Non-semisimple differential graded modular functors: Modular Functors

Programme(s):

• H2020-EU.1.3. - EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions 
• H2020-EU.1.3.2. - Nurturing excellence by means of cross-border and cross-sector mobility

Topic(s): MSCA-IF-2020 - Individual Fellowships

Call for proposal: H2020-MSCA-IF-2020

Funding Scheme: MSCA-IF-EF-ST - Standard EF

Grant agreement ID: 101022691

Objective

Non-semisimple differential graded modular functors: While semisimple modular categories can be entirely understood in terms of three-dimensional topological field theory, an equally satisfactory topological understanding of non-semisimple modular categories is not available. The proposed project will solve concrete problems related to the topological understanding of non-semisimple modular categories by unraveling within a homotopy coherent framework the relation between the homological algebra of a modular category (in particular, its Hochschild complex) and low-dimensional topology. The backbone of this approach is the differential graded modular functor associated to any modular category (a consistent system of projective mapping class group representations on chain complexes satisfying excision) that I have recently established in joint work with Schweigert. Among the concrete objectives is a generalization of the Verlinde formula to a statement about two compatible E_2-structures on the differential graded conformal block for the torus. This will naturally link the Verlinde formula to the Deligne conjecture. Moreover, rigidity requirements for categories that can be extracted from a modular functor will be studied systematically using cyclic and modular operads and results of Costello and Giansiracusa. This will lead to a vast generalization of existing string-net techniques, namely string-net complexes for any pivotal Grothendieck-Verdier category in the sense of Boyarchenko-Drinfeld. These string-net complexes can be used to compute differential graded conformal blocks for modular categories which are the Drinfeld center of a spherical pivotal finite tensor category and to create a link to Morrison-Walker blob homology.

The key techniques that I will learn during the fellowship involve graph models for mapping class group actions and multiplicative structures on Hochschild complexes. My host Nathalie Wahl is an expert in these areas.