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

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.

arXiv comment27 pages, tex file, no figures

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