SINGULAR VECTORS AND TOPOLOGICAL THEORIES FROM VIRASORO CONSTRAINTS VIA THE KONTSEVICH-MIWA TRANSFORM

We use the Kontsevich-Miwa transform to relate the different pictures describing matter coupled to topological gravity in two dimensions: to pological theories, Virasoro constraints on integrable hierarchies, an d a DDK-type formalism. With the help of the Kontsevich-Miwa transform , we solve the Virasoro constraints on the KP hierarchy in terms of mi nimal models dressed with a (free) Liouville-like scalar. The dressing prescription originates in a topological (twisted N = 2) theory. The Virasoro constraints are thus related to essentially the N = 2 null st ate decoupling equations. The N = 2 generators are constructed out of matter, the ''Liouville'' scalar, and c = -2 ghosts. By a ''dual'' con struction involving the reparametrization c = -26 ghosts, the DDK dres sing prescription is reproduced from the N = 2 symmetry. As a by-produ ct we thus observe that there are two ways to dress arbitrary d less-t han-or-equal-to 1 or d greater-than-or-equal-to 25 matter theory, whic h allow its embedding into a topological theory. By the Kontsevich-Miw a transform, which introduces an infinite set of ''time'' variables t( r), the equations ensuring the vanishing of correlators that involve B RST-exact primary states, factorize through the Virasoro generators ex pressed in terms of the t(r). The background charge of these Virasoro generators is determined in terms of the topological central charge c not-equal 3 as Q = square-root (3 - c)/3 - 2 square-root 3/(3 - c).