Rank-finiteness fails for non-semisimple modular categories

• Speaker: Quinn Kolt, University of California, Santa Barbara
• Time: 3:20pm to 3:30pm
• Abstract: The rank of a semisimple tensor category is the number of isomorphism classes of simple objects. For non-semisimple finite tensor categories, we also count the number of isomorphism classes of indecomposable projective objects. It is well-known that semisimple modular tensor categories have rank finiteness; there are finitely many modular tensor categories (up to the appropriate notion of equivalence) of any fixed rank. It is conjectured that the same is true for fusion categories. We construct two sequences of finite tensor categories from certain Hopf algebras. These families will demonstrate that rank finiteness does not hold for a variety of classes of non-semisimple finite tensor categories, including degenerate premodular categories (the analog of degenerate premodular tensor categories), and modular categories (the analog of modular tensor categories). This talk is based on joint work with L. Chang, Z. Wang, and Q. Zhang.

Boundary Algebras and Local Topological Order: An Example

• Speaker: Daniel Wallick, The Ohio State University
• Time: 3:20pm to 3:30pm
• Abstract: Topologically ordered quantum spin systems are physical systems whose low-energy excitations are modeled by braided fusion categories. Recently, we provided axioms for local topological order that give rise to a net of boundary algebras. From these boundary algebras, one can recover the braided fusion category of excitations in a bulk-boundary correspondence. In this talk, we illustrate the local topological order axioms using a specific example, namely the toric code. This is joint work with Corey Jones, Pieter Naaijkens, and David Penneys.