Duality of 2D Gravity as a Local Fourier Duality

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 us to view the p–q duality as a consequence of the duality of the relevant Virasoro constraints.