Hyperbolic compactification of M-theory and de Sitter quantum gravity

TL;DR

This paper proposes a mechanism for universe acceleration via hyperbolic manifold compactifications in M-theory, utilizing Casimir energy and fluxes to stabilize volume and achieve de Sitter space, with implications for holography and cosmology.

Contribution

It introduces a novel compactification approach using hyperbolic manifolds with Casimir energy and fluxes to realize de Sitter vacua in M-theory, supported by explicit solutions and stability analysis.

Findings

Stable de Sitter solutions with positive potential energy.

Explicit back-reacted solutions and perturbation analysis.

Connections established between hyperbolic geometry and string/M-theory compactifications.

Abstract

We present a mechanism for accelerated expansion of the universe in the generic case of negative-curvature compactifications of M-theory, with minimal ingredients. M-theory on a hyperbolic manifold with small closed geodesics supporting Casimir energy -- along with a single classical source (7-form flux) -- contains an immediate 3-term structure for volume stabilization at positive potential energy. Hyperbolic manifolds are well-studied mathematically, with an important rigidity property at fixed volume. They and their Dehn fillings to more general Einstein spaces exhibit explicit discrete parameters that yield small closed geodesics supporting Casimir energy. The off-shell effective potential derived by M. Douglas incorporates the warped product structure via the constraints of general relativity, screening negative energy. Analyzing the fields sourced by the localized Casimir energy and the available discrete choices of manifolds and fluxes, we find a regime where the net curvature, Casimir energy, and flux compete at large radius and stabilize the volume. Further metric and form field deformations are highly constrained by hyperbolic rigidity and warping effects, leading to calculations giving strong indications of a positive Hessian, and residual tadpoles are small. We test this via explicit back reacted solutions and perturbations in patches including the Dehn filling regions, initiate a neural network study of further aspects of the internal fields, and derive a Maldacena-Nunez style no-go theorem for Anti-de Sitter extrema. A simple generalization incorporating 4-form flux produces axion monodromy inflation. As a relatively simple de Sitter uplift of the large-N M2-brane theory, the construction applies to de Sitter holography as well as to cosmological modeling, and introduces new connections between mathematics and the physics of string/M theory compactifications.