19个涡旋方程及其可积性
Nineteen vortex equations and integrability
摘要：
将Manton最近讨论的五个可积涡方程推广到相对论BPS Chern-Simons涡，得到十九个涡方程。并不是所有的19个涡方程都是可积的，但发现了4个新的可积方程，我们将它们推广到无穷多组的4个可积涡方程，每组用它的整数阶$n$表示。它们的可积性类似于已知的情况，但产生了不同的（广义）巴普蒂斯塔几何，其中巴普蒂斯塔度规是由希格斯场对背景度规的共形重标度。特别是，巴普蒂斯塔流形有圆锥奇点。当Jackiw-Pi、Taubes、Popov和Ambjørn-Olesen涡在其Baptista流形中的每个涡零点处具有$2π$的锥形亏损时，这些方程的高阶推广也可以在较大的常曲率和每个涡零点处的$2πn$锥形亏损下积分。然后，我们推广叠加法，已知的Taubes涡如何添加涡一个已知的解决方案，所有的可积涡方程。我们发现，虽然Taubes方程和Popov方程是相互关联的，但Ambjørn-Olesen和Jackiw-Pi涡是通过使用Baptista度规和Popov方程叠加的。最后，我们发现涡方程之间的许多进一步的关系，例如，我们发现，陈-西蒙斯涡可以解释为Taubes涡的Baptista流形上的自己的解决方案。
英文摘要：
The class of five integrable vortex equations discussed recently by Manton is extended so it includes the relativistic BPS Chern-Simons vortices, yielding a total of nineteen vortex equations. Not all the nineteen vortex equations are integrable, but four new integrable equations are discovered and we generalize them to infinitely many sets of four integrable vortex equations, with each set denoted by its integer order $n$. Their integrability is similar to the known cases, but give rise to different (generalized) Baptista geometries, where the Baptista metric is a conformal rescaling of the background metric by the Higgs field. In particular, the Baptista manifolds have conical singularities. Where the Jackiw-Pi, Taubes, Popov and Ambjørn-Olesen vortices have conical deficits of $2π$ at each vortex zero in their Baptista manifolds, the higher-order generalizations of these equations are also integrable with larger constant curvatures and a $2πn$ conical deficit at each vortex zero. We then generalize a superposition law, known for Taubes vortices of how to add vortices to a known solution, to all the integrable vortex equations. We find that although the Taubes and the Popov equations relate to themselves, the Ambjørn-Olesen and Jackiw-Pi vortices are added by using the Baptista metric and the Popov equation. Finally, we find many further relations between vortex equations, e.g. we find that the Chern-Simons vortices can be interpreted as Taubes vortices on the Baptista manifold of their own solution.
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