Further results on Functional Determinants of Laplacians in Simplicial Complexes

TL;DR

This paper extends the analysis of the functional determinant of Laplacians to piece-wise flat 2D surfaces with conical singularities, revealing unique interaction and self-energy effects not present in smooth surfaces.

Contribution

It provides new results on the determinants of Laplacians on singular surfaces, highlighting differences from smooth cases in interaction and self-energies of singularities.

Findings

Different interaction energies between conical singularities.

Presence of self-energies for singularities.

Extension of determinant analysis to non-smooth surfaces.

Abstract

We investigate the functional determinant of the laplacian on piece-wise flat two-dimensional surfaces, with conical singularities in the interior and/or corners on the boundary. Our results extend earlier investigations of the determinants on smooth surfaces with smooth boundaries. The differences to the smooth case are: a) different ``interaction energies'' between pairs of conical singularities than one would expect from a naive extrapolation of the results for a smooth surface; and b) ``self-energies'' of the singularities.