Mirror of volume functionals on manifolds with special holonomy

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 Kähler, 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 $\mathrm{Spin}(7)$-manifold (a $\mathrm{Spin}(7)$-dDT connection), which are considered to be the "mirror" of special Lagrangian, (co)associative and Cayley submanifolds, respectively. When $X$ is a compact $\mathrm{Spin}(7)$-manifold, we prove the "mirror" of the Cayley equality, which implies the following. (a) Any $\mathrm{Spin}(7)$-dDT connection is a global minimizer of $V$ and its value is topological. (b) Any $\mathrm{Spin}(7)$-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 $\mathrm{Spin}(7)$-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 Kähler manifolds of dimension 3 or 4.