The Kitaev quantum double models are a family of topologically ordered spin models originally proposed to exploit the novel condensed matter phenomenology of topological phases for fault-tolerant quantum computation. Their physics is inherited from topological quantum field theories, while their underlying mathematical structure is based on a class of Hopf algebras. This structure is also seen across diverse fields of physics, and so allows connections to be made between the Kitaev models and topics as varied as quantum gauge theory and modified strong complementarity. This workshop will explore this shared mathematical structure and in so doing develop the connections between the fields of mathematical physics, quantum gravity, quantum information, condensed matter and quantum foundations.
Format results
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Semisimple Hopf algebras and fusion categories
Cesar Galindo - Universidad de los Andes
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The Hopf C*-algebraic quantum double models - symmetries beyond group theory
Andreas Bauer - Freie Universität Berlin
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Modular categories and the Witt group
Michael Mueger - Radboud Universiteit Nijmegen
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Topological Quantum Computation
Eric Rowell - Texas A&M University
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Gapped phases of matter vs. Topological field theories
Davide Gaiotto - Perimeter Institute for Theoretical Physics
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An Introduction to Hopf Algebra Gauge Theory
Derek Wise - University of Erlangen-Nuremberg
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Kitaev lattice models as a Hopf algebra gauge theory
Catherine Meusburger - University of Erlangen-Nuremberg
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Topological defects and higher-categorical structures
Jurgen Fuchs - Karlstad University
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Introduction to CQM
Ross Duncan - University of Oxford
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Interacting Hopf monoids and Graphical Linear Algebra
Pawel Sobocinski - University of Southampton
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Semisimple Hopf algebras and fusion categories
Cesar Galindo - Universidad de los Andes
This talk will be a short introduction to the semisimple Hopf algebras over an algebraically closed field of characteristic 0 and their representation theories. It is intended to outline the main basic results about structure and known methods for the construction of semisimple Hopf algebras… -
The Hopf C*-algebraic quantum double models - symmetries beyond group theory
Andreas Bauer - Freie Universität Berlin
I will give an introduction to the Kitaev quantum double models for Hopf C*-algebras. To this end I will introduce a graphical tensor-network notation to represent the algebraic objects and axioms. Using this notation I will then present the vertex- and plaquette symmetries of the model and discuss… -
Modular categories and the Witt group
Michael Mueger - Radboud Universiteit Nijmegen
The aim of this talk is to give an introduction to modular categories, touching both basics and recent developments. I will begin with a quick reminder concerning tensor categories, in particular braided and symmetric ones, and notions like duality, fusion and spherical categories. We'll meet… -
Topological Quantum Computation
Eric Rowell - Texas A&M University
The (Freedman-Kitaev) topological model for quantum computation is an inherently fault-tolerant computation scheme, storing information in topological (rather than local) degrees of freedom with quantum gates typically realized by braiding quasi-particles in two dimensional media. I will give an… -
Gapped phases of matter vs. Topological field theories
Davide Gaiotto - Perimeter Institute for Theoretical Physics
I will discuss the relation between topological field theories and gapped phases of matter. I will propose a general formalism to define a class of TFTs which can be realized by commuting projector Hamiltonians. This allows one to apply rigorous mathematical theorems about TFTs to gapped phases of… -
An Introduction to Hopf Algebra Gauge Theory
Derek Wise - University of Erlangen-Nuremberg
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… -
Kitaev lattice models as a Hopf algebra gauge theory
Catherine Meusburger - University of Erlangen-Nuremberg
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… -
Topological defects and higher-categorical structures
Jurgen Fuchs - Karlstad University
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… -
Symmetry-enriched topological order in tensor networks: Gauging and anyon condensation
Dominic Williamson - University of Vienna
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… -
Kitaev models based on unitary quantum groupoids
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… -
Introduction to CQM
Ross Duncan - University of Oxford
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… -
Interacting Hopf monoids and Graphical Linear Algebra
Pawel Sobocinski - University of Southampton
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)…