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  "body": "\n## Statement\n\nLet $M$ be a closed smooth manifold with a smooth positive density, and let $E$, $F$ be Hermitian vector bundles over $M$.\nLet $P \\colon C^\\infty(M; E) \\to C^\\infty(M; F)$ be an [[elliptic differential operator|MT9P2X85V8]] of positive order with formal adjoint $P^*$.\nThen for every $t \u003e 0$ the operators $e^{-tP^*P}$ and $e^{-tPP^*}$ are [[trace class|Z2CP530WDE]], and\n$$\\operatorname{ind} P = \\operatorname{tr} e^{-tP^*P} - \\operatorname{tr} e^{-tPP^*},$$\nwhere $\\operatorname{ind} P$ is the [[analytic index|44GBFZ16SG]].\n\n## Remarks\n\nIdea of the proof: $P^*P$ and $PP^*$ have discrete spectra, with $\\ker P^*P = \\ker P$ and $\\ker PP^* = \\ker P^*$.\nFor $\\lambda \u003e 0$, $P$ maps the $\\lambda$-eigenspace of $P^*P$ isomorphically onto the $\\lambda$-eigenspace of $PP^*$, with inverse $\\lambda^{-1} P^*$.\nSo all terms with $\\lambda \u003e 0$ cancel in the difference of the traces, and only $\\dim \\ker P - \\dim \\ker P^*$ remains.\nThe left side does not depend on $t$, so one may let $t \\to 0^+$.\nThen each trace is an integral of the [[heat kernel|327NP6S7H5]] on the diagonal, which has an asymptotic expansion in powers of $t$ with local coefficients.\nHence the index is the integral of a local density built from the coefficients of $P$, and the singular terms of the expansion cancel.\nExample: 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.\n\n## Explanations\n\n### L1 human\n\nThe index counts solutions of $Pu = 0$ minus solutions of $P^*v = 0$.\nThe heat operators $e^{-tP^*P}$ and $e^{-tPP^*}$ weight every eigenvalue by $e^{-t\\lambda}$.\nAll nonzero eigenvalues appear in pairs, one on each side, so they cancel, and only the zero modes, which give the index, survive.\n\n### L2 human\n\nThe formula is the starting point of the heat equation proof of the index theorem.\nFor small $t$ the heat traces are given by the [[Minakshisundaram–Pleijel expansion|45P2SEGGR1]], so the index is the integral of the constant term of a local expansion.\nThe local index theorem identifies this constant term with a characteristic form, which proves the Atiyah–Singer theorem for Dirac operators.\n",
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