The spinor heat kernel in maximally symmetric spaces

Abstract

The heat kernelK(x, x′, t) of the iterated Dirac operator on anN-dimensional simply connected maximally symmetric Riemannian manifold is calculated. On the odd-dimesional hyperbolic spacesK is a Minakshisundaram-DeWitt expansion which terminates to the coefficienta _N−1)/2 and is exact. On the odd spheres the heat kernel may be written as an image sum of WKB kernels, each term corresponding to a classical path (geodesic). In the even dimensional case the WKB approximation is not exact, but a closed form ofK is derived both in terms of (spherical) eigenfunctions and of a “sum over classical paths.” The spinor Plancherel measure μ(λ) and ζ function in the hyperbolic case are also calculated. A simple relation between the analytic structure of μ onH ^N and the degeneracies of the Dirac operator onS ^N is found.