Mirror of volume functionals on manifolds with special holonomy

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 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 G2-manifold (a G2-dDT connection), a deformed Donaldson–Thomas connection for a Spin(7)-manifold (a 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 Spin(7)-manifold, we prove the “mirror” of the Cayley equality, which implies the following. (a) Any Spin(7)-dDT connection is a global minimizer of V and its value is topological. (b) Any Spin(7)-dDT connection is flat on a flat line bundle. (c) If X is a product of S1 and a compact G2-manifold Y, any Spin(7)-dDT connection on the pullback of the Hermitian complex line bundle over Y is the pullback of a G2-dDT connection modulo closed 1-forms.We also prove analogous statements for G2-manifolds and Kähler manifolds of dimension 3 or 4.