866 research outputs found
The Whitham Deformation of the Dijkgraaf-Vafa Theory
We discuss the Whitham deformation of the effective superpotential in the
Dijkgraaf-Vafa (DV) theory. It amounts to discussing the Whitham deformation of
an underlying (hyper)elliptic curve. Taking the elliptic case for simplicity we
derive the Whitham equation for the period, which governs flowings of branch
points on the Riemann surface. By studying the hodograph solution to the
Whitham equation it is shown that the effective superpotential in the DV theory
is realized by many different meromorphic differentials. Depending on which
meromorphic differential to take, the effective superpotential undergoes
different deformations. This aspect of the DV theory is discussed in detail by
taking the N=1^* theory. We give a physical interpretation of the deformation
parameters.Comment: 35pages, 1 figure; v2: one section added to give a physical
interpretation of the deformation parameters, one reference added, minor
corrections; v4: minor correction
Classification of integrable two-component Hamiltonian systems of hydrodynamic type in 2+1 dimensions
Hamiltonian systems of hydrodynamic type occur in a wide range of
applications including fluid dynamics, the Whitham averaging procedure and the
theory of Frobenius manifolds. In 1+1 dimensions, the requirement of the
integrability of such systems by the generalised hodograph transform implies
that integrable Hamiltonians depend on a certain number of arbitrary functions
of two variables. On the contrary, in 2+1 dimensions the requirement of the
integrability by the method of hydrodynamic reductions, which is a natural
analogue of the generalised hodograph transform in higher dimensions, leads to
finite-dimensional moduli spaces of integrable Hamiltonians. In this paper we
classify integrable two-component Hamiltonian systems of hydrodynamic type for
all existing classes of differential-geometric Poisson brackets in 2D,
establishing a parametrisation of integrable Hamiltonians via
elliptic/hypergeometric functions. Our approach is based on the Godunov-type
representation of Hamiltonian systems, and utilises a novel construction of
Godunov's systems in terms of generalised hypergeometric functions.Comment: Latex, 34 page
On Integrable Systems and Supersymmetric Gauge Theories
The properties of the N=2 SUSY gauge theories underlying the Seiberg-Witten
hypothesis are discussed. The main ingredients of the formulation of the
finite-gap solutions to integrable equations in terms of complex curves and
generating 1-differential are presented, the invariant sense of these
definitions is illustrated. Recently found exact nonperturbative solutions to
N=2 SUSY gauge theories are formulated using the methods of the theory of
integrable systems and where possible the parallels between standard quantum
field theory results and solutions to integrable systems are discussed.Comment: LaTeX, 38 pages, no figures; based on the lecture given at INTAS
School on Advances in Quantum Field Theory and Statistical Mechanics, Como,
Italy, 1996; minor changes, few references adde
Self-stabilization of extra dimensions
We show that the problem of stabilization of extra dimensions in Kaluza-Klein
type cosmology may be solved in a theory of gravity involving high-order
curvature invariants. The method suggested (employing a slow-change
approximation) can work with rather a general form of the gravitational action.
As examples, we consider pure gravity with Lagrangians quadratic and cubic in
the scalar curvature and some more complex ones in a simple Kaluza-Klein
framework. After a transition to the 4D Einstein conformal frame, this results
in effective scalar field theories with certain effective potentials, which in
many cases possess positive minima providing stable small-size extra
dimensions. Estimates made in the original (Jordan) conformal frame show that
the problem of a small value of the cosmological constant in the present
Universe is softened in this framework but is not solved completely.}Comment: 10 pages, 4 figures, revtex4. Version with additions and corrections,
accepted at Phys. Rev.
Fractional Systems and Fractional Bogoliubov Hierarchy Equations
We consider the fractional generalizations of the phase volume, volume
element and Poisson brackets. These generalizations lead us to the fractional
analog of the phase space. We consider systems on this fractional phase space
and fractional analogs of the Hamilton equations. The fractional generalization
of the average value is suggested. The fractional analogs of the Bogoliubov
hierarchy equations are derived from the fractional Liouville equation. We
define the fractional reduced distribution functions. The fractional analog of
the Vlasov equation and the Debye radius are considered.Comment: 12 page
Rational Solutions of the Painleve' VI Equation
In this paper, we classify all values of the parameters , ,
and of the Painlev\'e VI equation such that there are
rational solutions. We give a formula for them up to the birational canonical
transformations and the symmetries of the Painlev\'e VI equation.Comment: 13 pages, 1 Postscript figure Typos fixe
On Separation of Variables for Integrable Equations of Soliton Type
We propose a general scheme for separation of variables in the integrable
Hamiltonian systems on orbits of the loop algebra
. In
particular, we illustrate the scheme by application to modified Korteweg--de
Vries (MKdV), sin(sinh)-Gordon, nonlinear Schr\"odinger, and Heisenberg
magnetic equations.Comment: 22 page
Bicomplexes and Integrable Models
We associate bicomplexes with several integrable models in such a way that
conserved currents are obtained by a simple iterative construction. Gauge
transformations and dressings are discussed in this framework and several
examples are presented, including the nonlinear Schrodinger and sine-Gordon
equations, and some discrete models.Comment: 17 pages, LaTeX, uses amssymb.sty and diagrams.st
Integrable equations in nonlinear geometrical optics
Geometrical optics limit of the Maxwell equations for nonlinear media with
the Cole-Cole dependence of dielectric function and magnetic permeability on
the frequency is considered. It is shown that for media with slow variation
along one axis such a limit gives rise to the dispersionless Veselov-Novikov
equation for the refractive index. It is demonstrated that the Veselov-Novikov
hierarchy is amenable to the quasiclassical DBAR-dressing method. Under more
specific requirements for the media, one gets the dispersionless
Kadomtsev-Petviashvili equation. Geometrical optics interpretation of some
solutions of the above equations is discussed.Comment: 33 pages, 7 figure
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