Fusion Categories Research Articles

Reflection fusion categories

We introduce the notion of a reflection fusion category, which is a type of a G-crossed category generated by objects of Frobenius–Perron dimension 1 and p, where p is an odd prime. We show that such categories correspond to orthogonal reflection groups over Fp. This allows us to use the known classification of irreducible reflection groups over finite fields to classify irreducible reflection fusion categories.

Eigenvalues of rotations and braids in spherical fusion categories

We give formulae for the multiplicities of eigenvalues of generalized rotation operators in terms of generalized Frobenius–Schur indicators in a semisimple spherical tensor category C. In particular, this implies that the entire collection of rotation eigenvalues for a fusion category can be computed from the fusion rules and the traces of rotation at finitely many tensor powers. We also establish a rigidity property for FS indicators of fusion categories with a given fusion ring via Jones's theory of planar algebras. If C is also braided, these formulae yield the multiplicities of eigenvalues for a large class of braids in the associated braid group representations. When C is modular, this allows one to determine the eigenvalues and multiplicities of braids in terms of just the S and T matrices.

The classification of $3^n$ subfactors and related fusion categories

We investigate a (potentially infinite) series of subfactors, called 3^n subfactors , including A_4 , A_7 , and the Haagerup subfactor as the first three members corresponding to n=1,2,3 . Generalizing our previous work for odd n , we further develop a Cuntz algebra method to construct 3^n subfactors and show that the classification of the 3^n subfactors and related fusion categories is reduced to explicit polynomial equations under a mild assumption, which automatically holds for odd n . In particular, our method with n=4 gives a uniform construction of 4 finite depth subfactors, up to dual, without intermediate subfactors of index 3+\sqrt{5} . It also provides a key step for a new construction of the Asaeda–Haagerup subfactor due to Grossman, Snyder, and the author.

On faithfulness of the lifting for Hopf algebras and fusion categories

We use a version of Haboush's theorem over complete local Noetherian rings to prove faithfulness of the lifting for semisimple cosemisimple Hopf algebras and separable (braided, symmetric) fusion categories from characteristic $p$ to characteristic zero (arXiv/math:0203060, Section 9), showing that, moreover, any isomorphism between such structures can be reduced modulo $p$. This fills a gap in arXiv/math:0203060, Subsection 9.3. We also show that lifting of semisimple cosemisimple Hopf algebras is a fully faithful functor, and prove that lifting induces an isomorphism on Picard and Brauer-Picard groups. Finally, we show that a subcategory or quotient category of a separable multifusion category is separable (resolving an open question from arXiv/math:0203060, Subsection 9.4), and use this to show that certain classes of tensor functors between lifts of separable categories to characteristic zero can be reduced modulo $p$.

Full dyon excitation spectrum in extended Levin-Wen models

In Levin-Wen (LW) models, a wide class of exactly solvable discrete models, for two dimensional topological phases, it is relatively easy to describe only single fluxon excitations, but not the charge and dyonic as well as many-fluxon excitations. To incorporate charged and dyonic excitations in (doubled) topological phases, an extension of the LW models is proposed in this paper. We first enlarge the Hilbert space with adding a tail on one of the edges of each trivalent vertex, to describe the internal charge degrees of freedom at the vertex. Then we study the full dyon spectrum of the extended LW models, including both quantum numbers and wave functions for dyonic quasiparticle excitations. The local operators associated with the dyonic excitations are shown to form the so-called tube algebra, whose representations (modules) form the quantum double (categoric center) of the input data (unitary fusion category). In physically relevant cases, the input data is from a finite or quantum group (with braiding $R$-matrices), we find that the elementary excitations (or dyon species), as well as any localized/isolated excited states, are characterized by three quantum numbers: charge, fluxon type, and twist. They provide a "complete basis" for many-body states in the enlarged Hilbert space. Concrete examples are presented and the relevance of our results to the electric-magnetic duality existing in the models is addressed.

Finiteness for mapping class group representations from twisted Dijkgraaf–Witten theory

We show that any twisted Dijkgraaf–Witten representation of a mapping class group of an orientable, compact surface with boundary has finite image. This generalizes work of Etingof et al. showing that the braid group images are finite [P. Etingof, E. C. Rowell and S. Witherspoon, Braid group representations from twisted quantum doubles of finite groups, Pacific J. Math. 234 (2008)(1) 33–42]. In particular, our result answers their question regarding finiteness of images of arbitrary mapping class group representations in the affirmative. Our approach is to translate the problem into manipulation of colored graphs embedded in the given surface. To do this translation, we use the fact that any twisted Dijkgraaf–Witten representation associated to a finite group [Formula: see text] and 3-cocycle [Formula: see text] is isomorphic to a Turaev–Viro–Barrett–Westbury (TVBW) representation associated to the spherical fusion category [Formula: see text] of twisted [Formula: see text]-graded vector spaces. The representation space for this TVBW representation is canonically isomorphic to a vector space of [Formula: see text]-colored graphs embedded in the surface [A. Kirillov, String-net model of Turaev-Viro invariants, Preprint (2011), arXiv:1106.6033 ]. By analyzing the action of the Birman generators [J. Birman, Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22 (1969) 213–242] on a finite spanning set of colored graphs, we find that the mapping class group acts by permutations on a slightly larger finite spanning set. This implies that the representation has finite image.

