superspace

Superspace is a framework in theoretical physics used to formulate supersymmetric theories. It extends ordinary spacetime by including fermionic coordinates that anticommute. In four-dimensional theories, the standard superspace coordinates are x^μ for spacetime and a pair of Weyl spinor coordinates θ^α and \bar{θ}_{\dot α}, which are Grassmann variables. The number of Grassmann coordinates reflects the amount of supersymmetry (N-extended supersymmetry).

Supercharges act as translations in superspace; in convenient representations they are realized as differential operators on x and θ, mixing bosonic and fermionic directions. A superfield is a function on superspace; expanding it in a finite series in the Grassmann variables yields a finite set of component fields with spins ranging from 0 to 1 (or higher for extended theories).

Chiral and vector superfields are common building blocks. A chiral superfield Φ satisfies D-bar Φ = 0 and can be expanded in θ, \bar{θ} with a complex scalar, a Weyl fermion, and auxiliary fields; a vector superfield V is real and contains gauge fields and gauginos. Interactions are typically written as integrals over superspace, such as a superpotential integral over chiral superspace or gauge interactions via vector superfields.

Beyond flat space, superspace concepts extend to curved superspace in supergravity. Mathematically, superspace is modeled as a supermanifold, with coordinates that anticommute; foundational work by Berezin and Leites formalized this viewpoint. Generalizations include harmonic and projective superspaces that provide alternative methods for handling extended supersymmetry.