scholarly journals Skyrmions from gravitational instantons

2013 ◽  
Vol 469 (2151) ◽  
pp. 20120576 ◽  
Author(s):  
Maciej Dunajski
Keyword(s):  
Topological Charge ◽  
Isometry Group ◽  
Mass Parameter ◽  
Left Translation ◽  
Spin Connection ◽  
Circle Bundle ◽  
Gravitational Instantons

We propose a construction of Skyrme fields from holonomy of the spin connection of gravitational instantons. The procedure is implemented for Atiyah–Hitchin and Taub–NUT instantons. The skyrmion resulting from the Taub–NUT is given explicitly on the space of orbits of a left translation inside the whole isometry group. The domain of the Taub–NUT skyrmion is a trivial circle bundle over the Poincaré disc. The position of the skyrmion depends on the Taub–NUT mass parameter, and its topological charge is equal to two.

scholarly journals Conformal geodesics on gravitational instantons

2021 ◽  
pp. 1-32
Author(s):  
MACIEJ DUNAJSKI ◽  
PAUL TOD
Keyword(s):  
Magnetic Field ◽  
Geodesic Flow ◽  
Isometry Group ◽  
Three Dimensional ◽  
First Integrals ◽  
Complete Integrability ◽  
Conformal Killing ◽  
Gravitational Instantons ◽  
Geodesic Equations ◽  
Jacobi Equations

Abstract We study the integrability of the conformal geodesic flow (also known as the conformal circle flow) on the SO(3)–invariant gravitational instantons. On a hyper–Kähler four–manifold the conformal geodesic equations reduce to geodesic equations of a charged particle moving in a constant self–dual magnetic field. In the case of the anti–self–dual Taub NUT instanton we integrate these equations completely by separating the Hamilton–Jacobi equations, and finding a commuting set of first integrals. This gives the first example of an integrable conformal geodesic flow on a four–manifold which is not a symmetric space. In the case of the Eguchi–Hanson we find all conformal geodesics which lie on the three–dimensional orbits of the isometry group. In the non–hyper–Kähler case of the Fubini–Study metric on $\mathbb{CP}^2$ we use the first integrals arising from the conformal Killing–Yano tensors to recover the known complete integrability of conformal geodesics.

scholarly journals Isometries of Riemannian and sub-Riemannian structures on three-dimensional Lie groups

2017 ◽  
Vol 25 (2) ◽  
pp. 99-135
Author(s):  
Rory Biggs
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Isometry Group ◽  
Three Dimensional ◽  
Isotropy Subgroup ◽  
Left Translation ◽  
Isometry Groups ◽  
Group Automorphism ◽  
Group Of Automorphisms ◽  
Simply Connected

Abstract We investigate the isometry groups of the left-invariant Riemannian and sub-Riemannian structures on simply connected three-dimensional Lie groups. More specifically, we determine the isometry group for each normalized structure and hence characterize for exactly which structures (and groups) the isotropy subgroup of the identity is contained in the group of automorphisms of the Lie group. It turns out (in both the Riemannian and sub-Riemannian cases) that for most structures any isometry is the composition of a left translation and a Lie group automorphism.

scholarly journals MATRIX MODEL FOR NONCOMMUTATIVE GRAVITY AND GRAVITATIONAL INSTANTONS

2004 ◽  
Vol 19 (02) ◽  
pp. 227-247 ◽  
Author(s):  
P. VALTANCOLI
Keyword(s):  
Gauge Group ◽  
Matrix Model ◽  
Equations Of Motion ◽  
Einstein Equations ◽  
Spin Connection ◽  
Metric Tensor ◽  
Gravitational Instantons ◽  
The Matrix ◽  
Noncommutative Gravity ◽  
Set Up

We introduce a matrix model for noncommutative gravity, based on the gauge group U (2)⊗ U (2). The vierbein is encoded in a matrix Yμ, having values in the coset space U (4)/( U (2)⊗ U (2)), while the spin connection is encoded in a matrix Xμ, having values in U (2)⊗ U (2). We show how to recover the Einstein equations from the θ→0 limit of the matrix model equations of motion. We stress the necessity of a metric tensor, which is a covariant representation of the gauge group in order to set up a consistent second order formalism. We finally define noncommutative gravitational instantons as generated by U (2)⊗ U (2) valued quasi-unitary operators acting on the background of the matrix model. Some of these solutions have naturally self-dual or anti-self-dual spin connections.

OCTONIONIC MULTI S8 CHIRAL AND GRAVITATIONAL INSTANTONS

1991 ◽  
Vol 06 (17) ◽  
pp. 1611-1614
Author(s):  
A. REŞIT DÜNDARER
Keyword(s):  
Topological Charge ◽  
Sigma Model ◽  
Gravitational Instantons ◽  
Self Duality ◽  
Poincaré Index ◽  
Poincare Index

An 8-dimensional generalization of the sigma model is given and it is shown that these fields have topological charge n and satisfy a self-duality equation for the octonionic mappings xn: S8 → S8. Furthermore the Euler–Poincaré index I E and the Pontryagin index I P are generalized to eight dimensions and it is shown that I E = n, I P = 0 for the above mappings.

scholarly journals Visualizing the strongly reshaped skyrmion Hall effect in multilayer wire devices

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Anthony K. C. Tan ◽  
Pin Ho ◽  
James Lourembam ◽  
Lisen Huang ◽  
Hang Khume Tan ◽  
...  
Keyword(s):  
Hall Effect ◽  
Topological Charge ◽  
Size Dependence ◽  
Particle Model ◽  
Model Simulations ◽  
Generation Computing ◽  
Magnetic Skyrmions ◽  
Transverse Deflection ◽  
Robust Framework ◽  
Geometric Effects

AbstractMagnetic skyrmions are nanoscale spin textures touted as next-generation computing elements. When subjected to lateral currents, skyrmions move at considerable speeds. Their topological charge results in an additional transverse deflection known as the skyrmion Hall effect (SkHE). While promising, their dynamic phenomenology with current, skyrmion size, geometric effects and disorder remain to be established. Here we report on the ensemble dynamics of individual skyrmions forming dense arrays in Pt/Co/MgO wires by examining over 20,000 instances of motion across currents and fields. The skyrmion speed reaches 24 m/s in the plastic flow regime and is surprisingly robust to positional and size variations. Meanwhile, the SkHE saturates at ∼22∘, is substantially reshaped by the wire edge, and crucially increases weakly with skyrmion size. Particle model simulations suggest that the SkHE size dependence — contrary to analytical predictions — arises from the interplay of intrinsic and pinning-driven effects. These results establish a robust framework to harness SkHE and achieve high-throughput skyrmion motion in wire devices.

scholarly journals General solutions in Chern-Simons gravity and $$ T\overline{T} $$-deformations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Eva Llabrés
Keyword(s):  
Variational Principle ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Dirichlet Boundary Conditions ◽  
Spin Connection ◽  
Departure Point ◽  
Boundary Stress ◽  
Chern Simons

Abstract We find the most general solution to Chern-Simons AdS3 gravity in Fefferman-Graham gauge. The connections are equivalent to geometries that have a non-trivial curved boundary, characterized by a 2-dimensional vielbein and a spin connection. We define a variational principle for Dirichlet boundary conditions and find the boundary stress tensor in the Chern-Simons formalism. Using this variational principle as the departure point, we show how to treat other choices of boundary conditions in this formalism, such as, including the mixed boundary conditions corresponding to a $$ T\overline{T} $$ T T ¯ -deformation.

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