Tensor network approach to electromagnetic duality in (3+1)d topological gauge models

Given the Hamiltonian realisation of a topological (3+1)d gauge theory with finite group $G$, we consider a family of tensor network representations of its ground state subspace. This family is indexed by gapped boundary conditions encoded into module 2-categories over the input spherical fusion 2-category. Individual tensors are characterised by symmetry conditions with respect to non-local operators acting on entanglement degrees of freedom. In the case of Dirichlet and Neumann boundary conditions, we show that the symmetry operators form the fusion 2-categories $\mathsf{2Vec}_G$ of $G$-graded 2-vector spaces and $\mathsf{2Rep}(G)$ of 2-representations of $G$, respectively. In virtue of the Morita equivalence between $\mathsf{2Vec}_G$ and $\mathsf{2Rep}(G)$ -- which we explicitly establish -- the topological order can be realised as the Drinfel'd centre of either 2-category of operators; this is a realisation of the electromagnetic duality of the theory. Specialising to the case $G = \mathbb Z_2$, we recover tensor network representations that were recently introduced, as well as the relation between the electromagnetic duality of a pure (3+1)d $\mathbb Z_2$ gauge theory and the Kramers-Wannier duality of a boundary (2+1)d Ising model

Gapped boundaries and string-like excitations in (3+1)d gauge models of topological phases

We study lattice Hamiltonian realisations of (3+1)d Dijkgraaf-Witten theory with gapped boundaries. In addition to the bulk loop-like excitations, the Hamiltonian yields bulk dyonic string-like excitations that terminate at gapped boundaries. Using a tube algebra approach, we classify such excitations and derive the corresponding representation theory. Via a dimensional reduction argument, we relate this tube algebra to that describing (2+1)d boundary point-like excitations at interfaces between two gapped boundaries. Such point-like excitations are well known to be encoded into a bicategory of module categories over the input fusion category. Exploiting this correspondence, we define a bicategory that encodes the string-like excitations ending at gapped boundaries, showing that it is a sub-bicategory of the centre of the input bicategory of group-graded 2-vector spaces. In the process, we explain how gapped boundaries in (3+1)d can be labelled by so-called pseudo-algebra objects over this input bicategory

On tensor network representations of the (3+1)d toric code

We define two dual tensor network representations of the (3+1)d toric code ground state subspace. These two representations, which are obtained by initially imposing either family of stabilizer constraints, are characterized by different virtual symmetries generated by string-like and membrane-like operators, respectively. We discuss the topological properties of the model from the point of view of these virtual symmetries, emphasizing the differences between both representations. In particular, we argue that, depending on the representation, the phase diagram of boundary entanglement degrees of freedom is naturally associated with that of a (2+1)d Hamiltonian displaying either a global or a gauge $\mathbb Z_2$-symmetry

Allylic Oxidation of Alkenes Catalyzed by a Copper−Aluminum Mixed Oxide

A strategy for the allylic oxidation of cyclic alkenes with a copper−aluminum mixed oxide as catalyst is presented. The reaction involves the treatment of an alkene with a carboxylic acid employing tert-butyl hydroperoxide as the oxidant. In all cases, the corresponding allylic esters are obtained. When L-proline is employed, the allylic alcohol or ketone is obtained. The oxidation of cyclohexene and valencene has been optimized by design of experiments (DoE) statistical methodology

Acyloxylation of Cyclic Enones: Synthesis of Densely Oxygenated Guaianolides

The α′-acyloxylation of cyclic enones with linear carboxylic acids is described. The reaction is promoted by KMnO4 in the presence of a carboxylic acid and its corresponding carboxylic anhydride. The optimization of the reaction has been carried out using the statistical methodology known as design of experiments. The optimized reaction conditions have been evaluated in terms of substrate scope and compatibility with different functional groups. The methodology has been applied to the synthesis of densely oxygenated guaianes and guaianolides

Submarine record of volcanic island construction and collapse in the Lesser Antilles arc: First scientific drilling of submarine volcanic island landslides by IODP Expedition 340

IODP Expedition 340 successfully drilled a series of sites offshore Montserrat, Martinique and Dominica in the Lesser Antilles from March to April 2012. These are among the few drill sites gathered around volcanic islands, and the first scientific drilling of large and likely tsunamigenic volcanic island-arc landslide deposits. These cores provide evidence and tests of previous hypotheses for the composition and origin of those deposits. Sites U1394, U1399, and U1400 that penetrated landslide deposits recovered exclusively seafloor-sediment, comprising mainly turbidites and hemipelagic deposits, and lacked debris avalanche deposits. This supports the concepts that i/ volcanic debris avalanches tend to stop at the slope break, and ii/ widespread and voluminous failures of pre-existing low-gradient seafloor sediment can be triggered by initial emplacement of material from the volcano. Offshore Martinique (U1399 and 1400), the landslide deposits comprised blocks of parallel strata that were tilted or micro-faulted, sometimes separated by intervals of homogenized sediment (intense shearing), while Site U1394 offshore Montserrat penetrated a flat-lying block of intact strata. The most likely mechanism for generating these large-scale seafloor-sediment failures appears to be propagation of a decollement from proximal areas loaded and incised by a volcanic debris avalanche. These results have implications for the magnitude of tsunami generation. Under some conditions, volcanic island landslide deposits comprised of mainly seafloor sediment will tend to form smaller magnitude tsunamis than equivalent volumes of subaerial block-rich mass flows rapidly entering water. Expedition 340 also successfully drilled sites to access the undisturbed record of eruption fallout layers intercalated with marine sediment which provide an outstanding high-resolution dataset to analyze eruption and landslides cycles, improve understanding of magmatic evolution as well as offshore sedimentation processes. This article is protected by copyright. All rights reserved

Geotourism, iconic landforms and island-style speciation patterns in National Parks of East Africa:

Many of the national parks in East Africa are equally as famous for their iconic landforms as they are for their diversity and concentrations of fauna and flora. The newly formed Ngorongoro-Lengai Geopark in northern Tanzania is the first geopark to be established in the region, but there is remarkable potential for geotourism in the majority of the national parks. The most spectacular landforms have been shaped by the East African Rift System. Formation of the two major rifts in the region, the Albertine Rift (or western branch) and the Gregory Rift (or eastern branch), was accompanied, or in some cases preceded, by extensive alkaline volcanism. The rifting and volcanism are primarily Late Cenozoic phenomenon that dissected and overprinted the older regional plateaus. Rifting impacted the regional drainage and captured major rivers, including the Victoria Nile

Excitations in strict 2-group higher gauge models of topological phases

We consider an exactly solvable model for topological phases in (3+1)d whose input data is a strict 2-group. This model, which has a higher gauge theory interpretation, provides a lattice Hamiltonian realisation of the Yetter homotopy 2-type topological quantum field theory. The Hamiltonian yields bulk flux and charge composite excitations that are either point-like or loop-like. Applying a generalised tube algebra approach, we reveal the algebraic structure underlying these excitations and derive the irreducible modules of this algebra, which in turn classify the elementary excitations of the model. As a further application of the tube algebra approach, we demonstrate that the ground state subspace of the three-torus is described by the central subalgebra of the tube algebra for torus boundary, demonstrating the ground state degeneracy is given by the number of elementary loop-like excitations.Comment: 39 page