# Rigorous QFT on a Torus

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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
asked Jun 24, 2014
Did you look in the references of the paper you're quoting?

This post imported from StackExchange Physics at 2014-06-27 11:24 (UCT), posted by SE-user user1504
@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.

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

## 1 Answer

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If you read any of the papers on 4d Yang-Mills referenced in the article you quote -- e.g., [3] by Balaban or [29] by Magnen, Seneor, & Rivasseau -- you'll discover that they are concerned with Yang-Mills on a 4-torus. This is standard in the subject, since no one wants to think about the boundary conditions on a cube.

This post imported from StackExchange Physics at 2014-06-27 11:24 (UCT), posted by SE-user user1504
answered Jun 25, 2014 by (1,110 points)
So is the difficult in fully constructing the theory solely because a torus is compact (ie bounded)?

This post imported from StackExchange Physics at 2014-06-27 11:24 (UCT), posted by SE-user user47299
I'm sorry: Are you asking if its harder to construct YM on a torus than on $\mathbb{R}^4$?

This post imported from StackExchange Physics at 2014-06-27 11:24 (UCT), posted by SE-user user1504
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?

This post imported from StackExchange Physics at 2014-06-27 11:24 (UCT), posted by SE-user user47299
Yes. The problem is that the infinite volume limit leads to divergences not present in finite volume. These divergences reflect real physics; they tell you that the gluons are confined on long distance scales.

This post imported from StackExchange Physics at 2014-06-27 11:24 (UCT), posted by SE-user user1504
You can't avoid dealing with color confinement once the spacetime volume is large enough. This is the big obstacle, the one the Clay prize is aimed at.

This post imported from StackExchange Physics at 2014-06-27 11:24 (UCT), posted by SE-user user1504
Perfect, thanks.

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

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