scholarly journals The Role of Spin(9) in Octonionic Geometry

Axioms ◽  
2018 ◽  
Vol 7 (4) ◽  
pp. 72 ◽  
Author(s):  
Maurizio Parton ◽  
Paolo Piccinni
Keyword(s):  
Vector Fields ◽  
Hopf Fibration ◽  
Intrinsic Torsion ◽  
Quaternionic Geometry ◽  
Clifford Systems ◽  
Clifford Structures

Starting from the 2001 Thomas Friedrich’s work on Spin ( 9 ) , we review some interactions between Spin ( 9 ) and geometries related to octonions. Several topics are discussed in this respect: explicit descriptions of the Spin ( 9 ) canonical 8-form and its analogies with quaternionic geometry as well as the role of Spin ( 9 ) both in the classical problems of vector fields on spheres and in the geometry of the octonionic Hopf fibration. Next, we deal with locally conformally parallel Spin ( 9 ) manifolds in the framework of intrinsic torsion. Finally, we discuss applications of Clifford systems and Clifford structures to Cayley–Rosenfeld planes and to three series of Grassmannians.

scholarly journals The Role of Spin(9) in Octonionic Geometry

2018 ◽  
Author(s):  
Maurizio Parton ◽  
Paolo Piccinni
Keyword(s):  
Riemannian Manifolds ◽  
Vector Fields ◽  
Hopf Fibration ◽  
Intrinsic Torsion ◽  
Quaternionic Geometry ◽  
Clifford Systems ◽  
Clifford Structures

Starting from Thomas Friedrich’s work “Weak Spin(9) structures on 16-dimensional Riemannian manifolds”, we review several interactions between Spin(9) and geometries related to octonions. Several topics are discussed in this respect: explicit descriptions of the Spin(9) canonical 8-form and its analogies with quaternionic geometry, the role of Spin(9) both in the classical problems of vector fields on spheres and in the geometry of the octonionic Hopf fibration. Next, we deal with locally conformally parallel Spin(9) manifolds in the framework of intrinsic torsion. Finally, we discuss applications of Clifford systems and Clifford structures to Cayley-Rosenfeld planes and to three series of Grassmannians.

Symmetries from the solution manifold

2015 ◽  
Vol 12 (08) ◽  
pp. 1560016 ◽  
Author(s):  
Víctor Aldaya ◽  
Julio Guerrero ◽  
Francisco F. Lopez-Ruiz ◽  
Francisco Cossío
Keyword(s):  
Symplectic Structure ◽  
Vector Fields ◽  
Sigma Model ◽  
Canonical Quantization ◽  
General Concept ◽  
Noether Theorem ◽  
Preliminary Step ◽  
Hamiltonian Vector Fields ◽  
Solution Manifold

We face a revision of the role of symmetries of a physical system aiming at characterizing the corresponding Solution Manifold (SM) by means of Noether invariants as a preliminary step towards a proper, non-canonical, quantization. To this end, "point symmetries" of the Lagrangian are generally not enough, and we must resort to the more general concept of contact symmetries. They are defined in terms of the Poincaré–Cartan form, which allows us, in turn, to find the symplectic structure on the SM, through some sort of Hamilton–Jacobi (HJ) transformation. These basic symmetries are realized as Hamiltonian vector fields, associated with (coordinate) functions on the SM, lifted back to the Evolution Manifold through the inverse of this HJ mapping, that constitutes an inverse of the Noether Theorem. The specific examples of a particle moving on S3, at the mechanical level, and nonlinear SU(2)-sigma model in field theory are sketched.

Semilinear PDEs in the Heisenberg group: The role of the right invariant vector fields

2010 ◽  
Vol 72 (2) ◽  
pp. 987-997 ◽  
Author(s):  
Isabeau Birindelli ◽  
Fausto Ferrari ◽  
Enrico Valdinoci
Keyword(s):  
Heisenberg Group ◽  
Vector Fields ◽  
Invariant Vector ◽  
Semilinear Pdes ◽  
The Right ◽  
Invariant Vector Fields

scholarly journals LIOUVILLE THEORY AND SPECIAL COADJOINT VIRASORO ORBITS

1995 ◽  
Vol 10 (06) ◽  
pp. 785-799 ◽  
Author(s):  
A. GORSKY ◽  
A. JOHANSEN
Keyword(s):  
Quantum Mechanics ◽  
Type Theory ◽  
Vector Fields ◽  
Liouville Theory ◽  
Hamiltonian Reduction ◽  
Coadjoint Orbits ◽  
Point Particles ◽  
Witten Theory ◽  
Wrong Sign

