Minimal Models from W-Constrained Hierarchies via the Kontsevich-Miwa Transform

TL;DR

This paper establishes a direct link between 2d-quantum gravity conformal formalism and W-constrained KP hierarchy using the Kontsevich-Miwa transform, connecting minimal models with tau functions and W constraints.

Contribution

It provides a novel direct relation between conformal 2d-quantum gravity and W-constrained KP hierarchy without matrix model intermediates, via the Kontsevich-Miwa transform.

Findings

Identifies W constraints with Virasoro null vector decoupling equations.

Maps W^{(l)}-constrained KP hierarchy to minimal models.

Expresses tau functions as correlators of dressed operators.

Abstract

A direct relation between the conformal formalism for 2d-quantum gravity and the W-constrained KP hierarchy is found, without the need to invoke intermediate matrix model technology. The Kontsevich-Miwa transform of the KP hierarchy is used to establish an identification between W constraints on the KP tau function and decoupling equations corresponding to Virasoro null vectors. The Kontsevich-Miwa transform maps the $W^{(l)}$-constrained KP hierarchy to the $(p^\prime,p)$ minimal model, with the tau function being given by the correlator of a product of (dressed) $(l,1)$ (or $(1,l)$) operators, provided the Miwa parameter $n_i$ and the free parameter (an abstract $bc$ spin) present in the constraints are expressed through the ratio $p^\prime/p$ and the level $l$.