Stratified Fiber Bundles, Quinn Homology and Brane Stability of Hyperbolic Orbifolds

TL;DR

This paper investigates the stability of string vacua on hyperbolic orbifolds using advanced homotopy and K-homology methods, proposing new criteria for brane stability and charges in complex geometric backgrounds.

Contribution

It introduces a novel framework for analyzing brane stability on stratified fibered hyperbolic orbifolds using Quinn homology and algebraic K-theory, extending previous models.

Findings

Derived stability criteria using Atiyah-Hirzebruch spectral sequence.

Defined stratified charges inducing new additive structures on K-homology.

Analyzed brane stability in backgrounds with H-flux using twisted K-groups.

Abstract

We revisit the problem of stability of string vacua involving hyperbolic orbifolds using methods from homotopy theory and K-homology. We propose a definition of Type II string theory on such backgrounds that further carry stratified systems of fiber bundles, which generalise the more conventional orbifold and symmetric string backgrounds, together with a classification of wrapped branes by a suitable generalized homology theory. For spaces stratified fibered over hyperbolic orbifolds we use the algebraic K-theory of their fundamental groups and Quinn homology to derive criteria for brane stability in terms of an Atiyah-Hirzebruch type spectral sequence with its lift to K-homology. Stable D-branes in this setting carry stratified charges which induce new additive structures on the corresponding K-homology groups. We extend these considerations to backgrounds which support H-flux, where we use K-groups of twisted group algebras of the fundamental groups to analyse stability of locally symmetric spaces with K-amenable isometry groups, and derive stability conditions for branes wrapping the fibers of an Eilenberg-MacLane spectrum functor.