New Realisations of Minimal Models and the Structure of W-Strings

The quantization of a free boson whose momentum satisfies a cubic constraint leads to a $c=\ha$ conformal field theory with a BRST symmetry. The theory also has a $W_\infty$ symmetry in which all the generators except the stress-tensor are BRST-exact and so topological. The BRST cohomology includes states of conformal dimensions $0,\si,\ha$, together with \lq copies' of these states obtained by acting with picture-changing and screening operators. The 3-point and 4-point correlation functions agree with those of the Ising model, suggesting that the theory is equivalent to the critical Ising model. At tree level, the $W_3$ string can be viewed as an ordinary $c=26$ string whose conformal matter sector includes this realisation of the Ising model. The two-boson $W_3$ string is equivalent to the Ising model coupled to two-dimensional quantum gravity. Similar results apply for other W-strings and minimal models.