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 su
perconformal primaries and is based on supersymmetric localisation. The integrated correlators are functions of $N$ and $tau=theta/(2pi)+4pi i/g_{_{YM}}^2$, 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 $q=e^{2pi i tau}$ 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 relation of $n$-point MUV correlators to $(n-4)$-loop contributions to the four-point correlator. In particular, we argue that it is important to ensure the $SL(2, mathbb{Z})$-covariance even in the construction of perturbative loop integrands.
We present a novel expression for an integrated correlation function of four superconformal primaries in $SU(N)$ $mathcal{N}=4$ SYM. This integrated correlator, which is based on supersymmetric localisation, has been the subject of several recent dev
elopments. The correlator is re-expressed as a sum over a two dimensional lattice that is valid for all $N$ and all values of the complex Yang-Mills coupling $tau$. In this form it is manifestly invariant under $SL(2,mathbb{Z})$ Montonen-Olive duality. Furthermore, it satisfies a remarkable Laplace-difference equation that relates the $SU(N)$ to the $SU(N+1)$ and $SU(N-1)$ correlators. For any fixed value of $N$ the correlator is an infinite series of non-holomorphic Eisenstein series, $E(s;tau,bartau)$ with $sin mathbb{Z}$, and rational coefficients. The perturbative expansion of the integrated correlator is asymptotic and the $n$-loop coefficient is a rational multiple of $zeta(2n+1)$. The $n=1$ and $n=2$ terms agree precisely with results determined directly by integrating the expressions in one- and two-loop perturbative SYM. Likewise, the charge-$k$ instanton contributions have an asymptotic, but Borel summable, series of perturbative corrections. The large-$N$ expansion of the correlator with fixed $tau$ is a series in powers of $N^{1/2-ell}$ ($ellin mathbb{Z}$) with coefficients that are rational sums of $E_s$ with $sin mathbb{Z}+1/2$. This gives an all orders derivation of the form of the recently conjectured expansion. We further consider t Hooft large-$N$ Yang-Mills theory. The coefficient of each order can be expanded as a convergent series in $lambda$. For large $lambda$ this becomes an asymptotic series with coefficients that are again rational multiples of odd zeta values. The large-$lambda$ series is not Borel summable, and its resurgent non-perturbative completion is $O(exp(-2sqrt{lambda}))$.
An integrated correlator of four superconformal stress-tensor primaries of $mathcal{N}=4$ supersymmetric $SU(N)$ Yang-Mills theory (SYM), originally obtained by localisation, is re-expressed as a two-dimensional lattice sum that is manifestly invaria
nt under $SL(2,mathbb{Z})$ S-duality. This expression is shown to satisfy a novel Laplace equation in the complex coupling constant $tau$ that relates the $SU(N)$ integrated correlator to those of the $SU(N+1)$ and $SU(N-1)$ theories. The lattice sum is shown to precisely reproduce known perturbative and non-perturbative properties of $mathcal{N}=4$ SYM for any finite $N$, as well as extending previously conjectured properties of the large-$N$ expansion.
This paper concerns a special class of $n$-point correlation functions of operators in the stress tensor supermultiplet of $mathcal{N}=4$ supersymmetric $SU(N)$ Yang-Mills theory. These are maximal $U(1)_Y$-violating correlators that violate the bonu
s $U(1)_Y$ charge by a maximum of $2(n-4)$ units. We will demonstrate that such correlators satisfy $SL(2,mathbb{Z})$-covariant recursion relations that relate $n$-point correlators to $(n-1)$-point correlators in a manner analogous to the soft dilaton relations that relate the corresponding amplitudes in flat-space type IIB superstring theory. These recursion relations are used to determine terms in the large-$N$ expansion of $n$-point maximal $U(1)_Y$-violating correlators in the chiral sector, including correlators with four superconformal stress tensor primaries and $(n-4)$ chiral Lagrangian operators, starting from known properties of the $n=4$ case. We concentrate on the first three orders in $1/N$ beyond the supergravity limit. The Mellin representations of the correlators are polynomials in Mellin variables, which correspond to higher derivative contact terms in the low-energy expansion of type IIB superstring theory in $AdS_5 times S^5$ at the same orders as $R^4, d^4R^4$ and $d^6R^4$. The coupling constant dependence of these terms is found to be described by non-holomorphic modular forms with holomorphic and anti-holomorphic weights $(n-4,4-n)$ that are $SL(2, mathbb{Z})$-covariant derivatives of Eisenstein series and certain generalisations. This determines a number of non-leading contributions to $U(1)_Y$-violating $n$-particle interactions ($n>4$) in the low-energy expansion of type IIB superstring amplitudes in $AdS_5times S^5$.
We study modular invariants arising in the four-point functions of the stress tensor multiplet operators of the ${cal N} = 4$ $SU(N)$ super-Yang-Mills theory, in the limit where $N$ is taken to be large while the complexified Yang-Mills coupling $tau
$ is held fixed. The specific four-point functions we consider are integrated correlators obtained by taking various combinations of four derivatives of the squashed sphere partition function of the ${cal N} = 2^*$ theory with respect to the squashing parameter $b$ and mass parameter $m$, evaluated at the values $b=1$ and $m=0$ that correspond to the ${cal N} = 4$ theory on a round sphere. At each order in the $1/N$ expansion, these fourth derivatives are modular invariant functions of $(tau, bar tau)$. We present evidence that at half-integer orders in $1/N$, these modular invariants are linear combinations of non-holomorphic Eisenstein series, while at integer orders in $1/N$, they are certain generalized Eisenstein series which satisfy inhomogeneous Laplace eigenvalue equations on the hyperbolic plane. These results reproduce known features of the low-energy expansion of the four-graviton amplitude in type IIB superstring theory in ten-dimensional flat space and have interesting implications for the structure of the analogous expansion in $AdS_5times S^5$.
