A Finite Landscape?
Bobby S Acharya, Michael R Douglas
TL;DR
The paper investigates whether the number of string/$M$ theory vacua compatible with observation is finite. It combines explicit analyses of infinite topological sequences, such as Freund-Rubin vacua with varying seven-manifolds, with rigorous finiteness theorems from Riemannian geometry (notably Cheeger’s finiteness theorem and its orbifold extensions) to show that only a finite subset of topologies can satisfy physically motivated bounds on $|\Lambda|$, $V$, and $M_{KK}$. It further discusses infinite classes of solutions but argues that, when considering barriers, distances in field space, and convergence/precompactness of moduli spaces (via Gromov-Hausdorff and spectral metrics), the space of physically distinct vacua remains finite in the supergravity regime and under moduli-stabilization constraints. The work links geometric finiteness to physical observables and connects to broader ideas on moduli spaces, KK thresholds, and the structure of the string landscape, offering a rigorous route toward falsifiability and predictivity in string/$M$ theory. Overall, it provides a geometric framework in which infinite mathematical possibilities collapse to a finite, testable set of vacua under realistic physical bounds.
Abstract
We present evidence that the number of string/$M$ theory vacua consistent with experiments is a finite number. We do this both by explicit analysis of infinite sequences of vacua and by applying various mathematical finiteness theorems.