Fusion Algebra Research Articles

Integrable and conformal twisted boundary conditions forsl(2)A-D-Elattice models

We study integrable realizations of conformal twisted boundary conditions for sl(2) unitary minimal models on a torus. These conformal field theories are realized as the continuum scaling limit of critical G = A,D,E lattice models with positive spectral parameter u > 0 and Coxeter number g. Integrable seams are constructed by fusing blocks of elementary local face weights. The usual A-type fusions are labelled by the Kac labels (r, s) and are associated with the Verlinde fusion algebra. We introduce a new type of fusion in the two braid limits u->\pm i\infty associated with the graph fusion algebra, and labelled by nodes $a,b\in G$ respectively. When combined with automorphisms, they lead to general integrable seams labelled by $x=(r,a,b,\kappa) \in (A_{g-2}, H, H, Z_2 )$ where H is the graph G itself for Type I theories and its parent for Type II theories. Identifying our construction labels with the conformal labels of Petkova and Zuber, we find that the integrable seams are in one-to-one correspondence with the conformal seams. The distinct seams are thus associated with the nodes of the Ocneanu quantum graph. The quantum symmetries and twisted partition functions are checked numerically for $|G| \leq 6$. We also show, in the case of $D_{2l}$, that the non-commutativity of the Ocneanu algebra of seams arises because the automorphisms do not commute with the fusions.

HKR-type invariants of 4–thickenings of 2–dimensional CW complexes

The HKR (Hennings-Kauffman-Radford) framework is used to construct invariants of 4-thickenings of 2-dimensional CW complexes under 2-deformations (1- and 2- handle slides and creations and cancellations of 1-2 handle pairs). The input of the invariant is a finite dimensional unimodular ribbon Hopf algebra A and an element in a quotient of its center, which determines a trace function on A. We study the subset T^4 of trace elements which define invariants of 4-thickenings under 2-deformations. In T^4 two subsets are identified : T^3, which produces invariants of 4-thickenings normalizable to invariants of the boundary, and T^2, which produces invariants of 4-thickenings depending only on the 2-dimensional spine and the second Whitney number of the 4-thickening. The case of the quantum sl(2) is studied in details. We conjecture that sl(2) leads to four HKR-type invariants and describe the corresponding trace elements. Moreover, the fusion algebra of the semisimple quotient of the category of representations of the quantum sl(2) is identified as a subalgebra of a quotient of its center.

Orientation matters for NIMreps

The problem of finding boundary states in CFT, often rephrased in terms of “NIMreps” of the fusion algebra, has a natural extension to CFT on non-orientable surfaces. This provides extra information that turns out to be quite useful to give the proper interpretation to a NIMrep. We illustrate this with several examples. This includes a rather detailed discussion of the interesting case of the simple current extension of A 2 level 9, which is already known to have a rich structure. This structure can be disentangled completely using orientation information. In particular we find here and in other cases examples of diagonal modular invariants that do not admit a NIMrep, suggesting that there does not exist a corresponding CFT. We obtain the complete set of NIMreps (plus Moebius and Klein bottle coefficients) for many exceptional modular invariants of WZW models, and find an explanation for the occurrence of more than one NIMrep in certain cases. We also (re)consider the underlying formalism, emphasizing the distinction between oriented and unoriented string annulus amplitudes, and the origin of orientation-dependent degeneracy matrices in the latter.

A family of isomorphic fusion algebras of twisted quantum doubles of finite groups

Let D ω ( G ) be the twisted quantum double of a finite group, G , where ω∈Z 3 (G, C ∗ ) . For each n∈ N , there exists an ω such that D ( G ) and D ω ( E ) have isomorphic fusion algebras, where G is an extraspecial 2-group with 2 2 n +1 elements, and E is an elementary abelian group with | E |=| G |.

Novel construction of boundary states in coset conformal field theories

We develop a systematic method to solve the Cardy condition for the coset conformal field theory G/ H. The problem is equivalent to finding a non-negative integer valued matrix representation (NIM-rep) of the fusion algebra. Based on the relation of the G/ H theory with the tensor product theory G× H, we give a map from NIM-reps of G× H to those of G/ H. Our map provides a large class of NIM-reps in coset theories. In particular, we give some examples of NIM-reps not factorizable into the G and the H sectors. The action of the simple currents on NIM-reps plays an essential role in our construction. As an illustration of our procedure, we consider the diagonal coset SU(2) 5× SU(2) 3/ SU(2) 8 to obtain a new NIM-rep based on the conformal embedding su(2) 3⊕ su(2) 8⊂ sp(6) 1.

Discrete series of fusion algebras

AbstractWe show that the left regular representation of a countably infinite (discrete) group admits no finite-dimensional invariant subspaces. We also discuss a consequence of this fact, and the reason for our interest in this statement.We then formally state, as a ‘conjecture’, a possible generalisation of the above statement to the context of fusion algebras. We prove the validity of this conjecture in the case of the fusion algebra arising from the dual of a compact Lie group.We finally show, by example, that our conjecture is false as stated, and raise the question of whether there is a ‘good’ class of fusion algebras, which contains (a) the two ‘good classes’ discussed above, namely, discrete groups and compact group duals, and (b) only contains fusion algebras for which the conjecture is valid.

Characters of sl(4)k fusion algebra at the non-rational level

We construct the fusion ring of a quasi-rational $\hat{sl}(4)_k$ WZNW theory at generic level $k \not\in Q$. It is generated by commutative elements in the group ring $Z[\tilde{W}]$ of the affine Weyl group $\tilde{W}$ which extend polynomially the formal characters of finite dimensional representations of $sl(4)$.

