Fusion potentials. I

Francesco, P. Di; Zuber, Jean-Bernard

doi:10.1088/0305-4470/26/6/025

Cited by 14 publications

(35 citation statements)

References 15 publications

(15 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For example, the fusion-rank of A 1,k and A 2,k equals 1 for all k, with {w 1 } a fusion-generator. This result for A 2 was first obtained in [6], though by a more complicated argument. We also obtain, from Thm.…”

Section: Fusion-rank Of a Rksupporting

confidence: 56%

See 1 more Smart Citation

On Fusion Algebras and Modular Matrices

Gannon¹,

Walton²

1999

Communications in Mathematical Physics

View full text Add to dashboard Cite

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 $A_r$ 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.Comment: 28 pages, plain Te

show abstract

Section: Fusion-rank Of a Rksupporting

confidence: 56%

“…Question 1: For a given g and k, what is the fusion-rank R k (g), and what is a fusion-basis? This problem was studied by Di Francesco and Zuber [6]. For the applications it should suffice to get a reasonable upper bound for the fusion-rank, and to find a Γ which realizes that bound.…”

Section: Introductionmentioning

confidence: 99%

On Fusion Algebras and Modular Matrices

Gannon¹,

Walton²

1999

Communications in Mathematical Physics

View full text Add to dashboard Cite

show abstract

“…The fusion algebra of the minimal model M(p, q) is polynomially generated by two generators X and Y , which one can associate with the representatives of N (2,1) and N (1,2) [12]. The other elements of the algebra are explicitly given by Tchebychev polynomials 7) and the generators must satisfy three relations:…”

Section: )mentioning

confidence: 99%

Symmetric boundary conditions in boundary critical phenomena

Ruelle

1999

J. Phys. A: Math. Gen.

View full text Add to dashboard Cite

Conformally invariant boundary conditions for minimal models on a cylinder are classified by pairs of Lie algebras (A, G) of ADE type. For each model, we consider the action of its (discrete) symmetry group on the boundary conditions. We find that the invariant ones correspond to the nodes in the product graph A × G that are fixed by some automorphism. We proceed to determine the charges of the fields in the various Hilbert spaces, but, in a general minimal model, many consistent solutions occur. In the unitary models (A, A), we show that there is a unique solution with the property that the ground state in each sector of boundary conditions is invariant under the symmetry group. In contrast, a solution with this property does not exist in the unitary models of the series (A, D) and (A, E6). A possible interpretation of this fact is that a certain (large) number of invariant boundary conditions have unphysical (negative) classical boundary Boltzmann weights. We give a tentative characterization of the problematic boundary conditions.

show abstract

“…The matrix N f , on the other hand, satisfies its characteristic equation P (x) = 0, that is also its minimal equation [22].The constraint on N f is thus P (N f ) = 0 that can be integrated to yield a "potential ".…”

Section: The Characteristic Polynomialsmentioning

confidence: 99%