Exact results for duality-covariant integrated correlators in $\mathcal{N}=4$ SYM with general classical gauge groups

TL;DR

This paper derives exact duality-covariant expressions for integrated correlators in $ =4$ SYM with classical gauge groups, revealing their structure as lattice sums, Eisenstein series, and Laplace-difference equations, with consistency checks from perturbation and string theory.

Contribution

It provides the first exact, duality-invariant formulas for integrated correlators in $ =4$ SYM with all classical gauge groups, generalizing previous $SU(N)$ results and establishing new Laplace-difference relations.

Findings

Expressions as two-dimensional lattice sums.

Correlators written as sums of Eisenstein series.

Laplace-difference equations relate correlators across gauge groups.

Abstract

We present exact expressions for certain integrated correlators of four superconformal primary operators in the stress tensor multiplet of $\mathcal{N}=4$ supersymmetric Yang--Mills (SYM) theory with classical gauge group, $G_N$ $= SO(2N)$, $SO(2N+1)$, $USp(2N)$. These integrated correlators are expressed as two-dimensional lattice sums by considering derivatives of the localised partition functions, generalising the expression obtained for $SU(N)$ in our previous works. These expressions are manifestly covariant under Goddard-Nuyts-Olive duality. The integrated correlators can also be formally written as infinite sums of non-holomorphic Eisenstein series with integer indices and rational coefficients. Furthermore, the action of the hyperbolic Laplace operator with respect to the complex coupling $\tau=\theta/(2\pi) + 4\pi i /g^2_{_{YM}}$ on any integrated correlator for gauge group $G_N$ relates it to a linear combination of correlators with gauge groups $G_{N+1}$, $G_N$ and $G_{N-1}$. These "Laplace-difference equation" determine the expressions of integrated correlators for all classical gauge groups for any value of $N$ in terms of the correlator for the gauge group $SU(2)$. The perturbation expansions of these integrated correlators for any finite value of $N$ agree with properties obtained from perturbative Yang--Mills quantum field theory, together with various multi-instanton calculations which are also shown to agree with those determined by supersymmetric localisation. The coefficients of terms in the large-$N$ expansion are sums of non-holomorphic Eisenstein series with half-integer indices, which extend recent results and make contact with low order terms in the low energy expansion of type IIB superstring theory in an $AdS_5\times S^5/\mathbb{Z}_2$ background.