Super Minkowski space
In mathematics and physics, super Minkowski space or Minkowski superspace is a supersymmetric extension of Minkowski space, sometimes used as the base manifold for superfields. It is acted on by the super Poincaré algebra.
Informally, super Minkowski space can be thought of as the super Poincaré algebra modulo the algebra of the Lorentz group, in the same way that ordinary Minkowski spacetime can be viewed as the cosets of the ordinary Poincaré algebra modulo the action of the Lorentz algebra. The coset space is naturally affine, (lacking an origin) and a nilpotent anti-commuting behavior of the fermionic directions arises naturally from the Clifford algebra associated with the Lorentz group.
The underlying supermanifold of super Minkowski space is isomorphic to a super vector space given by the direct sum of ordinary Minkowski spacetime in d dimensions (often taken to be 4) and a number N of real spinor representations of the Lorentz algebra. (When d is 2 mod 4 this is slightly ambiguous because there are 2 different real spin representations, so one needs to replace N by a pair of integers N=N1+N2, though some authors use a different convention and take N copies of both spin representations.)
However this construction is misleading for two reasons: first, super Minkowski space is really an affine space over a group rather than a group, or in other words it has no distinguished "origin", and second, the underlying supergroup of translations is not a super vector space but a nilpotent supergroup of nilpotent length 2. This supergroup has the following Lie algebra. Suppose that M is Minkowski space, and S is a finite sum of irreducible real spinor representations. Then there is an invariant symmetric bilinear map [,] from S×S to M that is positive definite in the sense that the image of s×s is in the closed positive cone of M, and is nonzero if s is nonzero. This bilinear map is unique up to isomorphism. The Lie superalgebra has M as its even part, S as its odd or fermionic part, and the Lie bracket is given by [,] (and the Lie bracket of anything in M with anything is zero).
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