The Liouville geometry of N = 2 instantons and the moduli of punctured spheres

Abstract

We propose that the M = 2 SYM instanton moduli space maps to the one of punctured spheres, in the framework of the Liouville F-models. This leads to a bilinear recursion relation for the instanton contributions, resembling Hurwitz numbers, and predicts the asymptotic form of the GW invariants. The map is interpreted in the frameworks of the geometric engineering approach to the gauge theory, i.e. the topological A-model, and of the noncritical string theory. We speculate on the extension to the gravitational background and its relation to the uniformization program. Finally, we show an intriguing analogy with the gravitational version of the SU(2) self-dual YM equations where surprisingly the same Hauptmodule of the SW solution appears. © SISSA/ISAS 2004.