1,471 research outputs found
Lattice Gauge Fields Topology Uncovered by Quaternionic sigma-model Embedding
We investigate SU(2) gauge fields topology using new approach, which exploits
the well known connection between SU(2) gauge theory and quaternionic
projective sigma-models and allows to formulate the topological charge density
entirely in terms of sigma-model fields. The method is studied in details and
for thermalized vacuum configurations is shown to be compatible with
overlap-based definition. We confirm that the topological charge is distributed
in localized four dimensional regions which, however, are not compatible with
instantons. Topological density bulk distribution is investigated at different
lattice spacings and is shown to possess some universal properties.Comment: revtex4, 19 pages (24 ps figures included); replaced to match the
published version, to appear in PRD; minor changes, references adde
Topological Phenomena in the Real Periodic Sine-Gordon Theory
The set of real finite-gap Sine-Gordon solutions corresponding to a fixed
spectral curve consists of several connected components. A simple explicit
description of these components obtained by the authors recently is used to
study the consequences of this property. In particular this description allows
to calculate the topological charge of solutions (the averaging of the
-derivative of the potential) and to show that the averaging of other
standard conservation laws is the same for all components.Comment: LaTeX, 18 pages, 3 figure
Two-dimensional algebro-geometric difference operators
A generalized inverse problem for a two-dimensional difference operator is
introduced. A new construction of the algebro-geometric difference operators of
two types first considered by I.M.Krichever and S.P.Novikov is proposedComment: 11 pages; added references, enlarged introduction, rewritten abstrac
On the numerical evaluation of algebro-geometric solutions to integrable equations
Physically meaningful periodic solutions to certain integrable partial
differential equations are given in terms of multi-dimensional theta functions
associated to real Riemann surfaces. Typical analytical problems in the
numerical evaluation of these solutions are studied. In the case of
hyperelliptic surfaces efficient algorithms exist even for almost degenerate
surfaces. This allows the numerical study of solitonic limits. For general real
Riemann surfaces, the choice of a homology basis adapted to the
anti-holomorphic involution is important for a convenient formulation of the
solutions and smoothness conditions. Since existing algorithms for algebraic
curves produce a homology basis not related to automorphisms of the curve, we
study symplectic transformations to an adapted basis and give explicit formulae
for M-curves. As examples we discuss solutions of the Davey-Stewartson and the
multi-component nonlinear Schr\"odinger equations.Comment: 29 pages, 20 figure
On the algebraic structures connected with the linear Poisson brackets of hydrodynamics type
The generalized form of the Kac formula for Verma modules associated with
linear brackets of hydrodynamics type is proposed. Second cohomology groups of
the generalized Virasoro algebras are calculated. Connection of the central
extensions with the problem of quntization of hydrodynamics brackets is
demonstrated
Reciprocal transformations and local Hamiltonian structures of hydrodynamic type systems
We start from a hyperbolic DN hydrodynamic type system of dimension which
possesses Riemann invariants and we settle the necessary conditions on the
conservation laws in the reciprocal transformation so that, after such a
transformation of the independent variables, one of the metrics associated to
the initial system be flat. We prove the following statement: let in
the case of reciprocal transformations of a single independent variable or
in the case of transformations of both the independent variable; then
the reciprocal metric may be flat only if the conservation laws in the
transformation are linear combinations of the canonical densities of
conservation laws, {\it i.e} the Casimirs, the momentum and the Hamiltonian
densities associated to the Hamiltonian operator for the initial metric. Then,
we restrict ourselves to the case in which the initial metric is either flat or
of constant curvature and we classify the reciprocal transformations of one or
both the independent variables so that the reciprocal metric is flat. Such
characterization has an interesting geometric interpretation: the hypersurfaces
of two diagonalizable DN systems of dimension are Lie equivalent if
and only if the corresponding local hamiltonian structures are related by a
canonical reciprocal transformation.Comment: 23 pages; corrected typos, added counterexample in Remark 3.
Quantum Transport in Molecular Rings and Chains
We study charge transport driven by deformations in molecular rings and
chains. Level crossings and the associated Longuet-Higgins phase play a central
role in this theory. In molecular rings a vanishing cycle of shears pinching a
gap closure leads, generically, to diverging charge transport around the ring.
We call such behavior homeopathic. In an infinite chain such a cycle leads to
integral charge transport which is independent of the strength of deformation.
In the Jahn-Teller model of a planar molecular ring there is a distinguished
cycle in the space of uniform shears which keeps the molecule in its manifold
of ground states and pinches level crossing. The charge transport in this cycle
gives information on the derivative of the hopping amplitudes.Comment: Final version. 26 pages, 8 fig
On a Conjecture of Givental
These brief notes record our puzzles and findings surrounding Givental's
recent conjecture which expresses higher genus Gromov-Witten invariants in
terms of the genus-0 data. We limit our considerations to the case of a
projective line, whose Gromov-Witten invariants are well-known and easy to
compute. We make some simple checks supporting his conjecture.Comment: 13 pages, no figures; v.2: new title, minor change
Bi-Hamiltonian representation of St\"{a}ckel systems
It is shown that a linear separation relations are fundamental objects for
integration by quadratures of St\"{a}ckel separable Liouville integrable
systems (the so-called St\"{a}ckel systems). These relations are further
employed for the classification of St\"{a}ckel systems. Moreover, we prove that
{\em any} St\"{a}ckel separable Liouville integrable system can be lifted to a
bi-Hamiltonian system of Gel'fand-Zakharevich type. In conjunction with other
known result this implies that the existence of bi-Hamiltonian representation
of Liouville integrable systems is a necessary condition for St\"{a}ckel
separability.Comment: To appear in Physical Review
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