Unification of all string models with c<1

Abstract

A 1-matrix model is proposed, which nicely interpolates between doublescaling continuum limits of all multimatrix models. The interpolating partition function is always a KP τ -function and always obeys L −1 -constraint and string equation. Therefore this model can be considered as a natural unification of all models of 2d-gravity (string models) with c ≤ 1.

BGWM as second constituent of complex matrix model

“…Not surprisingly, such representations are not unique, and one can instead represent the same solutions in a very different integral form: of Kontsevich-Penner integrals over n × n Hermitian matrices with a peculiar Penner term N tr log φ in the action. This puts all the four models in the unifying context of GKM theory [23]. Direct relation between the two integral representations is provided by a version of Faddeev-Popov trick from [21].…”

“…Unfortunately, even the simplest of these τ -functions, associated with Hermitian [3] and Kontsevich [22][23][24][25] matrix models, are not yet systematically studied/tabulated and still can not be included into the special-functions textbookssee [26] for the first attempts in this direction. It is very important to realize that the world of such τ -functions is cognizable, and is, perhaps, actually finitely-generated: many (all?)…”

“…Unitary correlators play a crucially important role in description of gluons in lattice gauge-theory models [34], however, unitary matrix models are more complicated than Hermitian ones, they are in intermediate position between eigenvalue and non-eigenvalue models and remain under-investigated, see [35]- [38] for some crucial references. A modern matrix-model-theory approach to BGWM and its embedding into the set of generalized Kontsevich models (GKM) [23] was outlined in [20], but has not been developed any further since then. Hopefully reappearance of this model in the context of matrix-model M-theory will help to attract new attention to this unjustly-abandoned subject.…”

“…In order to check that (1.17) satisfies (3) one begins with the Ward identity for this integral [23], associated to the shift h → h + ǫ of the integration variable by a small arbitrary matrix ǫ:…”

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…”