Linearizing W-algebras

Abstract

We show that the Zamolodchikov's and Polyakov-Bershadsky nonlinear algebras W 3 and W (2) 3 can be embedded as subalgebras into some linear algebras with finite set of currents. Using these linear algebras we find new field realizations of W (2) 3 and W 3 which could be a starting point for constructing new versions of W -string theories. We also reveal a number of hidden relationships between W 3 and W (2) 3 . We conjecture that similar linear algebras can exist for other W -algebras as well.

Note on W3 realizations of the bosonic string

“…Recently Krivonos and Sorin [5] have constructed a linearized version of the W 3 algebra, which is denoted by W lin 3 and is given by…”

“…This is an implementation through similarity transformations of the procedure of eliminating G − , K by redefinitions given in ref. [5].…”

Null fields realizations of W3 from W(sl(4), sl(3)) and W(sl(3|1), sl(3)) algebras

“…A solution to this problem was proposed in [2][3][4]. The idea of this approach is to embed the given nonlinear algebra into a conformal linear one which contains the former nonlinear algebra as a subalgebra.…”

“…In order to complete our task, we need to construct the realization of W 3 modulo null fields explicitly. In the next Section we will show that a straightforward way to construct such realizations comes from the conformal linearization procedure [2][3][4] applied to the algebras under consideration.…”

“…One of the possible ways to construct the realizations of W algebras is the conformal linearization procedure [2][3][4]. The main idea of this approach is to embed nonlinear W algebra as a subalgebra in some linear conformal algebra W lin .…”