# N = 4 supersymmetry & Pati Salam

1. Aug 25, 2008

### Jim Kata

I was reading a little bit about N = 4 SUSY, and I couldn't help but think that it's like the Pati Salam GUT. To get an N = 4 SUSY theory you can take SO(1,9) and compactify it to four dimensions. In doing so, you get Spin(1,3)XSU(4), where SU(4) is the R symmetry of the theory and Spin(1,3) is just the Lorentz group. The lie algebra of the Lorentz group
su(2)_L x su(2)_R. So you have the lie algebra of the theory being su(4)xsu(2)_L x su(2)_R, this is spontaneously broken to su(3)xsu(2)_Lxu(1). I guess my question is can R symmetry be treated just a gauge symmetry? I have heard that some people have tried to use the su(2)_L of the Lorentz group to explain isospin, but I haven't read any papers about it. Is what I'm saying at all possible, and if not, why not?

2. Aug 25, 2008

### BenTheMan

I think you're mixing up space-time symmetries and gauge symmetries.

So far as I know (I'm an N=1 guy, so SU(4)_R sounds a bit weird!), the R symmetry is a symmetry admitted by the supercharges. You'll get the supersymmetries by the properties of your compact space---I think what you mean is that you have SU(4) (SO(4)?) holonomy. So, for example, if you start with a PS gauge group in ten dimensions (N=1), and compactify on a 6-torus, you should end up with N=4 supersymmetric Pati-Salam in 4 dimensions.

Either way, you have to start out with the PS gauge group at the beginning.

A better way to do things is to start with E_6 or SO(10) in 5 dimensions, and compactify on an orbifold. This kills some of your SUSYs, AND you can break the GUT to PS. See these two papers:
http://arxiv.org/abs/hep-ph/0403065
http://arxiv.org/abs/hep-ph/0409098
The first one outlines a general procedure, and the second paper starts with a 5-d E6 theory and compactifies on an orbifold. Then the authors show how to embed the whole thing into string theory.

Either way, I'm sure I've screwed up something, and blechman will be along to clean up my mess :)