Drinfeld Center Research Articles

Reduced tensor product on the Drinfeld center

The annulus comes with a “stacking” operation which glues two annuli into one. This provides a tensor product structure on the category of boundary values ZCY(Ann) associated to the annulus in an extended Crane-Yetter TQFT. It is known that ZCY(Ann)≃Z(A), the Drinfeld center of the premodular category A from which ZCY is constructed. We give an explicit formula for the tensor product on Z(A) that corresponds to the stacking operation on ZCY(Ann).

Centers of Categorified Endomorphism Rings

We prove that for a large class of well-behaved cocomplete categories \(\mathcal {C}\) the weak and strong Drinfeld centers of the monoidal category \(\mathcal {E}\) of cocontinuous endofunctors of \(\mathcal {C}\) coincide. This generalizes similar results in the literature, where \(\mathcal {C}\) is the category of modules over a ring A and hence \(\mathcal {E}\) is the category of A-bimodules.

Cardy Algebras, Sewing Constraints and String-Nets

In Schweigert and Yang (Methods Appl Symmetry Integr Geom, 2021) it was shown how string-net spaces for the Cardy bulk algebra in the Drinfeld center {mathsf {Z}}({mathsf {C}}) of a modular tensor category {mathsf {C}} give rise to a consistent set of correlators. We extend their results to include open-closed world sheets and allow for more general field algebras, which come in the form of ({mathsf {C}}|{mathsf {Z}}({mathsf {C}}))-Cardy algebras. To be more precise, we show that a set of fundamental string-nets with input data from a ({mathsf {C}}|{mathsf {Z}}({mathsf {C}}))-Cardy algebra gives rise to a solution of the sewing constraints formulated in Kong et al. (Adv Math 262:604–681, 2014) and that any set of fundamental string-nets solving the sewing constraints determine a ({mathsf {C}}|{mathsf {Z}}({mathsf {C}}))-Cardy algebra up to isomorphism. Hence we give an alternative proof of the results in Kong et al. (2014) in terms of string-nets.

Cohomology of finite tensor categories: Duality and Drinfeld centers

We consider the finite generation property for cohomology of a finite tensor category C \mathscr {C} , which requires that the self-extension algebra of the unit \operatorname {Ext}^\text {\tiny ∙ }_\mathscr {C}(\mathbf {1},\mathbf {1}) is a finitely generated algebra and that, for each object V V in C \mathscr {C} , the graded extension group \operatorname {Ext}^\text {\tiny ∙ }_\mathscr {C}(\mathbf {1},V) is a finitely generated module over the aforementioned algebra. We prove that this cohomological finiteness property is preserved under duality (with respect to exact module categories) and taking the Drinfeld center, under suitable restrictions on C \mathscr {C} . For example, the stated result holds when C \mathscr {C} is a braided tensor category of odd Frobenius-Perron dimension. By applying our general results, we obtain a number of new examples of finite tensor categories with finitely generated cohomology. In characteristic 0 0 , we show that dynamical quantum groups at roots of unity have finitely generated cohomology. We also provide a new class of examples in finite characteristic which are constructed via infinitesimal group schemes.

INTERNAL NATURAL TRANSFORMATIONS AND FROBENIUS ALGEBRAS IN THE DRINFELD CENTER

For ℳ and mathcal{N} finite module categories over a finite tensor category mathcal{C} , the category mathrm{mathcal{R}}{ex}_{mathcal{C}} (ℳ, mathcal{N} ) of right exact module functors is a finite module category over the Drinfeld center mathcal{Z} ( mathcal{C} ). We study the internal Homs of this module category, which we call internal natural transformations. With the help of certain integration functors that map mathcal{C} - mathcal{C} -bimodule functors to objects of mathcal{Z} ( mathcal{C} ), we express them as ends over internal Homs and define horizontal and vertical compositions. We show that if ℳ and mathcal{N} are exact mathcal{C} -modules and mathcal{C} is pivotal, then the mathcal{Z} ( mathcal{C} )-module mathrm{mathcal{R}}{ex}_{mathcal{C}} (ℳ, mathcal{N} ) is exact. We compute its relative Serre functor and show that if ℳ and mathcal{N} are even pivotal module categories, then mathrm{mathcal{R}}{ex}_{mathcal{C}} (ℳ, mathcal{N} ) is pivotal as well. Its internal Ends are then a rich source for Frobenius algebras in mathcal{Z} ( mathcal{C} ).

