3,612 research outputs found
Analytic families of quantum hyperbolic invariants
We organize the quantum hyperbolic invariants (QHI) of -manifolds into
sequences of rational functions indexed by the odd integers and
defined on moduli spaces of geometric structures refining the character
varieties. In the case of one-cusped hyperbolic -manifolds we generalize
the QHI and get rational functions depending on a
finite set of cohomological data called {\it weights}. These
functions are regular on a determined Abelian covering of degree of a
Zariski open subset, canonically associated to , of the geometric component
of the variety of augmented -characters of . New
combinatorial ingredients are a weak version of branchings which exists on
every triangulation, and state sums over weakly branched triangulations,
including a sign correction which eventually fixes the sign ambiguity of the
QHI. We describe in detail the invariants of three cusped manifolds, and
present the results of numerical computations showing that the functions
depend on the weights as ,
and recover the volume for some specific choices of the weights.Comment: 54 pages, 21 figures. New section with 3 examples; the results about
the reduced invariants are postponed to a separate paper. To appear on Alg.
Geom. Topo
Quantification of Nematic Cell Polarity in Three-dimensional Tissues
How epithelial cells coordinate their polarity to form functional tissues is
an open question in cell biology. Here, we characterize a unique type of
polarity found in liver tissue, nematic cell polarity, which is different from
vectorial cell polarity in simple, sheet-like epithelia. We propose a
conceptual and algorithmic framework to characterize complex patterns of
polarity proteins on the surface of a cell in terms of a multipole expansion.
To rigorously quantify previously observed tissue-level patterns of nematic
cell polarity (Morales-Navarette et al., eLife 8:e44860, 2019), we introduce
the concept of co-orientational order parameters, which generalize the known
biaxial order parameters of the theory of liquid crystals. Applying these
concepts to three-dimensional reconstructions of single cells from
high-resolution imaging data of mouse liver tissue, we show that the axes of
nematic cell polarity of hepatocytes exhibit local coordination and are aligned
with the biaxially anisotropic sinusoidal network for blood transport. Our
study characterizes liver tissue as a biological example of a biaxial liquid
crystal. The general methodology developed here could be applied to other
tissues or in-vitro organoids.Comment: 27 pages, 9 color figure
The GIST of Concepts
A unified general theory of human concept learning based on the idea that humans detect invariance patterns in categorical stimuli as a necessary precursor to concept formation is proposed and tested. In GIST (generalized invariance structure theory) invariants are detected via a perturbation mechanism of dimension suppression referred to as dimensional binding. Structural information acquired by this process is stored as a compound memory trace termed an ideotype. Ideotypes inform the subsystems that are responsible for learnability judgments, rule formation, and other types of concept representations. We show that GIST is more general (e.g., it works on continuous, semi-continuous, and binary stimuli) and makes much more accurate predictions than the leading models of concept learning difficulty,such as those based on a complexity reduction principle (e.g., number of mental models,structural invariance, algebraic complexity, and minimal description length) and those based on selective attention and similarity (GCM, ALCOVE, and SUSTAIN). GIST unifies these two key aspects of concept learning and categorization. Empirical evidence from three\ud
experiments corroborates the predictions made by the theory and its core model which we propose as a candidate law of human conceptual behavior
CP violation conditions in N-Higgs-doublet potentials
Conditions for CP violation in the scalar potential sector of general
N-Higgs-doublet models (NHDMs) are analyzed from a group theoretical
perspective. For the simplest two-Higgs-doublet model (2HDM) potential, a
minimum set of conditions for explicit and spontaneous CP violation is
presented. The conditions can be given a clear geometrical interpretation in
terms of quantities in the adjoint representation of the basis transformation
group for the two doublets. Such conditions depend on CP-odd pseudoscalar
invariants. When the potential is CP invariant, the explicit procedure to reach
the real CP-basis and the explicit CP transformation can also be obtained. The
procedure to find the real basis and the conditions for CP violation are then
extended to general NHDM potentials. The analysis becomes more involved and
only a formal procedure to reach the real basis is found. Necessary conditions
for CP invariance can still be formulated in terms of group invariants: the
CP-odd generalized pseudoscalars. The problem can be completely solved for
three Higgs-doublets.Comment: RevTeX4 used. Minor modifications, in particular, the parameter
counting of . v3: Eqs.(28)-(31) correcte
The dissimilarity map and representation theory of
We give another proof that -dissimilarity vectors of weighted trees are
points on the tropical Grassmanian, as conjectured by Cools, and proved by
Giraldo in response to a question of Sturmfels and Pachter. We accomplish this
by relating -dissimilarity vectors to the representation theory of Comment: 11 pages, 8 figure
Are neutrino masses modular forms?
We explore a new class of supersymmetric models for lepton masses and mixing
angles where the role of flavour symmetry is played by modular invariance. The
building blocks are modular forms of level N and matter supermultiplets, both
transforming in representations of a finite discrete group Gamma_N. In the
simplest version of these models, Yukawa couplings are just modular forms and
the only source of flavour symmetry breaking is the vacuum expectation value of
a single complex field, the modulus. In the special case where modular forms
are constant functions the whole construction collapses to a supersymmetric
flavour model invariant under Gamma_N, the case treated so far in the
literature. The framework has a number of appealing features. Flavon fields
other than the modulus might not be needed. Neutrino masses and mixing angles
are simultaneously constrained by the modular symmetry. As long as
supersymmetry is exact, modular invariance determines all higher-dimensional
operators in the superpotential. We discuss the general framework and we
provide complete examples of the new construction. The most economical model
predicts neutrino mass ratios, lepton mixing angles, Dirac and Majorana phases
uniquely in terms of the modulus vacuum expectation value, with all the
parameters except one within the experimentally allowed range. As a byproduct
of the general formalism we extend the notion of non-linearly realised
symmetries to the discrete case.Comment: 40 pages, 3 figures; added comments and a new section with an example
of normal ordering of neutrino masses; to appear in the book "From my vast
repertoire: the legacy of Guido Altarelli", S. Forte, A. Levy and G. Ridolfi,
ed
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
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