Relativistic Toda Chain Research Articles

Mirror symmetry for extended affine Weyl groups

We give a uniform, Lie-theoretic mirror symmetry construction for the Frobenius manifolds defined by Dubrovin--Zhang in arXiv:hep-th/9611200 on the orbit spaces of extended affine Weyl groups, including exceptional Dynkin types. The B-model mirror is given by a one-dimensional Landau--Ginzburg superpotential constructed from a suitable degeneration of the family of spectral curves of the affine relativistic Toda chain for the corresponding affine Poisson--Lie group. As applications of our mirror theorem we give closed-form expressions for the flat coordinates of the Saito metric and the Frobenius prepotentials in all Dynkin types, compute the topological degree of the Lyashko--Looijenga mapping for certain higher genus Hurwitz space strata, and construct hydrodynamic bihamiltonian hierarchies (in both Lax--Sato and Hamiltonian form) that are root-theoretic generalisations of the long-wave limit of the extended Toda hierarchy.

From quantum groups to Liouville and dilaton quantum gravity

We investigate the underlying quantum group symmetry of 2d Liouville and dilaton gravity models, both consolidating known results and extending them to the cases with mathcal{N} = 1 supersymmetry. We first calculate the mixed parabolic representation matrix element (or Whittaker function) of Uq( mathfrak{sl} (2, ℝ)) and review its applications to Liouville gravity. We then derive the corresponding matrix element for Uq( mathfrak{osp} (1|2, ℝ)) and apply it to explain structural features of mathcal{N} = 1 Liouville supergravity. We show that this matrix element has the following properties: (1) its q → 1 limit is the classical OSp+(1|2, ℝ) Whittaker function, (2) it yields the Plancherel measure as the density of black hole states in mathcal{N} = 1 Liouville supergravity, and (3) it leads to 3j-symbols that match with the coupling of boundary vertex operators to the gravitational states as appropriate for mathcal{N} = 1 Liouville supergravity. This object should likewise be of interest in the context of integrability of supersymmetric relativistic Toda chains. We furthermore relate Liouville (super)gravity to dilaton (super)gravity with a hyperbolic sine (pre)potential. We do so by showing that the quantization of the target space Poisson structure in the (graded) Poisson sigma model description leads directly to the quantum group Uq( mathfrak{sl} (2, ℝ)) or the quantum supergroup Uq( mathfrak{osp} (1|2, ℝ)).

Recursion and Hamiltonian operators for integrable nonabelian difference equations

In this paper, we carry out the algebraic study of integrable differential-difference equations whose field variables take values in an associative (but not commutative) algebra. We adapt the Hamiltonian formalism to nonabelian difference Laurent polynomials and describe how to obtain a recursion operator from the Lax representation of an integrable nonabelian differential-difference system. As an application, we study a family of integrable equations: the nonabelian Narita–Itoh–Bogoyavlensky lattice, for which we construct their recursion operators and Hamiltonian operators and prove the locality of infinitely many commuting symmetries generated from their highly nonlocal recursion operators. Finally, we discuss the nonabelian version of several integrable difference systems, including the relativistic Toda chain and Ablowitz–Ladik lattice.

E8 spectral curves

I provide an explicit construction of spectral curves for the affine $\mathrm{E}_8$ relativistic Toda chain. Their closed form expression is obtained by determining the full set of character relations in the representation ring of $\mathrm{E}_8$ for the exterior algebra of the adjoint representation; this is in turn employed to provide an explicit construction of both integrals of motion and the action-angle map for the resulting integrable system. I consider two main areas of applications of these constructions. On the one hand, I consider the resulting family of spectral curves in the context of the correspondences between Toda systems, 5d Seiberg-Witten theory, Gromov-Witten theory of orbifolds of the resolved conifold, and Chern-Simons theory to establish a version of the B-model Gopakumar-Vafa correspondence for the $\mathrm{sl}_N$ L\^e-Murakami-Ohtsuki invariant of the Poincar\'e integral homology sphere to all orders in $1/N$. On the other, I consider a degenerate version of the spectral curves and prove a 1-dimensional Landau-Ginzburg mirror theorem for the Frobenius manifold structure on the space of orbits of the extended affine Weyl group of type $\mathrm{E}_8$ introduced by Dubrovin-Zhang (equivalently, the orbifold quantum cohomology of the type-$\mathrm{E}_8$ polynomial $\mathbb{C} P^1$ orbifold). This leads to closed-form expressions for the flat co-ordinates of the Saito metric, the prepotential, and a higher genus mirror theorem based on the Chekhov-Eynard-Orantin recursion. I will also show how the constructions of the paper lead to a generalisation of a conjecture of Norbury-Scott to ADE $\mathbb{P}^1$-orbifolds, and a mirror of the Dubrovin-Zhang construction for all Weyl groups and choices of marked roots.

