21 research outputs found
Polynomial Lie algebra methods in solving the second-harmonic generation model: some exact and approximate calculations
We compare exact and SU(2)-cluster approximate calculation schemes to
determine dynamics of the second-harmonic generation model using its
reformulation in terms of a polynomial Lie algebra and related
spectral representations of the model evolution operator realized in
algorithmic forms. It enabled us to implement computer experiments exhibiting a
satisfactory accuracy of the cluster approximations in a large range of
characteristic model parameters.Comment: LaTex file, 13 pages, 3 figure
Metric Properties of the Fuzzy Sphere
The fuzzy sphere, as a quantum metric space, carries a sequence of metrics
which we describe in detail. We show that the Bloch coherent states, with these
spectral distances, form a sequence of metric spaces that converge to the round
sphere in the high-spin limit.Comment: Slightly shortened version, no major changes, two new references,
version to appear on Letters in Mathematical Physic
Supersymmetric null-surfaces
Single trace operators with the large R-charge in supersymmetric Yang-Mills
theory correspond to the null-surfaces in . We argue that the
moduli space of the null-surfaces is the space of contours in the
super-Grassmanian parametrizing the complex -dimensional subspaces of
the complex -dimensional space. The odd coordinates on this
super-Grassmanian correspond to the fermionic degrees of freedom of the
superstring.Comment: v4: added a reference to the earlier work; corrected the formula for
the stabilizer of the BMN vacuum; added the discussion of the complex
structure of the odd coordinates in Section 3.
Large spin limits of AdS/CFT and generalized Landau-Lifshitz equations
We consider AdS_5 x S^5 string states with several large angular momenta along AdS_5 and S^5 directions which are dual to single-trace Super-Yang-Mills (SYM) operators built out of chiral combinations of scalars and covariant derivatives. In particular, we focus on the SU(3) sector (with three spins in S^5) and the SL(2) sector (with one spin in AdS_5 and one in S^5), generalizing recent work hep-th/0311203 and hep-th/0403120 on the SU(2) sector with two spins in S^5. We show that, in the large spin limit and at the leading order in the effective coupling expansion, the string sigma model equations of motion reduce to matrix Landau-Lifshitz equations. We then demonstrate that the coherent-state expectation value of the one-loop SYM dilatation operator restricted to the corresponding sector of single trace operators is also effectively described by the same equations. This implies a universal leading order equivalence between string energies and SYM anomalous dimensions, as well as a matching of integrable structures. We also discuss the more general 5-spin sector and comment on SO(6) states dual to non-chiral scalar operators
More on quantum groups from the the quantization point of view
Star products on the classical double group of a simple Lie group and on
corresponding symplectic grupoids are given so that the quantum double and the
"quantized tangent bundle" are obtained in the deformation description.
"Complex" quantum groups and bicovariant quantum Lie algebras are discused from
this point of view. Further we discuss the quantization of the Poisson
structure on symmetric algebra leading to the quantized enveloping
algebra as an example of biquantization in the sense of Turaev.
Description of in terms of the generators of the bicovariant
differential calculus on is very convenient for this purpose. Finally
we interpret in the deformation framework some well known properties of compact
quantum groups as simple consequences of corresponding properties of classical
compact Lie groups. An analogue of the classical Kirillov's universal character
formula is given for the unitary irreducible representation in the compact
case.Comment: 18 page
SU(1,1) Coherent States For Position-Dependent Mass Singular Oscillators
The Schroedinger equation for position-dependent mass singular oscillators is
solved by means of the factorization method and point transformations. These
systems share their spectrum with the conventional singular oscillator. Ladder
operators are constructed to close the su(1,1) Lie algebra and the involved
point transformations are shown to preserve the structure of the
Barut-Girardello and Perelomov coherent states.Comment: 11 pages, 5 figures. This shortened version (includes new references)
has been adapted for its publication in International Journal of Theoretical
Physic
Supercoherent States, Super K\"ahler Geometry and Geometric Quantization
Generalized coherent states provide a means of connecting square integrable
representations of a semi-simple Lie group with the symplectic geometry of some
of its homogeneous spaces. In the first part of the present work this point of
view is extended to the supersymmetric context, through the study of the
OSp(2/2) coherent states. These are explicitly constructed starting from the
known abstract typical and atypical representations of osp(2/2). Their
underlying geometries turn out to be those of supersymplectic OSp(2/2)
homogeneous spaces. Moment maps identifying the latter with coadjoint orbits of
OSp(2/2) are exhibited via Berezin's symbols. When considered within
Rothstein's general paradigm, these results lead to a natural general
definition of a super K\"ahler supermanifold, the supergeometry of which is
determined in terms of the usual geometry of holomorphic Hermitian vector
bundles over K\"ahler manifolds. In particular, the supergeometry of the above
orbits is interpreted in terms of the geometry of Einstein-Hermitian vector
bundles. In the second part, an extension of the full geometric quantization
procedure is applied to the same coadjoint orbits. Thanks to the super K\"ahler
character of the latter, this procedure leads to explicit super unitary
irreducible representations of OSp(2/2) in super Hilbert spaces of
superholomorphic sections of prequantum bundles of the Kostant type. This work
lays the foundations of a program aimed at classifying Lie supergroups'
coadjoint orbits and their associated irreducible representations, ultimately
leading to harmonic superanalysis. For this purpose a set of consistent
conventions is exhibited.Comment: 53 pages, AMS-LaTeX (or LaTeX+AMSfonts
Lattice Pseudospin Model for Quantum Hall Bilayers
We present a new theoretical approach to the study of quantum Hall
bilayer that is based on a systematic mapping of the microscopic Hamiltonian to
an anisotropic SU(4) spin model on a lattice. To study the properties of this
model we generalize the Heisenberg model Schwinger boson mean field theory
(SBMFT) of Arovas and Auerbach to spin models with anisotropy. We calculate the
temperature dependence of experimentally observable quantities, including the
spin magnetization, and the differential interlayer capacitance. Our theory
represents a substantial improvement over the conventional Hartree-Fock picture
which neglects quantum and thermal fluctuations, and has advantages over
long-wavelength effective models that fail to capture important microscopic
physics at all realistic layer separations. The formalism we develop can be
generalized to treat quantum Hall bilayers at filling factor .Comment: 26 pages, 10 figures. The final version, to appear in PR
Scattering theory and ground-state energy of Dirac fermions in graphene with two Coulomb impurities
We study the physics of Dirac fermions in a gapped graphene monolayer containing two Coulomb impurities. For the case of equal impurity charges, we discuss the ground-state energy using the linear combination of atomic orbitals (LCAO) approach. For opposite charges of the Coulomb centers, an electric dipole potential results at large distances. We provide a nonperturbative analysis of the corresponding low-energy scattering problem
Solitons in a Grassmannian sigma-model Coupled to Chern-Simons Term
We propose an exactly solvable Grassmannian sigma-model coupled to the
Chern-Simons theory. In the presence of a novel topological term our model
admits exact self-dual vortex solutions which are identical to those of pure
Grassmannian model, but the topological charge has a physical meaning as a
magnetic flux since the gauge field is no longer auxiliary. We also extend the
theory to a noncommutative plane and analyze the BPS solutions.Comment: 10+1 pages, No figure, LaTeX; Reference added, Minor changes, to
appear in Phys. Rev.