Braided Fusion Research Articles

On G-crossed Frobenius ⋆-algebras and fusion rings associated with braided G-actions

For a finite group G, Turaev introduced the notion of a braided G-crossed fusion category. The classification of braided G-crossed extensions of braided fusion categories was studied by Etingof, Nikshych and Ostrik in terms of certain group cohomological data. In this paper we will define the notion of a G-crossed Frobenius ⋆-algebra and give a classification of (strict) G-crossed extensions of a commutative Frobenius ⋆-algebra R equipped with a given action of G, in terms of the second group cohomology H2(G,R×). Now suppose that B is a non-degenerate braided fusion category equipped with a braided action of a finite group G. We will see that the associated G-graded fusion ring is in fact a (strict) G-crossed Frobenius ⋆-algebra. We will describe this G-crossed fusion ring in terms of the classification of braided G-actions by Etingof, Nikshych, Ostrik and derive a Verlinde formula to compute its fusion coefficients.

Braided Zesting and Its Applications

We give a rigorous development of the construction of new braided fusion categories from a given category known as zesting. This method has been used in the past to provide categorifications of new fusion rule algebras, modular data, and minimal modular extensions of super-modular categories. Here we provide a complete obstruction theory and parameterization approach to the construction and illustrate its utility with several examples.

Braided Picard groups and graded extensions of braided tensor categories

We classify various types of graded extensions of a finite braided tensor category \(\mathcal {B}\) in terms of its 2-categorical Picard groups. In particular, we prove that braided extensions of \(\mathcal {B}\) by a finite group A correspond to braided monoidal 2-functors from A to the braided 2-categorical Picard group of \(\mathcal {B}\) (consisting of invertible central \(\mathcal {B}\)-module categories). Such functors can be expressed in terms of the Eilnberg-Mac Lane cohomology. We describe in detail braided 2-categorical Picard groups of symmetric fusion categories and of pointed braided fusion categories.

Extension Theory for Braided-Enriched Fusion Categories

AbstractFor a braided fusion category $\mathcal{V}$, a $\mathcal{V}$-fusion category is a fusion category $\mathcal{C}$ equipped with a braided monoidal functor $\mathcal{F}:\mathcal{V} \to Z(\mathcal{C})$. Given a fixed $\mathcal{V}$-fusion category $(\mathcal{C}, \mathcal{F})$ and a fixed $G$-graded extension $\mathcal{C}\subseteq \mathcal{D}$ as an ordinary fusion category, we characterize the enrichments $\widetilde{\mathcal{F}}:\mathcal{V} \to Z(\mathcal{D})$ of $\mathcal{D}$ that are compatible with the enrichment of $\mathcal{C}$. We show that G-crossed extensions of a braided fusion category $\mathcal{C}$ are G-extensions of the canonical enrichment of $\mathcal{C}$ over itself. As an application, we parameterize the set of $G$-crossed braidings on a fixed $G$-graded fusion category in terms of certain subcategories of its center, extending Nikshych’s classification of the braidings on a fusion category.

Fusion structure from exchange symmetry in (2+1)-dimensions

Until recently, a careful derivation of the fusion structure of anyons from some underlying physical principles has been lacking. In Shi et al. (2020), the authors achieved this goal by starting from a conjectured form of entanglement area law for 2D gapped systems. In this work, we instead start with the principle of exchange symmetry, and determine the minimal prescription of additional postulates needed to make contact with unitary ribbon fusion categories as the appropriate algebraic framework for modelling anyons. Assuming that 2D quasiparticles are spatially localised, we build a functor from the coloured braid groupoid to the category of finite-dimensional Hilbert spaces. Using this functor, we construct a precise notion of exchange symmetry, allowing us to recover the core fusion properties of anyons. In particular, given a system of n quasiparticles, we show that the action of a certain n-braid βn uniquely specifies its superselection sectors. We then provide an overview of the braiding and fusion structure of anyons in the usual setting of braided 6j fusion systems. By positing the duality axiom of Kitaev (2006) and assuming that there are finitely many distinct topological charges, we arrive at the framework of ribbon categories.

Defects in the 3-dimensional toric code model form a braided fusion 2-category

It was well known that there are e-particles and m-strings in the 3-dimensional (spatial dimension) toric code model, which realizes the 3-dimensional ℤ2 topological order. Recent mathematical result, however, shows that there are additional string-like topological defects in the 3-dimensional ℤ2 topological order. In this work, we construct all topological defects of codimension 2 and higher, and show that they form a braided fusion 2-category satisfying a braiding non-degeneracy condition.

