Heat kernel asymptotic expansion

The main part of my research is devoted to the elaboration of effective covariant methods for calculating the heat kernel and the functional determinants of Laplace type differential operators and application of these methods to various problems of the quantum field theory and quantum gravity. Let (M,g) be a smooth compact Riemannian manifold with a smooth boundary $\partial M$ and a metric g, V(M) be a smooth vector bundle over M with connection $\nabla$ and $Q\in C^\infty({\rm End}\,(V),M)$ be a smooth section of the endomorphism bundle. Then a Laplace type operator $F:\ C^\infty(V,M)\to C^\infty(V,M)$ is a second-order partial differential operator defined by $F \equiv \nabla^*\nabla+Q$. The Laplace type operator F together with the corresponding boundary conditions B is elliptic and self-adjoint and determines the heat semi-group $U(t)=\exp(-tF)$ for t>0. The heat kernel U(t|x,x') is a smooth function in a neighborhood of the diagonal of $M\times M$ outside the boundary $\partial M$ and has the asymptotic expansion for $t\to +0$

$$U(t\vert x,x')\sim (4\pi t)^{-n/2}\Delta^{1/2}\exp\left(-{\sigma\over 2t}\right) \sum_{k=0}^{\infty} t^{k/2} a_{k/2}(x,x'),$$

where n is the dimension of the manifold M, $\sigma(x,x')$ is equal to one half the square of the geodesic distance between x and x' and $\Delta(x,x')$ is the corresponding Van Vleck determinant. The two-point quantities a_k/2(x,x') are defined on $M\times M$ and are called the local heat kernel coefficients, or the Hadamard-Minackshisundaram-De Witt-Seeley (HMDS)-coefficients. The HMDS-coefficients of integer order, a_k, are regular near the diagonal of $M\times M$ and do not depend on the boundary conditions. All coefficients of half-integer order a_k+1/2=0 vanish outside the boundary. The L² trace of the heat kernel has the asymptotic expansion for $t\to +0$

$${\rm Tr}_{L^2}\, \exp(-tF)\sim (4\pi t)^{-n/2}\sum_{k=0}^{\infty} t^{k/2}A_{k/2},$$

where

$$A_{k/2}=\int_M {\rm tr}_V\, \,a_{k/2}^{\rm diag} +\int_{\partial M} {\rm tr}_V\, b_{k/2}$$

are the global heat kernel coefficients. Here $a_{k/2}^{\rm diag}$ are the diagonal values of the HMDS-coefficients that are polynomials in the jets of the Riemannian curvature R, curvature ${\cal R}$ of the bundle connection $\nabla^V$ and the endomorphism Q. The boundary parts of the heat kernel coefficients b_k/2 include additionally the jets of the extrinsic curvature of the boundary K and the coefficients of the boundary operator B. For manifolds with boundary these coefficients do not vanish also for half-integer orders.

In [1,2,3] I developed a new effective manifestly covariant technique for calculating the HMDS-coefficients, a_k, for a Laplace type operator in form of a covariant Taylor series near the diagonal. Using this technique the previous result of Gilkey for the third coefficient $a_3^{\rm diag}$ has been confirmed and for the first time the forth coefficient $a_4^{\rm diag}$ has been calculated.