= Short-time limit of a supersymmetric path integral
= Short-time limit
{synonym}
For a <Witten index>,
$$
I(\beta)=\operatorname{Tr}\!\left((-1)^F e^{-\beta H}\right),
$$
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,
$$
I(\beta)=\sum_{n=0}^\infty\left(e^{-\beta n}-e^{-\beta(n+1)}\right)=1,\qquad\beta>0.
$$
Pairing before taking $\beta\to0$ 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.
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