Asymptotics of the Heat Kernel on Rank 1 Locally Symmetric Spaces.pdfVIP

a r X i v : m a t h / 9 8 0 4 1 1 5 v 1 [ m a t h .S P ] 2 3 A p r 1 9 9 8 Asymptotics of the Heat Kernel on Rank 1 Locally Symmetric Spaces A.A. Bytsenko ? Departamento de Fisica, Universidade Estadual de Londrina, Caixa Postal 6001, Londrina-Parana, Brazil and F.L. Williams ? Department of Mathematics, University of Massachusetts, Amherst, Massachusetts 01003 April, 1998 Abstract We consider the heat kernel (and the zeta function) associated with Laplace type operators acting on a general irreducible rank 1 locally symmetric space X. The set of Minakshisundaram-Pleijel coefficients {Ak(X)} ∞ k=0 in the short-time asymptotic expansion of the kernel is calculated explicitly. 1 Introduction In the theory of quantum fields on curved background spaces, the short-time expansion of the heat kernel plays an extremely important role. In particular situations, for example, the coefficients in the expansion control the one-loop ?E-mail: abyts@fisica.uel.br On leave from Sankt-Petersburg State Technical University ?E-mail: williams@math.umass.edu 1 divergences of the effective action, and related quantities such as the stress energy momentum tensor. Some of these coefficients have been determined and appear in the physics and mathematical literature. Note the references [1, 2, 3, 4, 5, 6] for closed Riemannian manifolds and [7, 8] for manifolds with a smooth boundary. The literature on these matters is very vast. In Refs. [1, 2, 3], R. Miatello studies the case of a closed locally sym- metric rank 1 manifold X, using the representation theory of the group of isometries of X. We consider the same case in the present paper, but we use the spectral zeta function of X. By our approach we determine the expan- sion coefficients immediately and explicitly (essentially in one step), given the results of [9]. Recently the topological Casimir energy [10], the one-loop effective action, and the multiplicative and conformal anomaly [11, 12] as- sociated with Laplace type operator