 3 reproduce terms in the 1/N expansion in the large-N , fixed g 2 Y M limit of MUV integrated correlators that were considered in [21], and which have a close holographic connection to the leading terms in the low energy expansion of the MUV amplitudes in AdS 5 × S 5 in accord with leading terms in the expansion of the flat-space type IIB superstring amplitudes discussed in [20]. Integration over the operator positions is crucial for ensuring supersymmetry, which is an essential feature of the localisation arguments. In this paper, as in the w = 0 case studied in [1,2], our focus was on the integrated correlator defined with the measure in (A.6). A second measure (A.7) that was introduced in [10] can be used to define a different class of integrated n-point correlators of the form The first few leading terms in the large-N expansion of these correlators, both at fixed λ = g 2 Y M N and at fixed g 2 Y M , were determined in [11]. It remains a challenge to formulate such correlators as lattice sums, which would lead to a more thorough elucidation of their properties.
It would also be of interest to apply these ideas to study exact properties of two-point functions of BPS operators in N = 2 supersymmetric theories 10 at finite coupling and finite N , making use of the methods of supersymmetric localisation as, for example, in [44][45][46][47][48][49][50][51][52][53][54]. These two-point correlators are functions of complexified couplings τ andτ , which also transform properly under the modular transformation. Most of the study has been focused on perturbative expansions.
Finally, we would like to emphasise the arguments in section 4, which pointed important differences between partially integrated MUV correlators and the expressions obtained from the Lagrangian insertion method of determining the -loop integrand of the four-point correlator of O 2 's as formulated in [23]. We find that the L = (n − 4)-loop correction to O 2 (x 1 ) · · · O 2 (x 4 ) follows by inserting (n − 4) factors of dx i O τ (x i ), which corresponds to applying (n − 4) covariant derivatives to the four-point correlator, (as in (4.3)). By contrast, the procedure advocated in [23] is to apply (n − 4) ordinary derivatives of the form g 2 Y M ∂/∂g 2 Y M , which do not contain inhomogeneous terms. This results in the insertion of (n − 4) factors of d 4 x(O τ (x) +Ōτ (x)), which is not of the form (4.4). However, since in [23] the L-loop correlator was assumed to be of the form (4.4), their expressions are correct.
factor I 4 (U, V ; Y i ) encodes the dependence on the R-symmetry quantum numbers and it is independent of τ and N (see [55,56]). The factor T N (U, V ) is the nontrivial part of the correlator. It is independent of the R-symmetry and is the main consideration in the following. In these expressions the cross-ratios U and V are defined in the standard manner by where U = 1 + r 2 − 2r cos(θ) and V = r 2 . As discussed in [3] this expression arises by considering (1.1), when the R-symmetry charges of the four operators are chosen in a manner that sets I 4 (U, V ; Y i ) = V . The second example of an integrated correlator of the product of four O 2 (x, Y )'s that preserves supersymmetry was presented in [10] where it was shown to arise from instead of (1.1). The functionD 1111 (U, V ) is the so-called D-function appears in the context of AdS/CFT duality. This corresponds to a choice of measure that was described in [10]. 11 However, in this paper we will only consider properties of correlators that are based on the first integrated correlator, G N (τ,τ ), defined by (1.1) or, equivalently, by (A.6). The partition function of N = 2 * SYM on S 4 was determined in [4] in terms of SU (N ) gaussian matrix model integrals over the elements of the Lie algebra su(N ), which reduce to (N − 1)-dimensional integrals over eigenvalues of SU (N ) matrices. The N -dependence is therefore encoded in the dimensionality of the integrals, which obscures the analysis of G N (τ,τ ) for general values of N and τ . However, the considerations in [3,10] emphasised the large-N expansion. This led to interesting patterns in the properties of the expression for integrated correlator in the large-N expansion at fixed 't Hooft coupling, where instanton contributions are suppressed exponentially in N . In the large-N limit with fixed g Y M considered in [6,11], Yang-Mills instantons are an important element in ensuring the SL(2, Z) duality of the correlator.

