Free-field-behavior conjecture for Yang–Mills approximations

Let $G$ be a Lie group satisfying the condition given at the beginning of Section 3.1. Let $\mathfrak{g}$ be its Lie algebra, and let $\{\mathbf{A}^n\}_{n\geq 1}$ be a sequence of $\mathfrak{g}^3$-valued stochastic processes with smooth sample paths. The previously listed assumptions are the assumptions imposed before Theorem 4.1 on these processes. Free-field-behavior conjecture. For some suitable sequence $\{\mathbf{A}^n\}_{n\geq 1}$ of $\mathfrak{g}^3$-valued stochastic processes with smooth sample paths, whose laws approximate the Yang–Mills measure, the previously listed assumptions are satisfied.

If true, this conjecture supplies the stochastic-process approximations required by the theorem asserting uniform bounds on the Yang–Mills action and hence subsequential weak convergence. The cited theorem establishes the consequence conditional on the assumptions, while their realization by a suitable approximating sequence remains open.