Modular Categories with Transitive Galois Actions

We give a complete classification of modular categories of dimension p 3 m p^3m where p p is prime and m m is a square-free integer relatively prime to p p . When p p is odd, all such categories are pointed. For p = 2 p=2 one encounters modular categories with the same fusion ring as orthogonal quantum groups at certain roots of unity, namely S O ( 2 m ) 2 \mathrm {SO}(2m)_2 . We also classify the more general class of modular categories with the same fusion rules as S O ( 2 N ) 2 \mathrm {SO}(2N)_2 with N N odd.

An abelian k-linear semisimple category having a finite number of simple objects, and endowed with a ribbon structure, is called premodular. It is modular (in the sense of Turaev) if the so-called S-matrix is invertible. A modular category defines invariants of 3-manifolds and a TQFT ([T]). When is it possible to construct a modularisation of a given premodular category, i.e. a functor to a modular category preserving the structures and ‘dominant’ in a certain sense? It turns out (2.3) that this amounts essentially to making ‘transparent’ objects trivial. We give a full answer to this problem in the case when k is a field of char. 0 (as well as partial answers in char. p): under a few obvious hypotheses, a premodular category admits a modularisation, which is unique (th. 3.1, and cor. 3.5 in char. 0) The proof relies on two main ingredients: a new and very simple criterion for the S-matrix to be invertible (1.1) and Deligne's internal characterization of tannakian categories in char. 0 [D]. When simple transparent objects are invertible, the criterion is simpler (4.2) and the modularisation can be described more explicitly (prop. 4.4). We conclude with two examples: the premodular categories associated with quantum $\hbox{\sf SL}_N$ and $\hbox{\sf PGL}_N$ at roots of unity; in the first case, we obtain modular categories which were built independently by C. Blanchet [B]; in the second case, we obtain modularizations in all the cases where Y. Yokota [Y] found Reshetikhin-Turaev invariants of 3-manifolds, thereby improving as well as explaining Yokota's results.

The fusion rules in [Formula: see text] for a finite group [Formula: see text] can be computed in terms of character inner products. Using an explicit formula (Theorem 3.2) for these fusion rules, we show that [Formula: see text] is multiplicity free for two infinite families of finite groups: the Dihedral groups and the Dicyclic groups. In fact, we will compute all fusion rules in these categories. Multiplicity freeness is a desired property for modular tensor categories, since it greatly simplifies the computation of [Formula: see text]-matrices. Furthermore, we observe that the fusion rules for Dihedral groups [Formula: see text] with [Formula: see text] odd are extremely similar to the fusion rules of type-B level-[Formula: see text] fusion algebras of Wess–Zumino–Witten conformal field theories. Moreover, we give a proof of the fusion rule formula by using Mackey theory.

Let \mathcal{C}(\mathfrak{g},k) be the unitary modular tensor categories arising from the representation theory of quantum groups at roots of unity for arbitrary simple finite-dimensional complex Lie algebra \mathfrak{g} and positive integer levels k . Here we classify nondegenerate fusion subcategories of the modular tensor categories of local modules \mathcal{C}(\mathfrak{g},k)_R^0 where R is the regular algebra of Tannakian Rep (H)\subset\mathcal{C}(\mathfrak{g},k)_\text{pt} . We describe the decomposition of \mathcal{C}(\mathfrak{g},k)_R^0 into prime factors, and as an application we classify relations in the Witt group of nondegenerately braided fusion categories generated by the equivalency classes of \mathcal{C}(\mathfrak{so}_5,k) and \mathcal{C}(\mathfrak{g}_2,k) for k\in\mathbb{Z}_{\geq1} .

We describe a systematic way of producing fermionic topological phases using the technique of fermion condensation. We give a prescription for performing fermion condensation in bosonic topological phases which contain an emergent fermion. Our approach to fermion condensation can roughly be understood as coupling the parent bosonic topological phase to a phase of physical fermions, and condensing pairs of physical and emergent fermions. There are two distinct types of objects in fermionic theories, which we call “m-type” and “q-type” particles. The endomorphism algebras of q-type particles are complex Clifford algebras, and they have no analogues in bosonic theories. We construct a fermionic generalization of the tube category, which allows us to compute the quasiparticle excitations in fermionic topological phases. We then prove a series of results relating data in condensed theories to data in their parent theories; for example, if C is a modular tensor category containing a fermion, then the tube category of the condensed theory satisfies Tube(C/ψ) ≅ C × C/ψ. We also study how modular transformations, fusion rules, and coherence relations are modified in the fermionic setting, prove a fermionic version of the Verlinde dimension formula, construct a commuting projector lattice Hamiltonian for fermionic theories, and write down a fermionic version of the Turaev-Viro-Barrett-Westbury state sum.

