Classification of structure constants for W-algebras from highest weights

Abstract

We show that the structure constants of W-algebras can be grouped according to the lowest (bosonic) spin(s) of the algebra. The structure constants in each group are described by a unique formula, depending on a functional parameter h(c) that is characteristic for each algebra. As examples we give the structure constants C 4 33 and C 4 44 for the algebras of type W(2, 3, 4, . . .) (that include the WA n−1 -algebras) and the structure constant C 4 44 for the algebras of type W(2, 4, . . .), especially for all the algebras WD n , WB(0, n), WB n and WC n . It also includes the bosonic projection of the super-Virasoro algebra and a yet unexplained algebra of type W(2, 4, 6) found previously.

The OPEs of spin-4 Casimir currents in the holographic SO ( N ) coset minimal models

“…also have noticed the presence of a previous work by Hornfeck [11] independently before their paper came out.…”

“…We also analyze the similar OPE in the W B N−1 2 coset minimal model. As described in [11], the findings over there did not answer for what kinds of the field contents for W D 4 (or W B 3 ) algebra are present. In this paper, one sees the field contents for the higher spin currents, explicitly, living in the specific coset model we are considering.…”

“…During this computation, the spin-6 Casimir current arises on the right hand side of this OPE and the two unknown coefficients in the spin-4 current are fixed. We would like to generalize the above OPE for the general N by using two N-generalization coupling constants [11]. We also analyze the similar OPE in the W B N−1 2 coset minimal model.…”

“…Furthermore, we 1 In other words, the direct construction using the Jacobi identity (without knowing the realization of the higher spin currents) is not connected to the Casimir (coset) constuction directly. This implies that even though one has the self-coupling constant for the spin-4 current from the work of [11,12], the coset construction itself (a realization in the specific coset model) is interesting in order to identify a complete set of generating currents which will have larger symmetry. Once we determine the lower higher spin currents using the coset construction explicitly, then the next undetermined higher spin currents can be generated, in principle, by computing the OPEs between the known higher spin currents repeatedly (i.e., by analyzing the various singular terms).…”

W-algebras and their representations

“…(3) Several times during these lectures I have mentioned that there are a large number of possible identifications between W-algebras which occur for specific c values. The most systematic investigation of these identifications have been carried out by Blumenhagen et al and by Hornfeck in [8,9,[39][40][41]. During this work they also uncovered a large class of W-algebras which are not freely generated, i.e.…”