To study $S$-duality in more detail, i tried to read, Electric-magnetic duality and the Geometric Langlands Program. In section 2.1, there is a comment about 10d SYM. Excerpt from the paper above,

Ten dimensions is the maximum possible dimension for supersymmetric Yang-Mills theory by the virtue of Nahm's theorem.

I know that Nahm's theorem as a theorem for classfication of superconformal algebra. (Nahm's paper: SUPERSYMMETRIES AND THEIR REPRESENTATIONS). I just took a glance of this paper, but I couldn't find relevant physics behind its mathematical description.

I want to know how the ten dim, SYM is the maximum possible dimension. And know some physical application like SYM in 10d for Nahm's theorem.


The point, really, is that if you're trying to contruct supersymmetric gauge theories, you want the superpartners to have lower spin than the gauge bosons. If you have a QFT with higher spin fields, you'd call it a gravity theory, not a Yang-Mills theory.

Nahm's classification of representations tells us that in D > 10, gauge multiplets inevitably contain massless particles with spin > 1. Likewise, in D > 11, graviton multiplets contain massless particles of spin > 2. It's not easy to make sense of such things in standard QFT, so these are not counted as valid theories.

Nahm makes this argument at the end of Section 4, although he relies on the reader to know the folklore about high spin massless particles.


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