Related papers: Tau-functions and Dressing Transformations for Zer…
In this paper, we consider the higher Br\'ezin--Gross--Witten tau-functions, given by the matrix integrals. For these tau-functions we construct the canonical Kac--Schwarz operators, quantum spectral curves, and $W^{(3)}$-constraints. For…
In this paper we generalize the quantization procedure of Toda-mKdV hierarchies to the case of arbitrary affine (super)algebras. The quantum analogue of the monodromy matrix, related to the universal R-matrix with the lower Borel subalgebra…
We consider ramified coverings of P^1 with arbitrary ramification type over 0 and infinity and simple ramifications elsewhere and prove that the generating function for the numbers of such coverings is a tau-function for the Toda lattice…
We consider a generalization of the full symmetric Toda hierarchy where the matrix $\tilde {L}$ of the Lax pair is given by $\tilde {L}=LS$, with a full symmetric matrix $L$ and a nondegenerate diagonal matrix $S$. The key feature of the…
The generating function of cubic Hodge integrals satisfying the local Calabi-Yau condition is conjectured to be a tau function of a new integrable system which can be regarded as a fractional generalization of the Volterra lattice…
Okounkov and Pandharipande proved that the equivariant Toda hierarchy governs the equivariant Gromov-Witten theory of $\mathbb{CP}^1$. A technical clue of their method is a pair of dressing operators on the Fock space of 2D charged free…
We review the role played by tau functions of special type - called {\it Bergman} tau functions in various areas: theory of isomonodromic deformations, solutions of Einstein's equations, theory of Dubrovin-Frobenius manifolds, geometry of…
In this paper, we extend the matrix-resolvent method to the study of the Dubrovin--Zhang type tau-functions for the constrained KP hierarchy and the bigraded Toda hierarchy of $(M,1)$-type. We show that the Dubrovin--Zhang type tau-function…
$Q$-systems and $T$-systems are systems of integrable difference equations that have recently attracted much attention, and have wide applications in representation theory and statistical mechanics. We show that certain $\tau$-functions,…
We construct integrable discrete nonautonomous quad-equations as B\"acklund auto-transformations for known Volterra and Toda type semidiscrete equations, some of which are also nonautonomous. Additional examples of this kind are found by…
We construct a zero curvature formulation, in superspace, for the sTB-B hierarchy which naturally reduces to the zero curvature condition in terms of components, thus solving one of the puzzling features of this model. This analysis,…
An effective method for constructing explicit solutions to the Davey--Stewartson type integrable equations is discussed based on the use of a dressing chain. The application of the method is exemplified by the equation DS I, for which a new…
Dressing technique is used to construct commuting Lax operators which provide an integrable (canonical) structure behind Witten--Dijkgraaf--Verlinde--Verlinde equations. The commuting flows are related to the isomonodromic flows. Examples…
We generalize Harish-Chandra-Itzykson-Zuber and certain other integrals (Gross-Witten integral and integrals over complex matrices) using the notion of tau function of matrix argument. In this case one can reduce the matrix integral to the…
In [1], a generalized type of Darboux transformations defined in terms of a twisted derivation was constructed in a unified form. Such twisted derivations include regular derivations, difference operators, superderivatives and…
It is proved that the action for nonlinear Beltrami equation (quasiclassical dbar-problem) evaluated on its solution gives a tau-function for dispersionless KP hierarchy. Infinitesimal transformations of tau-function corresponding to…
Sato introduced the tau-function to describe solutions to a wide class of completely integrable differential equations. Later Segal-Wilson represented it in terms of the relevant integral operators on Hardy space of the unit disc. This…
We introduce a generalization of $A_{r}$-type Toda theory based on a non-abelian group G, which we call the $(A_{r},G)$-Toda theory, and its affine extensions in terms of gauged Wess-Zumino-Witten actions with deformation terms. In…
We introduce the notion of a classical dihedral affine Gaudin model, associated with an untwisted affine Kac-Moody algebra $\widetilde{\mathfrak{g}}$ equipped with an action of the dihedral group $D_{2T}$, $T \geq 1$ through (anti-)linear…
The theory of matrix models is reviewed from the point of view of its relation to integrable hierarchies. Determinantal formulas, relation to conformal field models and the theory of Generalized Kontsevich model are discussed in some…