All W3 algebras from SL(3) Chern-Simons theory

The physical state conditions of SL(3, R) Chern-Simons theory are shown to be equivalent to the Ward identities of the W3 algebras. Depending on the polarization we choose for quantizing the Chern-Simons theory, we obtain the Ward identities of the (standard) W3triple bondW3(1) or of the (less standard) W3(2) algebra. Indeed, once a specific polarization has been chosen, we can solve part of the Chern-Simons physical state conditions F(a)Ψ=0 while the remaining ones become the Ward identities satisfied by the conformal blocks of one or the other W3 algebra. It is a conjecture that this pattern generalizes to SL(n, R) Chern-Simons and Wn algebras. (orig.)