We apply the methods of homology and K-theory for branes wrapping spaces stratified fibered over hyperbolic orbifolds. In addition, we discuss the algebraic K-theory of any discrete co-compact Lie group in terms of appropriate homology and Atiyah-Hir
zebruch type spectral sequence with its non-trivial lift to K-homology. We emphasize the fact that the physical D-branes properties are completely transparent within the mathematical framework of K-theory. We derive criteria for D-brane stability in the case of strongly virtually negatively curved groups. We show that branes wrapping spaces stratified fibered over hyperbolic orbifolds carry charge structure and change the additive structural properties in K-homology.
We analyse the role of vertex operator algebra and 2d amplitudes from the point of view of the representation theory of infinite dimensional Lie algebras, MacMahon and Ruelle functions. A p-dimensional MacMahon function is the generating function of
p-dimensional partitions of integers. These functions can be represented as amplitudes of a two-dimensional c=1 CFT. In this paper we show that p-dimensional MacMahon functions can be rewritten in terms of Ruelle spectral functions, whose spectrum is encoded in the Patterson-Selberg function of three dimensional hyperbolic geometry.
