Surface observables in gauge theories, modular Painlev\'e tau functions and non-perturbative topological strings

TL;DR

This paper explores BPS surface observables in 4D $ ext{SU}(2)$ gauge theories, linking them to Painlevé tau functions and non-perturbative topological string partition functions through modular and holomorphic properties.

Contribution

It establishes a connection between gauge theory surface observables, Painlevé tau functions, and topological string partition functions, revealing their modular and holomorphic structures.

Findings

Painlevé tau functions solve blow-up equations in the Nekrasov-Shatashvili limit.

Derived BCOV holomorphic anomaly equations from modular properties.

Constructed non-perturbative topological string partition functions with holomorphic and modular features.

Abstract

We study BPS surface observables of $\mathcal{N}=2$ four dimensional $SU(2)$ gauge theory in gravitational $\Omega$-background at perturbative and at Argyres-Douglas superconformal fixed points. This is done by formulating the equivariant gauge theory on the blow-up of $\mathbb{C}^2$ and considering the decoupling Nekrasov-Shatashvili limit. We show that in this limit the blow-up equations are solved by corresponding Painlev\'e $\mathcal{T}$-functions and exploit operator/state correspondence to compute their expansion in an integer basis, given in terms of the moduli of the quantum Seiberg-Witten curve. We study the modular properties of these solutions and show that they do directly lead to BCOV holomorphic anomaly equations for the corresponding topological string partition function. The resulting $\mathcal{T}$-functions are holomorphic and modular and as such they provide a natural non-perturbative completion of topological strings partition functions.