Compactification on negatively curved manifolds

TL;DR

The paper analyzes string/M-theory compactifications on negatively curved internal manifolds and derives a pointwise no-go for obtaining maximally symmetric 4D space-times without either warping or stringy corrections. Using the 10D Einstein equations, it shows R4 is constrained by the internal curvature R6 and stress-energy traces, and that standard fluxes and localized sources tend to yield nonnegative R6 in the absence of warping, blocking everywhere-negative curvature. It then discusses three avenues to evade the no-go—stringy corrections, warp factors, and conformal factors—finding that each has caveats: corrections can contribute negatively but are typically suppressed; warping helps locally but not in a global integral sense; conformal rescalings tend to increase the integrated curvature. The work underscores the need for explicit higher-dimensional solutions or robust effective potential frameworks to realize de Sitter vacua from negative-curvature compactifications and cautions about potential tachyonic or symmetry-breaking issues in such constructions.

Abstract

We show that string/M theory compactifications to maximally symmetric space-times using manifolds whose scalar curvature is everywhere negative, must have significant warping, large stringy corrections, or both.