Since 2002 Perimeter Institute has been recording seminars, conference talks, and public outreach events using video cameras installed in our lecture theatres. Perimeter now has 7 formal presentation spaces for its many scientific conferences, seminars, workshops and educational outreach activities, all with advanced audio-visual technical capabilities. Recordings of events in these areas are all available On-Demand from this Video Library and on Perimeter Institute Recorded Seminar Archive (PIRSA). PIRSA is a permanent, free, searchable, and citable archive of recorded seminars from relevant bodies in physics. This resource has been partially modelled after Cornell University's arXiv.org.
The quantum double models are parametrized by a finite-dimensional semisimple Hopf algebra (over $\mathbb{C}$). I will introduce the graphical calculus of these Hopf algebras and sketch how it is equivalent to the calculus of two interacting symmetric Frobenius algebras. Since symmetric Frobenius algebras are extended 2D TQFTs, this suggests that there is a canonical way to 'lift' a compatible pair of 2D TQFTs to a 3D TQFT.
We present a general scheme for constructing topological lattice models in any space dimension using tensor networks. Our approach relies on finding "simplex tensors" that satisfy a finite set of tensor equations. Given any such tensor, we construct a discrete topological quantum field theory (TQFT) and local commuting projector Hamiltonians on any lattice. The ground space degeneracy of these models is a topological invariant that can be computed via the TQFT, and the ground states are locally indistinguishable when the ground space is nondegenerate on the sphere.
The interaction of Hopf monoids and Frobenius monoids is the productive nucleus of the ZX calculus, where famously each Frobenius monoid-comonoid pair corresponds to a complementary basis and the Hopf structure describes the interaction between the bases. The theory of Interacting Hopf monoids (IH), introduced by Bonchi, Sobocinski and Zanasi, features essentially the same Hopf-Frobenius interaction pattern.
Categorical quantum mechanics is a research programme which aims to axiomatise (finite dimensional) quantum theory as an algebraic theory inside an abstract symmetric monoidal category. The central idea is that quantum observables can be
axiomatised as certain Frobenius algebras, and that two observables are (strongly) complementary when their Frobenius algebras jointly form a Hopf algebra. The resulting theory is surprisingly powerful, especially when combined with its graphical notation. In this talk
Kitaev originally constructed his quantum double model based on finite groups and anticipated the extension based on Hopf algebras, which was achieved later by Buerschaper, etc. In this talk, we will present the work on the generalization of Kitaev model for quantum groupoids and discuss its ground states.
I will describe a framework for the study of symmetry-enriched topological order using graded matrix product operator algebras. The approach is based upon an explicit construction of the extrinsic symmetry defects, which facilitates the extraction of their physical properties. This allows for a simple analysis of dual phase transitions, induced by gauging a global symmetry, and condensation of a bosonic subtheory.
I will discuss some (higher-)categorical structures present in three-dimensional topological field theories that include topological defects of any codimension. The emphasis will be on two topics:
(1) For Reshetikhin-Turaev type theories, regarded as 3-2-1-extended TFTs, I will explain why codimension-1 boundaries and defects form bicategories of module categories over suitable fusion categories.
In 1977, Blandford and Znajek discovered a process by which a spinning
black hole can transfer rotational energy to a force-free plasma, offering a possible mechanism for energy and jet emissions from quasars and other astrophysical sources. This Blandford-Znajek (BZ) mechanism is a Penrose process, which exploits the presence of an ergosphere supporting negative energy states, and it involves currents of electrical charge sourcing the toroidal magnetic field component of the emitted Poynting flux.
We show that Kitaev's lattice model for a finite-dimensional semisimple Hopf algebra H is equivalent to the combinatorial quantisation of Chern-Simons theory for the Drinfeld double D(H). As a result, Kitaev models are a special case of combinatorial quantization of Chern-Simons theory by Alekseev, Grosse and Schomerus. This equivalence is an analogue of the relation between Turaev-Viro and Reshetikhin-Turaev TQFTs and relates them to the quantisation of moduli spaces of flat connections.
A variety of models, especially Kitaev models, quantum Chern-Simons theory, and models from 3d quantum gravity, hint at a kind of lattice gauge theory in which the gauge group is generalized to a Hopf algebra. However, until recently, no general notion of Hopf algebra gauge theory was available. In this self-contained introduction, I will cover background on lattice gauge theory and Hopf algebras, and explain our recent construction of Hopf algebra gauge theory on a ribbon graph (arXiv:1512.03966).
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