The lattice sum description of the integrated correlator
The N -dependence of G N (τ,τ ) was made explicit for all values of N by the reformulation of the integrated correlator in terms of a lattice sum as suggested in [1,2] and where Q N (t) is a polynomial of degree 2N − 1 that takes the form and P (α,β) N (z) is a Jacobi polynomial. It is significant that the function B N (t) satisfies the inversion condition as well as the integration conditions The function G N (τ,τ ) defined in equation (A.8) is manifestly invariant under the SL(2, Z) transformations which is in accord with the expectations of Montonen-Olive duality [7][8][9]. In fact, as shown in [2], the expression (A.8) can be re-expressed as a formal infinite sum of non-holomorphic Eisenstein series with integer indices, where the coefficients c The definition and properties of non-holomorphic Eisenstein series are reviewed in appendix B.
It was also shown in [1,2] that the integrated correlator satisfies the Laplace-difference equation which connects the integrated correlators for SU (N ) gauge group with those of SU (N + 1) and SU (N − 1). One consequence is that the expressions for G N (τ,τ ) with N > 2 can be determined iteratively in terms of G 2 (τ,τ ). As will be reviewed in appendix B, each non-holomorphic Eisenstein series E(s; τ,τ ) contains two perturbative zero mode terms, proportional to τ 1−s available up to three loops [57]. However another feature of exact results in [1,2] is that they demonstrate that non-planar contributions first enter at four loops, which is also a known feature of N = 4 SYM [28,39]. It also predicts the pattern of non-planar contributions at higher loops.
The large-N expansion of the correlator was also considered in the 't Hooft limit in which λ = g 2 Y M N is fixed and instantons are suppressed. The results confirm and extend the results in [3]. In particular they confirm the results of summing the expansion in powers of 1/λ at large values of λ. However, in [1,2] it was found that this expansion is not Borel summable and requires a resurgent completion that is of order e −2 √ λ . This may be interpreted in the holographic string theory dual as the effects of world-sheet instantons.
The results obtained in [3] concerning the large-N expansion with fixed g 2 Y M in which instanton contributions play a vital rôle, were also extended in [1,2]. In fact the Laplacedifference equation (A.16) determines all the terms of higher order in 1/N in the large-N expansion once the first two lowest order terms are given.

B Non-holomorphic Eisenstein modular forms
In order to discuss properties of maximal U (1)-violating correlators we will here review some features of the particular class of modular forms that arise in this context, which are extensions of the standard non-holomorphic Eisenstein series, E(s, τ,τ ).

Modular covariant derivatives
The vector space of modular forms, M (w,ŵ) , with holomorphic and anti-holomorphic weights (w,w) is defined by for all γ = ( a b c d ) ∈ SL(2, Z). Covariant derivatives can be defined to act on this space by changing the modular weights in the following manner, More explicitly, restricting to the space M (w,ŵ) , We will be interested in theŵ = −w case for which the SL will play the rôle of Laplacians. In the w = 0 case these reduce to the standard Laplacian on the hyperbolic plane, Another useful derivative is the Cauchy-Riemann derivative ∇ = 2iτ 2 2 ∂ τ . This acts by changing the modular weights in the following manner, where both sides are (n, −n) modular forms. In particular, when f is a non-holomorphic Eisenstein series we have (for n ∈ Z) D n−1 D n−2 · · · D 0 E(n, τ,τ ) = 1 (2τ 2 ) n ∇ n E(n, τ,τ ) = Γ(2n) Γ(n) with G 2n the holomorphic Eisenstein series, which is defined (when n ≥ 2) by where q = e 2πiτ . More generally, in order to analyse the modes of E(s; τ,τ ) we will make use of the relation with y = πτ 2 .

