Fluxes, Brane Charges and Chern Morphisms of Hyperbolic Geometry

TL;DR

This paper applies K-theory and Chern character methods to hyperbolic spaces, deriving explicit formulas for brane charges, RR-fields, and Dirac operator indices, with potential implications for physical theories.

Contribution

It provides explicit formulas and connections between K-theory, cohomology, and Dirac indices specifically for hyperbolic spaces, extending mathematical tools to physical applications.

Findings

Explicit formulas for Chern character and spectral invariants.

Relations between analytic and topological indices via K-theory.

Analysis of torsion charges using lower K-groups.

Abstract

The purpose of this paper is to provide the reader with a collection of results which can be found in the mathematical literature and to apply them to hyperbolic spaces that may have a role in physical theories. Specifically we apply K-theory methods for the calculation of brane charges and RR-fields on hyperbolic spaces (and orbifolds thereof). It is known that by tensoring K-groups with the rationals, K-theory can be mapped to rational cohomology by means of the Chern character isomorphisms. The Chern character allows one to relate the analytic Dirac index with a topological index, which can be expressed in terms of cohomological characteristic classes. We obtain explicit formulas for Chern character, spectral invariants, and the index of a twisted Dirac operator associated with real hyperbolic spaces. Some notes for a bivariant version of topological K-theory (KK-theory) with its connection to the index of the twisted Dirac operator and twisted cohomology of hyperbolic spaces are given. Finally we concentrate on lower K-groups useful for description of torsion charges.