On the mass of the neutrinos

The note makes a proposal for a theoretical formula for the mass of three known kinds of neutrinos using fusion algebra and E (∞) theory.

On a class of general theories for high energy particle physics

The present paper gives a comparative study of a class of general theories for high energy particle physics with special attention paid to four-dimensional fusion algebra, E (∞) theory, non-commutative geometry and related subjects. It is further shown that E (∞) may be used to generate all fundamental constants of the standard model and beyond.

On the exact mass spectrum of quarks

The present work proposes a general methodology for obtaining the exact E (∞) mass spectrum of current and constituent quarks. The theory is based upon a physical interpretation of four-dimensional fusion algebra, noncommutative geometry and related theories. All the so-obtained results were found to be in remarkable agreement with the generally accepted experimental and theoretical results found in the literature.

The exact mass of the electron via the transfinite way

Based on four-dimensional fusion algebra and the transfinite E (∞) theory of unification, the present note suggests an exact theoretical solution to the problem of determining the mass of the electron which is found to be equal to the second root of the inverse coupling constant of supersymmetric quantum gravity divided by 10. In other words m e=( α ̄ gs )/10= 26.18033989 =0.51166273 MeV, whereas the experimentally found value is m e ≃0.511 MeV.

Notes on the Quantum Tetrahedron

This is a set of notes describing several aspects of the space of paths on ADE Dynkin diagrams, with a particular attention paid to the graph E6. Many results originally due to A. Ocneanu are here described in a very elementary way (manipulation of square or rectangular matrices). We define the concept of essential matrices for a graph and describe their module properties with respect to right and left actions of fusion algebras. In the case of the graph E6, essential matrices build up a right module with respect to its fusion algebra but a left module with respect to the fusion algebra of A11. We present two original results: 1) We show how to recover the Ocneanu graph of quantum symmetries of the Dynkin diagram E6 from the natural multiplication defined in the tensor square of its fusion algebra (the tensor product should be taken over a particular subalgebra); this is the Cayley graph for the two generators of the twelve dimensional algebra E6 ⊗A3 E6 (here A3 and E6 refer to the commutative fusion algebras of the corresponding graphs). 2) One already knows how to associate, with every point of the graph of quantum symmetries, a particular matrix describing the “ torus structure” of the chosen Dynkin diagram (Ocneanu construction). In the case of E6, one obtains in this way twelve such matrices of dimension 11 ×11; one of them is a modular invariant and encodes the partition function of the corresponding conformal field theory. We introduce a very simple algorithm that allows one to compute these matrices.

Twisted partition functions for ADE boundary conformal field theories and Ocneanu algebras of quantum symmetries

For every ADE Dynkin diagram, we give a realization, in terms of usual fusion algebras (graph algebras), of the algebra of quantum symmetries described by the associated Ocneanu graph. We give explicitly, in each case, the list of the corresponding twisted partition functions.

Wild topology, hyperbolic geometry and fusion algebra of high energy particle physics

The relation between Wild Topology, Hyperbolic Geometry and Fusion Algebra on the one side and the charge and coupling constants of the standard model and quantum gravity on the other is examined. The close connection found between E (∞) theory and the Topological theory of four manifolds as well as the theory of fundamental groups is elucidated using various classical theories and recent results due to Antoine, Wada, Alexander, Klein, Kummer, Freedman, Kaufmann, Witten, Jones and Connes.

The many faces of Ocneanu cells

We define generalised chiral vertex operators covariant under the Ocneanu ``double triangle algebra'' {\cal A}, a novel quantum symmetry intrinsic to a given rational 2-d conformal field theory. This provides a chiral approach, which, unlike the conventional one, makes explicit various algebraic structures encountered previously in the study of these theories and of the associated critical lattice models, and thus allows their unified treatment. The triangular Ocneanu cells, the 3j-symbols of the weak Hopf algebra {\cal A}, reappear in several guises. With {\cal A} and its dual algebra {hat A} one associates a pair of graphs, G and {\tilde G}. While G are known to encode complete sets of conformal boundary states, the Ocneanu graphs {\tilde G} classify twisted torus partition functions. The fusion algebra of the twist operators provides the data determining {\hat A}. The study of bulk field correlators in the presence of twists reveals that the Ocneanu graph quantum symmetry gives also an information on the field operator algebra.

FUSION RULES FOR THE HAMMING CODE VERTEX OPERATOR ALGEBRA

In this paper, we compute the fusion rules for the Hamming code vertex operator algebra and determine its fusion algebra. Our method is based on the fact that the Hamming code VOA is isomorphic to a fixed point space of the lattice VOA V D 4 .

Fusion algebras for N = 1 superconformal field theories through coinvariants I: (1|2)-symmetry

Fusion algebras for N = 1 superconformal field theories through coinvariants I: (1|2)-symmetry

Realization of amenable bicategories and weak amenability of fusion algebras

In this paper, we consider the realization problem of amenable bicategories by using AFD-bimodules. As a corollary, we can realize an amenable standard lattice as that of an AFD II1-subfactor. This gives another proof of S. Popa's theorem. Weak amenability of fusion algebras is also discussed.

Harmonic Function Spaces of Probability Measures on Fusion Algebras

In this note, we study harmonic function spaces of probability measures on fusion algebras of C *-tensor categories and show that almost ergodicity is equivalent to ergodicity for amenable fusion algebras, which generalizes a result due to S. Popa for probability measures associated to subfactors.

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 generalization 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.