### Fall 2017 Seminar

##### Organizers: Stephen Bigelow, Colleen Delaney, James Tener

###### Back to current seminar schedule

November 16 | Christina Knapp (UCSB)

**Anyonic entanglement and topological entanglement entropy**

In this talk, I will discuss the properties of entanglement in two-dimensional topologically ordered phases of matter, with particular focus on the topological entanglement entropy (see Bonderson, Knapp, Patel, Annals of Physics Vol. 385, 2017). Topologically ordered phases support anyons, quasiparticles with exotic exchange statistics. The emergent nonlocal state spaces of anyonic systems admit a particular form of entanglement that does not exist in conventional quantum mechanical systems. We study this entanglement by adapting standard notions of entropy to anyonic systems. We use the algebraic theory of anyon models (modular tensor categories) to present a general method of deriving the universal topological contributions to the entanglement entropy for general system configurations of a topological phase, including surfaces of arbitrary genus, punctures, and quasiparticle content. In doing so, we find that the topological entanglement entropy emerges as a consequence of the conservation of topological charge.

November 9 | Zhu-Xi Luo (University of Utah)

**Topological Entanglement Entropy in Euclidean AdS_3 **

Topological entanglement entropy (TEE) has been widely used in condensed matter physics to characterize topological order. We will see how TEE arises in gravitational theory living in Euclidean Asymptotic AdS_3 space. The case of the BTZ black hole will be discussed in detail. After summing over classical geometries, we will construct a Moonshine double state given by a "maximally entangled" superposition of "anyons", which captures the TEE property.

November 2 | James Tener (UCSB)

**Fusion of representations of affine Lie algebras**

Affine Lie algebras at positive integral level provide fundamental examples of unitary 2d conformal field theories. A notion of `fusion product' of modules over these Lie algebras was originally introduced by physicists, and today there are many mathematical formalizations of this work. In this talk I will describe a new construction of the fusion product, and how construction leads to the first examples of unitary functorial conformal field theories as originally described by Graeme Segal in the 1980s. No background in conformal field theory will be assumed. This is joint work with Andre Henriques.

October 24 | Terry Gannon (University of Alberta)

**Moonshine - old and new **

Moonshine formally began almost 40 years ago with the observation that 196884=196883+1. About 5 years ago or so a new era of Moonshine began with the equation 90=45+45. My talk will introduce the spectrum of moonshines, describing how they are similar and how they differ.

October 19 | Eric Samperton (UC Davis)

**Coloring invariants are often intractable**

I'll introduce an NP-complete problem called ZSAT, which is a model for reversible circuits where the alphabet has a symmetry and all of the gates respect that symmetry. I'll then sketch a proof that ZSAT reduces to the problem of counting pointed G-covers of knots (i.e. G-colorings), whenever G is a fixed non-abelian finite simple group. A useful tool is the classifying space BG_C for concordance classes of C-branched G-covers, where C is a conjugacy invariant subset of G consisting of allowed branching types. Time permitting, I'll speculate on applications to the universality of G-equivariant (2+1)-D TQFTs.

October 5 | Wade Bloomquist (UCSB)

**Spiders and Asymptotic Faithfulness **

The spiders of Kuperberg provide a diagrammatic formulation for studying the representation theory of Lie algebras (and closely related quantum groups). In many ways these spiders encode the combinatorics of these representations. After briefly introducing spiders, we will see how they can be used to construct families of mapping class group representations and explore some exciting properties that some of these families possess.