395 research outputs found
Computing necessary integrability conditions for planar parametrized homogeneous potentials
Let V\in\mathbb{Q}(i)(\a_1,\dots,\a_n)(\q_1,\q_2) be a rationally
parametrized planar homogeneous potential of homogeneity degree . We design an algorithm that computes polynomial \emph{necessary} conditions
on the parameters (\a_1,\dots,\a_n) such that the dynamical system associated
to the potential is integrable. These conditions originate from those of
the Morales-Ramis-Sim\'o integrability criterion near all Darboux points. The
implementation of the algorithm allows to treat applications that were out of
reach before, for instance concerning the non-integrability of polynomial
potentials up to degree . Another striking application is the first complete
proof of the non-integrability of the \emph{collinear three body problem}.Comment: ISSAC'14 - International Symposium on Symbolic and Algebraic
Computation (2014
Integrability and non integrability of some n body problems
We prove the non integrability of the colinear and body problem, for
any masses positive masses. To deal with resistant cases, we present strong
integrability criterions for dimensional homogeneous potentials of degree
, and prove that such cases cannot appear in the body problem.
Following the same strategy, we present a simple proof of non integrability for
the planar body problem. Eventually, we present some integrable cases of
the body problem restricted to some invariant vector spaces.Comment: 28 pages, 11 figures, 19 reference
Non-integrability of the Armbruster-Guckenheimer-Kim quartic Hamiltonian through Morales-Ramis theory
We show the non-integrability of the three-parameter
Armburster-Guckenheimer-Kim quartic Hamiltonian using Morales-Ramis theory,
with the exception of the three already known integrable cases. We use
Poincar\'e sections to illustrate the breakdown of regular motion for some
parameter values.Comment: Accepted for publication in SIAM Journal on Applied Dynamical
Systems. Adapted version for arxiv with 19 pages and 11 figure
Hamiltonian systems of hydrodynamic type in 2 + 1 dimensions
We investigate multi-dimensional Hamiltonian systems associated with constant
Poisson brackets of hydrodynamic type. A complete list of two- and
three-component integrable Hamiltonians is obtained. All our examples possess
dispersionless Lax pairs and an infinity of hydrodynamic reductions.Comment: 34 page
N=4 SYM on S^3 with Near Critical Chemical Potentials
We study the N = 4 theory at weak coupling, on a three sphere in the grand
canonical ensemble with R symmetry chemical potentials. We focus attention on
near critical values for the chemical potentials, above which the classical
theory has no ground state. By computing a one loop effective potential for the
light degrees of freedom in this regime, we show the existence of flat
directions of complex dimension N, 2N and 3N for one, two and three critical
chemical potentials respectively; these correspond to one half, one quarter and
one-eighth BPS states becoming light respectively at the critical values. At
small finite temperature we show that the chemical potentials can be continued
beyond their classical limiting values to yield a deconfined metastable phase
with lifetime diverging in the large N limit. Our low temperaure analysis
complements the high temperature metastability found by Yamada and Yaffe. The
resulting phase diagram at weak coupling bears a striking resemblance to the
strong coupling phase diagram for charged AdS black holes. Our analysis also
reveals subtle qualitative differences between the two regimes.Comment: 34 pages, 4 figure
Smooth Wilson Loops in N=4 Non-Chiral Superspace
We consider a supersymmetric Wilson loop operator for 4d N=4 super Yang-Mills
theory which is the natural object dual to the AdS_5 x S^5 superstring in the
AdS/CFT correspondence. It generalizes the traditional bosonic 1/2 BPS
Maldacena-Wilson loop operator and completes recent constructions in the
literature to smooth (non-light-like) loops in the full N=4 non-chiral
superspace. This Wilson loop operator enjoys global superconformal and local
kappa-symmetry of which a detailed discussion is given. Moreover, the
finiteness of its vacuum expectation value is proven at leading order in
perturbation theory. We determine the leading vacuum expectation value for
general paths both at the component field level up to quartic order in