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].

The Brauer–Picard groups of fusion categories coming from the ADE subfactors

We compute the group of Morita auto-equivalences of the even parts of the [Formula: see text] subfactors, and Galois conjugates. To achieve this, we study the braided auto-equivalences of the Drinfeld centers of these categories. We give planar algebra presentations for each of these Drinfeld centers, which we leverage to obtain information about the braided auto-equivalences of the corresponding categories. We also perform the same calculations for the fusion categories constructed from the full [Formula: see text] subfactors. Of particular interest, the even part of the [Formula: see text] subfactor is shown to have Brauer–Picard group [Formula: see text]. We develop combinatorial arguments to compute the underlying algebra objects of these invertible bimodules.

Theta Functions for Lattices of SU(3) Hyper-Roots

ABSTRACTWe recall the definition of the hyper-roots that can be associated with modules-categories over fusion categories defined by the choice of a simple Lie group G together with a positive integer k. This definition was proposed in 2000, using another language, by Adrian Ocneanu. If G = SU(2), the obtained hyper-roots coincide with the usual roots for ADE Dynkin diagrams. We consider the associated lattices when G = SU(3) and determine their theta functions in a number of cases; these functions can be expressed as modular forms twisted by appropriate Dirichlet characters.

Bivariate analysis of 3D structure for stereoscopic image quality assessment

Human visual system (HVS) identifies each unit of stereo-pair into binocular or monocular perception depending on distortions and depth perception. However, these HVS properties still have a large room for exploration in stereoscopic image quality assessment (SIQA) research field. In this paper, a bivariate natural scene statistics (NSS) model is proposed to capture image quality by extracting features from binocular and monocular perception regions, respectively. In the implementation details, the stereo-pair is first segmented into various regions based on spatial information of its disparity. Then the regions are classified into categories of binocular fusion, binocular rivalry and binocular suppression. Bivariate statistics of spatially adjacent Gabor response of image are extracted from each category of regions, based on which features are calculated for image quality representation. In particular, the extraction strategy depends on the type of image patch. Experimental results show that the proposed model is promising at handling the task of SIQA on LIVE 3D Image Quality Database.

On finite symmetries and their gauging in two dimensions

It is well-known that if we gauge a ℤn symmetry in two dimensions, a dual ℤn symmetry appears, such that re-gauging this dual ℤn symmetry leads back to the original theory. We describe how this can be generalized to non-Abelian groups, by enlarging the concept of symmetries from those defined by groups to those defined by unitary fusion categories. We will see that this generalization is also useful when studying what happens when a non-anomalous subgroup of an anomalous finite group is gauged: for example, the gauged theory can have non-Abelian group symmetry even when the original symmetry is an Abelian group. We then discuss the axiomatization of two-dimensional topological quantum field theories whose symmetry is given by a category. We see explicitly that the gauged version is a topological quantum field theory with a new symmetry given by a dual category.

Fermionic 6j-symbols in superfusion categories

We describe how the study of superfusion categories (roughly speaking, fusion categories enriched over the category of super vector spaces) reduces to that of fusion categories oversVect_, in the sense of [4]. Following Brundan and Ellis [1], we give the construction of the underlying fusion category of a superfusion category, and give an explicit formula for the associator in this category in terms of 6j-symbols. As an application, we prove a version of Ocneanu rigidity for superfusion categories.

Classification of Pointed Fusion Categories of dimension 8 up to weak Morita equivalence

ABSTRACTIn this paper we give a complete classification of pointed fusion categories over ℂ of global dimension 8. We first classify the equivalence classes of pointed fusion categories of dimension 8, and then we proceed to determine which of these equivalence classes have equivalent categories of modules following the procedure presented in [9, 11]. The results of this paper permit to recover the classification of twisted quantum doubles of groups of order 8 up to gauge equivalence of braided quasi-Hopf algebras that was previously done in [6] and [5].

The core of a weakly group-theoretical braided 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.

