Fermion absorption cross section and topology of spherically symmetric black holes

In 1997, Liberati and Pollifrone in [S. Liberati, G. Pollifrone, Phys. Rev. D 56 (1997) 6458, hep-th/9708014] achieved a new formulation of the Bekenstein-Hawking formula, where the entropy and the Euler characteristic are related by S=χA/8. In this work we present a relation between the low-energy absorption cross section for minimally coupled fermions and the Euler characteristic of (3+1)-dimensional spherically symmetric black holes, i.e. σ=χgh−1A. Based on the relation, using the Gauss-Bonnet-Chern theorem and the ϕ-mapping method, an absorption cross section density is introduced to describe the topology of the absorption cross section. It is shown that the absorption cross section and its density are determined by the singularities of the timelike Killing vector field of the spacetime and these singularities carry the topological numbers, Hopf indices and Brouwer degrees, naturally.