anti-commuting coordinates and in the full non-chiral superspace in suitable
gauges. Finally, we discuss loops built from quadric splines joined in such a
way that the path derivatives are continuous at the intersection.Comment: 44 pages. v2 Added some clarifying comments. Matches the published
versio
Y-system for Scattering Amplitudes
We compute N=4 Super Yang Mills planar amplitudes at strong coupling by
considering minimal surfaces in AdS_5 space. The surfaces end on a null
polygonal contour at the boundary of AdS. We show how to compute the area of
the surfaces as a function of the conformal cross ratios characterizing the
polygon at the boundary. We reduce the problem to a simple set of functional
equations for the cross ratios as functions of the spectral parameter. These
equations have the form of Thermodynamic Bethe Ansatz equations. The area is
the free energy of the TBA system. We consider any number of gluons and in any
kinematic configuration.Comment: 69 pages, 19 figures, v2: references added, minor addition
The generalized cusp in ABJ(M) N = 6 Super Chern-Simons theories
We construct a generalized cusped Wilson loop operator in N = 6 super
Chern-Simons-matter theories which is locally invariant under half of the
supercharges. It depends on two parameters and interpolates smoothly between
the 1/2 BPS line or circle and a pair of antiparallel lines, representing a
natural generalization of the quark-antiquark potential in ABJ(M) theories. For
particular choices of the parameters we obtain 1/6 BPS configurations that,
mapped on S^2 by a conformal transformation, realize a three-dimensional
analogue of the wedge DGRT Wilson loop of N = 4. The cusp couples, in addition
to the gauge and scalar fields of the theory, also to the fermions in the
bifundamental representation of the U(N)xU(M) gauge group and its expectation
value is expressed as the holonomy of a suitable superconnection. We discuss
the definition of these observables in terms of traces and the role of the
boundary conditions of fermions along the loop. We perform a complete two-loop
analysis, obtaining an explicit result for the generalized cusp at the second
non-trivial order, from which we read off the interaction potential between
heavy 1/2 BPS particles in the ABJ(M) model. Our results open the possibility
to explore in the three-dimensional case the connection between localization
properties and integrability, recently advocated in D = 4.Comment: 53 pages, 10 figures, added references, this is the version appeared
on JHE
Quantal Andreev billiards: Semiclassical approach to mesoscale oscillations in the density of states
Andreev billiards are finite, arbitrarily-shaped, normal-state regions,
surrounded by superconductor. At energies below the superconducting gap,
single-quasiparticle excitations are confined to the normal region and its
vicinity, the essential mechanism for this confinement being Andreev
reflection. This Paper develops and implements a theoretical framework for the
investigation of the short-wave quantal properties of these
single-quasiparticle excitations. The focus is primarily on the relationship
between the quasiparticle energy eigenvalue spectrum and the geometrical shape
of the normal-state region, i.e., the question of spectral geometry in the
novel setting of excitations confined by a superconducting pair-potential.
Among the central results of this investigation are two semiclassical trace
formulas for the density of states. The first, a lower-resolution formula,
corresponds to the well-known quasiclassical approximation, conventionally
invoked in settings involving superconductivity. The second, a
higher-resolution formula, allows the density of states to be expressed in
terms of: (i) An explicit formula for the level density, valid in the
short-wave limit, for billiards of arbitrary shape and dimensionality. This
level density depends on the billiard shape only through the set of
stationary-length chords of the billiard and the curvature of the boundary at
the endpoints of these chords; and (ii) Higher-resolution corrections to the
level density, expressed as a sum over periodic orbits that creep around the
billiard boundary. Owing to the fact that these creeping orbits are much longer
than the stationary chords, one can, inter alia, hear the stationary chords of
Andreev billiards.Comment: 52 pages, 15 figures, 1 table, RevTe
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