Hyperbolic Space Forms and Orbifold Compactification in M-Theory

Abstract

We analyze solutions of string theory and supergravity which involve real hyperbolic spaces. Examples of string compactifications are given in terms of hyperbolic coset spaces of finite volume $\Gamma\backslash {\mathbb H}^N$, where $\Gamma$ is a discrete group of isometries of ${\mathbb H}^N$. We describe finite flux and the tensor kernel associated with hyperbolic spaces. The case of arithmetic geometry of $\Gamma = SL(2, {\mathbb Z}+i{\mathbb Z})/\{\pm Id\}$, where $Id$ is the identity matrix, is analyzed. We discuss supersymmetry surviving for supergravity solutions involving real hyperbolic space factors, string-supergravity correspondence and holography principle for a class of conformal field theories.