Deformation and quantisation condition of the $\mathscr{Q}$-top recursion

TL;DR

This paper studies how deformations of hyperelliptic spectral curves influence the refined topological recursion and establishes a link between deformation and quantisation conditions, with implications for supersymmetric gauge theories.

Contribution

It introduces a deformation framework for hyperelliptic spectral curves and connects deformation conditions with quantisation conditions in the context of the $\

Findings

Deformation effects appear in the hyperelliptic refined topological recursion.

A coincidence between deformation and quantisation conditions is established.

Relation to Nekrasov-Shatashvili superpotential is discussed.

Abstract

We consider a deformation of a family of hyperelliptic refined spectral curves and investigate how deformation effects appear in the hyperelliptic refined topological recursion as well as the $\mathscr{Q}$-top recursion. We then show a coincidence between a deformation condition and a quantisation condition in terms of the $\mathscr{Q}$-top recursion on a degenerate elliptic curve. We also discuss a relation to the corresponding Nekrasov-Shatashivili effective twisted superpotential.