# Limit on space-time dimension from susy

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

## 1 Answer

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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