It is well known that the low energy expansion of tree-level superstring scattering amplitudes satisfies a suitably defined version of maximum transcendentality. In this paper it is argued that there is a natural extension of this definition that app
lies to the genus-one four-graviton Type II superstring amplitude to all orders in the low-energy expansion. To obtain this result, the integral over the genus-one moduli space is partitioned into a region ${cal M}_R$ surrounding the cusp and its complement ${cal M}_L$, and an exact expression is obtained for the contribution to the amplitude from ${cal M}_R$. The low-energy expansion of the ${cal M}_R$ contribution is proven to be free of irreducible multiple zeta-values to all orders. The contribution to the amplitude from ${cal M}_L$ is computed in terms of modular graph functions up to order $D^{12} {cal R}^4$ in the low-energy expansion, and general arguments are used beyond this order to conjecture the transcendentality properties of the ${cal M}_L$ contributions. Maximal transcendentality of the full amplitude holds provided we assign a non-zero weight to certain harmonic sums, an assumption which is familiar from transcendentality assignments in quantum field theory amplitudes.
We continue our investigation of the modular graph functions and string invariants that arise at genus-two as coefficients of low energy effective interactions in Type II superstring theory. In previous work, the non-separating degeneration of a genu
s-two modular graph function of weight $w$ was shown to be given by a Laurent polynomial in the degeneration parameter $t$ of degree $(w,w)$. The coefficients of this polynomial generalize genus-one modular graph functions, up to terms which are exponentially suppressed in $t$ as $t to infty$. In this paper, we evaluate this expansion explicitly for the modular graph functions associated with the $D^8 {cal R}^4$ effective interaction for which the Laurent polynomial has degree $(2,2)$. We also prove that the separating degeneration is given by a polynomial in the degeneration parameter $ln (|v|)$ up to contributions which are power-behaved in $v$ as $v to 0$. We further extract the complete, or tropical, degeneration and compare it with the independent calculation of the integrand of the sum of Feynman diagrams that contributes to two-loop type II supergravity expanded to the same order in the low energy expansion. We find that the tropical limit of the string theory integrand reproduces the supergravity integrand as its leading term, but also includes sub-leading terms proportional to odd zeta values that are absent in supergravity and can be ascribed to higher-derivative stringy interactions.
The concept and the construction of modular graph functions are generalized from genus-one to higher genus surfaces. The integrand of the four-graviton superstring amplitude at genus-two provides a generating function for a special class of such func
tions. A general method is developed for analyzing the behavior of modular graph functions under non-separating degenerations in terms of a natural real parameter $t$. For arbitrary genus, the Arakelov Green function and the Kawazumi-Zhang invariant degenerate to a Laurent polynomial in $t$ of degree $(1,1)$ in the limit $ttoinfty$. For genus two, each coefficient of the low energy expansion of the string amplitude degenerates to a Laurent polynomial of degree $(w,w)$ in $t$, where $w+2$ is the degree of homogeneity in the kinematic invariants. These results are exact to all orders in $t$, up to exponentially suppressed corrections. The non-separating degeneration of a general class of modular graph functions at arbitrary genus is sketched and similarly results in a Laurent polynomial in $t$ of bounded degree. The coefficients in the Laurent polynomial are generalized modular graph functions for a punctured Riemann surface of lower genus.
This paper investigates the relations between modular graph forms, which are generalizations of the modular graph functions that were introduced in earlier papers motivated by the structure of the low energy expansion of genus-one Type II superstring
amplitudes. These modular graph forms are multiple sums associated with decorated Feynman graphs on the world-sheet torus. The action of standard differential operators on these modular graph forms admits an algebraic representation on the decorations. First order differential operators are used to map general non-holomorphic modular graph functions to holomorphic modular forms. This map is used to provide proofs of the identities between modular graph functions for weight less than six conjectured in earlier work, by mapping these identities to relations between holomorphic modular forms which are proven by holomorphic methods. The map is further used to exhibit the structure of identities at arbitrary weight.
In earlier work we studied features of non-holomorphic modular functions associated with Feynman graphs for a conformal scalar field theory on a two-dimensional torus with zero external momenta at all vertices. Such functions, which we will refer to
as modular graph functions, arise, for example, in the low energy expansion of genus-one Type II superstring amplitudes. We here introduce a class of single-valued elliptic multiple polylogarithms, which are defined as elliptic functions associated with Feynman graphs with vanishing external momenta at all but two vertices. These functions depend on a coordinate, $zeta$, on the elliptic curve and reduce to modular graph functions when $zeta$ is set equal to $1$. We demonstrate that these single-valued elliptic multiple polylogarithms are linear combinations of multiple polylogarithms, and that modular graph functions are sums of single-valued elliptic multiple polylogarithms evaluated at the identity of the elliptic curve, in both cases with rational coefficients. This insight suggests the many interrelations between modular graph functions (a few of which were established in earlier papers) may be obtained as a consequence of identities involving multiple polylogarithms, and explains an earlier observation that the coefficients of the Laurent polynomial at the cusp are given by rational numbers times single-valued multiple zeta values.
