A heat trace anomaly on polygons

Let ${\mathrm{\Omega }}_0$ be a polygon in ${\text{RR}}^2$, or more generally a compact surface with piecewise smooth boundary and corners. Suppose that ${\mathrm{\Omega }}_{\text{e}}$ is a family of surfaces with ${\text{calC}}^{\mathrm{\infty }}$ boundary which converges to ${\mathrm{\Omega }}_0$ smoothly away from the corners, and in a precise way at the vertices to be described in the paper. Fedosov \cite{Fe}, Kac \cite{K} and McKean-Singer \cite{MS} recognized that certain heat trace coefficients, in particular the coefficient of $t^0$, are not continuous as $\text{e}\searrow 0$. We describe this anomaly using renormalized heat invariants of an auxiliary smooth domain $Z$ which models the corner formation. The result applies both for Dirichlet and Neumann conditions. We also include a discussion of what one might expect in higher dimensions.