Related papers: Affine Solitons: A Relation Between Tau Functions,…
Two methods of constructing 2D Toda $\tau$-functions that are generating functions for certain geometrical invariants of a combinatorial nature are related. The first involves generation of paths in the Cayley graph of the symmetric group…
An elementary introduction into the Seiberg-Witten theory is given. Many efforts are made to get it as pedagogical as possible, within a reasonable size. The selection of the relevant material is heavily oriented towards graduate students.…
In this paper we develop a general method for constructing 3-point functions in conformal field theory with affine Lie group symmetry, continuing our recent work on 2-point functions. The results are provided in terms of triangular…
Duality in the integrable systems arising in the context of Seiberg-Witten theory shows that their tau-functions indeed can be seen as generating functions for the mutually Poisson-commuting hamiltonians of the {\em dual} systems. We…
We study the soliton-type solutions of the system introduced by B. Feigin and the author in in [EF]. We show that it reduces to a top-like system, and we study the behaviour of the solutions at the lattice infinity. We compute the…
The Hamiltonian methods of the theory of solitons are applied to gravisolitons in the dressing formalism. The Poisson bracket for the Lie-algebra valued one-form $A(\varsigma, \eta, \gamma)=\Psi_{,\gamma}\Psi^{-1}$, for gravisolitons in the…
This paper addresses the issue of integrable structure in a modified melting crystal model of topological string theory on the resolved conifold. The partition function can be expressed as the vacuum expectation value of an operator on the…
The soliton solutions of the Camassa-Holm equation are derived by the implementation of the dressing method. The form of the one and two soliton solutions coincides with the form obtained by other methods.
We define a function by means of the minimum weight flow on a planar graph and prove that this function solves the ultradiscrete Toda molecule equation, its B\"acklund transformation and the two dimensional Toda molecule equation. The…
This paper is a brief review of recent results on the concept of ``generalized $\tau$-function'', defined as a generating function of all the matrix elements in a given highest-weight representation of a universal enveloping algebra ${\cal…
We quantise the reduced theory obtained by substituting the soliton solutions of affine Toda theory into its symplectic form. The semi-classical S-matrix is found to involve the classical Euler dilogarithm.
This paper is devoted to revealing the relationship between matrix-valued $\theta$-deformed bi-orthogonal polynomials and non-commutative Toda-type hierarchies. In this procedure, Wronski quasi-determinants are widely used and play the role…
We consider correlation functions of topologically twisted, $\mathcal{N}=2$ supersymmetric Yang-Mills theory with gauge group ${\rm SU}(2)$ and $N_f\leq 3$ massive hypermultiplets in the fundamental representation. For a smooth, compact,…
We construct B\"acklund transformations (BTs) for the Kirchhoff top by taking advantage of the common algebraic Poisson structure between this system and the $sl(2)$ trigonometric Gaudin model. Our BTs are integrable maps providing an exact…
We construct the moduli spaces associated to the solutions of equations of motion (modulo gauge transformations) of the Poisson sigma model with target being an integrable Poisson manifold. The construction can be easily extended to a case…
The space spanned by the characters of twisted affine Lie algebras admit the action of certain congruence subgroups of $SL(2,\mathbb{Z})$. By embedding the characters in the space spanned by theta functions, we study an…
We restrict affine Toda field theory to the half-line by imposing certain boundary conditions at $x=0$. The resulting theory possesses the same spectrum of solitons and breathers as affine Toda theory on the whole line. The classical…
A class of rational solutions of Toda lattice satisfying certain Backlund transformations and a class of mixed rational-soliton solutions (quasisolitons) in wronskian formare obtained using the method of Ablowitz and Satsuma. Also an…
Two sets of super Riccati equations are presented which result in two linear problems of super sine-Gordon equation. The linear problems are then shown to be related to each other by a super gauge transformation and to the super…
We study Nekrasov's instanton partition function of four-dimensional N=2 gauge theories in the presence of surface operators. This can be computed order by order in the instanton expansion by using results available in the mathematical…