Mirror of volume functionals on manifolds with special holonomy

TL;DR

This paper introduces a new flow for Hermitian line bundles called the line bundle mean curvature flow, relates it to mirror symmetry concepts, and proves minimization and flatness results for special holonomy manifolds.

Contribution

It develops the line bundle mean curvature flow, relates volume functionals to mirror symmetry, and establishes minimization and flatness properties of deformed Donaldson-Thomas connections on special holonomy manifolds.

Findings

Short-time existence and uniqueness of the flow.

Deformed Donaldson-Thomas connections are global minimizers of the volume functional.

Connections are flat on flat line bundles and pullbacks in product manifolds.

Abstract

We can define the ``volume'' $V$ for Hermitian connections on a Hermitian complex line bundle over a Riemannian manifold $X$, which can be considered to be the ``mirror'' of the standard volume for submanifolds. This is called the Dirac-Born-Infeld (DBI) action in physics. In this paper, (1) we introduce the negative gradient flow of $V$, which we call the line bundle mean curvature flow. Then, we show the short-time existence and uniqueness of this flow. When $X$ is K\"ahler, we relate the negative gradient of $V$ to the angle function and deduce the mean curvature for Hermitian metrics on a holomorphic line bundle defined by Jacob and Yau. (2) We relate the functional $V$ to a deformed Hermitian Yang--Mills (dHYM) connection, a deformed Donaldson--Thomas connection for a $G_2$-manifold (a $G_2$-dDT connection), a deformed Donaldson--Thomas connection for a ${\rm Spin}(7)$-manifold (a ${\rm Spin}(7)$-dDT connection), which are considered to be the ``mirror'' of special Lagrangian, (co)associative and Cayley submanifolds, respectively. When $X$ is a compact ${\rm Spin}(7)$-manifold, we prove the ``mirror'' of the Cayley equality, which implies the following. (a) Any ${\rm Spin}(7)$-dDT connection is a global minimizer of $V$ and its value is topological. (b) Any ${\rm Spin}(7)$-dDT connection is flat on a flat line bundle. (c) If $X$ is a product of $S^1$ and a compact $G_2$-manifold $Y$, any ${\rm Spin}(7)$-dDT connection on the pullback of the Hermitian complex line bundle over $Y$ is the pullback of a $G_2$-dDT connection modulo closed 1-forms. We also prove analogous statements for $G_2$-manifolds and K\"ahler manifolds of dimension 3 or 4.