Rational Theories of 2D Gravity from the Two-Matrix Model

TL;DR

This paper explicitly establishes the correspondence between multicritical regimes of the two-matrix model and 2D gravity coupled to (p,q) rational matter, deriving minimal potentials, boundary operators, and correlators.

Contribution

It provides a detailed explicit construction of the (p,q) multicritical potentials and correlators, clarifying the relation to 2D gravity and boundary dualities.

Findings

Derived minimal (p,q) multicritical potentials as polynomials of degree p and q.

Showed loop averages satisfy Heisenberg relations and match canonical momenta.

Presented a closed form for two-loop correlators and their scaling limit.

Abstract

The correspondence claimed by M. Douglas, between the multicritical regimes of the two-matrix model and 2D gravity coupled to (p,q) rational matter field, is worked out explicitly. We found the minimal (p,q) multicritical potentials U(X) and V(Y) which are polynomials of degree p and q, correspondingly. The loop averages W(X) and \tilde W(Y) are shown to satisfy the Heisenberg relations {W,X} =1 and {\tilde W,Y}=1 and essentially coincide with the canonical momenta P and Q. The operators X and Y create the two kinds of boundaries in the (p,q) model related by the duality (p,q) - (q,p). Finally, we present a closed expression for the two two-loop correlators and interpret its scaling limit.