Imaginary mass lens space determinants

TL;DR

This paper computes functional determinants for a scalar field with negative mass squared on lens spaces, revealing oscillatory behavior of the Hartle--Hawking wave function and providing explicit results for certain three-sphere factors.

Contribution

It presents a novel evaluation of functional determinants on lens spaces, including explicit forms and graphical representations, without requiring degeneracy details.

Findings

Hartle--Hawking wave function oscillates with increasing amplitude

Explicit quadrature form for determinants provided

Results for binary tetrahedral, octahedral, and icosahedral factors

Abstract

Functional determinants for a single scalar field with negative mass squared are evaluated on homogeneous lens spaces. For example, on even order spaces, the Hartle--Hawking wave function oscillates about its zeros with increasing amplitude as the (imaginary) mass increases. I also present results for the binary tetrahedral, octahedral and icosahedral factors of the three--sphere. The final answer is given as a quadrature and some graphs are drawn. In the technical evaluation of the infinite sums, the explicit form of the degeneracies is not needed.