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 τ -function of KPhierarchy, subjected to a kind of 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 τ -function of K-reduced KP-hierarchy, obeying a set of W K -algebra constraints identical to those conjectured in [1] for double-scaling continuum limit of (K − 1)-matrix model. In the case of K = 2 the statement reduces to the early established [2] 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.

Lattice topological field theory in two dimensions

“…However, it will then be interesting to incorporate our lattice model in the Kontsevich model [14], and to investigate whether the resulting model is equivalent to the so-called generalized Kontsevich model given in ref. [15] (see also [16]). Investigations along these two lines would be interesting.…”

Exact Solution of Quantum Field Theory on Noncommutative Phase Spaces

“…These equations coincide with the Virasoro constraints of the KP hierarchy [46]. However, while all of this provides a nice characterization of the exact solvability of the underlying matrix model, it is not very convenient for dealing explicitly with its large N limit, which is what is needed for the precise connection with the original noncommutative field theory.…”

Brezin-Gross-Witten model as “pure gauge” limit of Selberg integrals

“…However, for the purpose of present paper this definition is not enough. The case when the integrand f (U ) depends only on the eigenvalues, does not cover all the eigenvalue models [76][77][78][79][113][114][115][116]. In particular, the main object of the present paper, which we use to describe the PGL of the Selberg integrals, has an integrand which is not a function of the eigenvalues of U only, it involves an "external field" matrix Ψ in the following way…”