The problem description for the Yang-Mills Existence and Mass Gap problem (http://www.claymath.org/sites/default/files/yangmills.pdf) says in its "Mathematical Perspective" section that

*Some results are known for Yang-Mills theory on a 4-torus $\mathbb{T}^{4}$ approximating $\mathbb{R}^{4}$ and, while the construction is not complete, there is ample indication that known methods could be extended to construct Yang–Mills theory on $\mathbb{T}^{4}$.*

*In fact, at present we do not know any non-trivial relativistic ﬁeld theory that satisﬁes the Wightman (or any other reasonable) axioms in four dimensions. So even having a detailed mathematical construction of Yang–Mills theory on a compact space would represent a major breakthrough. Yet, even if this were accomplished, no present ideas point the direction to establish the existence of a mass gap that is uniform in the volume. Nor do present methods suggest how to obtain the existence of the inﬁnite volume limit $\mathbb{T}^{4}\rightarrow\mathbb{R}^{4}$.*

Could someone point me in the direction of a paper that describes the use of compact torus manifolds to construct 4d Quantum Yang-Mills, or else describe some of these attempts? Also, is the difficulty alluded to by Witten and Jaffe solely that a toroidal space is compact whereas a Euclidean space is unbounded, or is there more to the story?

This post imported from StackExchange Physics at 2014-06-27 11:24 (UCT), posted by SE-user user47299