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

References in corpus (5)

• Lensing Properties of Cored Galaxy Models
• On the Magnfication Relations in Quadruple Lenses: A Moment Approach
• Lensing Properties of Scale-Free Galaxies
• Magnification relations in gravitational lensing via multidimensional residue integrals
• A Simple Model for Lensing by Black Holes in Galactic Nuclei

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• Mathematics of Gravitational Lensing: Multiple Imaging and Magnification
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• Orbifolds, the A, D, E Family of Caustic Singularities, and Gravitational Lensing
• Geometry of Universal Magnification Invariants
• A Universal Magnification Theorem II. Generic Caustics up to Codimension Five