Multi-instantons in R^4 and Minimal Surfaces in R^{2,1}

TL;DR

This paper explores the deep connection between multi-instantons in four-dimensional space and minimal surfaces in three-dimensional Minkowski space, extending the correspondence to include topological and geometric aspects of gauge configurations.

Contribution

It extends the instanton-minimal surface correspondence beyond equations of motion, incorporating topological charge, moduli space, and explicit solutions like the BPST instanton.

Findings

Representation of instantons as minimal surfaces with arbitrary charge

Detailed minimal surface descriptions of the trivial vacuum and BPST instanton

Discussion of BPS monopoles and geodesic structures in this framework

Abstract

It is known that self-duality equations for multi-instantons on a line in four dimensions are equivalent to minimal surface equations in three dimensional Minkowski space. We extend this equivalence beyond the equations of motion and show that topological number, instanton moduli space and anti-self-dual solutions have representations in terms of minimal surfaces. The issue of topological charge is quite subtle because the surfaces that appear are non-compact. This minimal surface/instanton correspondence allows us to define a metric on the configuration space of the gauge fields. We obtain the minimal surface representation of an instanton with arbitrary charge. The trivial vacuum and the BPST instanton as minimal surfaces are worked out in detail. BPS monopoles and the geodesics are also discussed.