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Different group structures which underline the integrable systems are considered. In some cases, the quantization of the integrable system can be provided with substituting groups by their quantum counterparts. However, some other group…
We explicitly construct families of integrable $\sigma$-model actions smoothly interpolating between exact CFTs. In the ultraviolet the theory is the direct product of two current algebras at levels $k_1$ and $k_2$. In the infrared and for…
Integrable lattice equations arising in the context of singular manifold equations for scalar, multicomponent KP hierarchies and 2D Toda lattice hierarchy are considered. These equation generate the corresponding continuous hierarchy of…
We build in this paper the algebra of q-deformed pseudo-differential operators shown to be an essential step towards setting a q-deformed integrability program. In fact, using the results of this q-deformed algebra, we derive the…
We summarize the recent results about complete solvability of Hermitian and rectangular complex matrix models. Partition functions have very simple character expansions with coefficients made from dimensions of representation of the linear…
We study an iso-spectral deformation of general matrix which is a natural generalization of the Toda lattice equation. We prove the integrability of the deformation, and give an explicit formula for the solution to the initial value…
We introduce a combinatorial property for finitely generated groups called stackable that implies the existence of an inductive procedure for constructing van Kampen diagrams with respect to a canonical finite presentation. We also define…
We reformulate the singularity confinement of the discrete Toda equation. We prove the co-primeness property, which has been introduced in our previous paper (arXiv:1311.0060) as one of the integrability criteria, for the discrete Toda…
The exact eigenvalues of the infinite set of conserved charges on the multi-particle states in affine Toda theories are determined. This is done by constructing a free field realization of the Zamolodchikov-Faddeev algebra in which the…
We prove the KP integrability of non-perturbative topological recursion, which can be considered as a formal $\hbar$-deformation of the Krichever construction of algebro-geometric solutions of the KP hierarchy. This property goes back to a…
This paper goes some way in explaining how to construct an integrable hierarchy of flows on the space of conformally immersed tori in n-space. These flows have first occured in mathematical physics -- the Novikov-Veselov and…
We extend the computation of the invariant $\eta(\omega,C,a)$ defined in arXiv:2409.01751 to special points on the line at infinity and show that, as in the affine case, its value is determined purely by the geometry of the integral curve…
Two consistency conditions for partition functions established by Akemann and Dam-gaard in their studies of the fermionic mass dependence of the QCD partition function at low energy ({\it a la} Leutwiller-Smilga-Verbaarschot) are…
In this article we review the Duistermaat-Heckman integration formula and the ensuing equivariant cohomology structure, in the finite dimensional case. In particular, we discuss the connection between equivariant cohomology and classical…
We review some recent results concerning integrable quantum field theories in 1+1 space-time dimensions which contain unstable particles in their spectrum. Recalling first the main features of analytic scattering theories associated to…
We construct the quantum versions of the monodromy matrices of KdV theory. The traces of these quantum monodromy matrices, which will be called as ``${\bf T}$-operators'', act in highest weight Virasoro modules. The ${\bf T}$-operators…
In this article we study the differentiability of Mather's $\beta$-function on closed surfaces and its relation to the integrability of the system.
We consider the deformations of ``monomial solutions'' to Generalized Kontsevich Model \cite{KMMMZ91a,KMMMZ91b} and establish the relation between the flows generated by these deformations with those of $N=2$ Landau-Ginzburg topological…
In Part I, we extend our analysis in [arXiv:0807.1107], and show that a mathematically conjectured geometric Langlands duality for complex surfaces in [1], and its generalizations -- which relate some cohomology of the moduli space of…
In the example of the Schr\"odinger/KdV equation we give elementary treatment of the theory of finite-gap integration. The concept is equivalent to two kinds of Liouvillian integrability: quadrature integrability of linear differential…