A monotonicity formula for minimal connections
Kotaro Kawai
TL;DR
The paper develops a variational framework for minimal connections, defined as critical points of a DBI-type volume functional on Hermitian line bundles, and introduces the volume $V_g(\beta)$ and normalized volume $V^0_g(\beta)$ to study their behavior under metric changes. By constructing the stress-energy tensor $S_{g,\beta}$ and a conservation-law condition $\mathrm{div}\,S_{g,\beta}=0$, the authors derive integration-by-parts formulas for the generalized Laplacian $\Delta_\beta$ and establish monotonicity formulas for radius-normalized and modified volume functionals. These monotonicity results hold under dimension and curvature hypotheses, with special, stronger conclusions in odd dimensions and in two dimensions, leading to vanishing theorems for 2-forms on Euclidean spaces. The work connects minimal connections to Yang–Mills theory through a large-radius limit and proves an existence theorem for minimal connections on compact manifolds with sufficiently large metrics; it also shows that deformed Donaldson–Thomas (dDT) connections on $G_2$-manifolds are minimal and enjoy stronger monotonicity and vanishing results, highlighting a mirror correspondence with calibrated submanifolds via the real Fourier–Mukai transform. Collectively, these results lay groundwork toward a Yang–Mills-type compactness theory and potential enumerative frameworks for dDT objects in special holonomy settings. The methods center on the operator $\Delta_\beta$, the tensor $G_\beta$, and energy-monotonicity arguments that parallel classical results for YM and minimal submanifolds while accommodating the DBI-type geometry of minimal connections.
Abstract
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).