Supersymmetry Algebra - Structure of A Supersymmetry Algebra

Structure of A Supersymmetry Algebra

The general supersymmetry algebra for spacetime dimension d, and with the fermionic piece consisting of a sum of N irreducible real spinor representations, has a structure of the form

(P×Z).Q.(L×B)

where

• P is a bosonic abelian vector normal subalgebra of dimension d, normally identified with translations of spacetime. It is a vector representation of L.
• Z is a scalar bosonic algebra in the center whose elements are called central charges.
• Q is an abelian fermionic spinor subquotient algebra, and is a sum of N real spinor representations of L. (When the signature of spacetime is divisible by 4 there are two different spinor representations of L, so there is some ambiguity about the structure of Q as a representation of L.) The elements of Q, or rather their inverse images in the supersymmetry algebra, are called supercharges. The subalgebra (P×Z).Q is sometimes also called the supersymmetry algebra and is nilpotent of length at most 2, with the Lie bracket of two supercharges lying in P×Z.
• L is a bosonic subalgebra, isomorphic to the Lorentz algebra in d dimensions, of dimension d(d–1)/2
• B is a scalar bosonic subalgebra, given by the Lie algebra of some compact group, called the group of internal symmetries. It commutes with P,Z, and L, but may act non-trivially on the supercharges Q.

The terms "bosonic" and "fermionic" refer to even and odd subspaces of the superalgebra.

The terms "scalar", "spinor", "vector", refer to the behavior of subalgebras under the action of the Lorentz algebra L.

The number N is the number of irreducible real spin representations. When the signature of spacetime is divisible by 4 this is ambiguous as in this case there are two different irreducible real spinor representations, and the number N is sometimes replaced by a pair of integers (N₁, N₂).

The supersymmetry algebra is sometimes regarded as a real super algebra, and sometimes as a complex algebra with a hermitian conjugation. These two views are essentially equivalent, as the real algebra can be constructed from the complex algebra by taking the skew-Hermitian elements, and the complex algebra can be constructed from the real one by taking tensor produect with the complex numbers.

The bosonic part of the superalgebra is isomorphic to the product of the Poincare algebra P.L with the algebra Z×B of internal symmetries.

When N>1 the algebra is said to have extended supersymmetry.

When Z is trivial, the subalgebra P.Q.L is the Super-Poincaré algebra.