Hyperbolic Spaces in String and M-Theory

TL;DR

This paper explores string and supergravity solutions involving hyperbolic spaces, presenting new non-supersymmetric compactifications, geometric constructions, and holographic duals with potential applications in theoretical physics.

Contribution

It introduces novel solutions with hyperbolic spaces in string theory, including non-supersymmetric compactifications and geometric transformations via T-duality.

Findings

New hyperbolic space solutions in string and supergravity theories

Construction of non-supersymmetric compactifications on hyperbolic spaces

Development of holographic duals for non-supersymmetric conformal field theories

Abstract

We describe string-theory and $d=11$ supergravity solutions involving symmetric spaces of constant negative curvature. Many examples of non-supersymmetric string compactifications on hyperbolic spaces $H_r$ of finite volume are given in terms of suitable cosets of the form $H_r/\Gamma $, where $\Gamma $ is a discrete group. We describe in some detail the cases of the non-compact hyperbolic spaces $F_2$ and $F_3$, representing the fundamental regions of $H_2$ and $H_3$ under $SL(2,Z)$ and the Picard group, respectively. By writing $AdS$ as a U(1) fibration, we obtain new solutions where $AdS_{2p+1}$ gets untwisted by T-duality to ${\bf R}\times SU(p,1)/(SU(p)\times U(1))$. Solutions with time-dependent dilaton field are also constructed by starting with a solution with NS5-brane flux over $H_3$. A new class of non-supersymmetric conformal field theories can be defined via holography.