# Rigorous QFT on a Torus

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?

• Did you look in the references of the paper you're quoting? Jun 24, 2014 at 18:55
• @user1504 Well, I found this paper to involve toroidal space: projecteuclid.org/download/pdf_1/euclid.cmp/1104114382 but it was only in 3 dimensions. Jun 24, 2014 at 19:00

• I'm sorry: Are you asking if its harder to construct YM on a torus than on $\mathbb{R}^4$? Jun 25, 2014 at 17:25
• Sorry if I'm unclear: the problem description states that if Quantum Yang-Mills were to be constructed on a $\mathbb{T}^{4}$ torus, it would be difficult to extend the solution to $\mathbb{R}^{4}$ because of the difficult of extending the torus to infinite boundaries. Is this the only reason that such a difficulty would be present? Jun 25, 2014 at 17:41