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  "body": "\n## Statement\n\nLet $M$ be a closed Riemannian manifold of even dimension $n$ with a [[spin structure|RSPZ9TCFQ8]] and spinor bundle $S = S^+ \\oplus S^-$.\nLet $W$ be a Hermitian vector bundle over $M$ with a unitary connection, and let $D_W$ be the [[Dirac operator|QMRAVX14BJ]] on $S \\otimes W$.\nLet $k_t(x, y)$ be the [[heat kernel|327NP6S7H5]] of $D_W^2$, and let $\\operatorname{str}$ be the supertrace on $(S \\otimes W)_x$, the trace on $S^+ \\otimes W$ minus the trace on $S^- \\otimes W$.\nThen, uniformly in $x \\in M$,\n$$\\lim_{t \\to 0^+} \\operatorname{str} k_t(x, x) \\, \\mathrm{vol}(x) = \\Big[ \\hat A(TM, \\nabla) \\operatorname{ch}(W, \\nabla^W) \\Big]_{n}(x).$$\nHere $\\hat A(TM, \\nabla)$ and $\\operatorname{ch}(W, \\nabla^W)$ are the [[Chern–Weil|82ZPRD39J7]] forms of the Levi-Civita connection and of the connection on $W$.\nThey are normalized to represent the $\\hat A$-class and the Chern character, and $[\\cdot]_n$ is the component of degree $n$.\n\n## Remarks\n\nThe [[McKean–Singer formula|F4WDYPQ9VP]] gives $\\operatorname{ind} D_W^+ = \\int_M \\operatorname{str} k_t(x, x) \\, \\mathrm{vol}(x)$ for every $t \u003e 0$.\nLetting $t \\to 0^+$ and using the theorem gives $\\operatorname{ind} D_W^+ = \\int_M \\hat A(TM) \\operatorname{ch}(W)$, which is the index theorem for twisted Dirac operators and contains [[the Â-genus formula|JR2M58RCW3]] for $W$ trivial.\nThe striking point is that the singular terms $t^{-n/2}, \\ldots, t^{-1}$ of the asymptotic expansion of the supertrace cancel pointwise, not only after integration.\nThe cancellation was first proved by Patodi for the de Rham complex and by Gilkey with invariance theory.\nGetzler gave a proof by a rescaling of the Clifford variables, and Bismut gave a probabilistic proof.\nExample: for $n = 2$ and $W$ trivial, $\\hat A$ has no component of degree $2$, so the limit is $0$, matching the vanishing index of the Dirac operator on a closed surface.\n\n## Explanations\n\n### L1 human\n\nThe index of a Dirac operator is a global count of solutions.\nThe local index theorem says that the heat kernel, a local object, already knows the curvature integrand that adds up to this count.\nAt every point, as time goes to zero, the heat flow reveals the characteristic form of the curvature.\n\n### L2 human\n\nThe theorem is a sharpened form of the heat equation proof of the index theorem.\nIt gives the index density pointwise, which is what one needs on manifolds with boundary, for families, and for the [[Atiyah–Patodi–Singer theorem|GGFAH1M80J]].\nIts proof by Getzler rescaling is now the standard route to the index theorem in courses on Dirac operators.\n",
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