Quinn\u2019s Formula and Abelian 3-Cocycles for Quadratic Forms

In pointed braided fusion categories knowing the self-symmetry braiding of simples is theoretically enough to reconstruct the associator and braiding on the entire category (up to twisting by a braided monoidal auto-equivalence). We address the problem to provide explicit associator formulas given only such input. This problem was solved by Quinn in the case of finitely many simples. We reprove and generalize this in various ways. In particular, we show that extra symmetries of Quinn’s associator can still be arranged to hold in situations where one has infinitely many isoclasses of simples.

Norm, trace, and formal codegrees of fusion categories

We prove several results in the theory of fusion categories using the product (norm) and sum (trace) of Galois conjugates of formal codegrees. First, we prove that finitely-many fusion categories exist up to equivalence whose global dimension has a fixed norm. Furthermore, with two exceptions, all formal codegrees of spherical fusion categories with square-free norm are rational integers. This implies, with three exceptions, that every spherical braided fusion category whose global dimension has prime norm is pointed. The reason exceptions occur is related to the classical Schur-Siegel-Smyth problem of describing totally positive algebraic integers of small absolute trace.

Prime decomposition of modular tensor categories of local modules of type D

Let \mathcal{C}(\mathfrak{g},k) be the unitary modular tensor categories arising from the representation theory of quantum groups at roots of unity for arbitrary simple finite-dimensional complex Lie algebra \mathfrak{g} and positive integer levels k . Here we classify nondegenerate fusion subcategories of the modular tensor categories of local modules \mathcal{C}(\mathfrak{g},k)_R^0 where R is the regular algebra of Tannakian Rep (H)\subset\mathcal{C}(\mathfrak{g},k)_\text{pt} . We describe the decomposition of \mathcal{C}(\mathfrak{g},k)_R^0 into prime factors, and as an application we classify relations in the Witt group of nondegenerately braided fusion categories generated by the equivalency classes of \mathcal{C}(\mathfrak{so}_5,k) and \mathcal{C}(\mathfrak{g}_2,k) for k\in\mathbb{Z}_{\geq1} .

On the braid group representations coming from weakly group-theoretical fusion categories

We prove that representations of the braid groups coming from weakly group-theoretical braided fusion categories have finite images.

Classification of topological phases with finite internal symmetries in all dimensions

We develop a mathematical theory of symmetry protected trivial (SPT) orders and anomaly-free symmetry enriched topological (SET) orders in all dimensions via two different approaches with an emphasis on the second approach. The first approach is to gauge the symmetry in the same dimension by adding topological excitations as it was done in the 2d case, in which the gauging process is mathematically described by the minimal modular extensions of unitary braided fusion 1-categories. This 2d result immediately generalizes to all dimensions except in 1d, which is treated with special care. The second approach is to use the 1-dimensional higher bulk of the SPT/SET order and the boundary-bulk relation. This approach also leads us to a precise mathematical description and a classification of SPT/SET orders in all dimensions. The equivalence of these two approaches, together with known physical results, provides us with many precise mathematical predictions.

RANK-FINITENESS FOR G-CROSSED BRAIDED FUSION CATEGORIES

We establish rank-finiteness for the class of G-crossed braided fusion categories, generalizing the recent result for modular categories and including the important case of braided fusion categories. This necessitates a study of slightly degenerate braided fusion categories and their centers, which are interesting for their own sake.

Conjugacy classes and centralizers for pivotal fusion categories

A criterion for the Muger centralizer of a fusion subcategory of a braided non-degenerate fusion category is given. Along the way we extend some identities on the space of class functions of a fusion category introduced by Shimizu (J Pure Appl Algebra 221(9):2338–2371, 2017). We also show that in a modular tensor category the product of two conjugacy class sums is a linear combination of conjugacy class sums with rational coefficients.

Integral metaplectic modular categories

A braided fusion category is said to have Property F if the associated braid group representations factor through a finite group. We verify integral metaplectic modular categories have property F by showing these categories are group-theoretical. For the special case of integral categories [Formula: see text] with the fusion rules of [Formula: see text] we determine the finite group [Formula: see text] for which [Formula: see text] is braided equivalent to [Formula: see text]. In addition, we determine the associated classical link invariant, an evaluation of the 2-variable Kauffman polynomial at a point.

