Alqahtani, LS and Speight, JM orcid.org/0000-0002-6844-9539 (2015) Ricci magnetic geodesic motion of vortices and lumps. Journal of Geometry and Physics, 98. pp. 556-574. ISSN 0393-0440
Abstract
Ricci magnetic geodesic (RMG) motion in a kähler manifold is the analogue of geodesic motion in the presence of a magnetic field proportional to the ricci form. It has been conjectured to model low-energy dynamics of vortex solitons in the presence of a Chern-Simons term, the kähler manifold in question being the n-vortex moduli space. This paper presents a detailed study of RMG motion in soliton moduli spaces, focusing on the cases of hyperbolic vortices and spherical CP1 lumps. It is shown that RMG flow localizes on fixed point sets of groups of holomorphic isometries, but that the flow
on such submanifolds does not, in general, coincide with their intrinsic RMG flow. For planar vortices, it is shown that RMG flow differs from an earlier reduced dynamics proposed by Kim and Lee, and that the latter flow is ill-defined on the vortex coincidence set. An explicit formula for the metric on the whole moduli space of hyperbolic two-vortices is computed (extending an old result of Strachan's), and RMG motion of
centred two-vortices is studied in detail. Turning to lumps, the moduli space of static n-lumps is Ratn, the space of degree n rational maps, which is known to be kähler and geodesically incomplete. It is proved that Rat1 is, somewhat surprisingly, RMG complete (meaning that that the initial value problem for RMG motion has a global solution for all
initial data). It is also proved that the submanifold of rotationally equivariant n-lumps, Rateq n , a topologically cylindrical surface of revolution, is intrinsically RMG incomplete for n = 2 and all n _ 5, but that the extrinsic RMG flow on Rateq 2 (defined by the inclusion Rateq
2 ,! Rat2) is complete.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2015, Published by Elsevier B.V. This is an author produced version of a paper published in Journal of Geometry and Physics. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | Magnetic geodesic; Ricci form; Chern–Simons vortices; Lumps |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 25 Aug 2015 11:18 |
Last Modified: | 10 May 2019 14:06 |
Published Version: | http://dx.doi.org/10.1016/j.geomphys.2015.07.008 |
Status: | Published |
Publisher: | Elsevier |
Identification Number: | 10.1016/j.geomphys.2015.07.008 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:87769 |