497 research outputs found
Explicit determination of a 727-dimensional root space of the hyperbolic Lie algebra
The 727-dimensional root space associated with the level-2 root \bLambda_1
of the hyperbolic Kac--Moody algebra is determined using a recently
developed string theoretic approach to hyperbolic algebras. The explicit form
of the basis reveals a complicated structure with transversal as well as
longitudinal string states present.Comment: 12 pages, LaTeX 2
On the fundamental representation of Borcherds algebras with one imaginary simple root
Borcherds algebras represent a new class of Lie algebras which have almost
all the properties that ordinary Kac-Moody algebras have, and the only major
difference is that these generalized Kac-Moody algebras are allowed to have
imaginary simple roots. The simplest nontrivial examples one can think of are
those where one adds ``by hand'' one imaginary simple root to an ordinary
Kac-Moody algebra. We study the fundamental representation of this class of
examples and prove that an irreducible module is given by the full tensor
algebra over some integrable highest weight module of the underlying Kac-Moody
algebra. We also comment on possible realizations of these Lie algebras in
physics as symmetry algebras in quantum field theory.Comment: 8 page
Precision spectroscopy by photon-recoil signal amplification
Precision spectroscopy of atomic and molecular ions offers a window to new
physics, but is typically limited to species with a cycling transition for
laser cooling and detection. Quantum logic spectroscopy has overcome this
limitation for species with long-lived excited states. Here, we extend quantum
logic spectroscopy to fast, dipole-allowed transitions and apply it to perform
an absolute frequency measurement. We detect the absorption of photons by the
spectroscopically investigated ion through the photon recoil imparted on a
co-trapped ion of a different species, on which we can perform efficient
quantum logic detection techniques. This amplifies the recoil signal from a few
absorbed photons to thousands of fluorescence photons. We resolve the line
center of a dipole-allowed transition in 40Ca+ to 1/300 of its observed
linewidth, rendering this measurement one of the most accurate of a broad
transition. The simplicity and versatility of this approach enables
spectroscopy of many previously inaccessible species.Comment: 25 pages, 6 figures, 1 table, updated supplementary information,
fixed typo
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Anelastic-like nature of the rejuvenation of metallic glasses by cryogenic thermal cycling
Cryogenic thermal cycling (CTC) is an effective treatment for improving the room-temperature plasticity and toughness in metallic glasses. Despite considerable attention to characterizing the effects of CTC, they remain poorly understood. A prominent example is that, contrary to expectation, the stored energy in a metallic glass first rises, and then decreases, as CTC progresses. In this work, CTC is applied to bulk metallic glasses based on Pd, Pt, Ti, or Zr. The effects on calorimetric and mechanical properties are evaluated. Critically, CTC-induced effects, at whatever stage, are found to decay over about one week at room temperature after CTC, returning the properties to those of the as-cast glass. A model is proposed for CTC-induced effects, treating them as analogous to the accumulation of anelastic strain. The implications for analysis of existing data, and for future research on CTC effects, are highlighted
Polytopality and Cartesian products of graphs
We study the question of polytopality of graphs: when is a given graph the
graph of a polytope? We first review the known necessary conditions for a graph
to be polytopal, and we provide several families of graphs which satisfy all
these conditions, but which nonetheless are not graphs of polytopes. Our main
contribution concerns the polytopality of Cartesian products of non-polytopal
graphs. On the one hand, we show that products of simple polytopes are the only
simple polytopes whose graph is a product. On the other hand, we provide a
general method to construct (non-simple) polytopal products whose factors are
not polytopal.Comment: 21 pages, 10 figure
Oriented Matroids -- Combinatorial Structures Underlying Loop Quantum Gravity
We analyze combinatorial structures which play a central role in determining
spectral properties of the volume operator in loop quantum gravity (LQG). These
structures encode geometrical information of the embedding of arbitrary valence
vertices of a graph in 3-dimensional Riemannian space, and can be represented
