N = 4 supersymmetry & Pati Salam

[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?

 

[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 :)

 

[Entropia]

I cannot provide a definite answer without further analysis and evidence. However, I can offer some insights and thoughts on the topic.

First, let's clarify some terms. N = 4 supersymmetry (SUSY) is a theoretical framework in particle physics that describes a relationship between fermions (particles with half-integer spin) and bosons (particles with integer spin). This theory predicts the existence of supersymmetric particles, which have not yet been observed experimentally.

Pati Salam refers to a Grand Unified Theory (GUT) proposed by physicists Jogesh Pati and Abdus Salam in the 1970s. GUTs aim to unify the three fundamental forces of nature (electromagnetism, strong nuclear force, and weak nuclear force) into one theory. Pati Salam GUT specifically proposes a symmetry between quarks and leptons, which can be described by the group SU(4) x SU(2)_L x SU(2)_R.

Now, to address your question about treating R symmetry as a gauge symmetry, we first need to understand what gauge symmetry means. In physics, gauge symmetry is a mathematical concept that describes the invariance of a physical theory under a certain transformation. In other words, the theory remains the same even if we change the variables or parameters used to describe it. This is a fundamental principle in modern physics.

In the case of N = 4 SUSY, the R symmetry is a type of gauge symmetry that describes the invariance of the theory under rotations in the superspace (a mathematical space that includes both ordinary space and superspace). So, in a sense, R symmetry is already being treated as a gauge symmetry in N = 4 SUSY.

As for using the su(2)_L of the Lorentz group to explain isospin, it is possible in principle, but it would require further analysis and evidence to support such a claim. It is also worth noting that the su(2)_L of the Lorentz group is already a well-established concept in particle physics, known as the weak isospin, and it has been successfully incorporated into the Standard Model of particle physics.

In summary, the relationship between N = 4 SUSY and Pati Salam GUT is an interesting topic for further exploration and research. However, it is important to approach it with caution and not make any definitive claims without sufficient evidence and analysis.