Winter 2018 Seminar
Organizers: Stephen Bigelow, Colleen Delaney, James Tener
March 7 | Dom Williamson (Yale)
Graded fusion categories in tensor network states with topological order and global symmetry (audio)
I will describe how we stumbled upon fusion categories while thinking about the internal symmetries of tensor network descriptions of quantum states. Next, I will explain how Ocneanu's tube algebra can be used to construct the topological superselection sectors in these states and to extract the relevant physical data. If time permits I will also describe the modifications required to incorporate a global symmetry into this approach.
February 28 | Matthew Brown (UCSB)
1+1 Yang Mills
I'll give a pedagogical overview of Yang-Mills theory in two spacetime dimensions, an "almost" topological field theory with a number of remarkable properties. Time permitting, I'll discuss its connections to volumes of moduli spaces of flat connections, the Verlinde formula, the large level limit of Chern-Simons theory, and string theory.
Motion Groups Workshop (Part II)
Motion Groups Workshop (Part I)
January 31 | Wade Bloomquist (UCSB)
What is Quantum Topology/Algebra?
We look to continue the "back to the basics" series of talks. This talk will introduce the fields of quantum topology and quantum algebra from a point of view starting at the braid group. This talk will be aimed at a non-expert audience, and will have an underlying theme of diagrammatic algebra.
January 24 | Karel Casteels (UCSB)
What can a mathematician mean when they say "Quantum Group"?
The notion of a quantum group arose in the 1980's in connection to mathematical physics, and has since taken a life of its own within mathematics. While there is no widespread consensus on what exactly these are, they are usually various noncommutative deformations of classical algebro-geometric objects depending on a parameter q such that when q=1, one recovers the original object.
This talk, which is aimed at a non-expert audience, will tour some of the classical constructions of quantum groups up to some more modern viewpoints.
January 17 | Chao-Ming Jian (Microsoft)Folding approach to topological orders enriched by mirror symmetry
In 2+1D, the symmetry-enriched topological orders (SET) with local unitary symmetry can be systematically studied using the framework of G-crossed braided tensor category. In contrast, the SETs with mirror reflection symmetry require different techniques due to the non-local and orientation-reversing nature of the symmetry action. In this talk, I will introduce a folding approach that maps the SET with mirror symmetry to a bilayer system local unitary Z2 symmetry. The essential information of the mirror symmetry fractionalization in the original system will be mapped to the topological data of gapped boundaries in the bilayer system which can be then systematically studied using the anyon condensation theory. Using this approach, I will discuss the physical constraints on data that describe mirror SETs as well as the mirror symmetry quantum anomaly.