Project description
Enhancing the topological understanding of non-semisimple modular categories
Modular categories are algebraic structures ubiquitous in many branches of mathematics including topological quantum field theory and conformal field theory. The theory of modular categories is used to study quantum symmetry and topological phases of matter. So-called semisimple modular categories are well described in terms of 3D topological field theory, but the same is not true of non-semisimple modular categories. With the support of the Marie Skłodowska-Curie Actions programme, the Modular Functors project is filling this gap, solving concrete problems related to the description of non-semisimple modular categories using 3D topological field theory.
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.
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Funding Scheme
MSCA-IF - Marie Skłodowska-Curie Individual Fellowships (IF)Coordinator
1165 Kobenhavn
Denmark