Abstract
Supersymmetric localisation has led to several modern developments in the study of integrated correlators in \( \mathcal{N} \) = 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(2N), and for all values of the complex coupling, τ = θ/(2π) + 4πi/\( {g}_{YM}^2 \). 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 Er (with r = 6, 7, 8), F4 and G2. These expressions are manifestly covariant under Goddard-Nuyts-Olive duality which for the cases of F4 and G2 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.
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Acknowledgments
We would like to particularly thank Axel Kleinschmidt and Boris Pioline for bringing to our attention the potential use of Vogel universality for the study of integrated correlators with exceptional gauge groups. We also thank Stefano Cremonesi, Marialuisa Frau, Alberto Lerda, Ruben Mkrtchyan and Alessandro Pini for useful discussions and Marialuisa Frau, Axel Kleinschmidt, Alberto Lerda and Congkao Wen for comments on the draft. DD would like to thank the Albert Einstein Institute, Golm, for the hospitality during the final stages of this project. PV would like to thank the Department of Mathematical Sciences of Durham University for the hospitality and support within the scientific visits scheme organised by the International Network on Quantum Fields and Strings (IRN:QFS). PV is partially supported by the MUR PRIN contract 2020KR4KN2 “String Theory as a bridge between Gauge Theories and Quantum Gravity” and by the INFN project ST&FI “String Theory & Fundamental Interactions”.
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Dorigoni, D., Vallarino, P. Exceptionally simple integrated correlators in \( \mathcal{N} \) = 4 supersymmetric Yang-Mills theory. J. High Energ. Phys. 2023, 203 (2023). https://doi.org/10.1007/JHEP09(2023)203
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DOI: https://doi.org/10.1007/JHEP09(2023)203