Axiomatic TQFT, Axiomatic DQFT, and Exotic 4-Manifolds

TL;DR

This paper demonstrates that traditional axiomatic 4D TQFTs cannot detect smooth structure changes, but a new axiomatic framework called differential QFT can, reconciling the ability of Donaldson-Witten theory to detect such changes.

Contribution

The paper introduces a new set of axioms for differential quantum field theories that enable detection of smooth structure changes, addressing limitations of existing axiomatic TQFTs.

Findings

Axiomatic 4D TQFTs cannot detect smooth structure changes.

Differential QFTs with modified axioms can detect smooth structure changes.

Resolution of the contradiction between TQFT limitations and Donaldson-Witten theory.

Abstract

In this article we prove that any unitary, axiomatic topological quantum field theory in four-dimensions can not detect changes in the smooth structure of M, a simply connected, closed (compact without boundary), oriented smooth manifold. However, as Donaldson-Witten theory (a topological quantum field theory but not an axiomatic one) is able to detect changes in the smooth structure of such an M, this seemingly leads to a contradiction. This seeming contradiction is resolved by introducing a new set of axioms for a "differential quantum field theory", which in truth only slightly modify the naturality and functoriality axioms of a topological quantum field theory, such that these new axioms allow for a theory to detect changes in smooth structure.