Abstract
Integrated correlation functions in \( \mathcal{N} \) = 4 supersymmetric Yang-Mills theory with gauge group SU(N) can be expressed in terms of the localised S4 partition function, ZN, deformed by a mass m. Two such cases are \( {\mathcal{C}}_N={\left(\operatorname{Im}\tau \right)}^2{\partial}_{\tau }{\partial}_{\overline{\tau}}{\partial}_m^2\log {\left.{Z}_N\right|}_{m=0} \) and \( {\mathcal{H}}_N={\partial}_m^4\log {\left.{Z}_N\right|}_{m=0} \), which are modular invariant functions of the complex coupling τ. While \( {\mathcal{C}}_N \) was recently written in terms of a two-dimensional lattice sum for any N and τ, \( {\mathcal{H}}_N \) has only been evaluated up to order 1/N3 in a large-N expansion in terms of modular invariant functions with no known lattice sum realisation. Here we develop methods for evaluating \( {\mathcal{H}}_N \) to any desired order in 1/N and finite τ. We use this new data to constrain higher loop corrections to the stress tensor correlator, and give evidence for several intriguing relations between \( {\mathcal{H}}_N \) and \( {\mathcal{C}}_N \) to all orders in 1/N. We also give evidence that the coefficients of the 1/N expansion of \( {\mathcal{H}}_N \) can be written as lattice sums to all orders. Lastly, these large N and finite τ results are used to accurately estimate the integrated correlators at finite N and finite τ.
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Acknowledgments
DD and MBG are grateful for the hospitality of the Pollica Physics Centre and to conversations with Don Zagier, Kim Klinger-Logan, Ksenia Fedosova and Boris Pioline, as well as other participants in the programme ‘New Connections Between Physics and Number Theory’. The work of LFA is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 787185). LFA is also supported in part by the STFC grant ST/T000864/1. SMC is supported by a Royal Society University Research Fellowship, URF\R\221310. CW is supported by a Royal Society University Research Fellowship, URF\R\221015 and a STFC Consolidated Grant, ST\T000686\1 “Amplitudes, strings & duality”.
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Alday, L.F., Chester, S.M., Dorigoni, D. et al. Relations between integrated correlators in \( \mathcal{N} \) = 4 supersymmetric Yang-Mills theory. J. High Energ. Phys. 2024, 44 (2024). https://doi.org/10.1007/JHEP05(2024)044
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DOI: https://doi.org/10.1007/JHEP05(2024)044