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.

Informal sketch

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, and a nilpotent anti-commuting behavior of the fermionic directions arises naturally from the Clifford algebra associated with the Lorentz group.

Definition

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 and a number N of real spinor representations of the Lorentz algebra.
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 .
The dimensions of the irreducible real spinor representations for various dimensions d of spacetime are given by the following table:
Spacetime dimension, dReal dimension of spinor representationsStructureBilinear form
11RealSymmetric
21, 1RealTwo dual representations
32RealAlternating
44Complex Alternating
58Quaternionic Symmetric
68, 8Quaternionic Two dual representations
716Quaternionic Alternating
816Complex Symmetric
916RealSymmetric
1016, 16RealTwo dual representations
1132RealAlternating
1264Complex Alternating

The table repeats whenever the dimension increases by 8, except that the dimensions of the spin representations are multiplied by 16.

Notation

In the physics literature, Minkowski spacetime is often specified by giving the dimension d of the even bosonic part, and the number of times N that each irreducible spinor representation occurs in the odd fermionic part. In mathematics, Minkowski spacetime is sometimes specified in the form Mm|n where m is the dimension of the even part and n the dimension of the odd part. The relation is as follows: the integer d in the physics notation is the integer m in the mathematics notation, while the integer n in the mathematics notation is a power of 2 times the integer N in the physics notation, where the power of 2 is the dimension of the irreducible real spinor representation. For example, the, Minkowski spacetime is M4|4 while the Minkowski spacetime is M4|8. When the dimension d or m is 2 mod 4 there are two different irreducible real spinor representations, and authors use various different conventions.
In physics the letter P is used for a basis of the even bosonic part of the Lie superalgebra, and the letter Q is often used for a basis of the complexification of the odd fermionic part, so in particular the structure constants of the Lie superalgebra may be complex rather than real. Often the basis elements Q come in complex conjugate pairs, so the real subspace can be recovered as the fixed points of complex conjugation.