by sign strings containing relative orientations of embedded edges. We
demonstrate that these signature factors are a special representation of the
general mathematical concept of an oriented matroid. Moreover, we show that
oriented matroids can also be used to describe the topology (connectedness) of
directed graphs. Hence the mathematical methods developed for oriented matroids
can be applied to the difficult combinatorics of embedded graphs underlying the
construction of LQG. As a first application we revisit the analysis of [4-5],
and find that enumeration of all possible sign configurations used there is
equivalent to enumerating all realizable oriented matroids of rank 3, and thus
can be greatly simplified. We find that for 7-valent vertices having no
coplanar triples of edge tangents, the smallest non-zero eigenvalue of the
volume spectrum does not grow as one increases the maximum spin \jmax at the
vertex, for any orientation of the edge tangents. This indicates that, in
contrast to the area operator, considering large \jmax does not necessarily
imply large volume eigenvalues. In addition we give an outlook to possible
starting points for rewriting the combinatorics of LQG in terms of oriented
matroids.Comment: 43 pages, 26 figures, LaTeX. Version published in CQG. Typos
corrected, presentation slightly extende
Signatures of partition functions and their complexity reduction through the KP II equation
A statistical amoeba arises from a real-valued partition function when the
positivity condition for pre-exponential terms is relaxed, and families of
signatures are taken into account. This notion lets us explore special types of
constraints when we focus on those signatures that preserve particular
properties. Specifically, we look at sums of determinantal type, and main
attention is paid to a distinguished class of soliton solutions of the
Kadomtsev-Petviashvili (KP) II equation. A characterization of the signatures
preserving the determinantal form, as well as the signatures compatible with
the KP II equation, is provided: both of them are reduced to choices of signs
for columns and rows of a coefficient matrix, and they satisfy the whole KP
hierarchy. Interpretations in term of information-theoretic properties,
geometric characteristics, and the relation with tropical limits are discussed.Comment: 42 pages, 11 figures. Section 7.1 has been added, the organization of
the paper has been change
Intersecting Solitons, Amoeba and Tropical Geometry
We study generic intersection (or web) of vortices with instantons inside,
which is a 1/4 BPS state in the Higgs phase of five-dimensional N=1
supersymmetric U(Nc) gauge theory on R_t \times (C^\ast)^2 \simeq R^{2,1}
\times T^2 with Nf=Nc Higgs scalars in the fundamental representation. In the
case of the Abelian-Higgs model (Nf=Nc=1), the intersecting vortex sheets can
be beautifully understood in a mathematical framework of amoeba and tropical
geometry, and we propose a dictionary relating solitons and gauge theory to
amoeba and tropical geometry. A projective shape of vortex sheets is described
by the amoeba. Vortex charge density is uniformly distributed among vortex
sheets, and negative contribution to instanton charge density is understood as
the complex Monge-Ampere measure with respect to a plurisubharmonic function on
(C^\ast)^2. The Wilson loops in T^2 are related with derivatives of the Ronkin
function. The general form of the Kahler potential and the asymptotic metric of
the moduli space of a vortex loop are obtained as a by-product. Our discussion
works generally in non-Abelian gauge theories, which suggests a non-Abelian
generalization of the amoeba and tropical geometry.Comment: 39 pages, 11 figure
A Single Laser System for Ground-State Cooling of 25-Mg+
We present a single solid-state laser system to cool, coherently manipulate
and detect Mg ions. Coherent manipulation is accomplished by
coupling two hyperfine ground state levels using a pair of far-detuned Raman
laser beams. Resonant light for Doppler cooling and detection is derived from
the same laser source by means of an electro-optic modulator, generating a
sideband which is resonant with the atomic transition. We demonstrate
ground-state cooling of one of the vibrational modes of the ion in the trap
using resolved-sideband cooling. The cooling performance is studied and
discussed by observing the temporal evolution of Raman-stimulated sideband
transitions. The setup is a major simplification over existing state-of-the-art
systems, typically involving up to three separate laser sources
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