Singular Vectors and Topological Theories from Virasoro Constraints via the Kontsevich-Miwa Transform

TL;DR

This paper explores the relationship between topological gravity, Virasoro constraints, and integrable hierarchies using the Kontsevich-Miwa transform, revealing new connections and dressing prescriptions for matter theories in two-dimensional quantum gravity.

Contribution

It introduces a novel application of the Kontsevich-Miwa transform to relate different descriptions of matter coupled to topological gravity, and derives new dressing prescriptions from N=2 symmetry.

Findings

Virasoro constraints solved via minimal models with Liouville dressing.

Two distinct dressing methods for matter theories with different central charges.

Factorization of correlator conditions through Virasoro generators in the transformed variables.

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 $d\leq1$ or $d\geq25$ matter theory, that allow its embedding into a topological theory. By th e 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 by the topological central charge.