Towards unified theory of $2d$ gravity

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 ${\mathcal{L}}_{-1}$-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 ${\mathcal{W}}_K$-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.

67 pages (October 1991)

References in corpus (2)

• Unification of All String Models with $c<1$
• On Equivalence of Topological and Quantum 2d Gravity

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• Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions
• Generalized Kontsevich Model Versus Toda Hierarchy and Discrete Matrix Models
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• Replica Limit of the Toda Lattice Equation
• Integrability of Hurwitz Partition Functions. I. Summary
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• The matrix model for dessins d'enfants
• Towards a proof of AGT conjecture by methods of matrix models
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• An action variable of the sine-Gordon model
• Brezin-Gross-Witten model as "pure gauge" limit of Selberg integrals
• Unitary And Hermitian Matrices In An External Field II: The Kontsevich Model And Continuum Virasoro Constraints
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• Quantum and Classical Aspects of Deformed $c=1$ Strings.
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