Singular vectors and topological theories from Virasoro constraints via the Kontsevich-Miwa transform

Abstract

We use the Kontsevich-Miwa transform to relate the different pictures describing matter coupled to topological gravity in two dimensions: topological theories, Virasoro constraints on integrable hierarchies, and a DDK-type formalism. With the help of the Kontsevich-Miwa transform, we solve the Virasoro constraints on the KP hierarchy in terms of minimal 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 state decoupling equations. The N = 2 generators are constructed out of matter, the “Liouville” scalar, and c = −2 ghosts. By a “dual” construction involving the reparametrization c = −26 ghosts, the DDK dressing prescription is reproduced from the N = 2 symmetry. As a by-product we thus observe that there are two ways to dress arbitrary d ⩽ 1 ∪ d ⩾ 25 matter theory, which allow its embedding into a topological theory. By the Kontsevich-Miwa transform, which introduces an infinite set of “time” variables t r , the equations ensuring the vanishing of correlators that involve BRST-exact primary states, factorize through the Virasoro generators expressed in terms of the t r . The background charge of these Virasoro generators is determined in terms of the topological central charge c ≠ 3 as Q = √(3 − c)/3−2√3/(3 − c).