Geometric flows and supersymmetry

TL;DR

This paper explores the connection between supersymmetry and geometric flows in heterotic supergravity, extending known flows to higher dimensions and analyzing their solutions and convergence properties.

Contribution

It introduces a unified framework linking supersymmetry with geometric flows on complex, G$_2$, and Spin(7) manifolds, including new scalar flow equations and convergence results.

Findings

Flow equations match anomaly flow on complex threefolds

Flow simplifies to a scalar equation on torus fibrations

Existence and convergence of flows imply supergravity solutions

Abstract

We study the relation between supersymmetry and geometric flows driven by the Bianchi identity for the three-form flux $H$ in heterotic supergravity. We describe how the flow equations can be derived from a functional that appears in a rewriting of the bosonic action in terms of squares of supersymmetry operators. On a complex threefold, the resulting equations match what is known in the mathematics literature as "anomaly flow". We generalise this to seven- and eight-manifolds with G$_2$ or Spin(7) structures and discuss examples where the manifold is a torus fibration over a K3 surface. In the latter cases, the flow simplifies to a single scalar equation, with the existence of the supergravity solution implied by the long-time existence and convergence of the flow. We also comment on the $\alpha'$ expansion and highlight the importance of using the proper connection in the Bianchi identity to ensure that the flow's fixed points satisfy the supergravity equations of motion.