Towards unified theory of $2d$ gravity

Details Towards unified theory of $2d$ gravity Towards unified theory of $2d$ gravity

1992-01-08

arxiv.org/abs/hep-th/9201013v1

Nucl.Phys.B380:181-240,1992

Physics

High Energy Physics

High Energy Physics - Theory

67 pages (October 1991)

Scientific paper

10.1016/0550-3213(92)90521-C

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.

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