Ivan G. Avramidi (Department of Mathematics, New Mexico Tech)

ABSTRACT:

We study the heat kernel for a Laplace type partial differential operator
acting on smooth sections of a complex vector bundle with the structure
group $G\times U(1)$ over a Riemannian manifold $M$ without boundary. The total
connection on the vector bundle naturally splits into a $G$-connection and a
$U(1)$-connection. By assuming that the curvature $F$ of the $U(1)$ connection
is parallel and of order $t^{-1}$ we find a new local asymptotic expansion as
$t\to 0$ of heat kernel $U(t;x,x')$ in the neighborhood of the diagonal of
$M\times M$. The coefficients of this expansion are polynomial in
the Riemann curvature tensor and the curvature of the $G$-connection and their
derivatives with universal coefficients depending in a non-polynomial but analytic
way on the curvature $F$, more precisely, on $tF$. We compute explicitly
the first three coefficients (both diagonal and off-diagonal) of this new
asymptotic expansion. They generate all terms in the usual heat kernel coefficients
which are linear and quadratic in the Riemann curvature and of arbitrary order in $F$.

Slides of the lecture: PDF, PS