Fusion Rule Algebras Research Articles

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.

Decomposition of products in nilpotent table algebras

Two theorems on the decomposition of products in nilpotent commutative table algebras are established. This generalizes several results of Adan-Bante on the decomposition of products of irreducible characters and of conjugacy classes of finite p-groups, and extends these results to fusion rule algebras and association schemes.

Sylow theory for table algebras, fusion rule algebras, and hypergroups

Sylow theory for table algebras, fusion rule algebras, and hypergroups

Quantum double construction for subfactors arising from periodic commuting squares

By generalizing Erlijman's method, we construct a subfactor from a fusion rule algebra with quantum 6j-symbols which produce periodic commuting squares. This construction produces the same subfactor as Ocneanu's asymptotic inclusion for the subfactor which is generated by the original periodic commuting square. This result can be applied to the quantum SU(n)k subfactors which is the same as Hecke algebra subfactors of type A of Wenzl for example, which shows that Erlijman's construction gives the same subfactor as the asymptotic inclusion.

On α-Induction, Chiral Generators and¶Modular Invariants for Subfactors

We consider a type III subfactor $N\subset M$ of finite index with a finite system of braided $N$-$N$ morphisms which includes the irreducible constituents of the dual canonical endomorphism. We apply $\alpha$-induction and, developing further some ideas of Ocneanu, we define chiral generators for the double triangle algebra. Using a new concept of intertwining braiding fusion relations, we show that the chiral generators can be naturally identified with the $\alpha$-induced sectors. A matrix $Z$ is defined and shown to commute with the S- and T-matrices arising from the braiding. If the braiding is non-degenerate, then $Z$ is a ``modular invariant mass matrix'' in the usual sense of conformal field theory. We show that in that case the fusion rule algebra of the dual system of $M$-$M$ morphisms is generated by the images of both kinds of $\alpha$-induction, and that the structural information about its irreducible representations is encoded in the mass matrix $Z$. Our analysis sheds further light on the connection between (the classifications of) modular invariants and subfactors, and we will construct and analyze modular invariants from $SU(n)_k$ loop group subfactors in a forthcoming publication, including the treatment of all $SU(2)_k$ modular invariants.

Chiral Structure of Modular Invariants for Subfactors

In this paper we further analyze modular invariants for subfactors, in particular the structure of the chiral induced systems of M-M morphisms. The relative braiding between the chiral systems restricts to a proper braiding on their “ambichiral” intersection, and we show that the ambichiral braiding is non-degenerate if the original braiding of the N-N morphisms is. Moreover, in this case the dimensions of the irreducible representations of the chiral fusion rule algebras are given by the chiral branching coefficients which describe the ambichiral contribution in the irreducible decomposition of α-induced sectors. We show that modular invariants come along naturally with several non-negative integer valued matrix representations of the original N-N Verlinde fusion rule algebra, and we completely determine their decomposition into its characters. Finally the theory is illustrated by various examples, including the treatment of all SU(2) k modular invariants.

Branes: from free fields to general backgrounds

Motivated by recent developments in string theory, we study the structure of boundary conditions in arbitrary conformal field theories. A boundary condition is specified by two types of data: first, a consistent collection of reflection coefficients for bulk fields on the disk; and second, a choice of an automorphism ω of the fusion rules that preserves conformal weights and can be implemented on the spaces of chiral blocks. Non-trivial automorphisms ω correspond to D-brane configurations for arbitrary conformal field theories. The choice of the fusion rule automorphism ω amounts to fixing the dimension and certain global topological features of the D-brane world volume and the background gauge field on it. We present evidence that for fixed choice of ω the boundary conditions are classified as the irreducible representations of some commutative associative algebra, a generalization of the fusion rule algebra. Each of these irreducible representations corresponds to a choice of the moduli for the world volume of the D-brane and the moduli of the flat connection on it.

Duality and Entropy for Finite Commutative Hypergroups and Fusion Rule Algebras

We develop a theory of harmonic analysis and duality for finite commutative hypergroups by considering somewhat more general objects called signed hypergroups. A notion of entropy is defined, and a Second Law of Thermodynamics is established. Applications to group theory and to the fusion rule algebras of conformal field theory are given.

