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<p>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 <i>G</i>-structure, analogous gravitational instantons and certain supersymmetric solutions of ten-dimensional supergravity, using their formulation as generalised <i>G</i>-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 (<i>p</i>, <i>q</i>)-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.</p>

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

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年10月22日

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  • Julian Kupka,
  • Charles Strickland-Constable,
  • Eirik Eik Svanes,
  • David Tennyson,
  • Fridrich Valach

摘要

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.