Quantum Hamiltonian Reduction and Conformal Field Theories

It is proved that irreducible representation of the Virasoro algebra can be extracted from an irreducible representation space of the SL(2, {cal R}) current algebra by putting a constraint on the latter using the BRST formalism. Thus there is a SL(2, {cal R}) symmetry in the Virasoro algebra which is gauged and hidden. This construction of the Virasoro algebra is the quantum analog of the Hamiltonian reduction. We then naturally lead to consider an SL(2, {cal R}) Wess-Zumino-Witten model. This system is related to the quantum field theory of the coadjoint orbit of the Virasoro group. Based on this result we present the canonical derivation of the SL(2, {cal R}) current algebra in Polyakov's theory of two dimensional gravity; it is manifestation of the SL(2, {cal R}) symmetry in the conformal field theory hidden by the quantum Hamiltonian reduction. We discuss the quantum Hamiltonian reduction of the SL(n, {cal R}) current algebra for the general type of constraints labeled by index 1 <=q l <=q (n - 1) and claim that it leads to the new extended conformal algebras W_sp{n }{l}. For l = 1 we recover the well known W_ n algebra introduced by A. Zamolodchikov. For SL(3, {cal R }) Wess-Zumino-Witten model there are two different possibilities of constraining it. The first possibility gives the W_3 algebra, while the second--leads to the new chiral algebra W_sp {3}{2} generated by the stress -energy tensor, two bosonic supercurrents with spins 3/2 and the U(1) current. We conjecture a Kac formula that describes the highly reducible representation for this algebra. We also make some speculations concerning the structure of W gravity.