Fusion Rings Research Articles

Cyclic extensions of fusion categories via the Brauer–Picard groupoid

We construct a long exact sequence computing the obstruction space, \pi_1\mathrm {BrPic}(\mathcal C_0) , to G -graded extensions of a fusion category \mathcal C_0 . The other terms in the\linebreak sequence can be computed directly from the fusion ring of \mathcal C_0 . We apply our result to several examples coming from small index subfactors, thereby constructing several new fusion categories as G -extensions. The most striking of these is a \mathbb{Z}/2\mathbb{Z} -extension of one of the Asaeda–Haagerup fusion categories, which is one of only two known 3 -supertransitive fusion categories outside the ADE series. In another direction, we show that our long exact sequence appears in exactly the way one expects: it is part of a long exact sequence of homotopy groups associated to a naturally occuring fibration. This motivates our constructions, and gives another example of the increasing interplay between fusion categories and algebraic topology.

𝒲-Algebras, False Theta Functions and Quantum Modular Forms, I

In this paper, we study certain partial and false theta functions in connection to vertex operator algebras and conformal eld theory. We prove a variety of results concerning the asymptotics of modied characters of irreducible modules of certain W-algebras of singlet type, which allows us in particular to determine their (analytic) quantum dimensions. Our results are fully consistent with the previous conjectures on fusion rings for these vertex algebras. More importantly, we prove quantum modularity ( a la Zagier) of the numerator part of irreducible characters of singlet algebra modules, thus demonstrating that quantum modular forms naturally appear in many \suciently nice irrational vertex algebras. It is interesting that quantum modularity persists on the whole set of rationals as in the original Zagier’s example coming from Kontsevich’s \strange series. In the last part, slightly independent of all this, we also discuss Nahm-type q-hypergometric series in connection to tails of colored Jones polynomials of certain torus knots and characters of modules for the (1;p)-singlet algebra.

A classification of SU(d)-type C*-tensor categories

Kazhdan and Wenzl classified all rigid tensor categories with fusion ring isomorphic to the fusion ring of the group SU(d). In this paper we consider the C*-analogue of this problem. Given a rigid C*-tensor category 𝒞 with fusion ring isomorphic to the fusion ring of the group SU(d), we can extract a constant q from 𝒞 such that there exists a *-representation of the Hecke algebra Hn(q) into 𝒞. The categorical trace on 𝒞 induces a Markov trace on Hn(q). Using this Markov trace and a representation of Hn(q) in [Formula: see text] we show that 𝒞 is equivalent to a twist of the category [Formula: see text]. Furthermore a sufficient condition on a C*-tensor category 𝒞 is given for existence of an embedding of a twist of [Formula: see text] in 𝒞.

Fusion (semi)rings arising from quantum groups

We study the fusion semirings arising from easy quantum groups. We classify all the possible free ones, answering a question of T. Banica and R. Vergnioux: these are exactly the fusion rings of quantum groups without any nontrivial one-dimensional representation. We then classify the possible groups of one-dimensional representations for general easy quantum groups associated to noncrossing partitions. As an application, we give a unified proof of the Haagerup property for a broad class of easy quantum groups, recovering as special cases previous results by M. Brannan and F. Lemeux. We end with some considerations on the description of the full fusion ring in the general case.

Fusion Rings for Quantum Groups

We study the fusion rings of tilting modules for a quantum group at a root of unity modulo the tensor ideal of negligible tilting modules. We identify them in type A with the combinatorial rings from Korff, C., Stroppel, C.: The ˆ sl(n)k-WZNW fusion ring: a combinato-rial construction and a realisation as quotient of quantum cohomology. Adv. Math. 225(1), 200-268, (2010) and give a similar description of the sp2n-fusion ring in terms of non-commutative symmetric functions. Moreover we give a presentation of all fusion rings in classical types as quotients of polynomial rings. Finally we also compute the fusion rings for type G2.

Fusion rings with few degrees

The main result characterizes fusion rings whose degree set consists of 1 and a fixed prime p. A well known theorem of Isaacs and Passman on finite groups whose irreducible character degrees are 1 and p follows as a corollary. Numerically compatible (NC) fusion rings are defined as a potential generalization of the Grothendieck rings of fusion categories; and the NC fusion rings of order p, p2, p3, pq, and pq2 (where p, q are primes) are classified in terms of certain standard table algebras, as another application of the main theorem.

Modular data and Verlinde formulae for fractional level WZW models II

This article gives a complete account of the modular properties and Verlinde formula for conformal field theories based on the affine Kac–Moody algebra slˆ(2) at an arbitrary admissible level k. Starting from spectral flow and the structure theory of relaxed highest weight modules, characters are computed and modular transformations are derived for every irreducible admissible module. The culmination is the application of a continuous version of the Verlinde formula to deduce non-negative integer structure coefficients which are identified with Grothendieck fusion coefficients. The Grothendieck fusion rules are determined explicitly. These rules reproduce the well-known “fusion rules” of Koh and Sorba, negative coefficients included, upon quotienting the Grothendieck fusion ring by a certain ideal.