Variational methods for characterizing matrix product operator symmetries

We present a method of extracting information about topological order from the ground state of a strongly correlated two-dimensional system represented by an infinite projected entangled pair state (iPEPS). As in Phys. Rev. B 101, 041108 (2020) and 102, 235112 (2020) we begin by determining symmetries of the iPEPS represented by infinite matrix product operators (iMPO) that map between the different iPEPS transfer matrix fixed points, to which we apply the fundamental theorem of MPS to find zipper tensors between products of iMPO's that encode fusion properties of the anyons. The zippers can be combined to extract topological $F$-symbols of the underlying fusion category, which unequivocally identify the topological order of the ground state. We bring the $F$-symbols to the canonical gauge, as well as compute the Drinfeld center of this unitary fusion category to extract the topological $S$ and $T$ matrices encoding mutual- and self-statistics of the emergent anyons. The algorithm is applied to Abelian toric code, double semion and twisted quantum double of $Z_3$, as well as to non-Abelian double Fibonacci, double Ising, and quantum double of $S_3$ and ${\rm Rep}(S_3)$ string net models.

CFT Correlators for Cardy Bulk Fields via String-Net Models

We show that string-net models provide a novel geometric method to construct invariants of mapping class group actions. Concretely, we consider string-net models for a modular tensor category C. We show that the datum of a specific commutative symmetric Frobenius algebra in the Drinfeld center Z(C) gives rise to invariant string-nets. The Frobenius algebra has the interpretation of the algebra of bulk fields of the conformal field theory in the Cardy case.

A mathematical theory of gapless edges of 2d topological orders. Part II

This is the second part of a two-part work on the unified mathematical theory of gapped and gapless edges of 2+1D topological orders. In Part I, we have developed the mathematical theory of chiral gapless edges. In Part II, we study boundary-bulk relation and non-chiral gapless edges. In particular, we explain how the notion of the center of an enriched monoidal category naturally emerges from the boundary-bulk relation. After the study of 0+1D gapless walls, we give the complete boundary-bulk relation for 2+1D topological orders with chiral gapless edges (including gapped edges) and 0d walls between edges. This relation is stated precisely and proved rigorously as a monoidal equivalence, which generalizes the functoriality of the usual Drinfeld center to an enriched setting. We also develop the mathematical theory of non-chiral gapless edges and 0+1D walls, and explain how to gap out certain non-chiral 1+1D gapless edges and 0+1D gapless walls categorically. In the end, we show that all anomaly-free 1+1D boundary-bulk rational CFT's can be recovered from 2d topological orders with chiral gapless edges via a dimensional reduction process. This provides physical meanings to some mysterious connections between mathematical results in fusion categories and those in rational CFT's.

The center of the categorified ring of differential operators

Let $\mathcal Y$ be a derived algebraic stack satisfying some mild conditions. The purpose of this paper is three-fold. First, we introduce and study $\mathbb H(\mathcal Y)$, a monoidal DG category that might be regarded as a categorification of the ring of differential operators on $\mathcal Y$. When $\mathcal Y = \mathrm {LS}\_G$ is the derived stack of $G$-local systems on a smooth projective curve, we expect $\mathbb H (\mathrm {LS}g)$ to act on both sides of the geometric Langlands correspondence, compatibly with the conjectural Langlands functor. Second, we construct a novel theory of D-modules on derived algebraic stacks. In contrast to usual D-modules, this new theory, to be denoted by $\mathcal D^{\mathrm{der}}$, is sensitive to the derived structure. Third, we identify the Drinfeld center of $\mathbb H(\mathcal Y)$ with $\mathcal D^{\mathrm{der}}(L\mathcal Y)$, the DG category of $\mathcal D^{\mathrm{der}}$-modules on the loop stack $L\mathcal Y$: = $\mathcal Y \times{\mathcal Y \times \mathcal Y}\mathcal Y$.

Pointed Drinfeld Center Functor

In this work, using the functoriality of Drinfeld center of fusion categories, we generalize an earlier result on the functoriality of full center of simple separable algebras in a fixed fusion category to all fusion categories. This generalization produces a new center functor, which involves both Drinfeld center and full center and will be called the pointed Drinfeld center functor. We prove that this pointed Drinfeld center functor is a symmetric monoidal equivalence. It turns out that this functor provides a precise and rather complete mathematical formulation of the boundary-bulk relation of 1+1D rational conformal field theories (RCFT). In this process, we solve an old problem of computing the fusion of two 0D (or 1D) wall CFT's along a non-trivial 1+1D bulk RCFT.

