Pluriclosed flow and the Hull-Strominger system

TL;DR

This paper introduces a new flow extending pluriclosed flow to construct solutions for the Hull-Strominger system, providing geometric formulations, a priori estimates, and convergence results, with implications for complex geometry and string theory.

Contribution

It defines a natural extension of pluriclosed flow linked to the Hull-Strominger system, using string algebroids, and establishes regularity and convergence results.

Findings

Established a priori $C^{ abla}$ estimates for solutions.

Proved smooth regularity of solutions to the Hull-Strominger system.

Demonstrated global existence and convergence of the flow on special backgrounds.

Abstract

We define a natural extension of pluriclosed flow aiming at constructing solutions of the Hull-Strominger system. We give several geometric formulations of this flow, which yield a series of a priori estimates for the flow and also for the Hull-Strominger system. The evolution equations are derived using the theory of string algebroids, a class of Courant algebroids which occur naturally in higher gauge theory. Using this, we interpret the flow as generalized Ricci flow and also as a higher/coupled version of Hermitian-Yang-Mills flow, proving furthermore that it is compatible with symmetry reduction. Regarding our main analytical results, we prove a priori $C^{\infty}$ estimates for uniformly parabolic solutions. This in particular settles the question of smooth regularity of uniformly elliptic solutions of the Hull-Strominger system, generalizing Yau's $C^3$ estimate for the complex Monge-Amp\`ere equation. We prove global existence and convergence results for the flow on special backgrounds, and discuss a conjectural relationship of the flow to the geometrization of Reid's fantasy.