Fusion Algebra Research Articles

Boundary conditions in rational conformal field theories

We develop further the theory of Rational Conformal Field Theories (RCFTs) on a cylinder with specified boundary conditions emphasizing the role of a triplet of algebras: the Verlinde, graph fusion and Pasquier algebras. We show that solving Cardy's equation, expressing consistency of a RCFT on a cylinder, is equivalent to finding integer valued matrix representations of the Verlinde algebra. These matrices allow us to naturally associate a graph G to each RCFT such that the conformal boundary conditions are labelled by the nodes of G. This approach is carried to completion for sl(2) theories leading to complete sets of conformal boundary conditions, their associated cylinder partition functions and the A- D- E classification. We also review the current status for WZW sl(3) theories. Finally, a systematic generalisation of the formalism of Cardy–Lewellen is developed to allow for multiplicities arising from more general representations of the Verlinde algebra. We obtain information on the bulk–boundary coefficients and reproduce the relevant algebraic structures from the sewing constraints.

Generalization of Longo–Rehren Construction to Subfactors of Infinite Depth and Amenability of Fusion Algebras

We extend the Longo–Rehren construction to an infinite system of bimodules (of sectors). The construction is similar to the crossed product. We also show that the Longo–Rehren construction gives an isomorphic subfactor to Popa's symmetric enveloping algebra. We discuss the relation between the Longo–Rehren subfactors and amenability of fusion algebras in the sense of Hiai and Izumi.

On Fusion Algebras and Modular Matrices

We consider the fusion algebras arising in e.g. Wess–Zumino–Witten conformal field theories, affine Kac–Moody algebras at positive integer level, and quantum groups at roots of unity. Using properties of the modular matrix S, we find small sets of primary fields (equivalently, sets of highest weights) which can be identified with the variables of a polynomial realization of the Ar fusion algebra at level k. We prove that for many choices of rank r and level k, the number of these variables is the minimum possible, and we conjecture that it is in fact minimal for most r and k. We also find new, systematic sources of zeros in the modular matrix S. In addition, we obtain a formula relating the entries of S at fixed points, to entries of S at smaller ranks and levels. Finally, we identify the number fields generated over the rationals by the entries of S, and by the fusion (Verlinde) eigenvalues.

Orbifold analysis of broken bulk symmetries

In two-dimensional conformal field theory, we analyze conformally invariant boundary conditions which break part of the bulk symmetries. When the subalgebra that is preserved by the boundary conditions is the fixed algebra under the action of a finite group G, orbifold techniques can be used to determine the structure of the space of such boundary conditions. We present explicit results for the case when G is abelian. In particular, we construct a classifying algebra which controls these symmetry breaking boundary conditions in the same way as the fusion algebra governs the boundary conditions that preserve the full bulk symmetry.

On finite-dimensional commutative non-hermitian fusion algebras

We characterize the three and four dimensional commutative non-hermitian fusion algebras and construct some new examples of these objects. These algebras arise naturally in the study of graphs, specially those associated with von Neumann algebras. Characterisations of hermitian fusion algebras have been given earlier by Sunder and Wildberger. Commutative finite-dimensional non-herimitian fusion algebras are algebraically isomorphic to certain special Cartan 'subalgebras of matrices. Every Cartan subalgebra of Mn is a conjugate of the standard Cartan algebra by an orthogonal matrix. We characterize the orthogonal matrices that can occur here and thus characterize the four dimensional non-hermitian fusion algebras. The three dimensional ones are parametrized by the hyperbola (x, y) : y2 − x2 = 1 and x, y > 0. By restricting to a special subclass of orthogonal matrices obtained by the above characterization, we construct a family of new commutative finite-dimensional non-hermitian fusion algebras.

Notes on Amenability of Commutative Fusion Algebras

A commutative fusion algebra is proved to be amenable if and only if the associated regular representation is bounded.

