New realisations of minimal models and the structure of W-strings

The quantisation of a free boson whose momentum satisfies a cubic constraint leads to a c = 12 conformal field theory with a BRST symmetry. The theory also has a W∞ 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, 116, 12, together with “copies” of these states obtained by acting with picture-changing and screening operators. The three-point and four-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 W3 string can be viewed as an ordinary c = 26 string whose conformal matter sector includes this realisation of the Ising model. The two-boson W3 string is equivalent to the Ising model coupled to two-dimensional quantum gravity. Similar results apply for other W-strings and minimal models.