Drinfeld Center Research Articles

The distinguished invertible object as ribbon dualizing object in the Drinfeld center

The distinguished invertible object as ribbon dualizing object in the Drinfeld center

Twisted Kuperberg invariants of knots and Reidemeister torsion via twisted Drinfeld doubles

In this paper, we consider the Reshetikhin-Turaev invariants of knots in the three-sphere obtained from a twisted Drinfeld double of a Hopf algebra, or equivalently, the relative Drinfeld center of the crossed product Rep ( H ) ⋊ Aut ( H ) \text {Rep}(H)\rtimes \text {Aut}(H) . These are quantum invariants of knots endowed with a homomorphism of the knot group to Aut ( H ) \text {Aut}(H) . We show that, at least for knots in the three-sphere, these invariants provide a non-involutory generalization of the Fox-calculus-twisted Kuperberg invariants of sutured manifolds introduced previously by the author, which are only defined for involutory Hopf algebras. In particular, we describe the S L ( n , C ) SL(n,\mathbb {C}) -twisted Reidemeister torsion of a knot complement as a Reshetikhin-Turaev invariant.

Local modules in braided monoidal 2-categories

Given an algebra in a monoidal 2-category, one can construct a 2-category of right modules. Given a braided algebra in a braided monoidal 2-category, it is possible to refine the notion of right module to that of a local module. Under mild assumptions, we prove that the 2-category of local modules admits a braided monoidal structure. In addition, if the braided monoidal 2-category has duals, we go on to show that the 2-category of local modules also has duals. Furthermore, if it is a braided fusion 2-category, we establish that the 2-category of local modules is a braided multifusion 2-category. We examine various examples. For instance, working within the 2-category of 2-vector spaces, we find that the notion of local module recovers that of braided module 1-category. Finally, we examine the concept of a Lagrangian algebra, that is a braided algebra with trivial 2-category of local modules. In particular, we completely describe Lagrangian algebras in the Drinfeld centers of fusion 2-categories, and we discuss how this result is related to the classifications of topological boundaries of (3 + 1)d topological phases of matter.

Twisted Drinfeld centers and framed string-nets

We discuss a string-net construction on 2 -framed surfaces, taking as algebraic input a finite, rigid tensor category, which is assumed to be neither pivotal nor semi-simple. It is shown that circle categories of our framed string-net construction essentially compute Drinfeld centers twisted by powers of the double dual functor.

Balmer spectra and Drinfeld centers

Balmer spectra and Drinfeld centers

Quantum current and holographic categorical symmetry

We establish the formulation for quantum current. Given a symmetry group G, let \mathcal{C}𝒞:=Rep G be its representation category. Physically, symmetry charges are objects of \mathcal{C}𝒞 and symmetric operators are morphisms in \mathcal{C}𝒞. The addition of charges is given by the tensor product of representations. For any symmetric operator O crossing two subsystems, the exact symmetry charge transported by O can be extracted. The quantum current is defined as symmetric operators that can transport symmetry charges over an arbitrary long distance. A quantum current exactly corresponds to an object in the Drinfeld center Z_1(\mathcal{C})Z1(𝒞). The condition for quantum currents to be is also specified, which corresponds to condensation of anyons in one higher dimension. To express the local conservation, the internal hom must be used to compute the charge difference, and the framework of enriched category is inevitable. To illustrate these ideas, we develop a rigorous scheme of renormalization in one-dimensional lattice systems and analyse the fixed-point models. It is proved that in the fixed-point models, superconducting quantum currents form a Lagrangian algebra in Z_1(\mathcal{C})Z1(𝒞) and the boundary-bulk correspondence is verified in the enriched setting. Overall, the quantum current provides a natural physical interpretation to the holographic categorical symmetry.

A lattice model for condensation in Levin-Wen systems

Levin-Wen string-net models provide a construction of (2+1)D topologically ordered phases of matter with anyonic localized excitations described by the Drinfeld center of a unitary fusion category. Anyon condensation is a mechanism for phase transitions between (2+1)D topologically ordered phases. We construct an extension of Levin-Wen models in which tuning a parameter implements anyon condensation. We also describe the classification of anyons in Levin-Wen models via representation theory of the tube algebra, and use a variant of the tube algebra to classify low-energy localized excitations in the condensed phase.

