Fermionic Fock space

The fermionic Fock space of $n$ modes is the exterior algebra ${\Lambda }^{\bullet }{\mathbb{C}}^n$. Creation acts by exterior multiplication and annihilation by contraction.

Fermionic creation and annihilation operators satisfy $\{a_i,a_j^{†}\}={\delta }_{ij}$ and vanishing equal-type anticommutators. These relations make each mode either empty or singly occupied.

The fermion number operator counts occupied fermionic modes. Its parity $(-1{)}^F$ is positive on even exterior degree and negative on odd exterior degree.

A supertrace is the trace weighted by fermion parity. Supersymmetry makes paired positive-energy states cancel in a supertrace.

An equivariant supertrace inserts a symmetry $f$ as well as fermion parity, $\mathrm{Tr}((-1{)}^{F}fA)$. In a path integral, $f$ produces boundary conditions twisted by its action.