A Lefschetz fixed point theorem in gravitational lensing

Details A Lefschetz fixed point theorem in gravitational lensing A Lefschetz fixed point theorem in gravitational lensing

2007-03-15

arxiv.org/abs/math-ph/0703050v2

J. Math. Phys. 48, 052501 (2007)

Physics

Mathematical Physics

14 pages. Version 2: typos corrected

Scientific paper

10.1063/1.2735443

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.

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