Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1993-04-26
Commun.Math.Phys. 162 (1994) 399-432
Physics
High Energy Physics
High Energy Physics - Theory
48 pages, plain TeX, BONN-HE-93-14, DIAS-STP-93-02
Scientific paper
10.1007/BF02102024
We clarify the notion of the DS --- generalized Drinfeld-Sokolov --- reduction approach to classical ${\cal W}$-algebras. We first strengthen an earlier theorem which showed that an $sl(2)$ embedding ${\cal S}\subset {\cal G}$ can be associated to every DS reduction. We then use the fact that a $\W$-algebra must have a quasi-primary basis to derive severe restrictions on the possible reductions corresponding to a given $sl(2)$ embedding. In the known DS reductions found to date, for which the $\W$-algebras are denoted by ${\cal W}_{\cal S}^{\cal G}$-algebras and are called canonical, the quasi-primary basis corresponds to the highest weights of the $sl(2)$. Here we find some examples of noncanonical DS reductions leading to $\W$-algebras which are direct products of ${\cal W}_{\cal S}^{\cal G}$-algebras and `free field' algebras with conformal weights $\Delta \in \{0, {1\over 2}, 1\}$. We also show that if the conformal weights of the generators of a ${\cal W}$-algebra obtained from DS reduction are nonnegative $\Delta \geq 0$ (which is
Feher Laszlo
O'Raifeartaigh Lochlain
Ruelle Philippe
Tsutsui Izumi
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