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  "body": "\n## Definition\n\nLet $(M, g)$ be a closed [[Riemannian manifold|7CBT4SM83Y]] of dimension $n$ and let $E$ be a Hermitian [[vector bundle|P4DZM975BV]] over $M$.\nLet $L$ be a Laplace-type operator on $E$, that is, a second-order operator with principal symbol $\\abs{\\xi}^2 \\id_E$.\nEquivalently, $L = \\nabla^* \\nabla + F$ for a unique [[connection|5ZYRRVE8BW]] $\\nabla$ on $E$ and a unique endomorphism field $F$.\nLet $K_L(t, x, y)$ be the [[heat kernel|327NP6S7H5]] of $L$.\nThere are unique smooth sections $a_k(L)$ of $\\End(E)$, $k = 0, 1, 2, \\ldots$, such that for every $N$\n$$K_L(t, x, x) = (4\\pi t)^{-n/2} \\left( \\sum_{k = 0}^{N} a_k(L)(x) \\, t^k + O(t^{N+1}) \\right) \\quad \\text{as } t \\to 0^+,$$\nuniformly in $x$.\nThe $a_k(L)$ are the *Seeley–DeWitt coefficients* of $L$.\n\n## Remarks\n\nExistence follows by the same parametrix construction as for the [[Minakshisundaram–Pleijel expansion|45P2SEGGR1]], and uniqueness from the uniqueness of asymptotic expansions.\nThe first two are $a_0 = \\id_E$ and $a_1 = \\tfrac16 S \\, \\id_E - F$, where $S$ is the [[scalar curvature|ZHX2VYQDZ3]].\nSome 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 $t^{k/2}$.\n\nExample: take the scalar Laplacian on a flat torus, so $F = 0$.\nIts 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 $k \\ge 1$.\n\nNon-example: a Dirac operator $D$ is of first order, so it is not Laplace-type and has no Seeley–DeWitt coefficients in this sense.\nIts square $D^2$ is Laplace-type, and the coefficients of $D^2$ are the ones used in index theory.\n",
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