Superfields and the Super Poincare Group:
I describe below the big picture of how supersymmetry is treated mathematically via superfields. One big goal of my research as a graduate student was to understand carefully what precisely a superfield is from a mathematical viewpoint.
One central research goal of mine is to investigate the mathematical basis for supersymmetric field theory. Ultimately, we'd like to better understand the geometry underlying the superfield construction. Superfields are functions with rather unusual domains. The domain of a superfield contains abstract numbers called Grassmanns. Grassmanns have the strange property that they can be non-zero and yet when you square them you get zero. The domain of a superfield is parametrized by variable constructed from infinite real or complex linear sums of products of Grassmans. Probably sounds worse than it is. The reason for studying such unusual spaces is that these superspaces admit a very natural action of the super Poincaire group. The super Poincaire group is the group formed by the super Poicaire algebra which is the only known physically reasonable extension of the Poicaire algebra. In addition to the usual translations, boosts, and rotations the super Poincaire group has what are called supersymmetry transformations. These supersymmetry transformations become supertranslations when realized on a particular superspace. This particular superspace is known as N=1 rigid superspace and is one of the central objects of interest in supersymmetric field theory. Supersmooth functions from rigid superspace form representations of the super Poicaire group. As such it is easy to construct models which have supersymmetry ( invariance under the super symmetry transformations ) with the help of superfields. This invariance falls out very naturally in the construction. What is beautiful about it is that what seems simple at the level of superfields can entail very messy things at the level of usual relativistic field theory. Superfield models reduce to ordinary relativistic field models (mappings from Minkowski space as opposed to superspace) at what is known as the "component field level". Component fields are the typical fields discussed in relativistic field theory ( scalar fields, Weyl spinors, vector fields,... ), a whole ensemble are hidden within a particular superfield. Without the superfield construction one would have to find how to couple the ensemble together under the supersymmetry transformations. These couplings are somewhat messy, yet fall out almost for free within the superfield construction. At this time I believe that my advisor from NCSU, R.O. Fulp, and I have worked out a reasonable and rigorous formulation for most of the constructions described above. Our work on supersymmetric Yang-Mills theory is still in progress, but nearing completion. My next project is to make sense of what happens for gauge theory coupled to matter, we have an idea but it needs to be worked out in more detail. If you would like to learn about these things either from a math or physics perspective I would be very happy to help you. Just stop by my office hours or send me an email. Don't be scared by the plethora of unknown terms, its like anything else, you have to take it one step at a time. I'll help you get started if you have interest. The interest is up to you.
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Last Modified: 7-16-08