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 $G(2,n)$ 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