671 research outputs found
Null twisted geometries
We define and investigate a quantisation of null hypersurfaces in the context
of loop quantum gravity on a fixed graph. The main tool we use is the
parametrisation of the theory in terms of twistors, which has already proved
useful in discussing the interpretation of spin networks as the quantization of
twisted geometries. The classical formalism can be extended in a natural way to
null hypersurfaces, with the Euclidean polyhedra replaced by null polyhedra
with space-like faces, and SU(2) by the little group ISO(2). The main
difference is that the simplicity constraints present in the formalims are all
first class, and the symplectic reduction selects only the helicity subgroup of
the little group. As a consequence, information on the shapes of the polyhedra
is lost, and the result is a much simpler, abelian geometric picture. It can be
described by an Euclidean singular structure on the 2-dimensional space-like
surface defined by a foliation of space-time by null hypersurfaces. This
geometric structure is naturally decomposed into a conformal metric and scale
factors, forming locally conjugate pairs. Proper action-angle variables on the
gauge-invariant phase space are described by the eigenvectors of the Laplacian
of the dual graph. We also identify the variables of the phase space amenable
to characterize the extrinsic geometry of the foliation. Finally, we quantise
the phase space and its algebra using Dirac's algorithm, obtaining a notion of
spin networks for null hypersurfaces. Such spin networks are labelled by SO(2)
quantum numbers, and are embedded non-trivially in the unitary,
infinite-dimensional irreducible representations of the Lorentz group.Comment: 22 pages, 3 figures. v2: minor corrections, improved presentation in
section 4, references update
Group Field theory and Tensor Networks: towards a Ryu-Takayanagi formula in full quantum gravity
We establish a dictionary between group field theory (thus, spin networks and
random tensors) states and generalized random tensor networks. Then, we use
this dictionary to compute the R\'{e}nyi entropy of such states and recover the
Ryu-Takayanagi formula, in two different cases corresponding to two different
truncations/approximations, suggested by the established correspondence.Comment: 54 pages, 10 figures; v2: replace figure 1 with a new version.
Matches submitted version. v3: remove Renyi entropy computation on the random
tensor network, focusing on GFT computation and interpretatio
Area Law from Loop Quantum Gravity
We explore the constraints following from requiring the Area Law in the
entanglement entropy in the context of loop quantum gravity. We find a unique
solution to the single link wave-function in the large j limit, believed to be
appropriate in the semi-classical limit. We then generalize our considerations
to multi-link coherent states, and find that the area law is preserved very
generically using our single link wave-function as a building block. Finally,
we develop the framework that generates families of multi-link states that
preserve the area law while avoiding macroscopic entanglement, the space-time
analogue of "Schroedinger cat". We note that these states, defined on a given
set of graphs, are the ground states of some local Hamiltonian that can be
constructed explicitly. This can potentially shed light on the construction of
the appropriate Hamiltonian constraints in the LQG framework.Comment: 6+5 pages, 2 figures, presentation improved, appendices added,
revised version accepted for publication in Physical Review
Integrated Deep and Shallow Networks for Salient Object Detection
Deep convolutional neural network (CNN) based salient object detection
methods have achieved state-of-the-art performance and outperform those
unsupervised methods with a wide margin. In this paper, we propose to integrate
deep and unsupervised saliency for salient object detection under a unified
framework. Specifically, our method takes results of unsupervised saliency
(Robust Background Detection, RBD) and normalized color images as inputs, and
directly learns an end-to-end mapping between inputs and the corresponding
saliency maps. The color images are fed into a Fully Convolutional Neural
Networks (FCNN) adapted from semantic segmentation to exploit high-level
semantic cues for salient object detection. Then the results from deep FCNN and
RBD are concatenated to feed into a shallow network to map the concatenated
feature maps to saliency maps. Finally, to obtain a spatially consistent
saliency map with sharp object boundaries, we fuse superpixel level saliency
map at multi-scale. Extensive experimental results on 8 benchmark datasets
demonstrate that the proposed method outperforms the state-of-the-art
approaches with a margin.Comment: Accepted by IEEE International Conference on Image Processing (ICIP)
201
Canonical blow-ups of Grassmannians II
We give a linear algebraic construction of the Lafforgue spaces associated to
the Grassmannians by blowing up certain explicitly defined monomial
ideals, which sharpens and generalizes a result of Faltings. As an application,
we provide a family of homogeneous varieties with high complexity and with nice
compactifications, which exhibits the notion of homeward compactification
introduced in our previous work in a non-spherical setting
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