BPS complexes and Chern--Simons theories from $G$-structures in gauge theory and gravity

TL;DR

This paper introduces a universal algebraic BPS complex for analyzing infinitesimal moduli of generalized instantons across various physical theories, linking gauge, gravity, and string systems through $G$-structures.

Contribution

It constructs a universal BPS complex that computes moduli spaces of instantons in diverse systems, connecting supersymmetric solutions, gauge theories, and string theories.

Findings

BPS complex computes infinitesimal moduli spaces as cohomologies.

The BPS complex becomes a double complex in supergravity, analogous to $(p,q)$-forms.

Associated linearised BV Chern--Simons theory reproduces classic gauge theory examples.

Abstract

We consider a variety of physical systems in which one has states that can be thought of as generalised instantons. These include Yang--Mills theories on manifolds with a torsion-free $G$-structure, analogous gravitational instantons and certain supersymmetric solutions of ten-dimensional supergravity, using their formulation as generalised $G$-structures on Courant algebroids. We provide a universal algebraic construction of a complex, which we call the BPS complex, that computes the infinitesimal moduli space of the instanton as one of its cohomologies. We call a class of these spinor type complexes, which are closely connected to supersymmetric systems, and show how their Laplacians have nice properties. In the supergravity context, the BPS complex becomes a double complex, in a way that corresponds to the left- and right-moving sectors of the string, and becomes much like the double complex of $(p,q)$-forms on a K\"ahler manifold. If the BPS complex has a symplectic inner product, one can write down an associated linearised BV Chern--Simons theory, which reproduces several classic examples in gauge theory. We discuss applications to (quasi-)topological string theories and heterotic superpotential functionals, whose quadratic parts can also be constructed naturally from the BPS complex.