The Liouville Geometry of N=2 Instantons and the Moduli of Punctured Spheres

TL;DR

This paper explores the geometric structure of N=2 instantons using moduli spaces of punctured spheres, revealing new recursive relations and connections to Liouville theory, Hurwitz numbers, and Gromov-Witten invariants.

Contribution

It establishes a novel mapping between instanton moduli spaces and punctured sphere moduli spaces, introducing a bilinear recursion relation for instanton coefficients.

Findings

Derived a bilinear recursion relation for instanton coefficients.

Expressed instanton contributions as integrals over moduli spaces of punctured spheres.

Predicted asymptotic behavior of Gromov-Witten invariants based on this geometric framework.

Abstract

We study the instanton contributions of N=2 supersymmetric gauge theory and propose that the instanton moduli space is mapped to the moduli space of punctured spheres. Due to the recursive structure of the boundary in the Deligne-Knudsen-Mumford stable compactification, this leads to a new recursion relation for the instanton coefficients, which is bilinear. Instanton contributions are expressed as integrals on M_{0,n} in the framework of the Liouville F-models. This also suggests considering instanton contributions as a kind of Hurwitz numbers and also provides a prediction on the asymptotic form of the Gromov-Witten invariants. We also interpret this map in terms of the geometric engineering approach to the gauge theory, namely the topological A-model, as well as in the noncritical string theory framework. We speculate on the extension to nontrivial gravitational background and its relation to the uniformization program. Finally we point out an intriguing analogy with the self-dual YM equations for the gravitational version of SU(2) where surprisingly the same Hauptmodule of the SW solution appears.