  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 given
topologically. (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 of 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.
