Three-point functions from integrability in $\mathcal{N}=2$ orbifold theories

TL;DR

This paper extends the hexagon formalism from $ ext{N}=4$ SYM to $ ext{N}=2$ orbifold theories, demonstrating its applicability through gauge theory tests and suggesting potential for full structure constant determination at finite coupling.

Contribution

The authors adapt the hexagon formalism to $ ext{N}=2$ orbifold theories and validate it with tree-level calculations, opening avenues for finite coupling analysis.

Findings

Hexagon formalism applies to $ ext{N}=2$ orbifold theories with minor modifications.

Tree-level gauge theory calculations match hexagon predictions.

Indications that full structure constants could be determined at finite coupling.

Abstract

Besides solving the spectral problem of $\mathcal{N}=4$ Super-Yang-Mills (SYM) theory, integrability also provides us with tools to compute the structure constants of the theory, most prominently through the hexagon formalism. We show that, with minor modifications, this formalism can also be applied to orbifolds of $\mathcal{N}=4$ SYM theory, which are integrable theories in their own right. To substantiate this claim, we test our results against a direct gauge-theory calculation at tree-level. We focus here on a family of $\mathcal{N}=2$ supersymmetric $\mathbb{Z}_M$-orbifold theories. BPS correlators in these theories have recently been investigated with independent localisation techniques and a structural matching with wrapping corrections in the hexagon formalism was observed. Together with our weak-coupling evidence, this suggests that a full determination of the structure constants of orbifold theories at finite coupling may be within reach.