Integrable boundaries, conformal boundary conditions and A-D-E fusion rules

The $sl(2)$ minimal theories are labelled by a Lie algebra pair $(A,G)$ where $G$ is of $A$-$D$-$E$ type. For these theories on a cylinder we conjecture a complete set of conformal boundary conditions labelled by the nodes of the tensor product graph $A\otimes G$. The cylinder partition functions are given by fusion rules arising from the graph fusion algebra of $A\otimes G$. We further conjecture that, for each conformal boundary condition, an integrable boundary condition exists as a solution of the boundary Yang-Baxter equation for the associated lattice model. The theory is illustrated using the $(A_4,D_4)$ or 3-state Potts model.

On the classification of bulk and boundary conformal field theories

The classification of rational conformal field theories is reconsidered from the standpoint of boundary conditions. Solving Cardy's equation expressing the consistency condition on a cylinder is equivalent to finding integer valued representations of the fusion algebra. A complete solution not only yields the admissible boundary conditions but also gives valuable information on the bulk properties.

AMENABILITY AND STRONG AMENABILITY FOR FUSION ALGEBRAS WITH APPLICATIONS TO SUBFACTOR THEORY

The relationship between index theory and random walks on fusion algebras is discussed. Popa's notion of amenability is reformulated as a property of fusion algebras, and several equivalent conditions of amenability are obtained. A ratio limit theorem is proved as a characterization of amenability. A number of conditions, all equivalent to Popa's notion of strong amenability in the case of subfactors, of a pair of a fusion algebra and a probability measure are proposed, and their relationship is studied from the viewpoint of random walks and entropic densities. Fusion algebra homomorphisms and free products of fusion algebras are also discussed.

A classifying algebra for boundary conditions

We introduce a finite-dimensional algebra that controls the possible boundary conditions of a conformal field theory. For theories that are obtained by modding out a Z_2 symmetry (corresponding to a so-called D_odd-type, or half-integer spin simple current, modular invariant), this classifying algebra contains the fusion algebra of the untwisted sector as a subalgebra. Proper treatment of fields in the twisted sector, so-called fixed points, leads to structures that are intriguingly close to the ones implied by modular invariance for conformal field theories on closed orientable surfaces.

On fusion algebra of chiral models

We discuss some algebraic setting of chiral models in terms of the statistical dimensions of their fields. In particular, the conformal dimensions and the central charge of the chiral models are calculated from their braid matrices. Furthermore, at level K = 2, we present the characteristic polynomials of their fusion matrices in a factored form.

Open descendants in conformal field theory

Open descendants extend Conformal Field Theory to unoriented surfaces with boundaries. The construction rests on two types of generalizations of the fusion algebra. The first is needed even in the relatively simple case of diagonal models. It leads to a new tensor that satisfies the fusion algebra, but whose entries are signed integers. The second is needed when dealing with non-diagonal models, where Cardy's ansatz does not apply. It leads to a new tensor with positive integer entries, that satisfies a set of polynomial equations and encodes the classification of the allowed boundary operators.

Modular invariant partition functions in the quantum Hall effect

We study the partition function for the low-energy edge excitations of the incompressible electron fluid. On an annular geometry, these excitations have opposite chiralities on the two edges; thus, the partition function takes the standard form of rational conformal field theories. In particular, it is invariant under modular transformations of the toroidal geometry made by the angular variable and the compact Euclidean time. The Jain series of plateaus have been described by two types of edge theories: the minimal models of the W 1+∞ algebra of quantum area-preserving diffeomorphisms, and their non-minimal version, the theories with ▪ affine algebra. We find modular invariant partition functions for the latter models. Moreover, we relate the Wen topological order to the modular transformations and the Verlinde fusion algebra. We find new, non-diagonal modular invariants which describe edge theories with extended symmetry algebra; their Hall conductivities match the experimental values beyond the Jain series.

On fusion algebras associated to finite group actions

In our previous paper (see Kosaki and Yamagami), four kinds of bimodules naturally attached to crossed products P o G ⊇ P o H determined by a group-subgroup pair G ⊇ H were identified with certain vector bundles equipped with group actions. In the present paper we will describe the structure of the fusion algebra of vector bundles and clarify a relationship to fusion algebras appearing in other contexts. Some applications to automorphism analysis for subfactors will be also given.