Laplace eigenvalue equations
Non-holomorphic Eisenstein series are solutions of the equation where the hyperbolic Laplacian is defined by ∆ τ = 4τ 2 2 (∂ τ ∂τ ). The function E(s; τ,τ ) is a SL(2, Z) modular function that satisfies the asymptotic moderate growth condition lim τ 2 →∞ E(s; τ,τ ) < τ a 2 , where a is a real number. The solution has the form 12 where the zero Fourier mode consists of the sum of two power behaved terms, and the non-zero modes are D-instanton contributions, which are proportional to K-Bessel functions, where the divisor sum is defined by and we sum over the positive divisors d of k. We are generally interested in correlators of n = 4 + w operators in the stress tensor supermultiplet, which are proportional to nonholomorphic Eisenstein modular forms of weight (w, −w) that are defined by where E (0) (s; τ,τ ) := E(s; τ,τ ) and (s) w = Γ(s + w)/Γ(s) is the Pochhammer symbol. It is straightforward to show that this implies This expression has arisen previously in the context of the low energy expansion of superstring amplitudes [29] and the large-N expansion of MUV correlators [21]. These modular forms satisfy the Laplace equations 27) or, equivalently, We also note that making use of (B.14) and (B.15) when w ≥ s E (w) (s; τ,τ ) can be expressed as E (w) (s; τ,τ ) = 2 w Γ(s) Γ(s + w) 1 (2τ 2 ) w ∇ w E(s; τ,τ ) = Γ(2s) π s Γ(s + w) τ −w 2 ∇ w−s τ 2s 2 G 2s (τ ) , (B.29) where G 2s (τ ) is a holomorphic Eisenstein series defined in (B.16).
The following integral representation for E (w) (s; τ,τ ) is used in the main text [29], 2 w D w−1 D w−2 · · · D 0 E(s; τ,τ ) = (s) w E (w) (s; τ,τ ) = (m,n) =(0,0) (s) w m + nτ (i) n = 0. This gives a contribution to the coefficient of the τ s 2 term in the zero mode, which will be denoted F (w)(i) 0 (s, τ 2 ). When w = 0 the (m, n) = (0, 0) term vanishes and it is useful to Poisson resum the m variable, giving This sector gives a contribution to the sum over instantons with charges k =mn. This includes them = 0 terms that contribute to the coefficient of the τ 1−s 2 term in the zero mode, which will be denoted F (w)(i) 0 (s, τ 2 ). This has the form In this case the integral is somewhat more complicated. After the change of variables α → α/(−in √ πτ 2 ) we have (s, τ 2 ) is absent when w ≥ s.

Non-zero Fourier modes of Eisenstein modular forms
We will now consider the non-zero modes in the Fourier expansion (B.31). In the w = 0 case the correlator is a real function and the mode expansion is an expansion in a series of cos(2πkτ 1 ) functions, so instantons and anti-instantons contribute equally. The instanton contributions are contained in them = 0 terms in the sum in (B.34). We have to distinguish the contributions of instantons with k =mn > 0, and anti-instantons with k =mn < 0. Performing the integral and making a trivial change of variable α →α = in √ πτ 2 α we arrive at F We note, in particular, that both for k > 0 and k < 0 the Bessel function will produce the expected exponentially suppressed factor e −2π|k|τ 2 which will combine with e 2πikτ 1 to produce q k for k > 0 andq |k| for k < 0. Secondly it is once again possible to show that theq contribution vanishes identically for all s ≤ w, while in the case s = w the q andq contributions simplify dramatically to To illustrate this structure the following is a list of a few terms with small values of s and w where as usual y = πτ 2 : These examples illustrate the fact that when w ≥ s the anti-instanton contributions are absent, and when w = s the result reduces to (B.41) (using the fact that k 2s−1 σ 1−2s (k) = σ 2s−1 (k)). Furthermore, the leading 5instanton contribution is of order τ w 2 as τ 2 → ∞, while the leading anti-instanton contribution is of order τ −w 2 .