Duality of 2D gravity as a local Fourier duality

TL;DR

This paper presents a novel interpretation of the p-q duality in 2D quantum gravity as a local Fourier duality of D-modules, linking it to Virasoro constraints and providing a new geometric perspective.

Contribution

It introduces a new approach to p-q duality by framing it as a local Fourier duality of D-modules, connecting it to irregular connections and Virasoro constraints.

Findings

p-q duality can be realized as a local Fourier duality of D-modules

The duality is linked to a local Fourier duality between irregular connections

Virasoro constraints are related to the duality through Kac-Schwarz operators

Abstract

The p - q duality is a relation between the (p,q) model and the (q,p) model of two-dimensional quantum gravity. Geometrically this duality corresponds to a relation between the two relevant points of the Sato Grassmannian. Kharchev and Marshakov have expressed such a relation in terms of matrix integrals. Some explicit formulas for small p and q have been given in the work of Fukuma-Kawai-Nakayama. Already in the duality between the (2,3) model and the (3,2) model the formulas are long. In this work a new approach to p - q duality is given: It can be realized in a precise sense as a local Fourier duality of D-modules. This result is obtained as a special case of a local Fourier duality between irregular connections associated to Kac-Schwarz operators. Therefore, since these operators correspond to Virasoro constraints, this allows to view the p - q duality as a consequence of the duality of the relevant Virasoro constraints.