Five-dimensional Chern-Simons terms and Nekrasov's instanton counting

TL;DR

The paper addresses incorporating the five-dimensional Chern-Simons term into Nekrasov's instanton counting for 5D $U(N)$ SYM and demonstrates exact agreement with all-genus topological string amplitudes on local toric Calabi-Yau manifolds $X^m_N$ across all $m$. It achieves this by twisting the instanton-moduli-space spin bundle with the determinant line bundle, yielding a CS-phase modified fixed-point sum that reproduces the topological-string results. This work strengthens the gauge/string correspondence in five dimensions and suggests a pathway toward a universal gauge-vertex framework for CY geometries. It also clarifies how geometric data (triple intersections) control physical couplings via the CS term in the Nekrasov counting setup.

Abstract

We extend the graviphoton-corrected prepotential of five-dimensional pure U(N) super Yang-Mills, which was originally proposed by Nekrasov, by incorporating the effect of the five-dimensional Chern-Simons term. This extension allows us to reproduce by a gauge theory calculation the partition functions of corresponding topological A-model on local toric Calabi-Yau manifolds X^m_N for all m=0,1,...,N. The original proposal corresponds to the case m=0.