A monotonicity formula for minimal connections

For Hermitian connections on a Hermitian complex line bundle over a Riemannian manifold $(X,g)$, 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 under certain conditions on $\dim X$ and the curvature of $g$. These formulas would be important in bubbling analysis. As a corollary, we obtain the vanishing theorem for minimal connections on the odd dimensional Euclidean space. (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).