# Modular Categories--Their Representations, Classification, and Applications (16w5049)

Arriving in Oaxaca, Mexico Sunday, August 14 and departing Friday August 19, 2016

## Organizers

Zhenghan Wang (University of California, Santa Barabra)

Eric Rowell (Texas A&M University)

Siu-Hung Ng (Louisiana State University)

Dmitri Nikshych (University of New Hampshire)

## Objectives

The main objective is to understand the landscape of modular categories both as pure mathematical structures and as generalized group-like symmetries of quantum systems.

As mathematical structures, we are interested in their classification and representation. A recent breakthrough is the proof of the rank-finiteness conjecture: for a fixed rank, there are only finitely many equivalence classes of modular categories. The number theoretical tools that are developed for the proof can be used to advance several directions such as the classification of low rank modular categories. First we will pursue the extension of rank-finiteness to pre-modular categories and unitary fusion categories. Two more major open questions will be the property F conjecture for weakly integral modular categories and whether or not the modular S, T matrices determine a modular category uniquely. Another direction is the interplay of symmetries and modular categories as inspired by symmetry enriched topological phases of matter. The famous example of such a symmetry protected topological phase of matter is the new materials---topological insulators. Critical to both classification and finite group action on modular categories is the study of their representations---module categories. Module categories arise in physics as topological defects of topological phases of matter, and as topological boundary conditions. For applications, it is important to parametrize all module categories over a given modular category.

Application of modular categories to physics and quantum computation continues to inspire new mathematical problems and deepen our understanding of their structures. One direction is the close connection between modular categories and conformal field theories. The partition function on the tori for a conformal field theory is a modular form like object. So we will explore the possibility of attaching a modular form like object to a modular category with some extra data, potentially just a module category. Another direction is the promotion of a group symmetry of a modular category to a local gauge symmetry---so-called gauging. Gauging is the inverse of the procedure called taking the core by Drinfeld-Gelaki- Nikshych-Ostrik, and can also be regarded as a construction of new modular categories from an old one together with a finite group action. In particular, it leads to a large class of weakly integral modular categories by gauging finite group symmetries of pointed modular categories.

A long-term objective is to establish a robust, cohesive community of researchers working in modular categories and their applications. Vital to this objective is the inclusion of junior researchers and researchers in underrepresented groups in the effort, with both applied and theoretical backgrounds. Our workshop will emphasize this broad participation. We expect to find new approaches to old problems and open new directions on a mathematical structure that sits in the triple juncture of mathematics, physics, and computer science.

As mathematical structures, we are interested in their classification and representation. A recent breakthrough is the proof of the rank-finiteness conjecture: for a fixed rank, there are only finitely many equivalence classes of modular categories. The number theoretical tools that are developed for the proof can be used to advance several directions such as the classification of low rank modular categories. First we will pursue the extension of rank-finiteness to pre-modular categories and unitary fusion categories. Two more major open questions will be the property F conjecture for weakly integral modular categories and whether or not the modular S, T matrices determine a modular category uniquely. Another direction is the interplay of symmetries and modular categories as inspired by symmetry enriched topological phases of matter. The famous example of such a symmetry protected topological phase of matter is the new materials---topological insulators. Critical to both classification and finite group action on modular categories is the study of their representations---module categories. Module categories arise in physics as topological defects of topological phases of matter, and as topological boundary conditions. For applications, it is important to parametrize all module categories over a given modular category.

Application of modular categories to physics and quantum computation continues to inspire new mathematical problems and deepen our understanding of their structures. One direction is the close connection between modular categories and conformal field theories. The partition function on the tori for a conformal field theory is a modular form like object. So we will explore the possibility of attaching a modular form like object to a modular category with some extra data, potentially just a module category. Another direction is the promotion of a group symmetry of a modular category to a local gauge symmetry---so-called gauging. Gauging is the inverse of the procedure called taking the core by Drinfeld-Gelaki- Nikshych-Ostrik, and can also be regarded as a construction of new modular categories from an old one together with a finite group action. In particular, it leads to a large class of weakly integral modular categories by gauging finite group symmetries of pointed modular categories.

A long-term objective is to establish a robust, cohesive community of researchers working in modular categories and their applications. Vital to this objective is the inclusion of junior researchers and researchers in underrepresented groups in the effort, with both applied and theoretical backgrounds. Our workshop will emphasize this broad participation. We expect to find new approaches to old problems and open new directions on a mathematical structure that sits in the triple juncture of mathematics, physics, and computer science.