Modular Invariants and Twisted EquivariantK-theory II: Dynkin diagram symmetries

AbstractThe most basic structure ofchiralconformal field theory (CFT) is the Verlinde ring. Freed-Hopkins-Teleman have expressed the Verlinde ring for the CFT's associated to loop groups, as twisted equivariant K-theory. We build on their work to express K-theoretically the structures offullCFT. In particular, the modular invariant partition functions (which essentially parametrise the possible full CFTs) have a rich interpretation within von Neumann algebras (subfactors), which has led to the developments of structures of full CFT such as the full system (fusion ring of defect lines), nimrep (cylindrical partition function), alpha-induction etc. Modular categorical interpretations for these have followed. For the generic families of modular invariants (i.e. those associated to Dynkin diagram symmetries), we provide aK-theoretic framework for these other CFT structures, and show how they relate to D-brane charges and charge-groups. We also study conformal embeddings and themodular invariant of SU(2), as well as some families of finite group doubles. This newK-theoretic framework allows us to simplify and extend the less transparent, more ad hoc descriptions of these structures obtained previously within CFT.

Takiff superalgebras and conformal field theory

A class of non-semisimple extensions of Lie superalgebras is studied. They are obtained by adjoining to the superalgebra its adjoint representation as an Abelian ideal. When the superalgebra is of affine Kac–Moody type, a generalization of Sugawara’s construction is shown to give rise to a copy of the Virasoro algebra and so, presumably, to a conformal field theory. Evidence for this is detailed for the extension of the affinization of the superalgebra : its highest weight irreducible modules are classified using spectral flow, the irreducible supercharacters are computed and a continuum version of the Verlinde formula is verified to give non-negative integer structure coefficients. Interpreting these coefficients as those of the Grothendieck ring of fusion, partial results on the true fusion ring and its indecomposable structures are deduced.

Fusion Rings of Loop Group Representations

We compute the fusion rings of positive energy representations of the loop groups of the simple, simply connected Lie groups.

Fixed points and D-branes

The affine Kac-Moody algebras give rise to rational conformal field theories (RCFTs) called the Wess-Zumino-Witten (WZW) models. An important component of an RCFT is its fusion ring, whose structure constants are given by the associated S-matrix. We apply a fixed point property possessed by the WZW models ("fixed point factorization") to calculate nonnegative integer matrix representations of the fusion ring, allowing for the calculation of D-brane charges in string theory.

Conformal Nets and KK-Theory

Given a completely rational conformal net A on the circle, its fusion ring acts faithfully on the K_0-group of a certain universal C*-algebra associated to A, as shown in a previous paper. We prove here that this action can actually be identified with a Kasparov product, thus paving the way for a fruitful interplay between conformal field theory and KK-theory.

Free unitary groups are (almost) simple

We show that the quotients of Wang and Van Daele's universal quantum groups by their centers are simple in the sense that they have no normal quantum subgroups, thus providing the first examples of simple compact quantum groups with non-commutative fusion rings.

On intermediate conformal nets

Abstract. Motivated by our subfactor generalization of Wall's conjecture and interest about completely rational conformal nets in general, in this paper we show that there is a one-to-one correspondence between intermediate conformal nets in a pair of completely rational nets 𝒜 ⊂ ℬ $\mathcal {A}\subset \mathcal {B}$ with finite index and certain class of fusion rings. As a consequence we show that our conjecture for the pair 𝒜 ⊂ ℬ $\mathcal {A}\subset \mathcal {B}$ is reduced to a conjecture about fusion rings, and our result also gives a method of enumerating all intermediate conformal nets.

Interactions between surround suppression and interocular suppression in human vision.

Several types of suppression phenomena have been observed in the visual system. For example, the ability to detect a target stimulus is often impaired when the target is embedded in a high-contrast surround. This contextual modulation, known as surround suppression, was formerly thought to occur only in the periphery. Another type of suppression phenomena is interocular suppression, in which the sensitivity to a monocular target is reduced by a superimposed mask in the opposite eye. Here, we explored how the two types of suppression operating across different spatial regions interact with one another when they simultaneously exert suppressive influences on a common target presented at the fovea. In our experiments, a circular target grating presented to the fovea of one eye was suppressed interocularly by a noise pattern of the same size in the other eye. The foveal stimuli were either shown alone or surrounded by a monocular annular grating. The orientation and eye-of-origin of the surround grating were varied. We found that the detection of the foveal target subjected to interocular suppression was severely impaired by the addition of the surround grating, indicating strong surround suppression in the fovea. In contrast, when the interocular suppression was released by superimposing a binocular fusion ring onto both the target and the dichoptic mask, the surround suppression effect was found to be dramatically decreased. In addition, the surround suppression was found to depend on the contrast of the dichoptic noise with the greatest surround suppression effect being obtained only when the noise contrast was at an intermediate level. These findings indicate that surround suppression and interocular suppression are not independent of each other, but there are strong interactions between them. Moreover, our results suggest that strong surround suppression may also occur at the fovea and not just the periphery.

