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

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 Kähler 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.