Faithful simple objects, orders and gradings of fusion categories

Reconstructing braided subcategories of SU(N)k

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

On Müger's centralizer in braided equivariantized fusion categories

Let \({\mathcal C}\) be an integral fusion category. We study some graphs, called the prime graph and the common divisor graph, related to the Frobenius-Perron dimensions of simple objects in the category \({\mathcal C}\), that extend the corresponding graphs associated to the irreducible character degrees and the conjugacy class sizes of a finite group. We describe these graphs in several cases, among others, when \({\mathcal C}\) is an equivariantization under the action of a finite group, a \(2\)-step nilpotent fusion category, and the representation category of a twisted quantum double. We prove generalizations of known results on the number of connected components of the corresponding graphs for finite groups in the context of braided fusion categories. In particular, we show that if \({\mathcal C}\) is any integral non-degenerate braided fusion category, then the prime graph of \({\mathcal C}\) has at most \(3\) connected components, and it has at most \(2\) connected components if \({\mathcal C}\) is in addition solvable. As an application we prove a classification result for weakly integral braided fusion categories all of whose simple objects have prime power Frobenius-Perron dimension.

In this work, we use Ising chain and Kitaev chain to check the validity of an earlier proposal in arXiv:2011.02859 that enriched fusion (higher) categories provide a unified categorical description of all gapped/gapless quantum liquid phases, including symmetry-breaking phases, topological orders, SPT/SET orders and CFT-type gapless quantum phases. In particular, we show explicitly that, in each gapped phase realized by these two models, the spacetime observables form a fusion category enriched in a braided fusion category such that its monoidal center is trivial. We also study the categorical descriptions of the boundaries of these models. In the end, we obtain a classification of and the categorical descriptions of all 1-dimensional (spatial dimension) gapped quantum phases with a bosonic/fermionic finite onsite symmetry.

For 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.

Nilpotent fusion categories

The tensor functor called \alpha -induction produces a new unitary fusion category from a Frobenius algebra object, or a Q -system, in a braided unitary fusion category. In the operator algebraic language, it gives extensions of endomorphism of N to M arising from a subfactor N\subset M of finite index and finite depth, which gives a braided fusion category of endomorphisms of N . It is also understood in terms of Ocneanu’s graphical calculus. We study this \alpha -induction for bi-unitary connections, which provides a characterization of finite-dimensional nondegenerate commuting squares, and present certain 4 -tensors appearing in recent studies of 2 -dimensional topological order. We show that the resulting \alpha -induced bi-unitary connections are flat if we start with a commutative Frobenius algebra, or a local Q -system. Examples related to chiral conformal field theory and the Dynkin diagrams are presented.

We determine the fusion rules of the equivariantization of a fusion category \documentclass[12pt]{minimal}\begin{document}${\mathcal {C}}$\end{document}C under the action of a finite group G in terms of the fusion rules of \documentclass[12pt]{minimal}\begin{document}${\mathcal {C}}$\end{document}C and group-theoretical data associated to the group action. As an application we obtain a formula for the fusion rules in an equivariantization of a pointed fusion category in terms of group-theoretical data. This entails a description of the fusion rules in any braided group-theoretical fusion category.

We show that the core of a weakly group-theoretical braided fusion category [Formula: see text] is equivalent as a braided fusion category to a tensor product [Formula: see text], where [Formula: see text] is a pointed weakly anisotropic braided fusion category, and [Formula: see text] or [Formula: see text] is an Ising braided category. In particular, if [Formula: see text] is integral, then its core is a pointed weakly anisotropic braided fusion category. As an application we give a characterization of the solvability of a weakly group-theoretical braided fusion category. We also prove that an integral modular category all of whose simple objects have Frobenius–Perron dimension at most 2 is necessarily group-theoretical.

We prove that every braiding over a unitary fusion category is unitary and every unitary braided fusion category admits a unique unitary ribbon structure.

We show that a weakly integral braided fusion category ${{\mathcal C}}$ such that every simple object of ${{\mathcal C}}$ has Frobenius-Perron dimension ≤ 2 is solvable. In addition, we prove that such a fusion category is group-theoretical in the extreme case where the universal grading group of ${{\mathcal C}}$ is trivial.

We apply the yoga of classical homotopy theory to classification problems of G -extensions of fusion and braided fusion categories, where G is a finite group. Namely, we reduce such problems to classification (up to homotopy) of maps from BG to classifying spaces of certain higher groupoids. In particular, to every fusion category \mathcal C we attach the 3-groupoid \underline{\underline{\mathrm{BrPic}}}(\mathcal C) of invertible \mathcal C -bimodule categories, called the Brauer–Picard groupoid of \mathcal C , such that equivalence classes of G -extensions of \mathcal C are in bijection with homotopy classes of maps from BG to the classifying space of \underline{\underline{\mathrm{BrPic}}}(\mathcal C) . This gives rise to an explicit description of both the obstructions to existence of extensions and the data parametrizing them; we work these out both topologically and algebraically. One of the central results of the article is that the 2-truncation of \underline{\underline{\mathrm{BrPic}}}(\mathcal C) is canonically equivalent to the 2-groupoid of braided auto-equivalences of the Drinfeld center \mathcal Z(\mathcal C) of \mathcal C . In particular, this implies that the Brauer–Picard group \mathrm{BrPic}(\mathcal C) (i.e., the group of equivalence classes of invertible \mathcal C -bimodule categories) is naturally isomorphic to the group of braided auto-equivalences of \mathcal Z(\mathcal C) . Thus, if \mathcal C = \mathrm{Vec}_A , where A is a finite abelian group, then \mathrm{BrPic}(\mathcal C) is the orthogonal group \mathrm{O}(A \oplus A^*) . This allows one to obtain a rather explicit classification of extensions in this case; in particular, in the case G = \mathbb Z_2 , we re-derive (without computations) the classical result of Tambara and Yamagami. Moreover, we explicitly describe the category of all (\mathrm{Vec}_{A_1},\mathrm{Vec}_{A_2}) -bimodule categories (not necessarily invertible ones) by showing that it is equivalent to the hyperbolic part of the category of Lagrangian correspondences.

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.

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.