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I read an argument saying that it would be impossible to write down a super-symmetric theory in more than 11 dimensions, this limit coming from the dimension of the Clifford algebra that goes as $2^{\frac{N}{2}}$ or $2^{\frac{N-1}{2}}$ for $N$ even or odd, respectively.

I haven't studied a lot of susy and I don't see how it wouldn't be possible to create a super-symmetric multiplet in higher dimensions as long as we add enough scalar fields (${\cal{N}} =1$ in my example) to match the fermionic degrees of freedom.

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You can go up to 12 dimensions, if you have 2 time dimensions. – Ron Maimon Nov 2 '12 at 17:36

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Well, if we want massless supermultiplets, for instance massless gravitons, then the multiplet will also have to contain massless particles with spin greater than 2. Such particles have to be associated with a gauge symmetry, but it's not Yang-Mills as in massless spin-1, diffeomorphisms as in massless spin-2, or SUGRA as in massless 3/2. So what is that gauge symmetry physically?

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There's an even stronger result if I recall correctly (the name escapes me at the moment): you can't have an interacting relativistic qft with a finite number of fundamental particles with spin greater than 2. (String theory evades this restriction by having an infinite tower of string modes.) – Michael Brown Jan 1 at 11:45

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