Finite and Infinite W Algebras and their Applications

In this paper we present a systematic study of $W$ algebras from the Hamiltonian reduction point of view. The Drinfeld-Sokolov (DS) reduction scheme is generalized to arbitrary $sl_2$ embeddings thus showing that a large class of W algebras can be viewed as reductions of affine Lie algebras. The hierarchies of integrable evolution equations associated to these classical W algebras are constructed as well as the generalized Toda field theories which have them as Noether symmetry algebras. The problem of quantising the DS reductions is solved for arbitrary $sl_2$ embeddings and it is shown that any W algebra can be embedded into an affine Lie algebra. This also provides us with an algorithmic method to write down free field realizations for arbitrary W algebras. Just like affine Lie algebras W algebras have finite underlying structures called `finite W algebras'. We study the classical and quantum theory of these algebras, which play an important role in the theory of ordinary W algebras, in detail as well as some aspects of their representation theory. The symplectic leaves (or W-coadjoint orbits) associated to arbitrary finite W algebras are determined as well as their realization in terms of bosoic oscillators. Apart from these technical aspects we also review the potential applications of W symmetry to string theory, 2-dimensional critical phenomena, the quantum Hall effect and solitary wave phenomena. This work is based on the Ph.D. thesis of the author.