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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?

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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.

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  • $\begingroup$ So is the difficult in fully constructing the theory solely because a torus is compact (ie bounded)? $\endgroup$ – user47299 Jun 25 '14 at 16:08
  • $\begingroup$ I'm sorry: Are you asking if its harder to construct YM on a torus than on $\mathbb{R}^4$? $\endgroup$ – user1504 Jun 25 '14 at 17:25
  • $\begingroup$ 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? $\endgroup$ – user47299 Jun 25 '14 at 17:41
  • $\begingroup$ 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. $\endgroup$ – user1504 Jun 25 '14 at 17:56
  • $\begingroup$ 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. $\endgroup$ – user1504 Jun 25 '14 at 18:26

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