Positive Topological Quantum Field Theories

TL;DR

This paper introduces a new framework for positive topological quantum field theories (TFTs) based on semiring completeness, enabling the derivation of combinatorial, topological, and number-theoretic invariants through state sum identities.

Contribution

It develops a novel notion of positivity for TFTs using semiring completeness, unifying various phenomena and providing a foundation for future invariants.

Findings

Derived Polya's counting theory from positive TFT state sums

Connected Novikov signatures to topological invariants

Proposed a framework for detecting exotic smooth structures

Abstract

We propose a new notion of positivity for topological field theories (TFTs), based on S. Eilenberg's concept of completeness for semirings. We show that a complete ground semiring, a system of fields on manifolds and a system of action functionals on these fields determine a positive TFT. The main feature of such a theory is a semiring-valued topologically invariant state sum that satisfies a gluing formula. The abstract framework has been carefully designed to cover a wide range of phenomena. For instance, we derive Polya's counting theory in combinatorics from state sum identities in a suitable positive TFT. Several other concrete examples are discussed, among them Novikov signatures of fiber bundles over spacetimes and arithmetic functions in number theory. In the future, we will employ the framework presented here in constructing a new differential topological invariant that detects exotic smooth structures on spheres.