涡旋和团块的Ricci磁测地线运动
Ricci magnetic geodesic motion of vortices and lumps
摘要：
Kähler流形中的Ricci磁测地线（RMG）运动是存在与Ricci形式成比例的磁场时测地线运动的模拟。在讨论$n$ -涡旋模空间中的kähler流形时，考虑了Chern-Simons项，建立了涡旋孤子的低能动力学模型。本文对孤子模空间中的RMG运动进行了详细的研究，重点讨论了双曲涡旋和球形$\mathbb{C}P^1$块的情况。它表明，RMG流局部化的不动点集的全纯同构群，但在这样的子流形上的流量不，在一般情况下，符合其内在的RMG流。对于平面涡，它表明，RMG流不同于较早的约化动力学提出的Kim和Lee，后者的流动是不明确的涡重合设置。本文计算了双曲双涡在整个模空间上度规的一个显式公式（推广了Strachan的一个旧结果），并详细研究了双曲双涡的RMG运动。转向块，静态$n$ -块的模空间是$Rat_n$，度$n$有理映射的空间，已知它是kähler和测地线不完全的。证明了$Rat_1$是RMG完备的（这意味着RMG运动的初值问题对于所有初始数据都有全局解）。本文还证明了旋转等变$n$ -块的子流形$Rat_n^{eq}$，一个拓扑圆柱旋转曲面，对于$n=2$和所有$n\geq 5$都是固有RMG不完全的，但$Rat_2^{eq}$（由包含体$Rat_2^{eq}\hookrightarrow Rat_2$定义）上的外在RMG流是完备的.
英文摘要：
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 $\mathbb{C}P^1$ 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 kähler 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.
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