The Darboux transformation is used to obtain multisoliton solutions of the chiral model in two dimensions. The matrix solutions of the principal chiral model and its Lax pair are expressed in terms of quasideterminants. The iteration of the Darboux transformation gives the quasideterminant multisoliton solutions of the model. It has been shown that the quasideterminant multisoliton solution of the chiral model is the same as obtained by Zakharov and Mikhailov using the dressing method based on the matrix Riemann-Hilbert problem.
Matrix-valued Cauchy bi-orthogonal polynomials were proposed in this paper, together with its quasideterminant expression. It is shown that the coefficients in four-term recurrence relation for matrix-valued Cauchy bi-orthogonal polynomials should satisfy a novel noncommutative integrable system, whose Lax pair is given by fractional differential operators with non-abelian variables.
This is an introductory survey written in Japanese on classical integrability from the viewpoint of symmetry reduction of 4-dimensional anti-self-dual Yang-Mills equations. Twistor construction methods are discussed in detail in 4-dimension. Exact local solutions are, in this article, presented in terms of quasideterminants in a compact form. Some integrable aspects in lower-dimension are clarified in the context of the reduction. Noncommutative extension and higher-dimensional extension are briefly discussed.
At roots of unity the $N$-state integrable chiral Potts model and the six-vertex model descend from each other with the $τ_2$ model as the intermediate. We shall discuss how different gauge choices in the six-vertex model lead to two different quantum group constructions with different $q$-Pochhammer symbols, one construction only working well for $N$ odd, the other equally well for all $N$. We also address the generalization based on the sl$(m,n)$ vertex model.
Integrable deformations of principal chiral model from solutions of associative Yang-Baxter equation
Institute for Theoretical and Mathematical Physics, Lomonosov Moscow State University（莫斯科国立大学理论物理与数学物理研究所）；National Research University Higher School of Economics（国立高等经济大学）；National Research Centre ”Kurchatov Institute”（库尔恰托夫国家研究中心）；Institute for Information Transmission Problems RAS (Kharkevich Institute)（俄罗斯科学院信息传输问题研究所（哈尔凯维奇研究所））；Steklov Mathematical Institute of Russian Academy of Sciences（俄罗斯科学院斯捷克洛夫数学研究所）
It was demonstrated recently [Dunajski, Ferapontov and Kruglikov (2014)] that the Manakov-Santini system describes a local form of general Lorentzian Einstein-Weyl geometry. We introduce integrable matrix extension of the Manakov-Santini system and show that it describes (2+1)-dimensional integrable chiral model in Einstein-Weyl space. We develop a dressing scheme for the extended MS system and define an extended hierarchy. Matrix extension of Toda type system connected with another local form of Einstein-Weyl geometry is also considered.
Integrable hydrodynamic equations for initial chiral currents and infinite hydrodynamic chains from WZNW model and string model of WZNW type with $SU(2),$ $SO(3),$ $SP(2),$ $SU(\infty)$, $SO(\infty)$, $SP(\infty)$ constant torsions
The goal of this paper is to make first steps towards the quantisation of integrable non-linear sigma models using the formalism of affine Gaudin models, by approaching these theories through their conformal limits. We focus mostly on the example of the Klimčík model, which is a two-parameter deformation of the Principal Chiral Model on a Lie group $G$. We show that the UV fixed point of this theory is described classically by two decoupled chiral affine Gaudin models, encoding its left- and right-moving degrees of freedom, and give a detailed analysis of the chiral and integrable structures o…
Computational homological methods for integrable field theories
Università di Genova（热那亚大学）；INFN, Sezione di Genova（意大利国家核物理研究所热那亚分部）；University of York（约克大学）；Università di Trento（特伦托大学）；INFN-TIFPA（意大利国家核物理研究所-理论物理与应用物理中心）
In integrable field theories in two dimensions, the Bethe ansatz can be used to compute exactly the ground state energy in the presence of an external field coupled to a conserved charge. We generalize previous results by Volin and we extract analytic results for the perturbative expansion of this observable, up to very high order, in various asymptotically free theories: the non-linear sigma model and its supersymmetric extension, the Gross--Neveu model, and the principal chiral field. We study the large order behavior of these perturbative series and we give strong evidence that, as expected…
We study four-dimensional Chern-Simons theory on $D \times \mathbb{C}$ (where $D$ is a disk), which is understood to describe rational solutions of the Yang-Baxter equation from the work of Costello, Witten and Yamazaki. We find that the theory is dual to a boundary theory, that is a three-dimensional analogue of the two-dimensional chiral WZW model. This boundary theory gives rise to a current algebra that turns out to be an "analytically-continued" toroidal Lie algebra. In addition, we show how certain bulk correlation functions of two and three Wilson lines can be captured by boundary corre…
We investigate the homogeneous chiral edge theory of the filling $ν=4/3$ fractional quantum Hall state, which is parameterized by a Luttinger liquid velocity matrix and an electron tunneling amplitude (ignoring irrelevant terms). We identify two solvable cases: one case where the theory gives two free chiral boson modes, and the other case where the theory yields one free charge $\frac{2e}{\sqrt{3}}$ chiral fermion and two free chiral Bogoliubov (Majorana) fermions. For generic parameters, the energy spectrum from our exact diagonalization shows Poisson level spacing statistics (LSS) in each c…
We consider modified gravity cosmological models that can be transformed into two-field chiral cosmological models by the conformal metric transformation. For the $R^2$ gravity model with an additional scalar field and the corresponding two-field model with the cosmological constant and nonstandard kinetic part of the action, the general solutions have been obtained in the spatially flat FLRW metric. We analyze the correspondence of the cosmic time solutions obtained and different possible evolutions of the Hubble parameters in the Einstein and Jordan frames.
