The Skyrme model with a U*(1) gauged Wess‐Zumino term in a noncommutative spacetime

The Skyrme model is generalized for a noncommutative spacetime with the Weyl‐operators of SU(2) matrices and the corresponding star‐product. The unitary condition and the topological current can be extended to star‐exponential matrices. The Wess‐Zumino term which breaks unphysical symmetries of the Skyrme action is gauged with the U_*(1) group to allow for electromagnetic processes in a noncommutative spacetime. Apart from corrections to the anomalous decay γ→π⁰π⁺π^– in commuting spacetime, the additional anomalous process γ→π⁰π⁰π⁰ is found in the U_*(1) gauged Wess‐Zumino action for a noncommutative spacetime.     

Extended Discrete KP Hierarchy and its Reductions from a Geometric Viewpoint

We interpret the recently suggested extended discrete KP (Toda lattice) hierarchy from a geometrical point of view. We show that the latter corresponds to the union of invariant submanifolds S ₀ ⁿ of the system which is a chain of infinitely many copies of Darboux–KP hierarchy, while the intersections yields a number of reduction s to l-field lattices.     

Solitons on Noncommutative Orbifold T 2/Z N

Following the construction of the projection operators on T ² presented by Gopakumar, Headrick and Spradlin, we construct a set of projection operators on the integral noncommutative orbifold T ²/G(G=Z _N , N=2, 3, 4, 6) which correspond to a set of solitons on T ²/Z _N in noncommutative field theory. In this way, we derive an explicit form of projector on T ²/Z ₆ as an example. We also construct a complete set of projectors on T ²/Z _N by series expansions for integral case.     

On Symmetry Flows of Noncommutative Kadomtsev-Petviashvili Hierarchy

We discuss symmetry flows of noncommutative Kadomtsev-Petviashvili （NCKP） hierarchy. An operatorbased formulation, alternative to the star-product approach of extended symmetry flows is presented. Noncommutative additional symmetry flows of the NCKP hierarchy are formulated. A rescaling symmetry flow which is associated with the rescaling of whole coordinates is introduced.     

Quasi-classical limit of Toda hierarchy andW-infinity symmetries

Previous results on quasi-classical limit of the KP hierarchy and itsW-infinity symmetries are extended to the Toda hierarchy. The Planck constant now emerges as the spacing unit of difference operators in the Lax formalism. Basic notions, such as dressing operators, Baker-Akhiezer functions, and tau function, are redefined.W _{1 +} symmetries of the Toda hierarchy are realized by suitable rescaling of the Date-Jimbo-Kashiara-Miwa vertex operators. These symmetries are contracted tow _{1 +} symmetries of the dispersionless hierarchy through their action on the tau function.     

The Application of Extended Tanh-Function Approach in Toda Lattice Equations

In this paper, we generalize the extended tanh-function approach, which used to find new exact travelling wave solutions of nonlinear partial differential equations (NPDES) or coupled nonlinear partial differential equations, to nonlinear differential-difference equations (NDDES). As illustration, we discuss some Toda lattice equations, and solitary wave and periodic wave solutions of these Toda lattice equations are obtained by means of the extended tanh-function approach. PACS numbers: 05.45.Yv, 02.30.Jr, 02.30.Ik.     

WDVV equation and triple-product relation

We study the relation between the WDVV equations and the τ-function of the noncommutative KP (NCKP) hierarchy. WDVV-like equations (Hirota triple-product relation) in the noncommutative context appear as a consequence of the nontrivial equation for τ-function of the NC KP hierarchy, while the prepotential in the Seiberg–Witten (SW) theory has been identified to the τ-function of the Whitham hierarchy. We show that the spectral curve for the SW theory is the same as the Toda-chain hierarchy. We also show explicitly that Whitham hierarchy includes commutative Toda/KP hierarchy. Further, we comment on the origin of the Hirota triple-product relation in the context of the SW theory.     

