Instantons, Three Dimensional Gauge Theories and Monopole Moduli Spaces

TL;DR

This work computes instanton corrections to three-dimensional $N=4$ and $N=8$ supersymmetric gauge theories with $SU(n)$ gauge groups, and links these corrections to the monopole moduli spaces of $SU(2)$. For $N=4$, instantons aligned with roots induce leading exponential corrections to the $n$-monopole metric, producing non-pairwise interactions that are resolved by a constrained instanton approach, with CMS playing a crucial role. In the $N=8$ case, enhanced supersymmetry lifts extra zero modes, yielding a clean, pairwise structure that matches Matrix theory scattering of membranes, with the Euler character of the relative moduli spaces governing the instanton contributions. The analysis unifies low-energy gauge dynamics, monopole moduli space geometry, and Matrix theory intuition, highlighting how three-dimensional instantons encode intricate inter-monopole interactions and their stability structure.

Abstract

We calculate instanton corrections to three dimensional gauge theories with N=4 and N=8 supersymmetry and SU(n) gauge groups. The N=4 results give new information about the moduli space of n BPS SU(2) monopoles, including the leading order non-pairwise interaction terms. A few comments are made on the relationship of the N=8 results to membrane scattering in matrix theory.

Figures (1)

• Figure 1: For triangle (a), the height of the root cooresponding to the interaction between the first and second monopoles is 2. For triangle (b), the root is simple.