Ricci magnetic geodesic motion of vortices and lumps

Abstract

Ricci magnetic geodesic (RMG) motion in a k\"ahler 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 k\"ahler 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 $\mathbb{C}P^1$ 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 $Rat_n$, the space of degree $n$ rational maps, which is known to be k\"ahler and geodesically incomplete. It is proved that $Rat_1$ 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, $Rat_n^{eq}$, a topologically cylindrical surface of revolution, is intrinsically RMG incomplete for $n=2$ and all $n\geq 5$, but that the extrinsic RMG flow on $Rat_2^{eq}$ (defined by the inclusion $Rat_2^{eq}\hookrightarrow Rat_2$) is complete.