Curve counting, instantons and McKay correspondences

TL;DR

This paper explores the connections between instanton counting in gauge theories, enumerative geometry of holomorphic curves, and algebraic structures like affine Lie algebras and McKay correspondence, providing new formulas and insights.

Contribution

It presents new formulas for partition functions on ALE spaces, rigorously treats equivariant partition functions on Hirzebruch surfaces, and links McKay correspondence to instanton counting on specific spaces.

Findings

Partition functions on ALE spaces expressed as affine characters

Rigorous treatment of equivariant partition functions on Hirzebruch surfaces

Proposed connection between McKay correspondence and instanton counting on Hirzebruch-Jung spaces

Abstract

We survey some features of equivariant instanton partition functions of topological gauge theories on four and six dimensional toric Kahler varieties, and their geometric and algebraic counterparts in the enumerative problem of counting holomorphic curves. We discuss the relations of instanton counting to representations of affine Lie algebras in the four-dimensional case, and to Donaldson-Thomas theory for ideal sheaves on Calabi-Yau threefolds. For resolutions of toric singularities, an algebraic structure induced by a quiver determines the instanton moduli space through the McKay correspondence and its generalizations. The correspondence elucidates the realization of gauge theory partition functions as quasi-modular forms, and reformulates the computation of noncommutative Donaldson-Thomas invariants in terms of the enumeration of generalized instantons. New results include a general presentation of the partition functions on ALE spaces as affine characters, a rigorous treatment of equivariant partition functions on Hirzebruch surfaces, and a putative connection between the special McKay correspondence and instanton counting on Hirzebruch-Jung spaces.