Computing modular data for pointed fusion categories

A formula for the modular data of $\mathcal{Z}(Vec^{\omega}G)$ was given by Coste, Gannon and Ruelle in arXiv:hep-th/0001158, but without an explicit proof for arbitrary 3-cocycles. This paper presents a derivation using the representation category of the quasi Hopf algebra $D^{\omega}G$. Further, we have written code to compute this modular data for many pairs of small finite groups and 3-cocycles. This code is optimised using Galois symmetries of the S and T matrices. We have posted a database of modular data for the Drinfeld center of every Morita equivalence class of pointed fusion categories of dimension less than 64.

Infinite Families of Potential Modular Data Related to Quadratic Categories

We present several infinite families of potential modular data motivated by examples of Drinfeld centers of quadratic categories. In each case, the input is a pair of involutive metric groups with Gauss sums differing by a sign, along with some conditions on the fixed points of the involutions and the relative sizes of the groups. From this input we construct $S$ and $T$ matrices which satisfy the modular relations and whose Verlinde coefficients are non-negative integers. We also check certain restrictions coming from Frobenius-Schur indicators. These families generalize Evans and Gannon's conjectures for the modular data associated to generalized Haagerup and near-group categories for odd groups, and include the modular data of the Drinfeld centers of almost all known quadratic categories. In addition to the subfamilies which are conjecturally realized by centers of quadratic categories, these families include many examples of potential modular data which do not correspond to known types of modular tensor categories.

The construction of braided tensor categories from Hopf braces

ABSTRACT Let be a Hopf brace. We first show that the category of left compatible modules over is a tensor category. Next, we show that its Drinfeld centre is equivalent to the category of left Yetter–Drinfeld modules over as a braided tensor category. Finally, we show that Radford's biproduct and constitute a Hopf brace, where R is a Hopf algebra in . This presents some new method to construct Hopf braces.

Computing defects associated to bounded domain wall structures: the case

We discuss domain walls and defects in topological phases occurring as the Drinfeld center of some fusion category. Domain walls between such phases correspond to bimodules between the fusion categories. Point defects correspond to functors between the bimodules. A domain wall structure consists of a planar graph with faces labeled by fusion categories. Edges are labeled by bimodules. When the vertices are labeled by point defects we get a compound defect. We present an algorithm, called the domain wall structure algorithm, for computing the compound defect. We apply this algorithm to show that the bimodule associator, related to the O3 obstruction of Etingof et al (2010 Quantum Topol. 1 209), is trivial for all domain walls of . In the language of this paper, the ground states of the Levin–Wen model are compound defects. We use this to define a generalization of the Levin–Wen model with domain walls and point defects. The domain wall structure algorithm can be used to compute the ground states of these generalized Levin–Wen type models.

String-net models for nonspherical pivotal fusion categories

A string-net model associates a vector space to a surface in terms of graphs decorated by objects and morphisms of a pivotal fusion category modulo local relations. String-net models are usually considered for spherical fusion categories, and in this case, the vector spaces agree with the state spaces of the corresponding Turaev–Viro topological quantum field theory. In the present work, some effects of dropping the sphericality condition are investigated. In one example of nonspherical pivotal fusion categories, the string-net space counts the number of [Formula: see text]-spin structures on a surface and carries an isomorphic representation of the mapping class group. Another example concerns the string-net space of a sphere with one marked point labeled by a simple object [Formula: see text] of the Drinfeld center. This space is found to be nonzero iff [Formula: see text] is isomorphic to a nonunit simple object determined by the nonspherical pivotal structure. The last example mirrors the effect of deforming the stress tensor of a two-dimensional conformal field theory, such as in the topological twist of a supersymmetric theory.

