Taming the Landscape of Effective Theories

TL;DR

This paper proposes a Tameness Conjecture suggesting all effective theories compatible with quantum gravity have scalar field spaces and couplings describable by tame geometry, supported by evidence from string theory and F-theory.

Contribution

It introduces a new structural principle based on tame geometry that constrains effective theories in quantum gravity, with evidence from string compactifications and flux vacua.

Findings

Effective theories are labeled by definable parameter spaces.

Scalar fields and couplings are tame definable functions.

Finiteness of self-dual flux vacua in F-theory is established.

Abstract

We introduce a generalized notion of finiteness that provides a structural principle for the set of effective theories that can be consistently coupled to quantum gravity. More concretely, we propose a Tameness Conjecture that states that all valid effective theories are labelled by a definable parameter space and must have scalar field spaces and coupling functions that are definable using the tame geometry built from an o-minimal structure. We give a brief introduction to tame geometry and describe how it restricts sets, manifolds, and functions. We then collect evidence for the Tameness Conjecture by studying various effective theories arising from string theory compactifications by using some recent advances in tame geometry. In particular, we will exploit the fact that coset spaces and period mappings are definable in an o-minimal structure and argue for non-trivial tameness results in higher-supersymmetric theories and in Calabi-Yau compactifications. As strongest evidence for the Tameness Conjecture over a discrete parameter space, we then discuss a recent theorem stating that the locus of self-dual flux vacua of F-theory admits a tame geometry even if one allows for any flux choice satisfying the tadpole constraint. This result implies the finiteness of self-dual flux vacua in F-theory.