820 research outputs found
Cluster ensembles, quantization and the dilogarithm
Cluster ensemble is a pair of positive spaces (X, A) related by a map p: A ->
X. It generalizes cluster algebras of Fomin and Zelevinsky, which are related
to the A-space. We develope general properties of cluster ensembles, including
its group of symmetries - the cluster modular group, and a relation with the
motivic dilogarithm. We define a q-deformation of the X-space. Formulate
general duality conjectures regarding canonical bases in the cluster ensemble
context. We support them by constructing the canonical pairing in the finite
type case.
Interesting examples of cluster ensembles are provided the higher Teichmuller
theory, that is by the pair of moduli spaces corresponding to a split reductive
group G and a surface S defined in math.AG/0311149.
We suggest that cluster ensembles provide a natural framework for higher
quantum Teichmuller theory.Comment: Version 7: Final version. To appear in Ann. Sci. Ecole Normale. Sup.
New material in Section 5. 58 pages, 11 picture
Position space versions of Magueijo-Smolin doubly special relativity proposal and the problem of total momentum
We present and discuss two different possibilities to construct position
space version for Magueijo-Smolin (MS) doubly special relativity proposal. The
first possibility is to start from ordinary special relativity and then to
define conserved momentum in special way. It generates MS invariant as well as
nonlinear MS transformations on the momentum space, leading to consistent
picture for one-particle sector of the theory. The second possibility is based
on the following observation. Besides the nonlinear MS transformations, the MS
energy-momentum relation is invariant also under some inhomogeneous linear
transformations. The latter are induced starting from linearly realized Lorentz
group in five-dimensional position space. Particle dynamics and kinematics are
formulated starting from the corresponding five-dimensional interval. There is
no problem of total momentum in the theory. The formulation admits two observer
independent scales, the speed of light, , and with dimension of
velocity. We speculate on different possibilities to relate with
fundamental constants. In particular, expression of in terms of vacuum
energy suggests emergence of (minimum) quantum of mass.Comment: Latex twice, 14 pages, revised in accordance with the version
publishedin Phys. Rev.
Asymptotic behaviours of the heat kernel in covariant perturbation theory
The trace of the heat kernel is expanded in a basis of nonlocal curvature
invariants of th order. The coefficients of this expansion (the nonlocal
form factors) are calculated to third order in the curvature inclusive. The
early-time and late-time asymptotic behaviours of the trace of the heat kernel
are presented with this accuracy. The late-time behaviour gives the criterion
of analyticity of the effective action in quantum field theory. The latter
point is exemplified by deriving the effective action in two dimensions.Comment: 22 pages, REVTeX, Alberta Thy 45-9
Low-Energy Effective Action in Non-Perturbative Electrodynamics in Curved Spacetime
We study the heat kernel for the Laplace type partial differential operator
acting on smooth sections of a complex spin-tensor bundle over a generic
-dimensional Riemannian manifold. Assuming that the curvature of the U(1)
connection (that we call the electromagnetic field) is constant we compute the
first two coefficients of the non-perturbative asymptotic expansion of the heat
kernel which are of zero and the first order in Riemannian curvature and of
arbitrary order in the electromagnetic field. We apply these results to the
study of the effective action in non-perturbative electrodynamics in four
dimensions and derive a generalization of the Schwinger's result for the
creation of scalar and spinor particles in electromagnetic field induced by the
gravitational field. We discover a new infrared divergence in the imaginary
part of the effective action due to the gravitational corrections, which seems
to be a new physical effect.Comment: LaTeX, 42 page
Black hole puncture initial data with realistic gravitational wave content
We present improved post-Newtonian-inspired initial data for non-spinning
black-hole binaries, suitable for numerical evolution with punctures. We
revisit the work of Tichy et al. [W. Tichy, B. Bruegmann, M. Campanelli, and P.
Diener, Phys. Rev. D 67, 064008 (2003)], explicitly calculating the remaining
integral terms. These terms improve accuracy in the far zone and, for the first
time, include realistic gravitational waves in the initial data. We investigate
the behavior of these data both at the center of mass and in the far zone,
demonstrating agreement of the transverse-traceless parts of the new metric
with quadrupole-approximation waveforms. These data can be used for numerical
evolutions, enabling a direct connection between the merger waveforms and the
post-Newtonian inspiral waveforms.Comment: 13 pages, 7 figures; replaced with published versio
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