# Supersymmetric Chern-Simons theories in $d=3$

I am reading up on Chern-Simons matter theories in $d=3$. Here is the quote (from http://thesis.library.caltech.edu/7111 page 15) that I am having trouble with:

One could also add a supersymmetric Chern-Simons term for the gauge multiplet. This restricts one to at most N = 3 supersymmetry. However, in the presence of a Chern-Simons term alone (i.e., no Yang-Mills term), the gauge field is nonpropagating, and with a clever choice of matter content and superpotential, one can obtain very large amounts of supersymmetry

Context: The discussion is about possible supersymmetric constructions using vector multiplets. There are two things I do not understand:

1. Is it true that in the presence of a CS term alone (No Yang-Mills, no matter), the supersymmetry is limited to ${\cal N}=3$. If so, why?

2. I understand that when we add matter in a clever way one can obtain extended supersymmetry (as in the ABJM paper http://arxiv.org/abs/0806.1218). What I don't understand is that what does have to do with the gauge field being non-propagating??

I guess that the answer to my second question lies in the answer to the first. Any help?

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## 2 Answers

To your second question, the answer is that the chern-simons theory is a topological field theory, and TFTs are explicitly background independent. Now, what is a gauge field? From H.Gomes' thesis, which you can get on the arxiv, you can see that a gauge field is a section of the fibre bundle formed (the fibre bundle is usually assumed to be spacetime, I.e., the ADM split). Quite obviously, the background independence (BI) of the CS shows us that, it is not a "gauge field" in the sense of the above. This shows us that it does not propagate WITH RESPECT TO THE BACKGROUND SPACE. This is (probably) what the thesis' author meant. If we couple the Yang Mills to it, then it is background dependent, and thus it does propagate w.r.t the background space.

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OK. I think I understand that. But, why do we need a nonpropagating gauge field to extend the supersymmetry? Can we not make a clever choice of matter content with a Yang Mills gauge field and extend SUSY? –  Prahar Aug 17 at 16:10
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See that if we have $\mathcal{N}$ supersymmetry generators, then we have $2^\mathcal{N}$ other supersymmetric particles. Given $\mathcal{N}=3$, we see that we must have $8$ superparticles associated to the gauge multiplet in question. A supersymmetry generator is basically a $1$-form (the exterior derivative of a $0$-form, i.e., the "normal derivative" of a function) on a manifold, which "points away" from the manifold.

If the gauge field is not propagating with respect to the background space, then the SUSY can be extended - but only up to a certain limit - this limit depends on the number of particles in the gauge multiplet. In this case, the limit is $3$ because of the aspects of the Chern-Simons.

Of course, the Yang-Mills model has to be coupled to the clever choice of matter content to do what you have talked about. This can extent the SUSY, but even then, the superpotential does most of the extension of the SUSY. This is because, as can be noted by studying the superpotential, it is one of the parameters of SUSY.

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