Short-time limit of a supersymmetric path integral

For a Witten index,the path integral of supersymmetric quantum mechanics has periodic fermionic boundary conditions. They differ from the antiperiodic conditions for an ordinary thermal partition function. Under suitable trace and regulator assumptions, positive-energy bosonic and fermionic partners cancel, permitting a short-time evaluation from the surviving zero modes. A symmetry insertion instead imposes the twisted boundary conditions of an equivariant supertrace.

For the supersymmetric oscillator with bosonic energies $E_n^B=n$ and fermionic energies $E_n^F=n+1$ in units with unit oscillator gap,Pairing before taking $\beta \to 0$ preserves the single unpaired ground state. Taking separate short-time limits of the two traces gives divergent quantities and obscures that cancellation. A continuum spectrum or ill-defined trace requires additional analysis; supersymmetry alone does not justify exchanging all limits and integrals.