Fixed points and fusion rings. Part 1

In the first of this two-part series, we find `fixed point factorisation' formulas, towards an understanding of the fusion ring of WZW models. Fixed-point factorisation refers to the simplifications in the data of a CFT involving primary fields fixed by simple-currents. Until now, it has been worked out only for SU(n), where it has developed into a powerful tool for understanding the fusion rings of WZW models of CFT -- e.g. it has lead to closed formulas for NIM-reps and D-brane charges and charge-groups. In this paper, we generalise these formulas to the other classical algebras, laying the groundwork for future applications to fusion rings (Part 2). We also discuss connections with the twining characters of Fuchs-Schellekens-Schweigert.

Value-based Care in the Management of Spinal Disorders: A Systematic Review of Cost-utility Analysis

Spinal disorders are a major cause of disability and compromise in health-related quality of life. The direct and indirect costs of treating spinal disorders are estimated at more than $100 billion per year. With limited resources, the cost-utility of interventions is important for allocating resources. We therefore performed a systematic review of the literature on cost-utility for nonoperative and operative interventions for treating spinal disorders. We searched four databases for cost-utility analysis studies on low back pain management and identified 1004 items. The titles and abstracts of 752 were screened before selecting 27 studies for inclusion; full texts of these 27 studies were individually evaluated by five individuals. Studies of nonoperative treatments demonstrated greater value for graded activity over physical therapy and pain management; spinal manipulation over exercise; behavioral therapy and physiotherapy over advice; and acupuncture and exercise over usual general practitioner care. Circumferential fusion and femoral ring allograft had greater value than posterolateral fusion and titanium cage, respectively. The relative cost-utility of operative versus nonoperative interventions was variable with the most consistent evidence indicating superior value of operative care for treating spinal disorders involving nerve compression and instability. The literature on cost-utility for treating spinal disorders is limited. Studies addressing cost-utility of nonoperative and operative management of low back pain encompass a broad spectrum of diagnoses and direct comparison of treatments based on cost-utility thresholds for comparative effectiveness is limited by diversity among disorders and methods to assess cost-utility. Future research will benefit from uniform methods and comparison of treatments in cohorts with well-defined pathology.

Twisted K-Homology and Group-Valued Moment Maps

Let G be a compact, simply connected Lie group. We develop a ‘quantization functor’ from prequantized quasi-Hamiltonian G-spaces (M,ω,Φ) at level k to the fusion ring (Verlinde algebra) R k (G). The quantization is defined as a push-forward in twisted equivariant K-homology. It may be computed by a fixed point formula, similar to the equivariant index theorem for Spin c -Dirac operators. Using the formula, we calculate in several examples.

-extended Kac representations and integrable boundary conditions in the logarithmic minimal models

We construct new Yang–Baxter integrable boundary conditions in the lattice approach to the logarithmic minimal model giving rise to reducible yet indecomposable representations of rank 1 in the continuum scaling limit. We interpret these -extended Kac representations as finitely generated -extended Feigin–Fuchs modules over the triplet W-algebra . The -extended fusion rules of these representations are inferred from the recently conjectured Virasoro fusion rules of the Kac representations in the underlying logarithmic minimal model . We also introduce the modules contragredient to the -extended Kac modules and work out the correspondingly extended fusion algebra. Our results are in accordance with the Kazhdan–Lusztig dual of tensor products of modules over the restricted quantum universal enveloping algebra at q = eπi/p. Finally, polynomial fusion rings isomorphic with the various fusion algebras are determined, and the corresponding Grothendieck ring of characters is identified.

ON SECOND COHOMOLOGY OF DUALS OF COMPACT GROUPS

We show that for any compact connected group G the second cohomology group defined by unitary invariant two-cocycles on Ĝ is canonically isomorphic to [Formula: see text]. This implies that the group of autoequivalences of the C*-tensor category Rep G is isomorphic to [Formula: see text]. We also show that a compact connected group G is completely determined by Rep G. More generally, extending a result of Etingof–Gelaki and Izumi–Kosaki we describe all pairs of compact separable monoidally equivalent groups. The proofs rely on the theory of ergodic actions of compact groups developed by Landstad and Wassermann and on its algebraic counterpart developed by Etingof and Gelaki for the classification of triangular semisimple Hopf algebras. We give a self-contained account of amenability of tensor categories, fusion rings and discrete quantum groups, and prove an analog of Radford's theorem on minimal Hopf subalgebras of quasitriangular Hopf algebras for compact quantum groups.