Geometric Flows and Supersymmetry

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 \(_2\) ${}_2$ 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 \(\alpha '\) ${\alpha }^{\prime }$ 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.