Global analysis, analysis on manifolds Index theory Theorem

Local index theorem

Builds on Dirac operator on a spin manifold Heat kernel on a Riemannian manifold McKean–Singer formula Â-genus as the index of the Dirac operator Chern–Weil homomorphism Spin structure

Statement

Let $M$ be a closed Riemannian manifold of even dimension $n$ with a spin structure and spinor bundle $S = S^+ \oplus S^-$ . Let $W$ be a Hermitian vector bundle over $M$ with a unitary connection, and let $D_W$ be the Dirac operator on $S \otimes W$ . Let $k_t(x, y)$ be the heat kernel 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 $S^- \otimes W$ . Then, uniformly in $x \in M$ , $\lim_{t \to 0^+} \operatorname{str} k_t(x, x) \, \mathrm{vol}(x) = \Big[ \hat A(TM, \nabla) \operatorname{ch}(W, \nabla^W) \Big]_{n}(x).$ Here $\hat A(TM, \nabla)$ and $\operatorname{ch}(W, \nabla^W)$ are the Chern–Weil forms of the Levi-Civita connection and of the connection on $W$ . They are normalized to represent the $\hat A$ -class and the Chern character, and $[\cdot]_n$ is the component of degree $n$ .

Remarks

The McKean–Singer formula gives $\operatorname{ind} D_W^+ = \int_M \operatorname{str} k_t(x, x) \, \mathrm{vol}(x)$ for every $t > 0$ . Letting $t \to 0^+$ 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 for $W$ trivial. The 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. The cancellation was first proved by Patodi for the de Rham complex and by Gilkey with invariance theory. Getzler gave a proof by a rescaling of the Clifford variables, and Bismut gave a probabilistic proof. Example: 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.

Going deeper

For a student

The theorem is a sharpened form of the heat equation proof of the index theorem. It gives the index density pointwise, which is what one needs on manifolds with boundary, for families, and for the Atiyah–Patodi–Singer theorem. Its proof by Getzler rescaling is now the standard route to the index theorem in courses on Dirac operators.

Reading path

To understand this page from the ground up, read these 15 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Analytic index concept
Chern–Weil homomorphism concept
Dirac operator on a spin manifold concept
Elliptic operator on a compact manifold concept
Heat kernel on a Riemannian manifold concept
Pontryagin class concept
Riemannian manifold concept
Scalar curvature concept
Spectrum of the Laplace–Beltrami operator concept
Spin structure concept
Trace class operator concept
Â-genus as the index of the Dirac operator statement
Minakshisundaram–Pleijel asymptotic expansion statement
McKean–Singer formula statement
Local index theorem statement

Tag	`1YKCH218MN`
URI	`m2://1YKCH218MN`
Kind	statement, theorem
Writing	W0 stub
MSC	58J20, 58J35, 53C27
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About	Dirac operator on a spin manifold, McKean–Singer formula
Uses	Dirac operator on a spin manifold, Heat kernel on a Riemannian manifold, McKean–Singer formula, Â-genus as the index of the Dirac operator, Chern–Weil homomorphism, Spin structure