Tame Complexity of Effective Field Theories in the Quantum Gravity Landscape

TL;DR

This paper proposes a unifying framework using tame geometry to understand the finiteness constraints of effective field theories in quantum gravity, suggesting a bound on their complexity.

Contribution

It introduces a novel approach employing tame geometry and o-minimal structures to formalize the complexity bounds of effective field theories in quantum gravity.

Findings

Examples show infinite expansions can have finite complexity descriptions

String compactifications support the conjecture through moduli space analysis

Mathematically defined measures on theory space are finite under the framework

Abstract

Effective field theories consistent with quantum gravity obey surprising finiteness constraints, appearing in several distinct but interconnected forms. In this work we develop a framework that unifies these observations by proposing that the defining data of such theories, as well as the landscape of effective field theories that are valid at least up to a fixed cutoff, admit descriptions with a uniform bound on complexity. To make this precise, we use tame geometry and work in sharply o-minimal structures, in which tame sets and functions come with two integer parameters that quantify their information content; we call this pair their tame complexity. Our Finite Complexity Conjectures are supported by controlled examples in which an infinite Wilsonian expansion nevertheless admits an equivalent finite-complexity description, typically through hidden rigidity conditions such as differential or recursion relations. We further assemble evidence from string compactifications, highlighting the constraining role of moduli space geometry and the importance of dualities. This perspective also yields mathematically well-defined notions of counting and volume measures on the space of effective theories, formulated in terms of effective field theory domains and coverings, whose finiteness is naturally enforced by the conjectures.