Four dimensional topological quantum field theories from $G$-crossed braided categories

We construct a state-sum type invariant of smooth closed oriented 4-manifolds out of a G -crossed braided spherical fusion category ( G -BSFC) for G a finite group. The construction can be extended to obtain a (3+1) -dimensional topological quantum field theory (TQFT). The invariant of 4-manifolds generalizes several known invariants in literature such as the Crane–Yetter invariant from a ribbon fusion category and Yetter's invariant from homotopy 2-types. If the G -BSFC is concentrated only at the sector indexed by the trivial group element, a cohomology class in H^4(G,U(1)) can be introduced to produce a different invariant, which reduces to the twisted Dijkgraaf–Witten theory in a special case. Although not proven, it is believed that our invariants are strictly different from other known invariants. It remains to be seen if the invariants are sensitive to smooth structures. It is expected that the most general input to the state-sum type construction of (3+1) -TQFTs is a spherical fusion 2-category. We show that a G -BSFC corresponds to a monoidal 2-category with certain extra structure, but that structure does not satisfy all the axioms of a spherical fusion 2-category given by M. Mackaay. Thus the question of what axioms properly define a spherical fusion 2-category is open.

Slightly Degenerate Categories and ℤ-Modular Data

AbstractGiven a slightly degenerate braided pivotal fusion category $\mathscr{C}$, we explain how it naturally gives rise to a $\mathbb{Z}$-modular data. We do not restrict to spherical categories and work with pivotal categories. Finally, we give an interpretation in this framework of the Bonnafé–Rouquier categorification of the $\mathbb{Z}$-modular datum associated to nontrivial family of the cyclic complex reflection group.

On braided almost square-free fusion categories with odd dimension

Let q be a odd prime number, d be an odd square-free natural number. We prove that a braided fusion category of Frobenius–Perron dimension dqn is solvable. If then we further prove that is tensor equivalent to the category of finite-dimensional representations of a semisimple quasitriangular Hopf algebra which is kG for some abelian group G, or a crossed product for some σ.

Fusion categories for affine vertex algebras at admissible levels

The main result is that the category of ordinary modules of an affine vertex operator algebra of a simply laced Lie algebra at admissible level is rigid and thus a braided fusion category. If the level satisfies a certain coprime property then it is even a modular tensor category. In all cases open Hopf links coincide with the corresponding normalized S-matrix entries of torus one-point functions. This is interpreted as a Verlinde formula beyond rational vertex operator algebras. A preparatory Theorem is a convenient formula for the fusion rules of rational principal W-algebras of any type.

Vanishing of Categorical Obstructions for Permutation Orbifolds

The orbifold construction $A\mapsto A^G$ for a finite group $G$ is fundamental in rational conformal field theory. The construction of $Rep(A^G)$ from $Rep(A)$ on the categorical level, often called gauging, is also prominent in the study of topological phases of matter. Given a non-degenerate braided fusion category $\mathcal{C}$ with a $G$-action, the key step in this construction is to find a braided $G$-crossed extension compatible with the action. The extension theory of Etingof-Nikshych-Ostrik gives two obstructions for this problem, $o_3\in H^3(G)$ and $o_4\in H^4(G)$ for certain coefficients, the latter depending on a categorical lifting of the action and is notoriously difficult to compute. We show that in the case where $G\le S_n$ acts by permutations on $\mathcal{C}^{\boxtimes n}$, both of these obstructions vanish. This verifies a conjecture of M\"uger, and constitutes a nontrivial test of the conjecture that all modular tensor categories come from vertex operator algebras or conformal nets.

Level bounds for exceptional quantum subgroups in rank two

There is a long-standing belief that the modular tensor categories [Formula: see text], for [Formula: see text] and finite-dimensional simple complex Lie algebras [Formula: see text], contain exceptional connected étale algebras (sometimes called quantum subgroups) at only finitely many levels [Formula: see text]. This premise has known implications for the study of relations in the Witt group of nondegenerate braided fusion categories, modular invariants of conformal field theories, and the classification of subfactors in the theory of von Neumann algebras. Here, we confirm this conjecture when [Formula: see text] has rank 2, contributing proofs and explicit bounds when [Formula: see text] is of type [Formula: see text] or [Formula: see text], adding to the previously known positive results for types [Formula: see text] and [Formula: see text].