Surface observables in gauge theories, modular Painlevé tau functions and non-perturbative topological strings

TL;DR

This work builds a unified framework connecting BPS surface observables in 4d ${\cal N}=2$ SU(2) gauge theory to Painlevé $\mathcal{T}$-functions via blowup equations in the Nekrasov–Shatashvili limit. It shows that the NS blowup factor becomes a Painlevé $\mathcal{T}$-function, expandable in a chiral-ring basis, with explicit modular structure that reproduces the BCOV holomorphic anomaly equations and furnishes a non-perturbative completion of topological strings. The authors derive Hurwitz-type, integer-coefficient expansions around zeros for PVI, PIV, PIII$_2$, PI, and PIII$_3$ (including Argyres-Douglas points), revealing a universal SW-data-driven organization of the coefficients and providing a bridge between gauge theory, isomonodromic deformations, and topological-string theory. The results suggest broad generalizations to higher rank and five-dimensional uplifts, and they offer a precise, modular, and integrality-rich non-perturbative formulation of topological strings in the toric/calabi–Yau setting studied.

Abstract

We study BPS surface observables of $\mathcal{N}=2$ four dimensional $SU(2)$ gauge theory in gravitational $Ω$-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 Painlevé $\mathcal{T}$-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.

Figures (2)

• Figure 1: Handle gluing operator and state/operator correspondence.
• Figure 2: Toric diagram of the blow down of the exceptional divisor $E$ (in red) and state/operator correspondence.