Homology and K--Theory Methods for Classes of Branes Wrapping Nontrivial Cycles

TL;DR

This paper explores the application of homology and K-theory to classify branes on nontrivial cycles, analyze T-duality, and topology change in string theory, with specific examples involving fluxes and geometric spaces.

Contribution

It introduces new methods for classifying branes using homology and K-theory, and demonstrates T-duality and topology change in complex geometric backgrounds.

Findings

Classification of four-geometries via compact stabilizers.

Proof of K-amenability for certain Lie groups.

Explicit T-duality examples with fluxes and topology change.

Abstract

We apply some methods of homology and K-theory to special classes of branes wrapping homologically nontrivial cycles. We treat the classification of four-geometries in terms of compact stabilizers (by analogy with Thurston's classification of three-geometries) and derive the K-amenability of Lie groups associated with locally symmetric spaces listed in this case. More complicated examples of T-duality and topology change from fluxes are also considered. We analyse D-branes and fluxes in type II string theory on ${\mathbb C}P^3\times \Sigma_g \times {\mathbb T}^2$ with torsion $H-$flux and demonstrate in details the conjectured T-duality to ${\mathbb R}P^7\times X^3$ with no flux. In the simple case of $X^3 = {\mathbb T}^3$, T-dualizing the circles reduces to duality between ${\mathbb C}P^3\times {\mathbb T}^2 \times {\mathbb T}^2$ with $H-$flux and ${\mathbb R}P^7\times {\mathbb T}^3$ with no flux.