Integral constraints for $\mathcal{N}=4$ super-Yang-Mills from a squashed sphere

TL;DR

This paper derives three new integral constraints on the four-point function in  super-Yang-Mills theory on a squashed sphere, showing they are implied by existing constraints, thus enriching the theoretical framework for studying the theory.

Contribution

It introduces three new integral constraints for  super-Yang-Mills theory on a squashed sphere and demonstrates they are implied by previously known constraints, advancing the theoretical understanding.

Findings

Three new integral constraints derived from the squashed sphere setup.

The new constraints are shown to be implied by existing constraints.

Development of techniques to handle complex derivations.

Abstract

Supersymmetric localization and Ward identities have been used in the past several years to derive two integral constraints on the four-point function of the stress-tensor multiplet in $\mathcal{N} = 4$ super-Yang-Mills theory. These constraints are powerful tools for studying the theory, especially when used in tandem with analytic and/or numerical bootstrap techniques. In this paper, we consider three additional integral constraints that can be derived starting from the $\mathcal{N} = 4$ super-Yang-Mills theory placed on a squashed four-sphere. These constraints are technically much more challenging to derive than the ones in the literature, and much of the paper is devoted to developing techniques to make this computation tractable. Our end result is that these three constraints are implied by the two constraints already appearing in the literature.