298 research outputs found
Category O over a deformation of the symplectic oscillator algebra
We discuss the representation theory of , which is a deformation of the
symplectic oscillator algebra , where is the
((2n+1)-dimensional) Heisenberg algebra. We first look at a more general setup,
involving an algebra with a triangular decomposition. Assuming the PBW theorem,
and one other hypothesis, we show that the BGG category is
abelian, finite length, and self-dual.
We decompose as a direct sum of blocks \calo(\la), and show
that each block is a highest weight category.
In the second part, we focus on the case for , where we prove all
these assumptions, as well as the PBW theorem.Comment: 42 pages, LaTeX, 11pt; Typos removed, references added, presentation
improved, minor corrections and additions, Section 16 modified, and Standing
Assumption added in Section 17; Final form, to appear in the Journal of Pure
and Applied Algebr
A Product Formula for the Normalized Volume of Free Sums of Lattice Polytopes
The free sum is a basic geometric operation among convex polytopes. This note
focuses on the relationship between the normalized volume of the free sum and
that of the summands. In particular, we show that the normalized volume of the
free sum of full dimensional polytopes is precisely the product of the
normalized volumes of the summands.Comment: Published in the proceedings of 2017 Southern Regional Algebra
Conferenc
Ground states of supersymmetric Yang-Mills-Chern-Simons theory
We consider minimally supersymmetric Yang-Mills theory with a Chern-Simons
term on a flat spatial two-torus. The Witten index may be computed in the weak
coupling limit, where the ground state wave-functions localize on the moduli
space of flat gauge connections. We perform such computations by considering
this moduli space as an orbifold of a certain flat complex torus. Our results
agree with those obtained previously by instead considering the moduli space as
a complex projective space. An advantage of the present method is that it
allows for a more straightforward determination of the discrete electric 't
Hooft fluxes of the ground states in theories with non-simply connected gauge
groups. A consistency check is provided by the invariance of the results under
the mapping class group of a (Euclidean) three-torus.Comment: 18 page
Minkowski superspaces and superstrings as almost real-complex supermanifolds
In 1996/7, J. Bernstein observed that smooth or analytic supermanifolds that
mathematicians study are real or (almost) complex ones, while Minkowski
superspaces are completely different objects. They are what we call almost
real-complex supermanifolds, i.e., real supermanifolds with a non-integrable
distribution, the collection of subspaces of the tangent space, and in every
subspace a complex structure is given.
An almost complex structure on a real supermanifold can be given by an even
or odd operator; it is complex (without "always") if the suitable superization
of the Nijenhuis tensor vanishes. On almost real-complex supermanifolds, we
define the circumcised analog of the Nijenhuis tensor. We compute it for the
Minkowski superspaces and superstrings. The space of values of the circumcised
Nijenhuis tensor splits into (indecomposable, generally) components whose
irreducible constituents are similar to those of Riemann or Penrose tensors.
The Nijenhuis tensor vanishes identically only on superstrings of
superdimension 1|1 and, besides, the superstring is endowed with a contact
structure. We also prove that all real forms of complex Grassmann algebras are
isomorphic although singled out by manifestly different anti-involutions.Comment: Exposition of the same results as in v.1 is more lucid. Reference to
related recent work by Witten is adde
Is there a need and another way to measure the Cosmic Microwave Background temperature more accurately?
The recombination history of the Universe depends exponentially on the
temperature, T_0, of the cosmic microwave background. Therefore tiny changes of
T_0 are expected to lead to significant changes in the free electron fraction.
Here we show that even the current 1sigma-uncertainty in the value of T_0
results in more than half a percent ambiguity in the ionization history, and
more than 0.1% uncertainty in the TT and EE power spectra at small angular
scales. We discuss how the value of T_0 affects the highly redshifted
cosmological hydrogen recombination spectrum and demonstrate that T_0 could, in
principle, be measured by looking at the low frequency distortions of the
cosmic microwave background spectrum. For this no absolute measurements are
necessary, but sensitivities on the level of ~30nK are required to extract the
quasi-periodic frequency-dependent signal with typical Delta nu/nu~0.1 coming
from cosmological recombination. We also briefly mention the possibility of
obtaining additional information on the specific entropy of the Universe, and
other cosmological parameters.Comment: 4+epsilon pages, 4 Figures, accepted versio
Constraints on the cosmological density parameters and cosmic topology
A nontrivial topology of the spatial section of the universe is an
observable, which can be probed for all locally homogeneous and isotropic
universes, without any assumption on the cosmological density parameters. We
discuss how one can use this observable to set constraints on the density
parameters of the Universe by using a specific spatial topology along with type
Ia supenovae and X-ray gas mass fraction data sets.Comment: 11 pages, 4 figures. To appear in Int. J. Mod. Phys. D (2006).
Invited talk delivered at the 2nd International Workshop on Astronomy and
Relativistic Astrophysic
A Special Homotopy Continuation Method For A Class of Polynomial Systems
A special homotopy continuation method, as a combination of the polyhedral
homotopy and the linear product homotopy, is proposed for computing all the
isolated solutions to a special class of polynomial systems. The root number
bound of this method is between the total degree bound and the mixed volume
bound and can be easily computed. The new algorithm has been implemented as a
program called LPH using C++. Our experiments show its efficiency compared to
the polyhedral or other homotopies on such systems. As an application, the
algorithm can be used to find witness points on each connected component of a
real variety
Advanced Three Level Approximation for Numerical Treatment of Cosmological Recombination
New public numerical code for fast calculations of the cosmological
recombination of primordial hydrogen-helium plasma is presented. The code is
based on the three-level approximation (TLA) model of recombination and allows
us to take into account some fine physical effects of cosmological
recombination simultaneously with using fudge factors. The code can be found at
http://www.ioffe.ru/astro/QC/CMBR/atlant/atlant.htmlComment: 10 pages, 7 figures, 1 table, to be submitted to MNRA
Topology Change in Canonical Quantum Cosmology
We develop the canonical quantization of a midisuperspace model which
contains, as a subspace, a minisuperspace constituted of a
Friedman-Lema\^{\i}tre-Robertson-Walker Universe filled with homogeneous scalar
and dust fields, where the sign of the intrinsic curvature of the spacelike
hypersurfaces of homogeneity is not specified, allowing the study of topology
change in these hypersurfaces. We solve the Wheeler-DeWitt equation of the
midisuperspace model restricted to this minisuperspace subspace in the
semi-classical approximation. Adopting the conditional probability
interpretation, we find that some of the solutions present change of topology
of the homogeneous hypersurfaces. However, this result depends crucially on the
interpretation we adopt: using the usual probabilistic interpretation, we find
selection rules which forbid some of these topology changes.Comment: 23 pages, LaTex file. We added in the conclusion some comments about
path integral formalism and corrected litle misprinting
- …