Tightness conjecture for approximating Yang–Mills measures

Let $G$ be a Lie group satisfying the condition given at the beginning of Section 3.1, and let $\mathcal{X}$ be the state space of Yang–Mills heat-flow trajectories. For $t>0$, write $S_{\textup{YM}}(\mathbf{X}_n(t))$ for the Yang–Mills action evaluated on an $\mathcal{X}$-valued random variable $\mathbf{X}_n$. Tightness conjecture. For some suitable sequence of $\mathcal{X}$-valued random variables $\{\mathbf{X}_n\}_{n\geq 1}$, whose laws approximate the Yang–Mills measure, the sequence $\{S_{\textup{YM}}(\mathbf{X}_n(t))\}_{n\geq 1}$ is tight for every $t>0$.

This conjecture would provide the tightness input needed to obtain subsequential weak limits of approximating Yang–Mills measures on the state space. The paper's main theorem shows that this action tightness condition implies weak compactness, but the existence of a suitable approximating sequence satisfying it remains conjectural.