We give a direct proof of the relation between vacuum singular vectors and conservation laws for the quantum KdV equation or equivalently for $\Phi_{ (1,3)}$-perturbed conformal field theories. For each degree at which a classical conservation law exists, we find a quantum conserved quantity for a specific value of the central charge. Various generalizations ($N=1,2$ supersymmetric, Boussinesq) of this result are presented.
Categories: hep-th