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ONLINE: Semisimple topological quantum field theories and exotic smooth structure

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David Reutte
Tue, 2020-06-09 14:00 - 15:00

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. 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. In this context, `semisimplicity' is a certain algebraic condition applying to all currently known examples of vector-space-valued oriented 4-dimensional TQFTs, 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 TQFT on a simply connected 4-manifold and 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. This is based on arXiv:2001.02288.


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