Global analysis, analysis on manifolds Definition

Seeley–DeWitt coefficients

Builds on Riemannian manifold Vector bundle Connection on a vector bundle Heat kernel on a Riemannian manifold Minakshisundaram–Pleijel asymptotic expansion Scalar curvature

Definition

Let $(M, g)$ be a closed Riemannian manifold of dimension $n$ and let $E$ be a Hermitian vector bundle over $M$ . Let $L$ be a Laplace-type operator on $E$ , that is, a second-order operator with principal symbol $\abs{\xi}^2 \id_E$ . Equivalently, $L = \nabla^* \nabla + F$ for a unique connection $\nabla$ on $E$ and a unique endomorphism field $F$ . Let $K_L(t, x, y)$ be the heat kernel of $L$ . There are unique smooth sections $a_k(L)$ of $\End(E)$ , $k = 0, 1, 2, \ldots$ , such that for every $N$ $K_L(t, x, x) = (4\pi t)^{-n/2} \left( \sum_{k = 0}^{N} a_k(L)(x) \, t^k + O(t^{N+1}) \right) \quad \text{as } t \to 0^+,$ uniformly in $x$ . The $a_k(L)$ are the Seeley–DeWitt coefficients of $L$ .

Remarks

Existence follows by the same parametrix construction as for the Minakshisundaram–Pleijel expansion, and uniqueness from the uniqueness of asymptotic expansions. The first two are $a_0 = \id_E$ and $a_1 = \tfrac16 S \, \id_E - F$ , where $S$ is the scalar curvature. Some authors call the integrated traces $\int_M \operatorname{tr} a_k(L) \, dv$ the heat invariants, and on manifolds with boundary there are also half-integer powers $t^{k/2}$ .

Example: take the scalar Laplacian on a flat torus, so $F = 0$ . Its heat kernel on the diagonal is $(4\pi t)^{-n/2}$ times $1$ plus exponentially small terms, so $a_0 = 1$ and $a_k = 0$ for $k \ge 1$ .

Non-example: a Dirac operator $D$ is of first order, so it is not Laplace-type and has no Seeley–DeWitt coefficients in this sense. Its square $D^2$ is Laplace-type, and the coefficients of $D^2$ are the ones used in index theory.

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Connection on a vector bundle concept
Heat kernel on a Riemannian manifold concept
Riemannian manifold concept
Scalar curvature concept
Spectrum of the Laplace–Beltrami operator concept
Vector bundle concept
Minakshisundaram–Pleijel asymptotic expansion statement
Seeley–DeWitt coefficients definition

Tag	`SSCJWAQ7AX`
URI	`m2://SSCJWAQ7AX`
Kind	definition
Writing	W0 stub
MSC	58J35, 58J50, 81T20
Defines	Seeley–DeWitt coefficients
Uses	Riemannian manifold, Vector bundle, Connection on a vector bundle, Heat kernel on a Riemannian manifold, Minakshisundaram–Pleijel asymptotic expansion, Scalar curvature

Seeley–DeWitt coefficients

Definition

Remarks

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