Classifying fusion categories $$\otimes $$-generated by an object of small Frobenius–Perron dimension

The goal of this paper is to classify fusion categories $${\mathcal {C}}$$ which are $$\otimes $$-generated by an object X of Frobenius–Perron dimension less than 2, with the additional mild assumption that the adjoint subcategory of $${\mathcal {C}}$$ is $$\otimes $$-generated by the object $$X\otimes X^*$$. This classification has recently become accessible due to a result of Morrison and Snyder, showing that any such category must be a cyclic extension of a category of adjoint ADE type. Our main tools in this classification are the results of Etingof et al. (Quantum Topol 1(3);209–273, 2010. https://doi.org/10.4171/QT/6), classifying cyclic extensions of a given category in terms of data computed from the Brauer–Picard group, and Drinfeld centre of that category, and the results of Edie-Michell (Int. J. Math. 29(5):1850036, 2018. https://doi.org/10.1142/S0129167X18500362) which compute the Brauer–Picard group and Drinfeld centres of the categories of adjoint ADE type. Our classification includes the expected categories, constructed from cyclic groups and the categories of ADE type. More interestingly we have categories in our classification that are non-trivial de-equivariantizations of these expected categories. Most interesting of all, our classification includes three infinite families constructed from the exceptional quantum subgroups $${\mathcal {E}}_4$$ of $${\mathcal {C}}( \mathfrak {sl}_4, 4)$$, and $${\mathcal {E}}_{16,6}$$ of $${\mathcal {C}}( \mathfrak {sl}_2, 16)\boxtimes {\mathcal {C}}( \mathfrak {sl}_3,6)$$.

The Drinfeld centre of a symmetric fusion category is 2-fold monoidal

We show that the Drinfeld centre of a symmetric fusion category over an algebraically closed field of characteristic zero is a bilax 2-fold monoidal category. That is, it carries two monoidal structures, the convolution and symmetric tensor products, that are bilax monoidal functors with respect to each other. We additionally show that the braiding and symmetry for the convolution and symmetric tensor products are compatible with this bilax structure.We establish these properties without referring to Tannaka duality for the symmetric fusion category. This has the advantage that all constructions are done purely in terms of the fusion category structure, making the result easy to use in other contexts.

The symmetric tensor product on the Drinfeld centre of a symmetric fusion category

We define a symmetric tensor product on the Drinfeld centre of a symmetric fusion category, in addition to its usual tensor product. We examine what this tensor product looks like under Tannaka duality, identifying the symmetric fusion category with the representation category of a finite (super)-group. Under this identification, the Drinfeld centre is the category of equivariant vector bundles over the finite group (underlying the super-group, in the super case). In the non-super case, we show that the symmetric tensor product corresponds to the fibrewise tensor product of these vector bundles. In the super case, we define for each super-group structure on the finite group a super-version of the fibrewise tensor product. We show that the symmetric tensor product on the Drinfeld centre of the representation category of the resulting finite super-groups corresponds to this super-version of the fibrewise tensor product on the category of equivariant vector bundles over the finite group.

A mathematical theory of gapless edges of 2d topological orders. Part I

This is the first part of a two-part work on a unified mathematical theory of gapped and gapless edges of 2d topological orders. We analyze all the possible observables on the 1+1D world sheet of a chiral gapless edge of a 2d topological order, and show that these observables form an enriched unitary fusion category, the Drinfeld center of which is precisely the unitary modular tensor category associated to the bulk. This mathematical description of a chiral gapless edge automatically includes that of a gapped edge (i.e. a unitary fusion category) as a special case. Therefore, we obtain a unified mathematical description and a classification of both gapped and chiral gapless edges of a given 2d topological order. In the process of our analysis, we encounter an interesting and reoccurring phenomenon: spatial fusion anomaly, which leads us to propose the Principle of Universality at RG fixed points for all quantum field theories. Our theory also implies that all chiral gapless edges can be obtained from a so-called topological Wick rotations. This fact leads us to propose, at the end of this work, a surprising correspondence between gapped and gapless phases in all dimensions.

Higher Gauss sums of modular categories

The definitions of the $n^{th}$ Gauss sum and the associated $n^{th}$ central charge are introduced for premodular categories $\mathcal{C}$ and $n\in\mathbb{Z}$. We first derive an expression of the $n^{th}$ Gauss sum of a modular category $\mathcal{C}$, for any integer $n$ coprime to the order of the T-matrix of $\mathcal{C}$, in terms of the first Gauss sum, the global dimension, the twist and their Galois conjugates. As a consequence, we show for these $n$, the higher Gauss sums are $d$-numbers and the associated central charges are roots of unity. In particular, if $\mathcal{C}$ is the Drinfeld center of a spherical fusion category, then these higher central charges are 1. We obtain another expression of higher Gauss sums for de-equivariantization and local module constructions of appropriate premodular and modular categories. These expressions are then applied to prove the Witt invariance of higher central charges for pseudounitary modular categories.