Lax matrices for a 1-parameter subfamily of van Diejen–Toda chains

In this paper, we construct Lax matrices for certain relativistic open Toda chains endowed with a one-sided 1-parameter boundary interaction. Built upon the Lax representation of the dynamics, an algebraic solution algorithm is also exhibited. To our best knowledge, this particular 1-parameter subfamily of van Diejen–Toda chains has not been analyzed in earlier literature.

Exact relativistic Toda chain eigenfunctions from Separation of Variables and gauge theory

We provide a proposal, motivated by Separation of Variables and gauge theory arguments, for constructing exact solutions to the quantum Baxter equation associated to the N-particle relativistic Toda chain and test our proposal against numerical results. Quantum Mechanical non-perturbative corrections, essential in order to obtain a sensible solution, are taken into account in our gauge theory approach by considering codimension two defects on curved backgrounds (squashed S5 and degenerate limits) rather than flat space; this setting also naturally incorporates exact quantization conditions and energy spectrum of the relativistic Toda chain as well as its modular dual structure.

Exact solution of the relativistic quantum Toda chain

The relativistic quantum Toda chain model is studied with the generalized algebraic Bethe Ansatz method. By employing a set of local gauge transformations, proper local vacuum states can be obtained for this model. The exact spectrum and eigenstates of the model are thus constructed simultaneously.

Bethe/Gauge correspondence in odd dimension: modular double, non-perturbative corrections and open topological strings

Bethe/Gauge correspondence as it is usually stated is ill-defined in five dimensions and needs a "non-perturbative" completion; a related problem also appears in three dimensions. It has been suggested that this problem, probably due to incompleteness of Omega background regularization in odd dimension, may be solved if we consider gauge theory on compact $S^5$ and $S^3$ geometries. We will develop this idea further by giving a full Bethe/Gauge correspondence dictionary on $S^5$ and $S^3$ focussing mainly on the eigenfunctions of (open and closed) relativistic 2-particle Toda chain and its quantized spectral curve: these are most properly written in terms of non-perturbatively completed NS open topological strings. A key ingredient is Faddeev's modular double structure which is naturally implemented by the $S^5$ and $S^3$ geometries.

Q-bosons, Toda lattice, Pieri rules and Baxter q-operator**In honour of Rodney Baxter’s 75th birthday.

We use the Pieri rules to recover the q-boson model and show it is equivalent to a discretized version of the relativistic Toda chain. We identify its semi infinite transfer matrix and the corresponding Baxter Q-matrix with half vertex operators related by an ω-duality transformation. We observe that the scalar product of two higher spin XXZ wave functions can be expressed with a Gaudin determinant.

Cluster characters and the combinatorics of Toda systems

We survey some connections between Toda systems and cluster algebras. One of these connections is based on representation theory: it is known that Laurent expansions of cluster variables are generating functions of Euler characteristics of quiver Grassmannians, and the same turns out to be true of the Hamiltonians of the open relativistic Toda chain. Another connection is geometric: the closed nonrelativistic Toda chain can be regarded as a meromorphic Hitchin system and studied from the standpoint of spectral networks. From this standpoint, the combinatorial formulas for the Hamiltonians of the open relativistic system are sums of trajectories of differential equations defined by the closed nonrelativistic spectral curves.

Three semi-discrete integrable systems related to orthogonal polynomials and their generalized determinant solutions

In this paper, we present a generalized Toeplitz determinant solution for the generalized Schur flow and propose a mixed form of the two known relativistic Toda chains together with its generalized Toeplitz determinant solution. In addition, we also give a Hankel type determinant solution for a nonisospectral Toda lattice. All these results are obtained by technical determinant operations. As a bonus, we finally obtain some new combinatorial numbers based on the moment relations with respect to these semi-discrete integrable systems and give the corresponding combinatorial interpretations by means of the lattice paths.

On Lie groups and Toda lattices

We extend the construction of the relativistic Toda chains as integrable systems on the Poisson submanifolds in Lie groups beyond the case of the A-series. For the simply laced case this is just a direct generalization of the well-known relativistic Toda chains procedure, and we construct explicitly the set of Ad-invariant integrals of motion on symplectic leaves, which can be described using Poisson quivers, which are just blown-up Dynkin diagrams. We also demonstrate how to get the set of ‘minimal’ integrals of motion, using the co-multiplication rules for the corresponding Lie algebras. In the non-simply laced case the corresponding Bogoyavlensky–Coxeter–Toda systems are constructed using the Fock–Goncharov folding of the corresponding Poisson submanifolds. We discuss also how this procedure can be extended for the affine case beyond the A-series, and consider explicitly an example from the affine D-series.

