Towards unified theory of $2d$ gravity

TL;DR

This paper introduces a new matrix model with arbitrary potential that unifies various 2D quantum gravity models, showing its partition function's integrability and connection to KP hierarchy and ${ m W}_K$-constraints.

Contribution

It develops a generalized matrix model framework that interpolates between different 2D gravity models while maintaining integrability and specific algebraic constraints.

Findings

Partition function is a $ au$-function of KP hierarchy.

Model behaves smoothly as matrix size tends to infinity.

Special case reproduces known 2D quantum gravity relations.

Abstract

We introduce a new 1-matrix model with arbitrary potential and the matrix-valued background field. Its partition function is a $\tau$-function of KP-hierarchy, subjected to a kind of ${\cal L}_{-1}$-constraint. Moreover, partition function behaves smoothly in the limit of infinitely large matrices. If the potential is equal to $X^{K+1}$, this partition function becomes a $\tau$-function of $K$-reduced KP-hierarchy, obeying a set of ${\cal W} _K$-algebra constraints identical to those conjectured in \cite{FKN91} for double-scaling continuum limit of $(K-1)$-matrix model. In the case of $K=2$ the statement reduces to the early established \cite{MMM91b} relation between Kontsevich model and the ordinary $2d$ quantum gravity . Kontsevich model with generic potential may be considered as interpolation between all the models of $2d$ quantum gravity with $c<1$ preserving the property of integrability and the analogue of string equation.