901 research outputs found
Conformal Field Theories in Fractional Dimensions
We study the conformal bootstrap in fractional space-time dimensions,
obtaining rigorous bounds on operator dimensions. Our results show strong
evidence that there is a family of unitary CFTs connecting the 2D Ising model,
the 3D Ising model, and the free scalar theory in 4D. We give numerical
predictions for the leading operator dimensions and central charge in this
family at different values of D and compare these to calculations of phi^4
theory in the epsilon-expansion.Comment: 11 pages, 4 figures - references updated - one affiliation modifie
Bootstrapping 3D Fermions with Global Symmetries
We study the conformal bootstrap for 4-point functions of fermions in parity-preserving 3d CFTs, where
transforms as a vector under an global symmetry. We compute
bounds on scaling dimensions and central charges, finding features in our
bounds that appear to coincide with the symmetric Gross-Neveu-Yukawa
fixed points. Our computations are in perfect agreement with the
expansion at large and allow us to make nontrivial predictions at small
. For values of for which the Gross-Neveu-Yukawa universality classes
are relevant to condensed-matter systems, we compare our results to previous
analytic and numerical results.Comment: 29 pages, 7 figure
Gas sensing based on optical fibre coupled diode laser spectroscopy : a new approach to sensor systems for safety monitoring
We describe an entirely passive fibre optic network which senses, amongst other species, CH¬4¬ and CO¬¬2 , with sensitivity and selectivity compatible with safety sensing in the mine environment. The basic principle is that a single laser diode source targeted to a particular species addresses up to 200 sensing points which may be spread over an area of dimensions ten or more km. The detection and processing electronics is typically located with the laser source. Several laser sources can be introduced in parallel to enable monitoring multiple species. The network itself, entirely linked through optical fibre, is inherently intrinsically safe. It is self checking for faults at the sensing location and continuously self calibrating. In the methane sensing mode its sensitivity is sub 100ppm and it responds accurately up to 100% methane. It is therefore capable of detecting extremely hazardous gas pockets which are completely missed by other sensor technologies. The network has demonstrated stability with zero maintenance or recalibration over periods in excess of two years. We believe that this system offers unique benefits in the context of mine safety and ventilation system monitoring
Bootstrapping 3D Fermions
We study the conformal bootstrap for a 4-point function of fermions
in 3D. We first introduce an embedding
formalism for 3D spinors and compute the conformal blocks appearing in fermion
4-point functions. Using these results, we find general bounds on the
dimensions of operators appearing in the OPE, and also on
the central charge . We observe features in our bounds that coincide with
scaling dimensions in the Gross-Neveu models at large . We also speculate
that other features could coincide with a fermionic CFT containing no relevant
scalar operators.Comment: 45 pages, 8 figures; V2: added references and small clarifications to
match JHEP versio
Fermion-Scalar Conformal Blocks
We compute the conformal blocks associated with scalar-scalar-fermion-fermion
4-point functions in 3D CFTs. Together with the known scalar conformal blocks,
our result completes the task of determining the so-called `seed blocks' in
three dimensions. Conformal blocks associated with 4-point functions of
operators with arbitrary spins can now be determined from these seed blocks by
using known differential operators.Comment: 25 pages; V2: added small clarifications to match JHEP versio
Discrete complex analysis on planar quad-graphs
We develop a linear theory of discrete complex analysis on general
quad-graphs, continuing and extending previous work of Duffin, Mercat, Kenyon,
Chelkak and Smirnov on discrete complex analysis on rhombic quad-graphs. Our
approach based on the medial graph yields more instructive proofs of discrete
analogs of several classical theorems and even new results. We provide discrete
counterparts of fundamental concepts in complex analysis such as holomorphic
functions, derivatives, the Laplacian, and exterior calculus. Also, we discuss
discrete versions of important basic theorems such as Green's identities and
Cauchy's integral formulae. For the first time, we discretize Green's first
identity and Cauchy's integral formula for the derivative of a holomorphic
function. In this paper, we focus on planar quad-graphs, but we would like to
mention that many notions and theorems can be adapted to discrete Riemann
surfaces in a straightforward way.
In the case of planar parallelogram-graphs with bounded interior angles and
bounded ratio of side lengths, we construct a discrete Green's function and
discrete Cauchy's kernels with asymptotics comparable to the smooth case.
Further restricting to the integer lattice of a two-dimensional skew coordinate
system yields appropriate discrete Cauchy's integral formulae for higher order
derivatives.Comment: 49 pages, 8 figure
A discrete Laplace-Beltrami operator for simplicial surfaces
We define a discrete Laplace-Beltrami operator for simplicial surfaces. It
depends only on the intrinsic geometry of the surface and its edge weights are
positive. Our Laplace operator is similar to the well known finite-elements
Laplacian (the so called ``cotan formula'') except that it is based on the
intrinsic Delaunay triangulation of the simplicial surface. This leads to new
definitions of discrete harmonic functions, discrete mean curvature, and
discrete minimal surfaces. The definition of the discrete Laplace-Beltrami
operator depends on the existence and uniqueness of Delaunay tessellations in
piecewise flat surfaces. While the existence is known, we prove the uniqueness.
Using Rippa's Theorem we show that, as claimed, Musin's harmonic index provides
an optimality criterion for Delaunay triangulations, and this can be used to
prove that the edge flipping algorithm terminates also in the setting of
piecewise flat surfaces.Comment: 18 pages, 6 vector graphics figures. v2: Section 2 on Delaunay
triangulations of piecewise flat surfaces revised and expanded. References
added. Some minor changes, typos corrected. v3: fixed inaccuracies in
discussion of flip algorithm, corrected attributions, added references, some
minor revision to improve expositio
Functional Maps Representation on Product Manifolds
We consider the tasks of representing, analyzing and manipulating maps
between shapes. We model maps as densities over the product manifold of the
input shapes; these densities can be treated as scalar functions and therefore
are manipulable using the language of signal processing on manifolds. Being a
manifold itself, the product space endows the set of maps with a geometry of
its own, which we exploit to define map operations in the spectral domain; we
also derive relationships with other existing representations (soft maps and
functional maps). To apply these ideas in practice, we discretize product
manifolds and their Laplace--Beltrami operators, and we introduce localized
spectral analysis of the product manifold as a novel tool for map processing.
Our framework applies to maps defined between and across 2D and 3D shapes
without requiring special adjustment, and it can be implemented efficiently
with simple operations on sparse matrices.Comment: Accepted to Computer Graphics Foru
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