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, (lacking an origin) 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 (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).

The dimensions of the irreducible real spinor representations for various dimensions d of spacetime are given by the following table:

Spacetime dimension, d	Real dimension of spinor representations	Structure	Bilinear form
1	1	Real	Symmetric
2	1, 1	Real	Two dual representations
3	2	Real	Alternating
4	4	Complex (dimension 2)	Alternating
5	8	Quaternionic (dimension 2)	Symmetric
6	8, 8	Quaternionic (dimension 2, 2)	Two dual representations
7	16	Quaternionic (dimension 4)	Alternating
8	16	Complex (dimension 8)	Symmetric
9	16	Real	Symmetric
10	16, 16	Real	Two dual representations
11	32	Real	Alternating
12	64	Complex (dimension 32)	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 M^m|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 (or twice this if there are two irreducible real spinor representations). For example, the d = 4, N = 1 Minkowski spacetime is M^4|4 while the N = 2 Minkowski spacetime is M^4|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.

References

• Deligne, Pierre; Morgan, John W. (1999), "Notes on supersymmetry (following Joseph Bernstein)", in Deligne, Pierre; Etingof, Pavel; Freed, Daniel S.; Jeffrey, Lisa C.; Kazhdan, David; Morgan, John W.; Morrison, David R.; Witten., Edward (eds.), Quantum fields and strings: a course for mathematicians, Vol. 1, Providence, R.I.: American Mathematical Society, pp. 41–97, ISBN 978-0-8218-1198-6, MR 1701597