2-dimensional topological field theories via the genus filtration

          @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.