The Witt group of non-degenerate braided fusion categories
Alexei Davydov, Michael Mueger, Dmitri Nikshych, Victor Ostrik
TL;DR
The paper links the classification of fusion categories to braided, non-degenerate structures by characterizing Drinfeld centers as centers of fusion categories containing a Lagrangian algebra, and shows a 2-groupoid equivalence with quantum Manin pairs. It develops a Witt-group framework for non-degenerate braided fusion categories, including completely anisotropic representatives and connections to metric groups and VOAs, and demonstrates how central charge and conformal embeddings yield explicit Witt relations. The results provide a conceptual reduction of fusion category classification to braided data, and offer a toolkit for generating new examples and relationships via equivariantization/de-equivariantization, modules, and VOA/coset constructions. The paper also outlines key open questions about the structure, generation, and torsion in the Witt group, highlighting links to conformal field theory and arithmetic aspects of the theory.
Abstract
We give a characterization of Drinfeld centers of fusion categories as non-degenerate braided fusion categories containing a Lagrangian algebra. Further we study the quotient of the monoid of non-degenerate braided fusion categories modulo the submonoid of the Drinfeld centers and show that its formal properties are similar to those of the classical Witt group.