I'm not very experienced in this topic, but I read a short description of the YangMills existence and mass gap problem, and as long as I understood it has mainly physical consequences and implications. Therefore, I was just curious what it has to do with mathematics? And what are the mathematical and general consequences of a possible solution to it?

3$\begingroup$ YangMills asks for a proof that a certain mathematical model exists with certain mathematical properties. This is a mathematical question. $\endgroup$– Qiaochu YuanJul 25, 2012 at 23:15

$\begingroup$ There are many mathematical questions that are born from investigation of mathematical models of physical theories. That is essentially a characterization of the entire field of mathematical physics, which is often regarded as a subfield of mathematics. Certainly, a resolution of the YangMills mass gap problem would be a major advance within a subfield of mathematical physics, with lots of implications for other similar questions within the same field. Is this kind of answer sufficient or are you interested specifically in what these questions are? $\endgroup$– Igor KhavkineJul 26, 2012 at 1:02

$\begingroup$ Actually, I was looking for a more specific answers. Regarding both mathematical consequences and consequences to other branches of science. $\endgroup$– terettJul 26, 2012 at 11:24
3 Answers
There is a long, long list of mathematical subjects that were either pioneered or significantly inspired by results in quantum field theory. However, while physicists may trust the manipulations they do in QFT, and the results of those manipulations have been spectacularly successful, for almost every interesting quantum field theory, there isn't even a rigorous definition or existence proof, much less a justification behind the manipulations that led to the invention of, for example, SeibergWitten invariants.
Solving the mass gap problem in YangMills would represent the successful rigorous existence of a very nontrivial quantum field theory and the demonstration of a very nontrivial result about that field theory (that hasn't even been adequately demonstrated using physical techniques). While there probably aren't many direct mathematical consequences to the existence of a mass gap, the techniques involved would almost assuredly be applicable towards the large number of QFT inspired results in mathematics.
Mathematical physics is the study of physical questions from the point of view of full mathematical rigor. Physical questions are phrased as welldefined mathematical problems, to be attacked with methods from differential geometry, functional analysis, Lie groups, topology, etc..
The mathematically rigorous construction of interacting 4dimensional relativistic quantum field theories is one of the hardest problems of mathematical physics, unsolved since over 50 years, in spite of the continuous efforts of many excellent people. The YangMills quantum field theories are believed to be the most benign ones to be constructed, but the current functional analytic tools were so far not sufficient to produce the bounds needed to show the existence of the appropriate limits.
The techniques developed for an affirmative answer of the existence problem for a YangMills quantum field theory (in the sense required by the Clay Millennium problem) would open a door to a constructive mathematical approach to many other 4dimensional relativistic quantum field theories, including those describing realistic elementary particle physics.
Stochastic partial differential equations in 4 dimensions (e.g., fluid flow) pose closely related problems, connected through analytic continuation to imaginary time. Thus the new techniques would also find applications there.
For a synopsis of what in particular needs to be achieved see my review http://physicsoverflow.org/21846 of a failed attempt to solve the problem.
A geometry (differential say) is essentially about the study of a class of connections on a differentiable fiber area (the E. Cartan's viewpoint). This class is defined in general by imposing that the associated "covariant derivation" preserves a structure; the structure itself is very often represented by a tensor (said structure tensor); the best example being the metric tensor defining a pseudoriemannian structure on a smooth manifold (when the topological conditions are matched).
The problem may then arise to seek the canonical form of the associated quadratic form. This problem is essentially equivalent to the search for a suitable frame; i.e, the search for a gauge potential for the group of the rotations of the metric.
The YangMills equations system is of EulerLagrange type. The action density being a "norm" of the curvature of a class of connections. As a result, the YangMills potential field (in fact, the YangMills connection 1form), solution of the YangMills equations is an "extremalization" of a connection in the class of connections, producing by the associated covariant derivation, the YangMills Strength Field.
This is an account about the fact that YangMills is linked to the search for the best connection (from my viewpoint).
An extremalized connection 1form necessarily makes it easier, to approach a "canonical form". With welldesigned constraints, it is possible to obtain in principle (I think), the canonical form itself.
This is an account about the fact that YangMills is linked to the search for a canonical form of a given (or even looking for) geometry, would only it be in the case where the geometry is defined by a pseudoriemannian metric structure. This understood in the sense that we give in paragraph 1.
The answer given by @user1504 to the question [1] illustrates in a clear way, (being a profane in quantum theory, I could be mistaken), the link between the problem of MassGap and the mathematics, including the holomony, the theory of hilbertian measured spaces...
However, continuing with the example of the geometry defined by a metric, the quantisation of such theory corresponds to the quantisation of gravitation (at least when the metric is lorentzian). In this context, a link between the MassGap problem and mathematics is the proof by an effective method (quantification method), of the positive mass theorem. The positivity of the mass is intimately linked to the problem of the classification of differential varieties. This classification involves the most sofisticated devices of geometric analysis such as: the AtiyahSinger index theorem, the Dirac operators, SeibergWitten operators, the variation calculus, the Ricci Flow, ... Cf [2; 3; 4].
Note. The positive mass theorem of J. Lohkamp asserts that: under the hypothesis of the dominant energy condition, the total mass for any nonvacuous isolated relativistic gravitational systems should be positive. Cf introduction of [3] for a bright presentation of all of concepts (and very more). It's at this point, I think (and if I well understand the answer of @user1504 given to the question [1]), that a constructive proof (quantised proof) of a positive mass theorem (perhaps without any hypothesis on the energy regime) is linked (is equivalent?) to the Massgap problem for the quantised gravity (if any...).
I would like to thank very much @user1504 for his answer to question [1] and for his remark on the previous version of this answer.
The quantum theory seems so fascinating, unfortunately, it falls out of my field of competence (for now).
As I said in the previous version of this answer, these ideas are borrowed from the works of so many geometers that I can not precisely say of whom these ideas are (if I have grasped something about these works).
However, I will name a few, without any intention to underestimate someone (by omission): A. Banyaga, R. Bryant, Y. ChoquetBruhat, D. Christodoulou, P. T. Chrusciel, S. Donaldson, M. Gromov (The FatherChristmas in problems and results of geometry), N. Hitchin, S. Klainerman, C. LeBrun, P. Michor, R. Penrose, S. T. Yau.
That they all, all my recognition and my admiration.
[1 ]Statement of Millenium Problem: YangMills Theory and Mass Gap
[2 ]https://arxiv.org/pdf/1704.05490

$\begingroup$ It's just to bring some light about a comment about my previous answer. $\endgroup$ Dec 8, 2021 at 20:05

$\begingroup$ I hope that it's least zigzagging (disconnectedness of ideas) compared to the answer given previously $\endgroup$ Dec 8, 2021 at 20:11

2$\begingroup$ This answer looks to me like it only concerns the classical YangMills theory, which does not exhibit a mass gap. To answer the question asked above, you'd need to somehow address quantum YangMills theory. $\endgroup$– user1504Dec 8, 2021 at 23:56

$\begingroup$ @user1504 Thanks for your remarks. You are right. I was mainly about the classic side since it's that understand better. Nevertheless, I will fill the gap $\endgroup$ Dec 10, 2021 at 5:15

$\begingroup$ I hope that it fully answer the question by filling in the gap as pointed out by @user1504 $\endgroup$ Jan 1, 2022 at 11:43