WZW Fusion Rings in the Limit of Infinite Level

We show that the WZW fusion rings at finite levels form a projective system with respect to the partial ordering provided by divisibility of the height, i.e. the level shifted by a constant. From this projective system we obtain WZW fusion rings in the limit of infinite level. This projective limit constitutes a mathematically well-defined prescription for the `classical limit' of WZW theories which replaces the naive idea of `sending the level to infinity'. The projective limit can be endowed with a natural topology, which plays an important role for studying its structure. The representation theory of the limit can be worked out by considering the associated fusion algebra; this way we obtain in particular an analogue of the Verlinde formula.

Open Descendants in Conformal Field Theory

Open descendants extend Conformal Field Theory to unoriented surfaces with boundaries. The construction rests on two types of generalizations of the fusion algebra. The first is needed even in the relatively simple case of diagonal models. It leads to a new tensor that satisfies the fusion algebra, but whose entries are signed integers. The second is needed when dealing with non-diagonal models, where Cardy's ansatz does not apply. It leads to a new tensor with positive integer entries, that satisfies a set of polynomial equations and encodes the classification of the allowed boundary operators.

N=1 SUPERCONFORMAL MINIMAL MODEL CORRELATION FUNCTIONS ON THE TORUS

The Coulomb gas formalism is employed to construct contour integral representations of two-point correlation functions on the torus for the N=1 superconformal unitary discrete series, characterized by the single integer p. (For the particular case of the tricritical Ising model, these include the energy and vacancy density operators.) Modular and monodromy properties of the superconformal blocks are examined, and the generalization to superconformal theories of Verlinde’s results on modular transformations and the fusion algebra are discussed in some detail. For p odd the relevant modular matrix is (with respect to a particular basis) symmetric and unitary, as in ordinary rational conformal theory. However, for p even, there appears to be an obstruction due to the Ramond vacuum state.

Classification of Paragroup Actions on Subfactors

We define “a crossed product by a paragroup action on a subfactor” as a certain commuting square of type \mathrm{II}_1 factors and give their complete classification in a strongly amenable case (in the sense of S. Popa) in terms of a new combinatorial object which generalizes Ocneanu's paragroup. As applications, we show that the subfactor N ⊂ M of Goodman–de la Harpe–Jones with index 3 + \sqrt 3 is not conjugate to its dual M ⊂ M_1 by showing the fusion algebras of N - N bimodules and M - M bimodules are different, although the principal graph and the dual principal graph are the same. We also make an analogue of the coset construction in RCFT for subfactors in our settings.

On the classification of modular fusion algebras

We introduce the notion of (nondegenerate) strong-modular fusion algebras. Here strongly-modular means that the fusion algebra is induced via Verlinde's formula by a representation of the modular group SL(2,Z) whose kernel contains a congruence subgroup. Furthermore, nondegenerate means that the conformal dimensions of possibly underlying rational conformal field theories do not differ by integers. Our main result is the classification of all strongly-modular fusion algebras of dimension two, three and four and the classification of all nondegenerate strongly-modular fusion algebras of dimension less than 24. We use the classification of the irreducible representations of the finite groups SL(2,Z_{p^l}) where p is a prime and l a positive integer. Finally, we give polynomial realizations and fusion graphs for all simple nondegenerate strongly-modular fusion algebras of dimension less than 24.

Relevant perturbations of the SU (1, 1) /U (1) coset

It is shown that the space of cohomology classes of the SU(1,1) /U(1) coset at negative level k contains states of relevant conformal dimensions. These states correspond to the energy density operator of the associated nonlinear sigma model. We exhibit that there exists a subclass of relevant operators forming a closed fusion algebra. We make use of these operators to perform renormalizable perturbations of the SU(1,1)/U(1) coset. In the infra-red limit, the perturbed theory flows to another conformal model. We identify one of the perturbative conformal points with the SU(2)/U(1) coset at positive level. From the point of view of the string target space geometry, the given renormalization group flow maps the noncompact geometry described by the SU(1,1)/U(1) coset into the sphere described by the SU(2)/U(1) coset. This exhibits a new mechanism of topology change in the space of string compactifications.