In theoretical physics, a supermultiplet is a representation of a supersymmetry algebra, possibly with extended supersymmetry. Then a superfield is a field on superspace which is valued in such a representation. Naïvely, or when considering flat superspace, a superfield can simply be viewed as a function on superspace. Formally, it is a section of an associated supermultiplet bundle. Phenomenologically, superfields are used to describe particles. It is a feature of supersymmetric field theories that particles form pairs, called superpartners where bosons are paired with fermions. These supersymmetric fields are used to build supersymmetric quantum field theories, where the fields are promoted to operators.
History Superfields were introduced by Abdus Salam and J. A. Strathdee in a 1974 article. Operations on superfields and a partial classification were presented a few months later by Sergio Ferrara, Julius Wess and Bruno Zumino.
Naming and classification The most commonly used supermultiplets are vector multiplets, chiral multiplets (in d = 4 , N = 1 {\displaystyle d=4,{\mathcal {N}}=1} supersymmetry for example), hypermultiplets (in d = 4 , N = 2 {\displaystyle d=4,{\mathcal {N}}=2} supersymmetry for example), tensor multiplets and gravity multiplets. The highest component of a vector multiplet is a gauge boson, the highest component of a chiral or hypermultiplet is a spinor, the highest component of a gravity multiplet is a graviton. The names are defined so as to be invariant under dimensional reduction, although the organization of the fields as representations of the Lorentz group changes. The use of these names for the different multiplets can vary in literature. A chiral multiplet (whose highest component is a spinor) may sometimes be referred to as a scalar multiplet, and in d = 4 , N = 2 {\displaystyle d=4,{\mathcal {N}}=2} SUSY, a vector multiplet (whose highest component is a vector) can sometimes be referred to as a chiral multiplet.
Superfields in d = 4, N = 1 supersymmetry Conventions in this section follow the notes by Figueroa-O'Farrill (2001). A general complex superfield Φ ( x , θ , θ ¯ ) {\displaystyle \Phi (x,\theta ,{\bar {\theta }})} in d = 4 , N = 1 {\displaystyle d=4,{\mathcal {N}}=1} supersymmetry can be expanded as
Φ ( x , θ , θ ¯ ) = ϕ ( x ) + θ χ ( x ) + θ ¯ χ ¯ ′ ( x ) + θ ¯ σ μ θ V μ ( x ) + θ 2 F ( x ) + θ ¯ 2 F ¯ ′ ( x ) + θ ¯ 2 θ ξ ( x ) + θ 2 θ ¯ ξ ¯ ′ ( x ) + θ 2 θ ¯ 2 D ( x ) {\displaystyle \Phi (x,\theta ,{\bar {\theta }})=\phi (x)+\theta \chi (x)+{\bar {\theta }}{\bar {\chi }}'(x)+{\bar {\theta }}\sigma ^{\mu }\theta V_{\mu }(x)+\theta ^{2}F(x)+{\bar {\theta }}^{2}{\bar {F}}'(x)+{\bar {\theta }}^{2}\theta \xi (x)+\theta ^{2}{\bar {\theta }}{\bar {\xi }}'(x)+\theta ^{2}{\bar {\theta }}^{2}D(x)} , where ϕ , χ , χ ¯ ′ , V μ , F , F ¯ ′ , ξ , ξ ¯ ′ , D {\displaystyle \phi ,\chi ,{\bar {\chi }}',V_{\mu },F,{\bar {F}}',\xi ,{\bar {\xi }}',D} are different complex fields. This is not an irreducible supermultiplet, and so different constraints are needed to isolate irreducible representations.
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