On Symmetry Flows of Noncommutative Kadomtsev-Petviashvili Hierarchy

We discuss symmetry flows of noncommutative Kadomtsev-Petviashvili (NCKP) hierarchy. An operatorbased formulation, alternative to the star-product approach of extended symmetry flows is presented. Noncommutative additional symmetry flows of the NCKP hierarchy are formulated. A rescaling symmetry flow which is associated with the rescaling of whole coordinates is introduced.     

Gauge Theory on a Discrete Noncommutative Space

In this paper, we apply Connes' noncommutative geometry and the Seiberg—Witten map to a discrete noncommutative space consisting of n copies of a given noncommutative space R ^m . The explicit action functional of gauge fields on this discrete noncommutative space is obtained.     

Generalizations of the finite nonperiodic Toda lattice and its Darboux transformation

In this paper, we construct Hamiltonian systems for 2 N particles whose force depends on the distances between the particles. We obtain the generalized finite nonperiodic Toda equations via a symmetric group transformation. The solutions of the generalized Toda equations are derived using the tau functions. The relationship between the generalized nonperiodic Toda lattices and Lie algebras is then be discussed and the generalized Kac-van Moerbeke hierarchy is split into generalized Toda lattices, whose integrability and Darboux transformation are studied.     

Exact solutions of integrable 2D contour dynamics

A class of exact solutions of the dispersionless Toda hierarchy constrained by a string equation is obtained. These solutions represent deformations of analytic curves with a finite number of nonzero harmonic moments. The corresponding τ-functions are determined and the emergence of cusps is studied.     

Hamiltonian and gradient structures in the Toda flows

In this paper we consider gradient structures in the dynamics and geometry of the asymmetri nonperiodic tridiagonal and full Toda flow equations. We compare and contrast a number of formulations of the nonperiodic Toda equations. In the case of the full Kostant (asymmetric) Toda flow we explain the role of noncommutative integrability in its qualitative behavior. We describe the relationship between the asymmetric Toda flows and the symmetric and indefinite Toda flows, and prove in particular that one may conjugate from the full Kostant Toda flows to the full symmetric Toda flows via a Poisson map.     

The Hamiltonian Structures of the Two-Dimensional Toda Lattice and <Emphasis Type="Italic">R</Emphasis>-Matrices

We construct the tri-Hamiltonian structure of the two-dimensional Toda hierarchy using the R-matrix theory.Mathematical Subject Classifications (1991). 37K10.     

Spin-Hall Effect with Quantum Group Symmetries

We construct a model of spin-Hall effect on a noncommutative four sphere S ⁴ _Θ with isospin degrees of freedom, coming from a noncommutative instanton, and invariance under a quantum group SO _θ. The corresponding representation theory allows to explicitly diagonalize the Hamiltonian and construct the ground state; there are both integer and fractional excitations. Similar models exist on higher dimensional spheres S _Θ ^N and projective spaces . Dedicated to Rafael Sorkin with friendship and respect.     

Tri-Hamiltonian Toda Lattice and a Canonical Bracket for Closed Discrete Curves

Flows on (or variations of) discrete curves in give rise to flows on a subalgebra of functions on that curve. For a special choice of flows and a certain subalgebra this is described by the Toda lattice hierarchy. Here it is shown that the canonical symplectic structure on , which can be interpreted as the phase space of closed discrete curves in with length N, induces Poisson commutation relations on the above-mentioned subalgebra which yield the tri-Hamiltonian poisson structure of the Toda lattice hierarchy.     