Zn clock models and chains of so(n)2 non-Abelian anyons: symmetries, integrable points and low energy properties

We study two families of quantum models which have been used previously to investigate the effect of topological symmetries in one-dimensional correlated matter. Various striking similarities are observed between certain quantum clock models, spin chains generalizing the Ising model, and chains of non-Abelian anyons constructed from the so(n)2 fusion category for odd n, both subject to periodic boundary conditions. In spite of the differences between these two types of quantum chains, e.g. their Hilbert spaces being spanned by tensor products of local spin states or fusion paths of anyons, the symmetries of the lattice models are shown to be closely related. Furthermore, under a suitable mapping between the parameters describing the interaction between spins and anyons the respective Hamiltonians share part of their energy spectrum (although their degeneracies may differ). This spin-anyon correspondence can be extended by fine-tuning of the coupling constants leading to exactly solvable models. We show that the algebraic structures underlying the integrability of the clock models and the anyon chain are the same. For n = 3,5,7 we perform an extensive finite size study—both numerical and based on the exact solution—of these models to map out their ground state phase diagram and to identify the effective field theories describing their low energy behaviour. We observe that the continuum limit at the integrable points can be described by rational conformal field theories with extended symmetry algebras which can be related to the discrete ones of the lattice models.

The Relative Drinfeld Commutant of a Fusion Category and α-Induction

Abstract We establish a correspondence among simple objects of the relative commutant of a full fusion subcategory in a larger fusion category in the sense of Drinfeld, irreducible half-braidings of objects in the larger fusion category with respect to the fusion subcategory, and minimal central projections in the relative tube algebra. Based on this, we explicitly compute certain relative Drinfeld commutants of fusion categories arising from α-induction for braided subfactors. We present examples arising from chiral conformal field theory.

Third cohomology and fusion categories

It was observed recently that for a fixed finite group $G$, the set of all Drinfeld centres of $G$ twisted by 3-cocycles form a group, the so-called group of modular extensions (of the representation category of $G$), which is isomorphic to the third cohomology group of $G$. We show that for an abelian $G$, pointed twisted Drinfeld centres of $G$ form a subgroup of the group of modular extensions. We identify this subgroup with a group of quadratic extensions containing $G$ as a Lagrangian subgroup, the so-called group of Lagrangian extensions of $G$. We compute the group of Lagrangian extensions, thereby providing an interpretation of the internal structure of the third cohomology group of an abelian $G$ in terms of fusion categories.

Boundary Hamiltonian theory for gapped topological phases on an open surface

In this paper we propose a Hamiltonian approach to gapped topological phases on open surfaces. Our setting is an extension of the Levin-Wen model to a 2d graph on an open surface, whose boundary is part of the graph. We systematically construct a series of boundary Hamiltonians such that each of them, when combined with the usual Levin-Wen bulk Hamiltonian, gives rise to a gapped energy spectrum which is topologically protected. It is shown that the corresponding wave functions are robust under changes of the underlying graph that maintain the spatial topology of the system. We derive explicit ground-state wavefunctions of the system on a disk as well as on a cylinder. For boundary quasiparticle excitations, we are able to construct their creation, annihilation, measuring and hopping operators etc. Given a bulk string-net theory, our approach provides a classification scheme of possible types of gapped boundary conditions by Frobenius algebras (modulo Morita equivalence) of the bulk fusion category; the boundary quasiparticles are characterized by bimodules of the pertinent Frobenius algebras. Our approach also offers a set of concrete tools for computations. We illustrate our approach by a few examples.

State sum invariants of three manifolds from spherical multi-fusion categories

We define a family of quantum invariants of closed oriented [Formula: see text]-manifolds using spherical multi-fusion categories (SMFCs). The state sum nature of this invariant leads directly to [Formula: see text]-dimensional topological quantum field theories ([Formula: see text]s), which generalize the Turaev–Viro–Barrett–Westbury ([Formula: see text]) [Formula: see text]s from spherical fusion categories. The invariant is given as a state sum over labeled triangulations, which is mostly parallel to, but richer than the [Formula: see text] approach in that here the labels live not only on [Formula: see text]-simplices but also on [Formula: see text]-simplices. It is shown that a multi-fusion category in general cannot be a spherical fusion category in the usual sense. Thus, we introduce the concept of a SMFC by imposing a weakened version of sphericity. Besides containing the [Formula: see text] theory, our construction also includes the recent higher gauge theory [Formula: see text]-[Formula: see text]s given by Kapustin and Thorngren, which was not known to have a categorical origin before.

Dichromatic State Sum Models for Four-Manifolds from Pivotal Functors

A family of invariants of smooth, oriented four-dimensional manifolds is defined via handle decompositions and the Kirby calculus of framed link diagrams. The invariants are parametrised by a pivotal functor from a spherical fusion category into a ribbon fusion category. A state sum formula for the invariant is constructed via the chain-mail procedure, so a large class of topological state sum models can be expressed as link invariants. Most prominently, the Crane-Yetter state sum over an arbitrary ribbon fusion category is recovered, including the nonmodular case. It is shown that the Crane-Yetter invariant for nonmodular categories is stronger than signature and Euler invariant. A special case is the four-dimensional untwisted Dijkgraaf–Witten model. Derivations of state space dimensions of TQFTs arising from the state sum model agree with recent calculations of ground state degeneracies in Walker-Wang models. Relations to different approaches to quantum gravity such as Cartan geometry and teleparallel gravity are also discussed.