Supergravity and a Bogomol'nyi Bound in Three Dimensions

TL;DR

This paper explores the coupling of the Abelian Higgs model to N=2 supergravity in 2+1 dimensions, deriving a Bogomol'nyi bound and first-order equations for gravitating vortices, linking supersymmetry to classical solutions.

Contribution

It constructs the supercharge algebra in 2+1 dimensions with supergravity and derives a Bogomol'nyi bound for classical solutions, including gravitating vortices.

Findings

Mass of solutions is bounded by topological charge.

Half of supersymmetry is broken at the bound.

Self-duality equations include Einstein equations.

Abstract

We discuss the $2+1$ dimensional Abelian Higgs model coupled to $N=2$ supergravity. We construct the supercharge algebra and, from it, we show that the mass of classical static solutions is bounded from below by the topological charge. As it happens in the global case, half of the supersymmetry is broken when the bound is attained and Bogomol'nyi equations, resulting from the unbroken supersymmetry, hold. These equations, which correspond to gravitating vortices, include a first order self-duality equation whose integrability condition reproduces the Einstein equation.