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One of the famous "millennium prize problems" is the "Yang–Mills existence and mass gap" problem, which in its official description by E. Witten and A. Jaffe has the following form:

"Prove that for any compact simple gauge group $G$, a non-trivial quantum Yang–Mills theory exists on $\mathbb{R}^{4}$ and has a mass gap $∆>0$. Existence includes establishing axiomatic properties at least as strong as those cited in [Wightman, Streater: "PCT, Spin and Statistics, and All That"] and [Osterwalder, Schrader: "Axioms for Euclidean Green’s functions"]."

I would like to know about the current status of this problem. Since the official problem announcement around 2000, there have been many successes in mathematical QFT. I am not only talking about axiomatic and constructive field theory, but also about modern approaches, like algebraic QFT (and is generalizations, i.e. perturbative AQFT and locally covariant QFT) as well as in functorial (and topological) QFT. So what is the current status, especially with respect to all this recent mathematical developments?

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    $\begingroup$ More on mathematical aspects of Yang-Mills. $\endgroup$
    – Qmechanic
    Oct 9, 2021 at 10:24
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    $\begingroup$ Current status: still unsolved. Many unsuccessful attempts published every year. $\endgroup$ Oct 9, 2021 at 12:26
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    $\begingroup$ @J.G. Asking if any weaker results for the YMG existing might be a useful question in it's own right if properly phrased. Not exactly my area of expertise (which is pizza consumption). :-) $\endgroup$ Oct 9, 2021 at 17:58
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    $\begingroup$ I am aware of the fact that it is not solved. I guess, we would have heard of it. My question was rather if there are some advances on this question or minor achievements in the past years, especially with respect to modern mathematical approaches of QFT. $\endgroup$ Oct 9, 2021 at 22:22
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    $\begingroup$ Of course this is still far from a solution to the Clay YM problem, but there is some progress in this general area. Namely, it is the development of new rigorous techniques based on stochastic quantization. See this article by Chandra, Chevyrev, Hairer and Shen arxiv.org/abs/2006.04987 One of the main unsolved problem with this approach is being able to handle just renormalizable theories. $\endgroup$ Oct 11, 2021 at 14:45

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Of course this is still far from a solution to the Clay YM problem, but there is some progress in this general area. Namely, it is the development of new rigorous techniques based on stochastic quantization. See this article by Chandra, Chevyrev, Hairer and Shen https://arxiv.org/abs/2006.04987

One of the main unsolved problem with this approach is being able to handle just renormalizable theories. This would presumably need to be addressed on $\phi^4$ in 4d first before YM in 4d.

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