The Tameness of Quantum Field Theory, Part II -- Structures and CFTs

TL;DR

This paper explores the mathematical structure of quantum field theories using tame geometry, formalizing their logical foundations and investigating the tameness of conformal field theories, with implications for quantum gravity.

Contribution

It formalizes the connection between quantum field theories and logical structures and investigates the tameness of conformal field theories and effective theories.

Findings

Quantum field theories define logical structures that can be analyzed for tameness.

Conjectures about the tameness of conformal field theory observables are proposed.

Examples support the potential universality of tameness constraints in CFTs.

Abstract

Tame geometry originated in mathematical logic and implements strong finiteness properties by defining the notion of tame sets and functions. In part I we argued that observables in a wide class of quantum field theories are tame functions and that the tameness of a theory relies on its UV definition. The aims of this work are (1) to formalize the connection between quantum field theories and logical structures, and (2) to investigate the tameness of conformal field theories. To address the first aim, we start from a set of quantum field theories and explain how they define a logical structure that is subsequently extended to a second structure by adding physical observables. Tameness, or o-minimality, of the two structures is then a well-defined property, and sharp statements can be made by identifying these with known examples in mathematics. For the second aim we quantify our expectations on the tameness of the set of conformal field theories and effective theories that can be coupled to quantum gravity. We formulate tameness conjectures about conformal field theory observables and propose universal constraints that render spaces of conformal field theories to be tame sets. We test these conjectures in several examples and highlight first implications.