Covariant techniques for computation of the heat kernel

Details Covariant techniques for computation of the heat kernel Covariant techniques for computation of the heat kernel

1997-04-21

arxiv.org/abs/hep-th/9704166v1

Rev.Math.Phys.11:947-980,1999

Physics

High Energy Physics

High Energy Physics - Theory

31 pages, LaTeX, no figures, Invited Lecture at the University of Iowa, Iowa City, April, 1997

Scientific paper

10.1142/S0129055X99000295

The heat kernel associated with an elliptic second-order partial differential operator of Laplace type acting on smooth sections of a vector bundle over a Riemannian manifold, is studied. A general manifestly covariant method for computation of the coefficients of the heat kernel asymptotic expansion is developed. The technique enables one to compute explicitly the diagonal values of the heat kernel coefficients, so called Hadamard-Minackshisundaram-De Witt-Seeley coefficients, as well as their derivatives. The elaborated technique is applicable for a manifold of arbitrary dimension and for a generic Riemannian metric of arbitrary signature. It is very algorithmic, and well suited to automated computation. The fourth heat kernel coefficient is computed explicitly for the first time. The general structure of the heat kernel coefficients is investigated in detail. On the one hand, the leading derivative terms in all heat kernel coefficients are computed. On the other hand, the generating functions in closed covariant form for the covariantly constant terms and some low-derivative terms in the heat kernel coefficients are constructed by means of purely algebraic methods. This gives, in particular, the whole sequence of heat kernel coefficients for an arbitrary locally symmetric space.

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