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Semisimple 4-dimensional topological field theories cannot detect exotic smooth structure

[Moved Online] Tensor categories and topological quantum field theories March 16, 2020 - March 20, 2020

March 16, 2020 (02:00 PM PDT - 03:00 PM PDT)
Speaker(s): David Reutter (Max-Planck-Institut für Mathematik)
  • topological field theory

  • ribbon fusion categories

  • 4-manifolds

  • stable diffeomorphisms

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification



A major open problem in quantum topology is the construction of an oriented 4-dimensional topological field theory in the sense of Atiyah-Segal which is sensitive to exotic smooth structure. In this talk, I will sketch a proof that no semisimple field theory can achieve this goal and that such field theories are only sensitive to the homotopy types of simply connected 4-manifolds. This applies to all currently known examples of oriented 4-dimensional TFTs valued in the category of vector spaces, including unitary field theories and once-extended field theories which assign algebras or linear categories to 2-manifolds. If time permits, I will give a concrete expression for the value of a semisimple TFT on a simply connected 4-manifold, explain how the presence of `emergent fermions’ in a field theory is related to its potential sensitivity to more than the homotopy type of a non-simply connected 4-manifold, and comment on implications for the theory of ribbon fusion categories. This is based on arXiv:2001.02288.

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H.264 Video 917_28187_8231_3-Reutter.mp4
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