Braided Fusion Categories Research Articles

Braided Zestings of Verlinde Modular Categories and Their Modular Data

Zesting of braided fusion categories is a procedure that can be used to obtain new modular categories from a modular category with non-trivial invertible objects. In this paper, we classify and construct all possible braided zesting data for modular categories associated with quantum groups at roots of unity. We produce closed formulas, based on the root system of the associated Lie algebra, for the modular data of these new modular categories.

Electric-magnetic duality and Z2 symmetry enriched Abelian lattice gauge theory

Kitaev’s quantum double model is a lattice realization of Dijkgraaf–Witten topological quantum field theory. Its topologically protected ground state space has broad applications for topological quantum computation and topological quantum memory. We investigate the Z2 symmetry enriched generalization of the model for the Abelian group in a categorical framework and present an explicit Hamiltonian construction. This model provides a lattice realization of the Z2 symmetry of the topological phase. We discuss in detail the categorical symmetry of the phase, for which the electric-magnetic (EM) duality symmetry is a special case. The aspects of symmetry defects are investigated using the G-crossed unitary braided fusion category. By determining the corresponding anyon condensation, the gapped boundaries and boundary-bulk duality are also investigated. In the last part, an explicit lattice realization of EM duality is discussed.

$\alpha$-induction for bi-unitary connections

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.

Evaluating TQFT invariants from $G$-crossed braided spherical fusion categories via Kirby diagrams with 3-handles

A family of TQFTs parametrised by G -crossed braided spherical fusion categories has been defined recently as a state sum model and as a Hamiltonian lattice model. Concrete calculations of the resulting manifold invariants are scarce because of the combinatorial complexity of triangulations, if nothing else. Handle decompositions, and in particular Kirby diagrams are known to offer an economic and intuitive description of 4-manifolds. We show that if 3-handles are added to the picture, the state sum model can be conveniently redefined by translating Kirby diagrams into the graphical calculus of a G -crossed braided spherical fusion category. This reformulation is very efficient for explicit calculations, and the manifold invariant is calculated for several examples. It is also shown that in most cases, the invariant is multiplicative under connected sum, which implies that it does not detect exotic smooth structures.

G‐crossed braided zesting

AbstractFor a finite group , a ‐crossed braided fusion category is a ‐graded fusion category with additional structures, namely, a ‐action and a ‐braiding. We develop the notion of ‐crossed braided zesting: an explicit method for constructing new ‐crossed braided fusion categories from a given one by means of cohomological data associated with the invertible objects in the category and grading group . This is achieved by adapting a similar construction for (braided) fusion categories recently described by the authors. All ‐crossed braided zestings of a given category are ‐extensions of their trivial component and can be interpreted in terms of the homotopy‐based description of Etingof, Nikshych, and Ostrik. In particular, we explicitly describe which ‐extensions correspond to ‐crossed braided zestings.

Reconstructing braided subcategories of SU(N)k

Ocneanu rigidity implies that there are finitely many (braided) fusion categories with a given set of fusion rules. While there is no method for determining all such categories up to equivalence, there are a few cases for which one can. For example, Kazhdan and Wenzl described all fusion categories with fusion rules isomorphic to those of SU(N)k. In this paper we extend their results to a statement about braided fusion categories, and obtain similar results for certain subcategories of SU(N)k.

Minimal nondegenerate extensions

We prove that every slightly degenerate braided fusion category admits a minimal nondegenerate extension, and hence that every pseudo-unitary super modular tensor category admits a minimal modular extension. This completes the program of characterizing minimal nondegenerate extensions of braided fusion categories. Our proof relies on the new subject of fusion 2 2 -categories. We study in detail the Drinfel’d centre Z ( M o d - B ) \mathcal {Z}({_{}\mathrm {Mod}\text {-}\mathcal {B}}) of the fusion 2 2 -category M o d - B {_{}\mathrm {Mod}\text {-}\mathcal {B}} of module categories of a braided fusion 1 1 -category B \mathcal {B} . We show that minimal nondegenerate extensions of B \mathcal {B} correspond to certain trivializations of Z ( M o d - B ) \mathcal {Z}({_{}\mathrm {Mod}\text {-}\mathcal {B}}) . In the slightly degenerate case, such trivializations are obstructed by a class in H 5 ( K ( Z 2 , 2 ) ; k × ) H^5(K(\mathbb {Z}_2, 2); \mathbb {k}^\times ) and we use a numerical invariant—defined by evaluating a certain two-dimensional topological field theory on a Klein bottle—to prove that this obstruction always vanishes. Along the way, we develop techniques to explicitly compute in braided fusion 2 2 -categories which we expect will be of independent interest. In addition to the model of Z ( M o d - B ) \mathcal {Z}({_{}\mathrm {Mod}\text {-}\mathcal {B}}) in terms of braided B \mathcal {B} -module categories, we develop a computationally useful model in terms of certain algebra objects in B \mathcal {B} . We construct an S S -matrix pairing for any braided fusion 2 2 -category, and show that it is nondegenerate for Z ( M o d - B ) \mathcal {Z}({_{}\mathrm {Mod}\text {-}\mathcal {B}}) . As a corollary, we identify components of Z ( M o d - B ) \mathcal {Z}({_{}\mathrm {Mod}\text {-}\mathcal {B}}) with blocks in the annular category of B \mathcal {B} and with the homomorphisms from the Grothendieck ring of the Müger centre of B \mathcal {B} to the ground field.

