A geometric notion of complete integrability

A system of partial differential equations which can be described as a harmonic mapping of riemannian manifolds is called completely integrable when the corresponding n-dimensional manifold of fields admits 2n?1 independent Killing vector fields. It is conjectured that, for systems of two independent variables, complete integrability in the present sense implies the existence of a Lax pair for the system, for which the theory of the inverse scattering method is applicable. The stationary axisymmetric Einstein and Einstein-Maxwell equations, the SU(n) self-dual Yang-Mills fields in 1+1 dimensions, and the two-dimensional non-linear σ-models are shown to satisfy the conjecture; the conjecture is also proved for any system of n = 2 and n = 3 partial differential equations for n unknown scalar fields.     

On the complete integrability of completely integrable systems

The question of complete integrability of evolution equations associated ton×n first order isospectral operators is investigated using the inverse scattering method. It is shown that forn>2, e.g. for the three-wave interaction, additional (nonlinear) pointwise flows are necessary for the assertion of complete integrability. Their existence is demonstrated by constructing action-angle variables. This construction depends on the analysis of a natural 2-form and symplectic foliation for the groupsGL(n) andSU(n).Research supported by NSF grants DMS-8916968 and DMS 8901607     

First integrals versus configurational invariants and a weak form of complete integrability

A confusion over the concept of first integrals, which has been created in a recent paper by Hall [13] is clarified. The clear distinction between first integrals and functions which are first integrals only on a specific, fixed hypersurface is discussed. Hall's terminology of configurational invariants is adopted for the latter case. The possible relevance of knowing configurational invariants for a Hamiltonian system is illustrated by results concerning a weak form of the theory on complete integrability.     

The algebraic complete integrability of geodesic flow onSO(N)

We study for which left invariant diagonal metrics λ onSO(N), the Euler-Arnold equations $$\dot X = [x,\lambda (X)], X = (x_{ij} ) \in so(N), \lambda _{ij} x_{ij} , \lambda _{ij} = \lambda _{ji} $$ can be linearized on an abelian variety, i.e. are solvable by quadratures. We show that, merely by requiring that the solutions of the differential equations be single-valued functions of complex timet∈?, suffices to prove that (under a non-degeneracy assumption on the metric λ) the only such metrics are those which satisfy Manakov's conditions λ _ij =(b _i ?b _j ) (a _i ?a _j )^?1. The case of degenerate metrics is also analyzed. ForN=4, this provides a new and simpler proof of a result of Adler and van Moerbeke [3].     

New coupled Liouville system: Prolongation structure,soliton solution,and complete integrability

We suggest a new coupled Liouville equation which is exactly solvable. We obtain the Lax pair through a prolongation analysis and also obtain the exact one-soliton-like solution by a direct procedure. We confirm our result through a Painlevé analysis of the similarity reduced systems.     

Non-Abelian plane waves

The non-Abelian analogues of electromagnetic plane waves are constructed.     

Non-Abelian magnetic monopoles

It is shown that a general, irreducible, SU(n), Sp(n), SO(2n) monopole with maximal symmetry breaking is determined by its spectral data. For SU(n) with minimal symmetry breaking the spectral data is defined and also shown to determine the monopole.Research supported in part by NSF Grant 8120790     

Non-Abelian anyon superconductivity

Non-Abelian anyons exist in certain spin models and may exist in quantum Hall systems at certain filling fractions. In this work, we studied the ground state of dynamical SU(2) level-kappa Chern-Simons non-Abelian anyons at finite density and no external magnetic field. We find that, in the large-kappa limit, the topological interaction induces a pairing instability and the ground state is a superconductor with d+id gap symmetry. We also develop a picture of pairing for the special value kappa=2 and argue that the ground state is a superfluid of pairs for all values of kappa.     

Non-Abelian superconducting pumps

Cooper pair pumping is a coherent process. We derive a general expression for the adiabatic pumped charge in superconducting nanocircuits in the presence of level degeneracy and relate it to non-Abelian holonomies of Wilczek and Zee. We discuss an experimental system where the non-Abelian structure of the adiabatic evolution manifests in the pumped charge.     

Non-Abelian Dynamics and Heavy Multiquarks

A brief review is first presented of attempts to predict stable multiquark states within current models of hadron spectroscopy. Then a model combining flip?Cflop and connected Steiner trees is introduced and shown to lead to stable multiquarks, in particular for some configurations involving several heavy quarks and bearing exotic quantum numbers.     

Non-Abelian higher-derivative Chern-Simons theories

The canonical and path-integral quantization of the non-Abelian higher-derivative Chem-Simons model in three dimensions coupled to a matter field is constructed. The expression of the gauge field propagator in the momentum space for this higher-derivative model is computed. In the framework of the perturbative formalism, the diagrammatic and the Feynman rules are analyzed. Among other results, we conclude that higher-derivative terms added to the Lagrangian improve the ultraviolet behavior, rendering the model less divergent.     

Octonionic Non-Abelian Gauge Theory

We have made an attempt to describe the octonion formulation of Abelian and non-Abelian gauge theory of dyons in terms of 2×2 Zorn vector matrix realization. As such, we have discussed the U(1) _e ×U(1) _m Abelian gauge theory and U(1)×SU(2) electroweak gauge theory and also the SU(2) _e ×SU(2) _m non-Abelian gauge theory in term of 2×2 Zorn vector matrix realization of split octonions. It is shown that SU(2) _e characterizes the usual theory of the Yang Mill’s field (isospin or weak interactions) due to presence of electric charge while the gauge group SU(2) _m may be related to the existence of ’t Hooft-Polyakov monopole in non-Abelian Gauge theory. Accordingly, we have obtained the manifestly covariant field equations and equations of motion.