Hamiltonian Reduction and Classical Extended Superconformal Algebras

We present a systematic construction of classical extended superconformal algebras from the hamiltonian reduction of a class of affine Lie superalgebras, which include an even subalgebra sl(2). In particular, we obtain the doubly extended N = 4 superconformal algebra Ãγ from the hamiltonian reduction of the exceptional Lie superalgebra D(2|1; γ/(1− γ)). We also find the Miura transformation for these algebras and give the free field representation. A W -algebraic generalization is discussed. In the last few years it has been known that various higher-spin extension of the Virasoro algebra including Zamolodchikov’s W algebra[1] can be constructed using the Drinfeld-Sokolov type hamiltonian reduction of affine Lie algebras (or the associated integrable systems) [2],[3] and its generalization [4]-[7]. Supersymmetric extension of the Virasoro algebra is important from the viewpoint of application to superstrings and topological field theory. It has been shown that N = 1 and N = 2 super W algebras can be obtained from the hamiltonian reduction (or related super Toda field theories) of a special class of affine Lie superalgebras which have the purely odd simple root system [8]. The purpose of this paper is to present a systematic construction of the classical extended superconformal algebra (ESA) from the hamiltonian reduction viewpoint. ESAs with u(N) and so(N) Kac-Moody symmetries, which have been constructed by Knizhnik and Bershadsky [9], have in general non-linear properties similar to those of the W algebra. Some previous works describe the relation between N = 1 and 2 superconformal algebras and Lie superalgebras osp(N |2) [10]. In particular Mathieu has shown that the hamiltonian structure of the osp(N |2) KdV equations leads to the classical so(N) ESA [11], although in his formulation the Lie superalgebraic structure is not obvious. In the present paper we shall discuss a class of Lie superalgebras which include even subalgebras g⊕sl(2), where g is a semi-simple Lie algebra. Using Kac’s notation [12], such basic classical Lie superalgebras are classified as follows (see Table 1): A(n|1) (n ≥ 1), A(1|0) B(n|1), D(n|1), B(1|n), D(2|1;α), F (4) and G(3). By considering the hamiltonian reduction of an even subalgebra sl(2), we will find ESAs with An ⊕ u(1) (A1 for n = 1), u(1), Bn, Dn, Cn, A1⊕A1, spin(7), G2 Kac-Moody symmetry . In particular forD(2|1;α) the corresponding ESA is shown to be the N = 4 doubly extended superconformal algebra [14]. We will also show that the Miura transformation can be naturally obtained by connecting the Drinfeld-Sokolov gauge and the “diagonal” gauge. The present classical For B(1|n) the corresponding algebra has spin 0, 1 and 2 fermionic currents with Cn Kac-Moody symmetry.