Multi-Instantons, Three-Dimensional Gauge Theory, and the Gauss-Bonnet-Chern Theorem

TL;DR

This paper computes multi-instanton effects in a 3D supersymmetric gauge theory, linking them to topological invariants of monopole moduli spaces and exploring their implications for M(atrix) theory.

Contribution

It establishes a connection between multi-instanton contributions and the Gauss-Bonnet-Chern integral on monopole moduli spaces, providing explicit calculations and conjectures.

Findings

For k=2, the integral equals the Euler characteristic of the moduli space.

Multi-instanton effects are proportional to topological invariants of monopole moduli spaces.

Conjecture that boundary terms in the integral vanish, simplifying the calculation.

Abstract

We calculate multi-instanton effects in a three-dimensional gauge theory with N=8 supersymmetry and gauge group SU(2). The k-instanton contribution to an eight-fermion correlator is found to be proportional to the Gauss-Bonnet-Chern integral of the Gaussian curvature over the centered moduli-space of charge-k BPS monopoles, \tilde{M}_{k}. For k=2 the integral can be evaluated using the explicit metric on \tilde{M}_{2} found by Atiyah and Hitchin. In this case the integral is equal to the Euler character of the manifold. More generally the integral is the volume contribution to the index of the Euler operator on \tilde{M}_{k}, which may differ from the Euler character by a boundary term. We conjecture that the boundary terms vanish and evaluate the multi-instanton contributions using recent results for the cohomology of \tilde{M}_{k}. We comment briefly on the implications of our result for a recently proposed test of M(atrix) theory.