# Superfield

In theoretical physics, one often analyzes theories with supersymmetry in which superfields play a very important role. A superfield is a function defined in superspace which properly packages the various fields of a supermultiplet, namely, the array of fermion and boson fields related among themselves by supersymmetry. Superfields were introduced by Abdus Salam and John Strathdee in 1979.[1]

## Overview

In four dimensions, the simplest example—namely, the minimal N = 1 supersymmetry—may be written as a vector in a superspace with four extra fermionic coordinates $\theta^1,\theta^2,\bar\theta^1,\bar\theta^2$, transforming as a two-component spinor and its conjugate. More generally, there are 4N extra fermionic (Grassmann number) coordinates .

A more coordinate-free description of the superspace is that it's the quotient space of the super-Poincaré group divided by the Lorentz group.

Every superfield, i.e. a field that depends on all coordinates of the superspace (or in other words, an element of a module of the algebra of functions over superspace), may be expanded with respect to the new fermionic coordinates. There exists a special kind of superfields, the so-called chiral superfields, that, in the chiral representation of supersymmetry, depend only on the variables θ but not their conjugates. The last term in the corresponding expansion, namely $F \theta^1\theta^2$, is called the F-term. Other superfields include vector superfields.

There also exist superfields in theories with larger supersymmetry.

Manifestly supersymmetric actions may also be written as integrals over the whole superspace. Some special terms, such as the superpotential, may be written as integrals over θs only. They are also referred to as F-terms, much like the terms in the ordinary potential that arise from these terms of the supersymmetric Lagrangian.

## References

1. ^ Salam, A.; Strathdee, J. (1974). "Super-gauge transformations". Nuclear Physics B 76 (3): 477–201. Bibcode:1974NuPhB..76..477S. doi:10.1016/0550-3213(74)90537-9.