Lattice Integrable System Research Articles

N-Dimensional Lattice Integrable Systems and Their bi-Hamiltonian Structure on the Time Scale Using the R-Matrix Approach

A time scale is a special measure chain that can unify continuous and discrete spaces, enabling the construction of integrable equations. In this paper, with the Lax operator generated by the displacement operator, N-dimensional lattice integrable systems on the time scale are given by the R-matrix approach. The recursion operators of the lattice systems are derived on the time scale. Finally, two integrable hierarchies of the discrete chain with a bi-Hamiltonian structure are obtained. In particular, we give the structure of two-field and four-field systems.

Gauge theory and boundary integrability

We study the mixed topological/holomorphic Chern-Simons theory of Costello, Witten and Yamazaki on an orbifold (Σ×ℂ)/ℤ2, obtaining a description of lattice integrable systems in the presence of a boundary. By performing an order ℏ calculation we derive a formula for the the asymptotic behaviour of K-matrices associated to rational, quasi-classical R-matrices. The ℤ2-action on Σ × ℂ fixes a line L, and line operators on L are shown to be labelled by representations of the twisted Yangian. The OPE of such a line operator with a Wilson line in the bulk is shown to give the coproduct of the twisted Yangian. We give the gauge theory realisation of the Sklyanin determinant and related conditions in the RTT presentation of the boundary Yang-Baxter equation.

Algebraic Structure of Discrete Zero Curvature Equations and Master Symmetries of Discrete Evolution Equations**Supported by the National Science Foundation of China under Grant No. 11371244 and the Applied Mathematical Subject of SSPU under Grant No. XXKPY1604

In this paper, based on a discrete spectral problem and the corresponding zero curvature representation, the isospectral and nonisospectral lattice hierarchies are proposed. An algebraic structure of discrete zero curvature equations is then established for such integrable systems. the commutation relations of Lax operators corresponding to the isospectral and non-isospectral lattice flows are worked out, the master symmetries of each lattice equation in the isospectral hierarchyand are generated, thus a τ-symmetry algebra for the lattice integrable systems is engendered from this theory.

Generating integrable lattice hierarchies by some matrix and operator Lie algebras

Two types of matrix Lie algebras are presented. We make use of the first loop algebra to obtain a new \((1+1)\)-dimensional integrable discrete hierarchy, which generalizes a result given by Gordoa et al., whose reduction is a discrete modified KdV system. Then we produce another new \((2+1)\)-dimensional integrable discrete hierarchy with three fields under a \((2+1)\)-dimensional non-isospectral linear problem. We again generalize the \((1+1)\)- and \((2+1)\)-dimensional discrete hierarchies to obtain a positive and negative integrable discrete hierarchy. In addition, we obtain a discrete integrable coupling system of the \((1+1)\)-dimensional discrete hierarchy presented in the paper by enlarging such the loop algebras. Next, we apply the second matrix loop algebra to introduce an isospectral problem and deduce a new integrable discrete hierarchy, whose quasi-Hamiltonian structure is derived from the trace identity proposed by Tu Guizhang, which can be reduced to some modified Toda lattice equations. A type of Darboux transformation of a reduced discrete system of the latter integrable discrete hierarchy is obtained as well. We introduce two types of operator-Lie algebras according to a given spectral problem by a matrix Lie algebra and apply the r-matrix theory to obtain a few lattice integrable systems, including two \((2+1)\)-dimensional lattice systems.

Separation of Variables for a Lattice Integrable System and the Inverse Problem

We investigate the relation between the local variables of a discrete integrable lattice system and the corresponding separation variables, derived from the associated spectral curve. In particular, we have shown how the inverse transformation from the separation variables to the discrete lattice variables may be factorised as a sequence of canonical transformations, following the procedure outlined by Kuznetsov.

Equivariant homology and K-theory of affine Grassmannians and Toda lattices

For an almost simple complex algebraic group G with affine Grassmannian $\text{Gr}_G=G(\mathbb{C}(({\rm t})))/G(\mathbb{C}[[{\rm t}]])$ , we consider the equivariant homology $H^{G(\mathbb{C}[[{\rm t}]])}(\text{Gr}_G)$ and K -theory $K^{G(\mathbb{C}[[{\rm t}]])}(\text{Gr}_G)$ . They both have a commutative ring structure with respect to convolution. We identify the spectrum of homology ring with the universal group-algebra centralizer of the Langlands dual group $\check G$ , and we relate the spectrum of K -homology ring to the universal group-group centralizer of G and of $\check G$ . If we add the loop-rotation equivariance, we obtain a noncommutative deformation of the ( K -)homology ring, and thus a Poisson structure on its spectrum. We relate this structure to the standard one on the universal centralizer. The commutative subring of $G(\mathbb{C}[[{\rm t}]])$ -equivariant homology of the point gives rise to a polarization which is related to Kostant's Toda lattice integrable system. We also compute the equivariant K -ring of the affine Grassmannian Steinberg variety. The equivariant K -homology of Gr G is equipped with a canonical basis formed by the classes of simple equivariant perverse coherent sheaves. Their convolution is again perverse and is related to the Feigin–Loktev fusion product of $G(\mathbb{C}[[{\rm t}]])$ -modules.

Lattice analogue of the W-infinity algebra and discrete KP-hierarchy

In the development of the activity started on lattice analogues of W algebras, we define the notion of a lattice W ∞ algebra, associated with a lattice integrable system with an infinite set of fields. Various kinds of reduction to lattice W N algebras, related to discrete N-KdV hierarchies are described. We also discuss the connection of our results with those obtained in the papers of Xiong [Phys. Lett. B 279 (1992) 347] and Bonora [Multimatrix models without continuous limit, preprint SISSA-ISAS 211/92/EP(1992)].

Lattice integrable systems of Haldane-Shastry type.

We present a new lattice integrable system in one dimension of the Haldane-Shastry type. It consists of spins positioned at the static equilibrium positions of particles in a corresponding classical Calogero system and interacting through an exchange term with strength inversely proportional to the square of their distance. We achieve this by viewing the Haldane-Shastry system as a high-interaction limit of the Sutherland system of particles with internal degrees of freedom and identifying the same limit in a corresponding Calogero system. The commuting integrals of motion of this system are found using the exchange operator formalism.

Time discretizations of lattice integrable systems

Time discretizations are found of lattice integrable systems of Lax type, including the infinite and periodic Toda lattices. The resulting systems are integrable. Considered as numerical schemes, they are implicit, but can be made explicit up to arbitrary order of the time step Δ t.

P-CELL GAUGE THEORIES, MANIFOLD SPACE, AND MULTI-DIMENSIONAL INTEGRABILITY

We construct lattice gauge theories where the gauge potentials live on p-cells, using ideas from the theory of multi-dimensional lattice integrability. The classical limit of these models can naturally be considered as chiral models in the space ΩPM of p-dimensional manifolds on M, or alternatively as gauge theories in ΩP−1M. The continuum models have an infinite set of functionally conserved currents in ΩPM, which are classical analogs of the simplex equations of lattice integrable systems.