  For Hermitian connections on a Hermitian complex line bundle over a
Riemannian manifold, we can define the ``volume", which can be considered to be
the ``mirror" of the standard volume for submanifolds. We call the critical
points minimal connections.
  In this paper, (1) we prove monotonicity formulas for minimal connections
with respect to some versions of volume functionals, which would be important
in bubbling analysis. As a corollary, we obtain the vanishing theorem for
minimal connections in the odd-dimensional case.
  (2) We see that the formal ``large radius limit" of the defining equation of
minimal connections is that of Yang-Mills connections. Then the existence
theorem of minimal connections is proved for a ``sufficiently large" metric.
  (3) We can consider deformed Donaldson-Thomas (dDT) connections on
$G_2$-manifolds as ``mirrors" of calibrated (associative) submanifolds. We show
that dDT connections are minimal connections, just as calibrated submanifolds
are minimal submanifolds. By the argument specific to dDT connections, we
obtain the stronger monotonicity formulas and vanishing theorem for dDT
connections than in (1).
