Axiomatic TQFT, Axiomatic DQFT, and Exotic 4-Manifolds

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.