Superconformal algebras from pseudoparticle mechanics

Abstract

We consider a one-dimensional Osp( N|2 M) pseudoparticle mechanical model which may be written as a phase space gauge theory. We show how the pseudoparticle model naturally encodes and explains the two-dimensional zero curvature approach to finding extended conformal symmetries. We describe a procedure of partial gauge fixing of the pseudoparticle model that yields theories in which the Lagrange multiplier gauge fields may be identified with the generators of superconformally extended W-algebras. The residual gauge transformations of the gauge fields give the superconformally extended W-algebra transformations of the generators, while those of the pseudoparticle matter give the transformations of matter under the superconformally extended W-algebra. Furthermore, the pseudoparticle model allows one to derive the finite versions of these generally nonlinear transformations. In particular, the partial gauge fixing of the Osp( N|2) pseudoparticle mechanical model allows one to obtain the SO( N) invariant N-extended superconformal symmetry algebras of Bershadsky and Knizhnik. These algebras are nonlinear for N ⩾ 3. We discuss in detail the cases of N = 1 and N = 2, giving two new derivations of the superschwarzian derivatives. Some comments are made in the N = 2 case on how twisted and topological theories represent a significant deformation of the original particle model. The particle model also allows one to interpret superconformal transformations as deformations of flags in super jet bundles over the associated super Riemann surface.