Multi-Instanton Calculus and Equivariant Cohomology

TL;DR

The paper develops a systematic multi-instanton calculus using ADHM equivariant cohomology and a supersymmetric localization of equivariant forms for $N=2$, $N=2^*$, and $N=4$ SYM. It constructs a deformed BRST operator that yields isolated fixed points and reduces the moduli-space integrals to sums over fixed points labeled by Young diagrams, producing explicit $Z_k$ expressions, including cases with fundamental matter and the $N=4$ limit. It provides $k=1,2$ explicit results and a generating-function structure for the pure $N=4$ theory, highlighting Euler characteristics and modular features. The framework bridges BRST-based instanton counting with Nekrasov's localization and the broader string-theoretic interpretation via brane constructions.

Abstract

We present a systematic derivation of multi-instanton amplitudes in terms of ADHM equivariant cohomology. The results rely on a supersymmetric formulation of the localization formula for equivariant forms. We examine the cases of N=4 and N=2 gauge theories with adjoint and fundamental matter.

Figures (1)

• Figure 1: Two generic Young diagrams denoted by the indices $\lambda, \tilde{\lambda}$ in the main text.