A Lefschetz fixed point theorem in gravitational lensing

Abstract

Topological invariants play an important rôle in the theory of gravitational lensing by constraining the image number. Furthermore, it is known that, for certain lens models, the image magnifications ${\mu }_i$ obey invariants of the form ${\sum }_i{\mu }_i=1$. In this paper, we show that this magnification invariant is the holomorphic Lefschetz number of a suitably defined complexified lensing map, and hence a topological invariant. We also provide a heat kernel proof of the holomorphic Lefschetz fixed point formula which is central to this argument, based on Kotake's proof of the more general Atiyah-Bott theorem. Finally, we present a new astronomically motivated lens model for which this invariant holds.

14 pages. Version 2: typos corrected