Let V be a simple vertex operator algebra satisfying the following conditions: (i) V(n)) = 0 for n < 0, V(0)=C1, and the contragredient module V' is isomorphic to V as a V-module; (ii) every N-gradable weak V-module is completely reducible; (iii) V is C(2)-cofinite. We announce a proof of the Verlinde conjecture for V, that is, of the statement that the matrices formed by the fusion rules among irreducible V-modules are diagonalized by the matrix given by the action of the modular transformation tau |--> -1/tau on the space of characters of irreducible V-modules. We discuss some consequences of the Verlinde conjecture, including the Verlinde formula for the fusion rules, a formula for the matrix given by the action of tau |--> -1/tau, and the symmetry of this matrix. We also announce a proof of the rigidity and nondegeneracy property of the braided tensor category structure on the category of V-modules when V satisfies in addition the condition that irreducible V-modules not equivalent to V have no nonzero elements of weight 0. In particular, the category of V-modules has a natural structure of modular tensor category.

The category of finite dimensional modules over the quantum superalgebra [Formula: see text] is not semi-simple and the quantum dimension of a generic [Formula: see text]-module vanishes. This vanishing happens for any value of [Formula: see text] (even when [Formula: see text] is not a root of unity). These properties make it difficult to create a fusion or modular category. Loosely speaking, the standard way to obtain such a category from a quantum group is: (1) specialize [Formula: see text] to a root of unity; this forces some modules to have zero quantum dimension, (2) quotient by morphisms of modules with zero quantum dimension, (3) show the resulting category is finite and semi-simple. In this paper, we show an analogous construction works in the context of [Formula: see text] by replacing the vanishing quantum dimension with a modified quantum dimension. In particular, we specialize [Formula: see text] to a root of unity, quotient by morphisms of modules with zero modified quantum dimension and show the resulting category is generically finite semi-simple. Moreover, we show the categories of this paper are relative [Formula: see text]-spherical categories. As a consequence, we obtain invariants of 3-manifold with additional structures.

We consider a pivotal monoidal functor whose domain is a modular tensor category (MTC). We show that the trace of such a functor naturally extends to a representation of the corresponding tube category. As irreducible representations of the tube category are indexed by pairs of simple objects in the underlying MTC, the simple multiplicities of this representation form a candidate modular invariant matrix. In general, this matrix will not be modular invariant, however it will always commute with the T-matrix. Furthermore, under certain additional conditions on the original functor, it is shown that the corresponding representation of the tube category is a haploid, symmetric, commutative Frobenius algebra. Such algebras are known to be connected to modular invariants, in particular a result of Kong and Runkel implies that the matrix of simple multiplicities commutes with the S-matrix if and only if the dimension of the algebra is equal to the dimension of the underlying MTC. Finally, we apply these techniques to certain pivotal monoidal functors arising from module categories over the Temperley-Lieb category and the associated MTC. This provides a novel explanation of the A-D-E pattern appearing in the classification of $ A_1^{(1)} $ modular invariants.

This is the second part of the paper (the first part is published in Jour. of AMS, vol.9, 1135--1170, q-alg/9508017). In the first part, we defined for every modular tensor category (MTC) inner products on the spaces of morphisms and proved that the inner product on the space $\Hom (\bigoplus X_i\otimes X^*_i, U)$ is modular invariant. Also, we have shown that in the case of the MTC arising from the representations of the quantum group $U_q \sln$ at roots of unity and $U$ being a symmetric power of the fundamental representation, this inner product coincides with so-called Macdonald's inner product on symmetric polynomials. In this paper, we apply the same construction to the MTC coming from the integrable representations of affine Lie algebras. In this case our construction immediately gives a hermitian form on the spaces of conformal blocks, and this form is modular invariant (Warning: we cannot prove that it is positive definite). We show that this form can be rewritten in terms of asymptotics of KZ equations, and calculate it for $sl_2$, in which case the formula is a natural affine analogue of Macdonald's inner product identities. We also formulate as a conjecture similar formula for $sl_n$.

From quantum groups to unitary modular tensor categories

Ribbon tensor structure on the full representation categories of the singlet vertex algebras

We determine the fusion rules of the equivariantization of a fusion category \documentclass[12pt]{minimal}\begin{document}${\mathcal {C}}$\end{document}C under the action of a finite group G in terms of the fusion rules of \documentclass[12pt]{minimal}\begin{document}${\mathcal {C}}$\end{document}C and group-theoretical data associated to the group action. As an application we obtain a formula for the fusion rules in an equivariantization of a pointed fusion category in terms of group-theoretical data. This entails a description of the fusion rules in any braided group-theoretical fusion category.

The authors construct families of solvable quantum spin chains, vertex and interactions round a face (IRF) models from representations of the Temperley-Lieb algebra associated with the quantum groups Uq(Xn) for Xn=A1, Bn, Cn and Dn. They determine a correlation measuring order in the IRF models via the fusion rule in the level l Xn(1). Wess-Zumino-Witten model.

Equivariant Kähler geometry and localization in the G/ G model

A comprehensive introduction to two-dimensional conformal field theory is given. The structure of the meromorphic subtheory is described in detail, and a number of examples are presented explicitly. Standard constructions such as the coset and the orbifold construction are explained. The concept of a representation of the meromorphic theory is introduced, and the role of Zhu's algebra in classifying highest weight representations is elucidated. The fusion product of two representations and the corresponding fusion rules are defined, and Verlinde's formula is explained. Finally, higher correlation functions are considered, and the polynomial relations of Moore and Seiberg and the quantum group structure of chiral conformal field theory are discussed. The treatment is relatively general and also allows for a description of less well known classes of theories such as logarithmic conformal field theories.