Paper
A parabolic flow for the large volume heterotic $G_2$ system
Authors
Mario Garcia-Fernandez, Andres J. Moreno, Alec Payne, Jeffrey Streets
Abstract
We introduce a geometric flow of conformally coclosed -structures, whose fixed points are large volume solutions of the heterotic system, with vanishing scalar torsion class . After conformal rescaling, it becomes a flow of coclosed -structures, related to Grigorian's modified coflow, which is coupled to a flow for a dilaton function. Our main results establish fundamental short-time existence and Shi-type smoothing properties of this flow, as well as a classification of its fixed points. By a classical rigidity result in the string theory literature, the fixed points on a compact manifold correspond to torsion-free -structures, that is, to metrics with holonomy contained in . Thus, we establish in the affirmative a folklore question in the special holonomy community, about the existence of a well-posed flow for coclosed -structures with fixed points given by torsion-free -structures. The flow also satisfies a monotonicity formula for the -dilaton functional (volume scale in string theory), which allows us to strengthen the rigidity result with an alternative proof. The monotonicity of the -dilaton functional, combined with the Shi-type estimates, leads to a general result on the convergence of nonsingular solutions. A dimension reduction analysis reveals an interesting link with natural flows for -structures, previously introduced in the literature.