TODA–HEISENBERG CHAIN: INTERACTING σ-FIELDS IN TWO DIMENSIONS

We study a (2 + 1)-dimensional system that can be viewed as an infinite number of O(3) σ-fields coupled by a nearest-neighbour Heisenberg-like interaction. We reduce the field equations of this model to an integrable system that is closely related to the two-dimensional relativistic Toda chain and the Ablowitz–Ladik equations. Using this reduction we obtain the dark-soliton solutions of our model.

Integrability conditions for an analogue of the relativistic Toda chain

We consider a class of discrete-differential equations that contains the relativistic Toda chain and is characterized by one arbitrary function of six variables. We derive three conditions that allow testing the integrability of any given equation in this class. In deriving these conditions, we use higher symmetries distinguishing the equations that are integrable via the inverse scattering method.

Relativistic Toda Chain with Boundary Interaction at Root of Unity

We apply the Separation of Variables method to obtain eigenvectors of commu- ting Hamiltonians in the quantum relativistic Toda chain at a root of unity with boundary interaction.

Linear operator pencils on Lie algebras and Laurent biorthogonal polynomials

We study operator pencils on generators of the Lie algebras sl2 and the oscillator algebra. These pencils are linear in a spectral parameter λ. The corresponding generalized eigenvalue problem gives rise to some sets of orthogonal polynomials and Laurent biorthogonal polynomials (LBP) expressed in terms of the Gauss 2F1 and degenerate 1F1 hypergeometric functions. For special choices of the parameters of the pencils, we identify the resulting polynomials with the Hendriksen–van Rossum LBP which are widely believed to be the biorthogonal analogues of the classical orthogonal polynomials. This places these examples under the umbrella of the generalized bispectral problem which is considered here. Other (non-bispectral) cases give rise to some 'nonclassical' orthogonal polynomials including Tricomi–Carlitz and random-walk polynomials. An application to solutions of relativistic Toda chain is considered.

Relativistic Toda Chains and Schlesinger Transformations

We construct the auto-Schlesinger transformations for all equations in the known list of integrable relativistic Toda chains. Our construction is essentially based on the equations being Lagrangian and on a standard transition to their Hamiltonian form; in this case, the transition is described by the changes of variables that are invertible but not pointwise. We discuss two examples of another type that has similar properties; these are also integrable Lagrangian equations allowing the Schlesinger transformation.

Quantization scheme for modular q-difference equations

Modular pairs of some second order q-difference equations are considered. These equations may be interpreted as a quantum mechanics of a sort of hyperelliptic pendulum. It is shown the quantization of a spectrum may be provided by the condition of the analyticity of the wave function. Baxter's t-Q equations for the quantum relativistic Toda chain in the ``strong coupling regime'' are related to the system considered, and the quantization condition for Q-operator is also considered.

Unifying scheme for generating discrete integrable systems including inhomogeneous and hybrid models

A unifying scheme based on an ancestor model is proposed for generating a wide range of integrable discrete and continuum as well as inhomogeneous and hybrid models. They include in particular discrete versions of sine-Gordon, Landau–Lifshitz, nonlinear Schrödinger (NLS), derivative NLS equations, Liouville model, (non-)relativistic Toda chain, Ablowitz–Ladik model, etc. Our scheme introduces the possibility of building a novel class of integrable hybrid systems including multicomponent models like massive Thirring, discrete self-trapping, two-mode derivative NLS by combining different descendant models. We also construct inhomogeneous systems like Gaudin model including new ones like variable mass sine-Gordon, variable coefficient NLS, Ablowitz–Ladik, Toda chains, etc. keeping their flows isospectral, as opposed to the standard approach. All our models are generated from the same ancestor Lax operator (or its q→1 limit) and satisfy the classical Yang–Baxter equation sharing the same r-matrix. This reveals an inherent universality in these diverse systems, which become explicit at their action-angle level.

Q operators for the simple quantum relativistic Toda Chain

We investigate the simple quantum relativistic Toda chain. The ultralocal simple Weyl algebra pair is associated with each site of the chain. Weyl’s q is considered to be inside a unit circle. Both independent Baxter operators Q are constructed explicitly as series in local Weyl generators. The operator-valued Wronskian of Q-s is also calculated.