5,993 research outputs found
Tri-hamiltonian vector fields, spectral curves and separation coordinates
We show that for a class of dynamical systems, Hamiltonian with respect to
three distinct Poisson brackets (P_0, P_1, P_2), separation coordinates are
provided by the common roots of a set of bivariate polynomials. These
polynomials, which generalise those considered by E. Sklyanin in his
algebro-geometric approach, are obtained from the knowledge of: (i) a common
Casimir function for the two Poisson pencils (P_1 - \lambda P_0) and (P_2 - \mu
P_0); (ii) a suitable set of vector fields, preserving P_0 but transversal to
its symplectic leaves. The frameworks is applied to Lax equations with spectral
parameter, for which not only it unifies the separation techniques of Sklyanin
and of Magri, but also provides a more efficient ``inverse'' procedure not
involving the extraction of roots.Comment: 49 pages Section on reduction revisite
Self-Dual Chern-Simons Theories
In these lectures I review classical aspects of the self-dual Chern-Simons
systems which describe charged scalar fields in dimensions coupled to a
gauge field whose dynamics is provided by a pure Chern-Simons Lagrangian. These
self-dual models have one realization with nonrelativistic dynamics for the
scalar fields, and another with relativistic dynamics for the scalars. In each
model, the energy density may be minimized by a Bogomol'nyi bound which is
saturated by solutions to a set of first-order self-duality equations. In the
nonrelativistic case the self-dual potential is quartic, the system possesses a
dynamical conformal symmetry, and the self-dual solutions are equivalent to the
static zero energy solutions of the equations of motion. The nonrelativistic
self-duality equations are integrable and all finite charge solutions may be
found. In the relativistic case the self-dual potential is sixth order and the
self-dual Lagrangian may be embedded in a model with an extended supersymmetry.
The self-dual potential has a rich structure of degenerate classical minima,
and the vacuum masses generated by the Chern-Simons Higgs mechanism reflect the
self-dual nature of the potential.Comment: 42 pages LaTe
Statistical Mechanics of 2+1 Gravity From Riemann Zeta Function and Alexander Polynomial:Exact Results
In the recent publication (Journal of Geometry and Physics,33(2000)23-102) we
demonstrated that dynamics of 2+1 gravity can be described in terms of train
tracks. Train tracks were introduced by Thurston in connection with description
of dynamics of surface automorphisms. In this work we provide an example of
utilization of general formalism developed earlier. The complete exact solution
of the model problem describing equilibrium dynamics of train tracks on the
punctured torus is obtained. Being guided by similarities between the dynamics
of 2d liquid crystals and 2+1 gravity the partition function for gravity is
mapped into that for the Farey spin chain. The Farey spin chain partition
function, fortunately, is known exactly and has been thoroughly investigated
recently. Accordingly, the transition between the pseudo-Anosov and the
periodic dynamic regime (in Thurston's terminology) in the case of gravity is
being reinterpreted in terms of phase transitions in the Farey spin chain whose
partition function is just a ratio of two Riemann zeta functions. The mapping
into the spin chain is facilitated by recognition of a special role of the
Alexander polynomial for knots/links in study of dynamics of self
homeomorphisms of surfaces. At the end of paper, using some facts from the
theory of arithmetic hyperbolic 3-manifolds (initiated by Bianchi in 1892), we
develop systematic extension of the obtained results to noncompact Riemannian
surfaces of higher genus. Some of the obtained results are also useful for 3+1
gravity. In particular, using the theorem of Margulis, we provide new reasons
for the black hole existence in the Universe: black holes make our Universe
arithmetic. That is the discrete Lie groups of motion are arithmetic.Comment: 69 pages,11 figures. Journal of Geometry and Physics (in press
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