Heat kernel expansion

A heat kernel expansion is the short-time asymptotic expansion of the trace or diagonal kernel of an elliptic operator. Its local coefficients are polynomials in background curvatures and field strengths; in quantum field theory they extract ultraviolet divergences and anomalies.

For the positive scalar Laplace-Beltrami operator on a compact manifold without boundary, the heat invariants are the integrated coefficients $a_j={\int }_M{u}_j(x,x)d{V}_g$ in the short-time heat kernel expansion. In particular $a_0=\mathrm{Vol}(M)$ and $a_1=\frac{1}{6}{\int }_M{R}_{g}d{V}_g$, with $R_g$ the scalar curvature. They are determined by the heat trace, hence by the spectrum with multiplicities. The leading positive coefficient recovers the dimension from ${\lim }_{t\downarrow 0}\log \mathrm{Tr}(e^{-t\Delta })/\log (1/t)=d/2$.

For the positive scalar Laplace-Beltrami operator on a closed manifold with a Riemannian metric, the coefficient of $t^2$ in the normalized heat trace is the displayed integral. The local coefficient additionally has a scalar-curvature divergence term whose integral vanishes. In dimension two, $R=2K$, $|\mathrm{Ric}{|}^2=2K^2$, $|\mathrm{Rm}{|}^2=4K^2$, giving $a_2=\frac{1}{15}{\int }_M{K}^{2}dA$. Together with volume and total Gaussian curvature, it determines the variance of curvature and detects constant curvature.

On a closed two-dimensional Riemannian manifold, the first three integrated scalar heat invariants are $A$, $\frac{1}{3}\int KdA$, and $\frac{1}{15}\int K^{2}dA$, where $K$ is Gaussian curvature. If its scalar spectrum agrees with that of a hyperbolic surface of area $A$, these integrals are $A,-A/3,A/15$. Expanding the displayed square then forces $K=-1$ everywhere. Thus constant negative curvature is spectrally determined in this setting, although the global hyperbolic metric need not be determined up to isometry.

For $L=-{\nabla }^a{\nabla }_a$ on scalar functions, the leading coefficient of the heat parametrix in normal coordinates is the inverse square root of the metric volume density: $u_0=1+R_{ab}{\xi }^a{\xi }^b/12+O(|\xi {|}^3)$. The heat-kernel transport equations give $u_1(x,x)={\nabla }^a{\nabla }_a{u}_0(x,x)=R(x)/6$. This gives the displayed diagonal expansion. The coefficient is specific to the minimally coupled scalar operator; a bundle connection or a curvature-dependent potential changes the general Laplace-type coefficients.

For $L=-{\partial }_j(g^{ij}{\partial }_i)$ in geodesic normal coordinates, the Gauss lemma gives $g^{ij}{x}_i=x_j$. Substituting $F=(4\pi t{)}^{-n/2}{e}^{-|x{|}^2/(4t)}{\sum }_{k\ge 0}{t}^k{B}_k$ into $({\partial }_t+L)F=0$ therefore yields $(x\cdot \nabla +k)B_k=-L{B}_{k-1}$, with $B_0=1$. Integration along radial rays gives the displayed formula. Every coefficient is smooth at the centre. Any smooth solution of $(x\cdot \nabla +k)h=0$ is zero for $k>0$, since $s^{k}h(sx)$ is constant and tends to zero at $s=0$. This proves uniqueness. In this special divergence-form case $L1=0$, so every $B_k$ for $k>0$ is zero: the Gaussian itself solves the equation exactly on the coordinate ball. The positive Laplace-Beltrami operator instead has a metric volume-density term and a nonconstant leading amplitude.