TOPOLOGICAL QUANTUM FIELD THEORY AND EXOTIC SMOOTH STRUCTURES.

          An axiomatic framework for Topological Quantum Field Theories (TQFT) was introduced by
          Atiyah in the 1980s. While these axioms are in some respects analogous to the
          Eilenberg-Steenrod axioms for homology, it is the stringent gluing axiom required of a TQFT
          that makes them much harder to construct than homology theories. To this day,
          examples of explicit TQFTs are very sparse, especially for high-dimensional manifolds.

          In our work, we first introduce an abstract framework of positive TQFTs, where positivity
          refers to the fact that these theories are defined over semirings rather than rings.
          The idea here is to use the observation of S. Eilenberg from the 1970s (in his work on
          automata theory) that certain semirings allow for an additive completeness which rings
          cannot possess. Using this completeness, one bypasses well-known measure theoretic
          difficulties in making the Feynman path integral rigorous. In the second step, we employed
          smooth maps to the plane, which are required to have only fold singularities, as the fields on bordisms.
          The action functional extracts the singular set of the map and uses representation theory
          of the Brauer category to arrive at matrices. The resulting state sum invariants are power series
          in one variable and we are able to prove that they are rational functions (once boundary conditions
          are fixed). We also prove that the invariants resulting from our approach are able
          to distinguish exotic smooth spheres from standard spheres.

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