C17029 - Hopf Algebras in Kitaev's Quantum Double Models: Mathematical Connections from Gauge Theory to Topological Quantum Computing and Categorical Quantum Mechanics

I will explain a general strategy to lift (2+1)D topological phases, in particular string nets, to (3+1)D models with line defects. This allows a systematic construction of (3+1)D topological theories with defects, including an improved version of the Walker-Wang Model. It has also an interesting application to quantum gravity as it leads to quantum geometry realizations for which all geometric operators have discrete and bounded spectra. I will furthermore comment on some interesting (self-) duality relations that emerge in these constructions.

Ground state degeneracy is an important characteristic of topological order. It is a natural question under what conditions such topological degeneracy extends to higher energy states or even to the full energy spectrum of a model, in such a way that the degeneracy is preserved when the Hamiltonian of the system is perturbed. It appears that Ising/Majorana wires have this property due to the presence of robust edge zero modes. Generalized wire models based on parafermions also have edge zero modes and topological degeneracy at special points in parameter space, but the stability of these modes is a much more intricate question. These models are related to Hopf algebras or tensor categories in several ways. In particular they are "golden chain" type models based on fusion categories for boundary defects of Abelian TQFTs. As such they are part of a much larger class of Hopf algebra based chain models with edge modes. It is natural to ask which of these have stable edge zero modes and/or full spectrum degeneracy.

It is well known that commutative Frobenius algebras can be represented as topological surfaces, using the graphical calculus of dualizable objects in monoidal 2-categories. We build on related ideas to show that the interacting Frobenius algebras of Duncan and Dunne, which have a Hopf algebra structure, arise naturally in a similar way, by requiring a single 3-morphism in a 3-category to be invertible. We show that this gives a purely geometrical proof of Mueger's version of Tannakian reconstruction of Hopf algebras from fusion categories equipped with a fibre functor. We also relate our results to the theory of lattice code surgery.

The Turaev-Viro invariant for a closed 3-manifold is defined as the contraction of a certain tensor network. The tensors correspond to tetrahedra in a triangulation of the manifold, with values determined by a fixed spherical category. For a manifold with boundary, the tensor network has free indices that can be associated to qudits, and its contraction gives the coefficients of a quantum error-correcting code. The code has local stabilizers determined by Levin and Wen. By studying braid group representations acting on equivalence classes of colored ribbon graphs embedded in a punctured sphere, we identify the anyons, and give a simple recipe for mapping fusion basis states of the doubled category to ribbon graphs. Combined with known universality results for anyonic systems, this provides a large family of schemes for quantum computation based on local deformations of stabilizer codes. These schemes may serve as a starting point for developing fault-tolerance schemes using continuous stabilizer measurements and active error-correction. This is joint work with Greg Kuperberg and Ben Reichardt.

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. Time permitting, I will also showcase how to rederive graphically parts of the Larson-Sweedler theorem, giving various equivalent characterizations of semisimplicity, thereby generalizing these results to arbitrary Hopf monoids in traced symmetric monoidal categories.

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. Any ground state can be realized by a tensor network obtained by contracting simplex tensors. Our models are exact renormalization fixed points, covering a broad range of models in the literature. We identify symmetries on the virtual level of the tensor networks of our models that generalize the topological invariance properties beyond fixed point models. This framework combined with recent tensor network techniques is convenient for studying excitations, their statistics, phase transitions, and ultimately for classification of gapped phases of many-body theories in 3+1 and higher dimensions.

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. The free symmetric monoidal category generated by IH is isomorphic to the category of linear relations over the field of rationals: thus the string diagrams of IH are an alternative graphical language for elementary concepts of linear algebra. IH has a modular construction via distibutive laws of props, and has been applied as a compositional language of signal flow graphs. In this talk I will outline the equational theory, its construction and applications, as well as report on ongoing and future work.

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 I'll introduce the main concepts and present some applications to quantum computation.

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 the case of defects separating three-dimensional regions supporting the same theory, the relevant fusion category $A$ is the modular tensor category underlying that theory, while for defects separating two theories of Turaec-Viro type with underlying fusion categories $A_1$ and $A_2$, respectively, $A$ is the the Deligne product $A_1 boxtimes A_2^{op}$. (2) I will indicate the building blocks of a generalization of the TV-BW state-sum construction to theories with defects. Making use of ends and coends, various aspects of this construction can be formulated without requiring semisimplicity.

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. We show that the topological invariants of the two models, the algebra of operators acting on the protected space of the Kitaev model and the quantum moduli algebra from the combinatorial quantisation formalism, are isomorphic. This is established in a gauge theoretical picture, in which both models appear as Hopf algebra valued lattice gauge theories.

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). The resulting theory parallels ordinary lattice gauge theory, generalizing its structure only as necessary to accommodate more general Hopf algebras. All of the key features of gauge theory, including gauge transformations, connections, holonomy and curvature, and observables, have Hopf algebra analogues, but with a richer structure arising from non-cocommuntativity, the key property distinguishing Hopf algebras from groups. Main results include topological invariance of algebras of observables, and a gauge theoretic derivation of algebras previously obtained in the combinatorial quantization of Chern-Simons theory.

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 matter. I will also discuss the role of generalized cohomology theories and spectra in the classification of SPT phases.

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 overview of this model, emphasizing the mathematical aspects.

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 algebras in tensor categories, categories of modules, module categories and their connection. I will then focus on modular categories and some basic structure theory. We will consider two ways of obtaining modular categories: modularization and the Drinfeld center. (The important third one, quantum groups at root of unity, is too complicated to be discussed in any depth.) The Drinfeld center will be used to define Witt equivalence of modular categories and the Witt group. Several equivalent characterization of Witt equivalence, using module categories, will be discussed. The Witt group will be crucial for any (future) classification of modular categories, as well as for application to physics in condensed matter physics and conformal field theory.

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 their interaction and the excitation structure they give rise to.

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: extensions, twisting, Tannakian reconstruction. Basic notions concerning tensor categories will be introduced: braided structures, center construction, fiber functors. Special emphasis is given to the notion of fusion category. At the same time, the relations between these notions and those of the Hopf algebras are studied.