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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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