W-algebras from Chern-Simons theory

We show that the physical state conditions of SL(3, R) Chern-Simons theory are equivalent to the Ward identities of the W(2)3-algebra. More precisely, choosing a specific polarization for quantizing the Chern-Simons theory, we can solve explicitly part of the Chern-Simons physical state conditions, while the remaining ones reduce to the Ward identities satisfied by the conformal blocks of the W(2)3-algebra extended conformal field theory (implying the OPEs). It is an obvious conjecture that a different choice of polarization leads to the Ward identities of W3triple bondW(1)3 and that these relations can be generalized to SL(n,R) Chern-Simons and Wn. (orig.)