Related papers: Discrete Painleve equation, Miwa variables, and st…
A recently formulated conjecture of Gamayun, Iorgov and Lisovyy gives an asymptotic expansion of the Jimbo--Miwa--Ueno isomonodromic $\tau$-function for certain Painlev\'e transcendents. The coefficients in this expansion are given in terms…
Conformal blocks and their AGT relations to LMNS integrals and Nekrasov functions are best described by "conformal" (or Dotsenko-Fateev) matrix models, but in non-Gaussian Dijkgraaf-Vafa phases, where different eigenvalues are integrated…
The Hirota-Miwa equation (also known as the discrete KP equation, or the octahedron recurrence) is a bilinear partial difference equation in three independent variables. It is integrable in the sense that it arises as the compatibility…
A direct relation between the conformal formalism for 2d-quantum gravity and the W-constrained KP hierarchy is found, without the need to invoke intermediate matrix model technology. The Kontsevich-Miwa transform of the KP hierarchy is used…
We use the Kontsevich--Miwa transform to relate the Virasoro constraints on integrable hierarchies with the David-Distler-Kawai formalism of gravity-coupled conformal theories. The derivation relies on evaluating the energy-momentum tensor…
We attempt a direct derivation of a conformal field theory description of 2D quantum gravity~+~matter from the formalism of integrable hierarchies subjected to Virasoro constraints. The construction is based on a generalization of the…
Even though matrix model partition functions do not exhaust the entire set of tau-functions relevant for string theory, they seem to be elementary building blocks for many others and they seem to properly capture the fundamental symplicial…
Matrix integrals used in random matrix theory for the study of eigenvalues of Hermitian ensembles have been shown to provide $\tau$-functions for several hierarchies of integrable equations. In this article, we extend this relation by…
In this paper, we study nonlinear integrable equations with three independent variables of the following types: Toda-type lattices, semi-discrete lattices, and fully discrete Hirota-Miwa type models. It is shown that integrable equations of…
In [1], Nakatsu and Takasaki have shown that the melting crystal model behind the topological strings vertex provides a tau-function of the KP hierarchy after an appropriate time deformation. We revisit their derivation with a focus on the…
Painlev\`e equation for conformal blocks is a combined corollary of integrability and Ward identities, which can be explicitly revealed in the matrix model realization of AGT relations. We demonstrate this in some detail, both for…
We discuss specific hypergeometric solutions of the conformal Ward identities (CWI's) of scalar 4-point functions of primary fields in momentum space, in $d$ spacetime dimensions. We determine such solutions using various dual conformal…
Multivariate orthogonal polynomials in $D$ real dimensions are considered from the perspective of the Cholesky factorization of a moment matrix. The approach allows for the construction of corresponding multivariate orthogonal polynomials,…
We introduce a collection of nonlinear integrable partial differential-difference equations that are satisfied by the one-point distribution functions of some classical integrable KPZ models. Moreover, these equations can be regarded as…
We investigate self-similar solutions of the extended discrete KP hierarchy. It is shown that corresponding ansatzes lead to purely discrete equations with dependence on some number of parameters together with equations governing…
Important illustration to the principle ``partition functions in string theory are $\tau$-functions of integrable equations'' is the fact that the (dual) partition functions of $4d$ $\mathcal{N}=2$ gauge theories solve Painlev\'e equations.…
Hirota's discrete KdV equation is an integrable partial difference equation on $\mathbb{Z}^2$, which approaches the Korteweg-de Vries (KdV) equation in a continuum limit. In this paper, we show that its multiplicative-discrete versions have…
The Cholesky factorization of the moment matrix is applied to discrete orthogonal polynomials on the homogeneous lattice. In particular, semiclassical discrete orthogonal polynomials, which are built in terms of a discrete Pearson equation,…
We analyze a variable coefficient coupled HI mKdV system that has shifted nonlocal reductions. The Weiss Tabor Carnevale test gives us coefficient restrictions to perform a time reparametrization to achieve an autonomous integrable model.…
We propose that the grand canonical topological string partition functions satisfy finite-difference equations in the closed string moduli. In the case of genus one mirror curve these are conjectured to be the q-difference Painlev\'e…