We describe the Hamiltonian reduction of the coadjoint Kac–Moody orbits to the Virasoro coadjoint orbits explicitly in terms of the Lagrangian approach for the Wess–Zumino–Novikov–Witten theory. While a relation of the coadjoint Virasoro orbit Diff S1/ SL (2, R) to the Liouville theory has already been studied, we analyze the role of special coadjoint Virasoro orbits Diff [Formula: see text]corresponding to stabilizers generated by the vector fields with double zeros. The orbits with stabilizers with single zeros do not appear in the model. We find an interpretation of zeros xi of the vector field of stabilizer [Formula: see text] and additional parameters qi, i = 1, …, n, in terms of quantum mechanics for n-point particles on the circle. We argue that the special orbits are generated by insertions of "wrong sign" Liouville exponential into the path integral. The additional parmeters qi are naturally interpreted as accessory parameters for the uniformization map. Summing up the contributions of the special Virasoro orbits we get the integrable sinh–Gordon type theory.

scholarly journals Skyrmions, Quantum Hall Droplets, and one current to rule them all

2020 ◽  
Vol 9 (1) ◽  
Author(s):  
Avner Karasik
Keyword(s):  
Vector Fields ◽  
Quantum Hall ◽  
Local Symmetry ◽  
Effective Theory ◽  
Vector Mesons ◽  
Low Energies ◽  
Baryon Current ◽  
Chern Simons ◽  
Natural Way

We introduce a novel Skyrme-like conserved current in the effective theory of pions and vector mesons based on the idea of hidden local symmetry. The associated charge is equivalent to the skyrmion charge for any smooth configuration. In addition, there exist singular configurations that can be identified as N_f=1Nf=1 baryons charged under the new symmetry. Under this identification, the vector mesons play the role of the Chern-Simons vector fields living on the quantum Hall droplet that forms the N_f=1Nf=1 baryon. We propose that this current is the correct effective expression for the baryon current at low energies. This proposal gives a unified picture for the two types of baryons and allows them to continuously transform one to the other in a natural way. In addition, Chern-Simons dualities on the droplet can be interpreted as a result of Seiberg-like duality between gluons and vector mesons.

scholarly journals A Divergence-Based Approach for the Identification of Atrial Fibrillation Focal Drivers From Multipolar Mapping: A Computational Study

2021 ◽  
Vol 12 ◽  
Author(s):  
Michela Masè ◽  
Alessandro Cristoforetti ◽  
Maurizio Del Greco ◽  
Flavia Ravelli
Keyword(s):  
Atrial Fibrillation ◽  
Vector Fields ◽  
Computational Study ◽  
Pulmonary Veins ◽  
Proof Of Concept ◽  
Activation Time ◽  
Mapping Data ◽  
Activation Patterns ◽  
Novel Approach

The expanding role of catheter ablation of atrial fibrillation (AF) has stimulated the development of novel mapping strategies to guide the procedure. We introduce a novel approach to characterize wave propagation and identify AF focal drivers from multipolar mapping data. The method reconstructs continuous activation patterns in the mapping area by a radial basis function (RBF) interpolation of multisite activation time series. Velocity vector fields are analytically determined, and the vector field divergence is used as a marker of focal drivers. The method was validated in a tissue patch cellular automaton model and in an anatomically realistic left atrial (LA) model with Courtemanche–Ramirez–Nattel ionic dynamics. Divergence analysis was effective in identifying focal drivers in a complex simulated AF pattern. Localization was reliable even with consistent reduction (47%) in the number of mapping points and in the presence of activation time misdetections (noise <10% of the cycle length). Proof-of-concept application of the method to human AF mapping data showed that divergence analysis consistently detected focal activation in the pulmonary veins and LA appendage area. These results suggest the potential of divergence analysis in combination with multipolar mapping to identify AF critical sites. Further studies on large clinical datasets may help to assess the clinical feasibility and benefit of divergence analysis for the optimization of ablation treatment.

scholarly journals The role of vector fields in modified gravity scenarios

2013 ◽  
Vol 2013 (11) ◽  
pp. 037-037 ◽  
Author(s):  
Gianmassimo Tasinato ◽  
Kazuya Koyama ◽  
Nima Khosravi
Keyword(s):  
Modified Gravity ◽  
Vector Fields

scholarly journals Holographic Projection of Electromagnetic Maxwell Theory

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1134
Author(s):  
Erica Bertolini ◽  
Nicola Maggiore
Keyword(s):  
Gauge Theory ◽  
Degrees Of Freedom ◽  
Vector Fields ◽  
Central Charge ◽  
Momentum Tensor ◽  
Boundary Theory ◽  
Coupling Constant ◽  
Maxwell Theory ◽  
Chern Simons

The 4D Maxwell theory with single-sided planar boundary is considered. As a consequence of the presence of the boundary, two broken Ward identities are recovered, which, on-shell, give rise to two conserved currents living on the edge. A Kaç-Moody algebra formed by a subset of the bulk fields is obtained with central charge proportional to the inverse of the Maxwell coupling constant, and the degrees of freedom of the boundary theory are identified as two vector fields, also suggesting that the 3D theory should be a gauge theory. Finally the holographic contact between bulk and boundary theory is reached in two inequivalent ways, both leading to a unique 3D action describing a new gauge theory of two coupled vector fields with a topological Chern-Simons term with massive coefficient. In order to check that the 3D projection of 4D Maxwell theory is well defined, we computed the energy-momentum tensor and the propagators. The role of discrete symmetries is briefly discussed.

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