Semisimple topological field theories in even dimensions

David Reutter (MPI Bonn)

19-Feb-2021, 17:00-18:00 (3 years ago)

Abstract: A major open problem in quantum topology is the construction of an oriented 4-dimensional topological quantum field theory (TQFT) in the sense of Atiyah-Segal which is sensitive to exotic smooth structure. More generally, how much manifold topology can a TQFT see?

In this talk, I will answer this question for semisimple field theories in even dimensions — I will sketch a proof that such field theories can at most see the stable diffeomorphism type of a manifold and conversely, that if two sufficiently finite manifolds are not stably diffeomorphic then they can be distinguished by semisimple field theories. In this context, `semisimplicity' is a certain algebraic condition applying to all currently known examples of vector-space-valued TQFTs, including `unitary field theories’, and `once-extended field theories' which assign algebras or linear categories to codimension 2 manifolds. I will discuss implications in dimension 4, such as the fact that oriented semisimple field theories cannot see smooth structure, while unoriented ones can.

Throughout, I will use the Crane-Yetter field theory associated to a ribbon fusion category, as a guiding example.

This is based on arXiv:2001.02288 and joint work in progress with Chris Schommer-Pries.

mathematical physicsalgebraic topologycategory theoryquantum algebra

Audience: researchers in the topic

( video )


Topological Quantum Field Theory Club (IST, Lisbon)

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