Related papers: Introduction to classical and quantum integrabilit…
The correspondence between the integrability of classical mechanical systems and their quantum counterparts is not a 1-1, although some close correspondencies exist. If a classical mechanical system is integrable with invariants that are…
We propose a Lax equation for the non-linear sigma model which leads directly to the conserved local charges of the system. We show that the system has two infinite sets of such conserved charges following from the Lax equation, much like…
The main results for the two-dimensional quantum gravity, conjectured from the matrix model or integrable approach, are presented in the form to be compared with the world-sheet or Liouville approach. In spherical limit the integrable side…
R-matrix is explicitly constructed for simplest representations of the Ding-Iohara-Miki algebra. The calculation is straightforward and significantly simpler than the one through the universal R-matrix used for a similar calculation in the…
We review various combinatorial problems with underlying classical or quantum integrable structures. (Plenary talk given at the International Congress of Mathematical Physics, Aalborg, Denmark, August 10, 2012.)
We present a statistical analysis of spectra of transfer matrices of classical lattice spin models; this continues the work on the eight-vertex model of the preceding paper. We show that the statistical properties of these spectra can serve…
We derive an explicit mapping between the spectra of conserved local operators of integrable quantum lattice models and the density distributions of their thermodynamic particle content. This is presented explicitly for the Heisenberg XXZ…
There are 13 equivalence classes of 2D second order quantum and classical superintegrable systems with nontrivial potential, each associated with a quadratic algebra of hidden symmetries. We study the finite and infinite irreducible…
The generalized open XXZ model at $q$ root of unity is considered. We review how associated models, such as the $q$ harmonic oscillator, and the lattice sine-Gordon and Liouville models are obtained. Explicit expressions of the local…
We relate a large class of classical spin models, including the inhomogeneous Ising, Potts, and clock models of q-state spins on arbitrary graphs, to problems in quantum physics. More precisely, we show how to express partition functions as…
The quantum field theory describing the massive O(2) nonlinear sigma-model is investigated through two non-perturbative constructions: The form factor bootstrap based on integrability and the lattice formulation as the XY model. The…
We present an introduction to the canonical quantization of gravity performed in loop quantum gravity, based on lectures held at the 3rd quantum geometry and quantum gravity school in Zakopane in 2011. A special feature of this introduction…
This paper collects and extends the lectures I gave at the "XXIV International Fall Workshop on Geometry and Physics" held in Zaragoza (Spain) August 31 - September 4, 2015. Within these lectures I review the formulation of Quantum…
We introduce and study conformal field theories specified by $W-$algebras commuting with certain set of screening charges. These CFT's possess perturbations which define integrable QFT's. We establish that these QFT's have local and…
We introduce and solvev a special family of integrable interacting vertex models that generalizes the well known six-vertex model. In addition to the usual nearest-neighbor interactions among the vertices, there exist extra hard-core…
We study integrable models in the context of the recently discovered Gauge/YBE correspondence, where the Yang-Baxter equation is promoted to a duality between two supersymmetric gauge theories. We study flavored elliptic genus of 2d…
The purpose of this talk is to address a couple of simple-sounding questions: what boundary conditions are compatible with (a) Classical integrability? (b) Quantum integrability?
In these notes we review the S-matrix theory in (1+1)-dimensional integrable models, focusing mainly on the relativistic case. Once the main definitions and physical properties are introduced, we discuss the factorization of scattering…
We investigate the integrable Yang-Baxter deformation of the 2d Principal Chiral Model with a Wess-Zumino term. For arbitrary groups, the one-loop beta functions are calculated and display a surprising connection between classical and…
A master equation expressing the classical integrability of two-dimensional non-linear sigma models is found. The geometrical properties of this equation are outlined. In particular, a closer connection between integrability and T-duality…