PIRSA:25050034 Scientific Series Mathematical physics

DOI10.48660/25050034

Series: Mathematical Physics

(2025). 2-dimensional topological field theories via the genus filtration. Perimeter Institute. https://pirsa.org/25050034

2-dimensional topological field theories via the genus filtration. Perimeter Institute, May. 22, 2025, https://pirsa.org/25050034 

@misc{ pirsa_PIRSA:25050034,
  doi = {10.48660/25050034},
  url = {https://pirsa.org/25050034},
  author = {},
  keywords = {Mathematical physics},
  language = {en},
  title = {2-dimensional topological field theories via the genus filtration},
  publisher = {Perimeter Institute},
  year = {2025},
  month = {may},
  note = {PIRSA:25050034 see, \url{https://pirsa.org}}
}

Abstract

By a folk theorem (non-extended) 2-dimensional TFTs valued in the category of vector spaces are equivalent to commutative Frobenius algebras. Upgrading the bordism category to an (infinity, 1)-category whose 2-morphism are diffeomorphisms, one can study 2D TFTs valued in higher categories, leading for example to (derived) modular functors and cohomological field theories.
 
I will explain how to describe such more general (non-extended) 2D TFTs as algebras over the modular infinity-operad of surfaces. In genus 0 this yields an E_2^{SO}-Frobenius algebra and I will outline an obstruction theory for inductively extending such algebras to higher genus. Specialising to invertible TFTs, this amounts to a genus filtration of the classifying space of the bordism category and hence the Madsen--Tillmann spectrum MTSO_2. The aforementioned obstruction theory identifies the associated graded in terms of curve complexes and thereby yields a spectral sequence starting with the unstable and converging to the stable cohomology of mapping class groups.

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