McKean–Singer formula

Remarks

Idea of the proof: $P^*P$ and $PP^*$ have discrete spectra, with $\ker P^*P = \ker P$ and $\ker PP^* = \ker P^*$. For $\lambda > 0$, $P$ maps the $\lambda$-eigenspace of $P^*P$ isomorphically onto the $\lambda$-eigenspace of $PP^*$, with inverse $\lambda^{-1} P^*$. So all terms with $\lambda > 0$ cancel in the difference of the traces, and only $\dim \ker P - \dim \ker P^*$ remains. The left side does not depend on $t$, so one may let $t \to 0^+$. Then each trace is an integral of the heat kernel on the diagonal, which has an asymptotic expansion in powers of $t$ with local coefficients. Hence the index is the integral of a local density built from the coefficients of $P$, and the singular terms of the expansion cancel. Example: for $P = d + d^*$ from even to odd forms, the formula reads $\chi(M) = \sum_k (-1)^k \operatorname{tr} e^{-t \Delta_k}$, where $\Delta_k$ is the Hodge Laplacian on $k$-forms.

Going deeper

For a student

The formula is the starting point of the heat equation proof of the index theorem. For small $t$ the heat traces are given by the Minakshisundaram–Pleijel expansion, so the index is the integral of the constant term of a local expansion. The local index theorem identifies this constant term with a characteristic form, which proves the Atiyah–Singer theorem for Dirac operators.