On unimodular module categories

Let C be a finite tensor category and M an exact left C-module category. We call M unimodular if the finite multitensor category RexC(M) of right exact C-module endofunctors of M is unimodular. In this article, we provide various characterizations, properties, and examples of unimodular module categories. As our first application, we employ unimodular module categories to construct (commutative) Frobenius algebra objects in the Drinfeld center of any finite tensor category. When C is a pivotal category, and M is a unimodular, pivotal left C-module category, the Frobenius algebra objects are symmetric as well. Our second application is a classification of unimodular module categories over the category of finite dimensional representations of a finite dimensional Hopf algebra; this answers a question of Shimizu [40]. Using this, we provide an example of a finite tensor category whose categorical Morita equivalence class does not contain any unimodular tensor category.

Interpolated family of non-group-like simple integral fusion rings of Lie type

This paper is motivated by the quest of a non-group irreducible finite index depth 2 maximal subfactor. We compute the generic fusion rules of the Grothendieck ring of Rep(PSL(2,[Formula: see text])), [Formula: see text] prime-power, by applying a Verlinde-like formula on the generic character table. We then prove that this family of fusion rings [Formula: see text] interpolates to all integers [Formula: see text], providing (when [Formula: see text] is not prime-power) the first example of infinite family of non-group-like simple integral fusion rings. Furthermore, they pass all the known criteria of (unitary) categorification. This provides infinitely many serious candidates for solving the famous open problem of whether there exists an integral fusion category which is not weakly group-theoretical. We prove that a complex categorification (if any) of an interpolated fusion ring [Formula: see text] (with [Formula: see text] non-prime-power) cannot be braided, and so its Drinfeld center must be simple. In general, this paper proves that a non-pointed simple fusion category is non-braided if and only if its Drinfeld center is simple; and also that every simple integral fusion category is weakly group-theoretical if and only if every simple integral modular fusion category is pointed.

Homotopy invariants of braided commutative algebras and the Deligne conjecture for finite tensor categories

It is easy to find algebras T∈C in a finite tensor category C that naturally come with a lift to a braided commutative algebra T∈Z(C) in the Drinfeld center of C. In fact, any finite tensor category has at least two such algebras, namely the monoidal unit I and the canonical end ∫X∈CX⊗X∨. Using the theory of braided operads, we prove that for any such algebra T the homotopy invariants, i.e. the derived morphism space from I to T, naturally come with the structure of a differential graded E2-algebra. This way, we obtain a rich source of differential graded E2-algebras in the homological algebra of finite tensor categories. We use this result to prove that Deligne's E2-structure on the Hochschild cochain complex of a finite tensor category is induced by the canonical end, its multiplication and its non-crossing half braiding. With this new and more explicit description of Deligne's E2-structure, we can lift the Farinati-Solotar bracket on the Ext algebra of a finite tensor category to an E2-structure at cochain level. Moreover, we prove that, for a unimodular pivotal finite tensor category, the inclusion of the Ext algebra into the Hochschild cochains is a monomorphism of framed E2-algebras, thereby refining a result of Menichi.

Support theory for Drinfeld doubles of some infinitesimal group schemes

Consider a Frobenius kernel G in a split semisimple algebraic group, in very good characteristic. We provide an analysis of support for the Drinfeld center Z(rep(G)) of the representation category for G, or equivalently for the representation category of the Drinfeld double of kG. We show that thick ideals in the corresponding stable category are classified by cohomological support, and calculate the Balmer spectrum of the stable category of Z(rep(G)). We also construct a $\pi$-point style rank variety for the Drinfeld double, identify $\pi$-point support with cohomological support, and show that both support theories satisfy the tensor product property. Our results hold, more generally, for Drinfeld doubles of Frobenius kernels in any smooth algebraic group which admits a quasi-logarithm, such as a Borel subgroup in a split semisimple group in very good characteristic.

Ribbon structures of the Drinfeld center of a finite tensor category

We classify the ribbon structures of the Drinfeld center $\mathcal{Z}(\mathcal{C})$ of a finite tensor category $\mathcal{C}$. Our result generalizes Kauffman and Radford's classification result of the ribbon elements of the Drinfeld double of a finite-dimensional Hopf algebra. Our result implies that $\mathcal{Z}(\mathcal{C})$ is a modular tensor category in the sense of Lyubashenko if $\mathcal{C}$ is a spherical finite tensor category in the sense of Douglas, Schommer-Pries and Snyder.

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.

