877 research outputs found
The quantum Casimir operators of \Uq and their eigenvalues
We show that the quantum Casimir operators of the quantum linear group
constructed in early work of Bracken, Gould and Zhang together with one extra
central element generate the entire center of \Uq. As a by product of the
proof, we obtain intriguing new formulae for eigenvalues of these quantum
Casimir operators, which are expressed in terms of the characters of a class of
finite dimensional irreducible representations of the classical general linear
algebra.Comment: 10 page
The Kazhdan-Lusztig conjecture for W-algebras
The main result in this paper is the character formula for arbitrary
irreducible highest weight modules of W algebras. The key ingredient is the
functor provided by quantum Hamiltonian reduction, that constructs the W
algebras from affine Kac-Moody algebras and in a similar fashion W modules from
KM modules. Assuming certain properties of this functor, the W characters are
subsequently derived from the Kazhdan-Lusztig conjecture for KM algebras. The
result can be formulated in terms of a double coset of the Weyl group of the KM
algebra: the Hasse diagrams give the embedding diagrams of the Verma modules
and the Kazhdan-Lusztig polynomials give the multiplicities in the characters.Comment: uuencoded file, 29 pages latex, 5 figure
Monotonic functions in Bianchi models: Why they exist and how to find them
All rigorous and detailed dynamical results in Bianchi cosmology rest upon
the existence of a hierarchical structure of conserved quantities and monotonic
functions. In this paper we uncover the underlying general mechanism and derive
this hierarchical structure from the scale-automorphism group for an
illustrative example, vacuum and diagonal class A perfect fluid models. First,
kinematically, the scale-automorphism group leads to a reduced dynamical system
that consists of a hierarchy of scale-automorphism invariant sets. Second, we
show that, dynamically, the scale-automorphism group results in
scale-automorphism invariant monotone functions and conserved quantities that
restrict the flow of the reduced dynamical system.Comment: 26 pages, replaced to match published versio
A new proof of the Bianchi type IX attractor theorem
We consider the dynamics towards the initial singularity of Bianchi type IX
vacuum and orthogonal perfect fluid models with a linear equation of state. The
`Bianchi type IX attractor theorem' states that the past asymptotic behavior of
generic type IX solutions is governed by Bianchi type I and II vacuum states
(Mixmaster attractor). We give a comparatively short and self-contained new
proof of this theorem. The proof we give is interesting in itself, but more
importantly it illustrates and emphasizes that type IX is special, and to some
extent misleading when one considers the broader context of generic models
without symmetries.Comment: 26 pages, 5 figure
Electrocardiogram of the Mixmaster Universe
The Mixmaster dynamics is revisited in a new light as revealing a series of
transitions in the complex scale invariant scalar invariant of the Weyl
curvature tensor best represented by the speciality index , which
gives a 4-dimensional measure of the evolution of the spacetime independent of
all the 3-dimensional gauge-dependent variables except for the time used to
parametrize it. Its graph versus time characterized by correlated isolated
pulses in its real and imaginary parts corresponding to curvature wall
collisions serves as a sort of electrocardiogram of the Mixmaster universe,
with each such pulse pair arising from a single circuit or ``complex pulse''
around the origin in the complex plane. These pulses in the speciality index
and their limiting points on the real axis seem to invariantly characterize
some of the so called spike solutions in inhomogeneous cosmology and should
play an important role as a gauge invariant lens through which to view current
investigations of inhomogeneous Mixmaster dynamics.Comment: version 3: 20 pages iopart style, 19 eps figure files for 8 latex
figures; added example of a transient true spike to contrast with the
permanent true spike example from the Lim family of true spike solutions;
remarks in introduction and conclusion adjusted and toned down; minor
adjustments to the remaining tex
Circular holonomy in the Taub-NUT spacetime
Parallel transport around closed circular orbits in the equatorial plane of
the Taub-NUT spacetime is analyzed to reveal the effect of the gravitomagnetic
monopole parameter on circular holonomy transformations. Investigating the
boost/rotation decomposition of the connection 1-form matrix evaluated along
these orbits, one finds a situation that reflects the behavior of the general
orthogonally transitive stationary axisymmetric case and indeed along Killing
trajectories in general.Comment: 9 pages, LaTeX iopart class, no figure
Quantum state transfer in a q-deformed chain
We investigate the quantum state transfer in a chain of particles satisfying
q-deformed oscillators algebra. This general algebraic setting includes the
spin chain and the bosonic chain as limiting cases. We study conditions for
perfect state transfer depending on the number of sites and excitations on the
chain. They are formulated by means of irreducible representations of a quantum
algebra realized through Jordan-Schwinger maps. Playing with deformation
parameters, we can study the effects of nonlinear perturbations or interpolate
between the spin and bosonic chain.Comment: 13 pages, 4 figure
On unitarizability in the case of classical p-adic groups
In the introduction of this paper we discuss a possible approach to the
unitarizability problem for classical p-adic groups. In this paper we give some
very limited support that such approach is not without chance. In a forthcoming
paper we shall give additional evidence in generalized cuspidal rank (up to)
three.Comment: This paper is a merged and revised version of ealier preprints
arXiv:1701.07658 and arXiv:1701.07662. The paper is going to appear in the
Proceedings of the Simons Symposium on Geometric Aspects of the Trace Formul
Factorization Structure of Gauge Theory Amplitudes and Application to Hard Scattering Processes at the LHC
Previous work on electroweak radiative corrections to high energy scattering
using soft-collinear effective theory (SCET) has been extended to include
external transverse and longitudinal gauge bosons and Higgs bosons. This allows
one to compute radiative corrections to all parton-level hard scattering
amplitudes in the standard model to NLL order, including QCD and electroweak
radiative corrections, mass effects, and Higgs exchange corrections, if the
high-scale matching, which is suppressed by two orders in the log counting, and
contains no large logs, is known. The factorization structure of the effective
theory places strong constraints on the form of gauge theory amplitudes at high
energy for massless and massive gauge theories, which are discussed in detail
in the paper. The radiative corrections can be written as the sum of
process-independent one-particle collinear functions, and a universal soft
function. We give plots for the radiative corrections to q qbar -> W_T W_T, Z_T
Z_T, W_L W_L, and Z_L H, and gg -> W_T W_T to illustrate our results. The
purely electroweak corrections are large, ranging from 12% at 500 GeV to 37% at
2 TeV for transverse W pair production, and increasing rapidly with energy. The
estimated theoretical uncertainty to the partonic (hard) cross-section in most
cases is below one percent, smaller than uncertainties in the parton
distribution functions (PDFs). We discuss the relation between SCET and other
factorization methods, and derive the Magnea-Sterman equations for the Sudakov
form factor using SCET, for massless and massive gauge theories, and for light
and heavy external particles.Comment: 44 pages, 30 figures. Refs added, typos fixed. ZL ZL plots removed
because of a possible subtlet
Spin precession in the Schwarzschild spacetime: circular orbits
We study the behavior of nonzero rest mass spinning test particles moving
along circular orbits in the Schwarzschild spacetime in the case in which the
components of the spin tensor are allowed to vary along the orbit, generalizing
some previous work.Comment: To appear on Classical and Quantum Gravity, 200
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