The quantum symmetry of rational field theories

We describe the role of Rational Hopf Algebras as the symmetries of rational field theories and discuss their relation with algebraic field theory, braided monoidal categories and modular fusion rule algebras.

Fusion Rules in Conformal Field Theory

Several aspects of fusion rings and fusion rule algebras, and of their manifestations in twodimensional (conformal) field theory, are described: diagonalization and the connection with modular invariance; the presentation in terms of quotients of polynomial rings; fusion graphs; various strategies that allow for a partial classification; and the role of the fusion rules in the conformal bootstrap programme.

TOWARDS A CLASSIFICATION OF FUSION RULE ALGEBRAS IN RATIONAL CONFORMAL FIELD THEORIES

We review the main topics concerning Fusion Rule Algebras (FRA) of Rational Conformal Field Theories. After an exposition of their general properties, we examine known results on the complete classification for low number of fields (≤4). We then turn our attention to FRA’s generated polynomially by one (real) fundamental field, for which a classification is known. Attempting to generalize this result, we describe some connections between FRA’s and Graph Theory. The possibility to get new results on the subject following this “graph” approach is briefly discussed.

Twisted sector representations of discrete background symmetries for two-dimensional orbifolds

We have investigated two-dimensional orbifold compactifications of the closed bosonic string theory subject to modular transformations of the complex background parameter π. The representations of the pair of generators s(π → 1 π ) and t(π → π − 1) , which amount to the duality and axionic shift operations respectively, have been determined for the various twisted sectors. This analysis may be performed in two different ways: either by an inspection of correlation functions of twist field ground states on the sphere (Yukawa couplings, etc.), or by deducing the finite symmetry group for fixed values of π which leaves the correlators of twist fields unaltered. The latter option is closely related to Verlinde's approach providing constraints on the fusion rule algebra. Our results take a form which is familiar from world-sheet modular transformations of characters related to highest weight states in rational conformal field theories.

MODULAR INVARIANCE AND FUSION RULE ALGEBRAS: CLASSIFICATION OF THE THREE-OPERATOR CASE

We show how to handle together constraints coming from associativity, modular invariance and analyticity in RCFT's. In particular, looking at the theories with three operators we classify fusion rule algebras, central charges and conformal weights fulfilling such constraints. We also show that all allowed theories with three operators are algebraic.

Analyticity, modular invariance and the classification of three operator fusion algebras

A way is suggested to enforce constraints from associativity, and modular invariance in RCFTs. In the case of three operators, a complete classification is obtained for fusion rule algebras, central charges and conformal weights which fulfill such constraints; all allowed theories are proved to be algebraic.

Covariance properties of higher genus twist correlators under [formula omitted] durability in orbifold conformal field theory

We study R↔ 1 R duality in simple orbifold models at arbitrary orders in string perturbation theory. It is shown that duality involves certain linear relations among twist correlators at dual radii. We derive general expressions for the coefficients of these relations and show that they can be expressed in terms of functions of the orbifold cosets obeying an algebra bearing a striking resemblance to Verlinde's fusion rule algebra. Finally, we study the behavior of the coefficients under factorization, in particular their dependence on the genus. A few examples are worked out.

Fusion rule algebras from graph theory

We describe a new class of fusion algebras related to graph theory which bear intriguing connections with group algebras. The structure constants and the matrix S, which diagonalizes the fusion rules, are explicitly computed in terms of SU(2) coupling coefficients.

Naturality in conformal field theory

We discuss constraints on the operator product coefficients in diagonal and nondiagonal rational conformal field theories. Nondiagonal modular invariants always arise from automorphisms of the fusion rule algebra or from extensions of the chiral algebra. Moreover, when the chiral algebra has been maximally extended a strong form of the naturality principle of field theory can be proven for rational conformal field theory: operator product coefficients vanish if and only if the corresponding fusion rules vanish; that is, if and only if the vanishing can be understood in terms of a symmetry. We illustrate these ideas with several examples. We also generalize our ideas about rational conformal field theories to a larger class of theories: “quasi-rational conformal field theories” and we explore some of their properties.