Drinfeld centers of fusion categories arising from generalized Haagerup subfactors

We consider generalized Haagerup categories such that 1 \oplus X admits a Q -system for every non-invertible simple object X . We show that in such a category, the group of order two invertible objects has size at most four. We describe the simple objects of the Drinfeld center and give partial formulas for the modular data. We compute the remaining corner of the modular data for several examples and make conjectures about the general case. We also consider several types of equivariantizations and de-equivariantizations of generalized Haagerup categories and describe their Drinfeld centers. In particular, we compute the modular data for the Drinfeld centers of a number of examples of fusion categories arising in the classification of small-index subfactors: the Asaeda–Haagerup subfactor; the 3^{\mathbb{Z}_4} and 3^{\mathbb{Z}_2 \times \mathbb{Z}_2} subfactors; the 2D2 subfactor; and the 4442 subfactor. The results suggest the possibility of several new infinite families of quadratic categories. A description and generalization of the modular data associated to these families in terms of pairs of metric groups is taken up in the accompanying paper [Comm. Math. Phys. 380 (2020), 1091–1150].

Constructing modular categories from orbifold data

The notion of an orbifold datum \operatorname{\mathbb{A}} in a modular fusion category \mathcal{C} was introduced as part of a generalised orbifold construction for Reshetikhin–Turaev TQFTs by Carqueville, Runkel, and Schaumann in 2018. In this paper, given a simple orbifold datum \operatorname{\mathbb{A}} in \mathcal{C} , we introduce a ribbon category \mathcal{C}_{\operatorname{\mathbb{A}}} and show that it is again a modular fusion category. The definition of \mathcal{C}_{\operatorname{\mathbb{A}}} is motivated by properties of Wilson lines in the generalised orbifold. We analyse two examples in detail: (i) when \operatorname{\mathbb{A}} is given by a simple commutative \Delta -separable Frobenius algebra A in \mathcal{C} ; (ii) when \operatorname{\mathbb{A}} is an orbifold datum in \mathcal{C} = \operatorname{Vect} , built from a spherical fusion category \mathcal{S} . We show that, in case (i), \mathcal{C}_{\operatorname{\mathbb{A}}} is ribbon-equivalent to the category of local modules of A , and, in case (ii), to the Drinfeld centre of \mathcal{S} . The category \mathcal{C}C_{\operatorname{\mathbb{A}}} thus unifies these two constructions into a single algebraic setting.

Pivotal structures of the Drinfeld center of a finite tensor category

We show that pivotal structures of the Drinfeld center of a finite tensor category C are in bijection with the set of equivalence classes of a pair (β,j) consisting of an invertible object β of C and a monoidal natural transformation jX:β⊗X⊗β⁎→X⁎⁎ (X∈C). As an application of techniques used to prove the main result, we also obtain a non-semisimple analogue of a categorification of the Rosenberg-Zelinsky exact sequence for fusion categories given by Galindo and Plavnik

Equivariant Morita theory for graded tensor categories

We extend categorical Morita equivalence to finite tensor categories gra\-ded by a finite group $G$. We show that two such categories are graded Morita equivalent if and only if their equivariant Drinfeld centers are equivalent as braided $G$-crossed tensor categories.

Fibonacci-type orbifold data in Ising modular categories

An orbifold datum is a collection A of algebraic data in a modular fusion category C. It allows one to define a new modular fusion category CA in a construction that is a generalisation of taking the Drinfeld centre of a fusion category. Under certain simplifying assumptions we characterise orbifold data A in terms of scalars satisfying polynomial equations and give an explicit expression which computes the number of isomorphism classes of simple objects in CA.In Ising-type modular categories we find new examples of orbifold data which – in an appropriate sense – exhibit Fibonacci fusion rules. The corresponding orbifold modular categories have 11 simple objects, and for a certain choice of parameters one obtains the modular category for sl(2) at level 10. This construction inverts the extension of the latter category by the E6 commutative algebra.

Centers of braided tensor categories

Let C be a finite braided multitensor category. Then the end B=∫X∈CX⊗X⁎ is natural a Hopf algebra in C. We show that the Drinfeld center of C is isomorphic to the category of left B-comodules in C, and the decomposition of B into a direct sum of indecomposable C-subcoalgebras leads to a decomposition of B-ComodC into a direct sum of indecomposable C-module subcategories.As an application, we present an explicit characterization of the structure of irreducible Yetter-Drinfeld modules over semisimple and cosemisimple quasi-triangular weak Hopf algebras. Our results generalize those results on finite groups and on quasi-triangular Hopf algebras.

The Symmetry Enriched Center Functor is Fully Faithful

In this work, inspired by some physical intuitions, we define a series of symmetry enriched categories to describe symmetry enriched topological (SET) orders, and define a new tensor product, called the relative tensor product, which describes the stacking of 2+1D SET orders. Then we choose and modify the domain and codomain categories, and manage to make the Drinfeld center a fully faithful symmetric monoidal functor. It turns out that this functor, named the symmetry enriched center functor, provides a precise and rather complete mathematical formulation of the boundary-bulk relation of SET orders. We also provide another description of the relative tensor product via a condensable algebra.