A heat trace anomaly on polygons

Let Ω0be a polygon in$\mathbb{R}$2, or more generally a compact surface with piecewise smooth boundary and corners. Suppose that Ωεis a family of surfaces with${\mathcal C}$∞boundary which converges to Ω0smoothly away from the corners, and in a precise way at the vertices to be described in the paper. Fedosov [6], Kac [8] and McKean–Singer [13] recognised that certain heat trace coefficients, in particular the coefficient oft0, are not continuous as ε ↘ 0. We describe this anomaly using renormalized heat invariants of an auxiliary smooth domainZwhich models the corner formation. The result applies to both Dirichlet and Neumann boundary conditions. We also include a discussion of what one might expect in higher dimensions.

HEAT KERNEL ON HOMOGENEOUS BUNDLES

We consider Laplacians acting on sections of homogeneous vector bundles over symmetric spaces. By using an integral representation of the heat semi-group we find a formal solution for the heat kernel diagonal that gives a generating function for the whole sequence of heat invariants. We argue that the obtained formal solution correctly reproduces the exact heat kernel diagonal after a suitable regularization and analytical continuation.

Combinatorics of the Heat Trace on Spheres

We present a concise explicit expression for the heat trace coefficients of spheres. Our formulas yield certain combinatorial identities which are proved following ideas of D. Zeilberger. In particular, these identities allow to recover in a surprising way some known formulas for the heat trace asymptotics. Our approach is based on a method for computation of heat invariants developed in [P].

Heat invariants for odd-dimensional hemispheres

We compute the heat content asymptotics of odd dimensional hemispheres in closed form.