Julia Yael Plavnik

On Gauging Symmetry of Modular Categories

Topological order of a topological phase of matter in two spacial dimensions is encoded by a unitary modular (tensor) category (UMC). A group symmetry of the topological phase induces a group symmetry of its corresponding UMC. Gauging is a well-known theoretical tool to promote a global symmetry to a local gauge symmetry. We give a mathematical formulation of gauging in terms of higher category formalism. Roughly, given a UMC with a symmetry group G, gauging is a 2-step process: first extend the UMC to a G-crossed braided fusion category and then take the equivariantization of the resulting category. Gauging can tell whether or not two enriched topological phases of matter are different, and also provides a way to construct new UMCs out of old ones. We derive a formula for the

Tensor functors between Morita duals of fusion categories

{H^4}

Low-dimensional representations of the three component loop braid group

H4-obstruction, prove some properties of gauging, and carry out gauging for two concrete examples.

On fusion categories with few irreducible degrees

We study spin and super-modular categories systematically as inspired by fermionic topological phases of matter, which are always fermion parity enriched and modelled by spin TQFTs at low energy. We formulate a

ON THE CLASSIFICATION OF WEAKLY INTEGRAL MODULAR CATEGORIES

16

Modular categories of dimension

-fold way conjecture for the minimal modular extensions of super-modular categories to spin modular categories, which is a categorical formulation of gauging the fermion parity. We investigate general properties of super-modular categories such as fermions in twisted Drinfeld doubles, Verlinde formulas for naive quotients, and explicit extensions of

p^3m

PSU(2)_{4m+2}

with

with an eye towards a classification of the low-rank cases.