Curve counting and instanton counting

TL;DR

This paper proves combinatorial results supporting Nekrasov's conjecture that relates string invariants in certain Calabi-Yau geometries to equivariant genera of instanton moduli spaces, bridging geometry and physics.

Contribution

It establishes key combinatorial foundations necessary for proving Nekrasov's conjecture linking string invariants and instanton moduli space genera.

Findings

Proved combinatorial results for Nekrasov's conjecture

Connected string invariants with equivariant $\hat{A}$-genera

Supported the geometric-physical correspondence in Calabi-Yau geometries

Abstract

We prove some combinatorial results required for the proof of the following conjecture of Nekrasov: The generating function of closed string invariants in local Calabi-Yau geometries obtained by appropriate fibrations of $A_N$ singularities over $P^1$ reproduce the generating function of equivariant $\hat{A}$-genera of moduli space of instants on $C^2$.