Fierz identities for real Clifford algebras and the number of supercharges

Abstract

One considers supersymmetric gauge theories in quantum mechanics with the bosons and fermions belonging to the adjoint representation of the gauge group. One shows that the supersymmetry constraints are related to the existence of certain Fierz identities for real Clifford algebras. These identities are valid when one has 2, 4, 8, and 16 supercharges.