Nondegenerate extensions of near-group braided fusion categories

Nondegenerate extensions of near-group braided fusion categories

Disentangling modular Walker-Wang models via fermionic invertible boundaries

Walker-Wang models are fixed-point models of topological order in $3+1$ dimensions constructed from a braided fusion category. For a modular input category $\mathcal M$, the model itself is invertible and is believed to be in a trivial topological phase, whereas its standard boundary is supposed to represent a $2+1$-dimensional chiral phase. In this work we explicitly show triviality of the model by constructing an invertible domain wall to vacuum as well as a disentangling generalized local unitary circuit in the case where $\mathcal M$ is a Drinfeld center. Moreover, we show that if we allow for fermionic (auxiliary) degrees of freedom inside the disentangling domain wall or circuit, the model becomes trivial for a larger class of modular fusion categories, namely those in the Witt classes generated by the Ising UMTC. In the appendices, we also discuss general (non-invertible) boundaries of general Walker-Wang models and describe a simple axiomatization of extended TQFT in terms of tensors.

The relative Deligne tensor product over pointed braided fusion categories

We give a formula for the relative Deligne tensor product of two indecomposable finite semisimple module categories over a pointed braided fusion category over an algebraically closed field.

Uniqueness of Unitary Structure for Unitarizable Fusion Categories

We show that every unitarizable fusion category, and more generally every semisimple textrm{C}^*-tensor category, admits a unique unitary structure. Our proof is based on a categorified polar decomposition theorem for monoidal equivalences between such categories. We prove analogous results for unitarizable braided fusion categories and module categories.

Three-Dimensional Quantum Cellular Automata from Chiral Semion Surface Topological Order and beyond

We construct a novel three-dimensional quantum cellular automaton (QCA) based on a system with short-range entangled bulk and chiral semion boundary topological order. We argue that either the QCA is nontrivial, i.e. not a finite-depth circuit of local quantum gates, or there exists a two-dimensional commuting projector Hamiltonian realizing the chiral semion topological order (characterized by $U(1)_2$ Chern-Simons theory). Our QCA is obtained by first constructing the Walker-Wang Hamiltonian of a certain premodular tensor category of order four, then condensing the deconfined bulk boson at the level of lattice operators. We show that the resulting Hamiltonian hosts chiral semion surface topological order in the presence of a boundary and can be realized as a non-Pauli stabilizer code on qubits, from which the QCA is defined. The construction is then generalized to a class of QCAs defined by non-Pauli stabilizer codes on ${2^n}$-dimensional qudits that feature surface anyons described by $U(1)_{2^n}$ Chern-Simons theory. Our results support the conjecture that the group of nontrivial three-dimensional QCAs is isomorphic to the Witt group of non-degenerate braided fusion categories.

Higher central charges and Witt groups

In this paper, we introduce the definitions of signatures of braided fusion categories, which are proved to be invariants of their Witt equivalence classes. These signature assignments define group homomorphisms on the Witt group. The higher central charges of pseudounitary modular categories can be expressed in terms of these signatures, which are applied to prove that the Ising modular categories have infinitely many square roots in the Witt group modulo the pointed part. This result is further applied to prove a conjecture of Davydov-Nikshych-Ostrik on the super-Witt group: the torsion subgroup generated by the completely anisotropic s-simple braided fusion categories has infinite rank.

Geometric perspective on Nichols algebras

We formulate the generation of finite dimensional pointed Hopf algebras by group-like elements and skew-primitives in geometric terms. This is done through a more general study of connected and coconnected Hopf algebras inside a braided fusion category C. We describe such Hopf algebras as orbits for the action of a reductive group on an affine variety. We then show that the closed orbits are precisely the orbits of Nichols algebras, and that all other algebras are therefore deformations of Nichols algebras. For the case where the category C is the category YDGG of Yetter-Drinfeld modules over a finite group G, this reduces the question of generation by group-like elements and skew-primitives to a geometric question about rigidity of orbits. Comparing the results of Angiono Kochetov and Mastnak, this gives a new proof for the generation of finite dimensional pointed Hopf algebras with abelian groups of group-like elements by skew-primitives and group-like elements. We show that if V is a simple object in C and B(V) is finite dimensional, then B(V) must be rigid. We also show that a non-rigid Nichols algebra can always be deformed to a pre-Nichols algebra or a post-Nichols algebra which is isomorphic to the Nichols algebra as an object of the category C.

One dimensional gapped quantum phases and enriched fusion categories

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.

Generalized parafermions of orthogonal type

There is an embedding of affine vertex algebras Vk(gln)↪Vk(sln+1), and the coset Ck(n)=Com(Vk(gln),Vk(sln+1)) is a natural generalization of the parafermion algebra of sl2. It was called the algebra of generalized parafermions by the third author and was shown to arise as a one-parameter quotient of the universal two-parameter W∞-algebra of type W(2,3,…). In this paper, we consider an analogous structure of orthogonal type, namely Dk(n)=Com(Vk(so2n),Vk(so2n+1))Z2. We realize this algebra as a one-parameter quotient of the two-parameter even spin W∞-algebra of type W(2,4,…), and we classify all coincidences between its simple quotient Dk(n) and the algebras Wℓ(so2m+1) and Wℓ(so2m)Z2. As a corollary, we show that for the admissible levels k=−(2n−2)+12(2n+2m−1) for soˆ2n the simple affine algebra Lk(so2n) embeds in Lk(so2n+1), and the coset is strongly rational. As a consequence, the category of ordinary modules of Lk(so2n+1) at such a level is a braided fusion category.

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.

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.

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.