Exceptionally simple integrated correlators in 𝒩 = 4 supersymmetric Yang-Mills theory

Supersymmetric localisation has led to several modern developments in the study of integrated correlators in 𝒩 = 4 supersymmetric Yang-Mills (SYM) theory. In particular, exact results have been derived for certain integrated four-point functions of superconformal primary operators in the stress tensor multiplet which are valid for all classical gauge groups, SU( N ), SO( N ), and USp(2 N ), and for all values of the complex coupling, τ = θ/ (2 π ) + 4 πi/ g_YM^2 . In this work we extend this analysis and provide a unified two-dimensional lattice sum representation valid for all simple gauge groups, in particular for the exceptional series E r (with r = 6 , 7 , 8), F 4 and G 2 . These expressions are manifestly covariant under Goddard-Nuyts-Olive duality which for the cases of F 4 and G 2 is given by particular Fuchsian groups. We show that the perturbation expansion of these integrated correlators is universal in the sense that it can be written as a single function of three parameters, called Vogel parameters, and a suitable ’t Hooft-like coupling. To obtain the perturbative expansion for the integrated correlator with a given gauge group we simply need substituting in this single universal expression specific values for the Vogel parameters. At the non-perturbative level we conjecture a formula for the one-instanton Nekrasov partition function valid for all simple gauge groups and for general Ω-deformation background. We check that our expression reduces in various limits to known results and that it produces, via supersymmetric localisation, the same one-instanton contribution to the integrated correlator as the one derived from the lattice sum representation. Finally, we consider the action of the hyperbolic Laplace operator with respect to τ on the integrated correlators with exceptional gauge groups and derive inhomogeneous Laplace equations very similar to the ones previously obtained for classical gauge groups.