A Lefschetz fixed point theorem in gravitational lensing

TL;DR

This paper links topological invariants in gravitational lensing to the holomorphic Lefschetz fixed point theorem, providing a new perspective on image magnification invariants and introducing a novel lens model.

Contribution

It demonstrates that the magnification invariant in gravitational lensing is a holomorphic Lefschetz number, connecting lensing invariants to topological fixed point theory.

Findings

Magnification invariants are holomorphic Lefschetz numbers.

A heat kernel proof of the Lefschetz fixed point formula is provided.

A new astronomically motivated lens model satisfying the invariant is introduced.

Abstract

Topological invariants play an important r\^{o}le 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.