ClassicalA n -W-geometry

By analyzing theextrinsic geometry of two dimensional surfaces chirally embedded inC P ⁿ (theC P ⁿ W-surface [1]), we give exact treatments in various aspects of the classical W-geometry in the conformal gauge: First, the basis of tangent and normal vectors are defined at regular points of the surface, such that their infinitesimal displacements are given by connections which coincide with the vector potentials of the (conformal)A _n -Toda Lax pair. Since the latter is known to be intrinsically related with the W symmetries, this gives the geometrical meaning of theA _n W-Algebra. Second, W-surfaces are put in one-to-one correspondence with solutions of the conformally-reduced WZNW model, which is such that the Toda fields give the Cartan part in the Gauss decomposition of its solutions. Third, the additional variables of the Toda hierarchy are used as coordinates ofC P ⁿ . This allows us to show that W-transformations may be extended as particular diffeomorphisms of this target-space. Higher-dimensional generalizations of the WZNW equations are derived and related with the Zakharov-Shabat equations of the Toda hierarchy. Fourth, singular points are studied from a global viewpoint, using our earlier observation [1] that W-surfaces may be regarded as instantons. The global indices of the W-geometry, which are written in terms of the Toda fields, are shown to be the instanton numbers for associated mappings of W-surfaces into the Grassmannians. The relation with the singularities of W-surface is derived by combining the Toda equations with the Gauss-Bonnet theorem.     

Representation theoretical meaning of the initial value problem for the Toda lattice hierarchy: I

The Toda lattice hierarchy is shown to have the Bruhat decomposition of the A group as its parameter space instead of the Grassmann manifold for the KP hierarchy. Takasaki's work on the initial value problem for the Toda lattice hierarchy is reinterpreted from this point of view.     

Virasoro Symmetries of the Extended Toda Hierarchy

We prove that the extended Toda hierarchy of [1] admits a nonabelian Lie algebra of infinitesimal symmetries isomorphic to half of the Virasoro algebra. The generators L_m, m–1 of the Lie algebra act by linear differential operators onto the tau function of the hierarchy. We also prove that the tau function of a generic solution to the extended Toda hierarchy is annihilated by a combination of the Virasoro operators and the flows of the hierarchy. As an application we show that the validity of the Virasoro constraints for the CP¹ Gromov-Witten invariants and their descendents implies that their generating function is the logarithm of a particular tau function of the extended Toda hierarchy.Acknowledgements.The research of B.D. was partially supported by Italian Ministry of Education research grant Cofin2001 Geometry of Integrable Systems. The research of Y.Z. was partially supported by the Chinese National Science Fund for Distinguished Young Scholars grant No.10025101 and the Special Funds of Chinese Major Basic Research Project Nonlinear Sciences. Y.Z. thanks Abdus Salam International Centre for Theoretical Physics and SISSA where part of the work was done for their hospitality. The authors are grateful to the referee for the suggested improvements of the paper.     

Combinatorics of Dispersionless Integrable Systems and Universality in Random Matrix Theory

It is well-known that the partition function of the unitary ensembles of random matrices is given by a τ-function of the Toda lattice hierarchy and those of the orthogonal and symplectic ensembles are τ-functions of the Pfaff lattice hierarchy. In these cases the asymptotic expansions of the free energies given by the logarithm of the partition functions lead to the dispersionless (i.e. continuous) limits for the Toda and Pfaff lattice hierarchies. There is a universality between all three ensembles of random matrices, one consequence of which is that the leading orders of the free energy for large matrices agree. In this paper, this universality, in the case of Gaussian ensembles, is explicitly demonstrated by computing the leading orders of the free energies in the expansions. We also show that the free energy as the solution of the dispersionless Toda lattice hierarchy gives a solution of the dispersionless Pfaff lattice hierarchy, which implies that this universality holds in general for the leading orders of the unitary, orthogonal, and symplectic ensembles. We also find an explicit formula for the two point function F _nm which represents the number of connected ribbon graphs with two vertices of degrees n and m on a sphere. The derivation is based on the Faber polynomials defined on the spectral curve of the dispersionless Toda lattice hierarchy, and \frac1nmF_nm{\frac{1}{nm}F_{nm}} are the Grunsky coefficients of the Faber polynomials.     

（2＋2）-Dimensional Discrete Soliton Equations and Integrable Coupling System

In this paper, we extend a （2＋2）-dimensional continuous zero curvature equation to （2＋2）-dimensional discrete zero curvature equation, then a new （2＋2）-dimensional cubic Volterra lattice hierarchy is obtained. Fhrthermore, the integrable coupling systems of the （2＋2）-dimensional cubic Volterra lattice hierarchy and the generalized Toda lattice soliton equations are presented by using a Lie algebraic system sl（4）.