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

Abstract

We present exact expressions for certain integrated correlators of four superconformal primary operators in the stress tensor multiplet of \mathcal{N}=4𝒩=4 supersymmetric Yang–Mills (SYM) theory with classical gauge group, G_NGN= SO(2N)=SO(2N), SO(2N+1)SO(2N+1), USp(2N)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)SU(N) {gauge group} 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}}τ=θ/(2π)+4πi/gYM2 on any integrated correlator for gauge group G_NGN relates it to a linear combination of correlators with gauge groups G_{N+1}GN+1, G_NGN and G_{N-1}GN−1. These ``Laplace-difference equations’’ determine the expressions of integrated correlators for all classical gauge groups for any value of NN in terms of the correlator for the gauge group SU(2)SU(2). The perturbation expansions of these integrated correlators for any finite value of NN 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-NN 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}_2AdS5×S5/ℤ2 background.