The Abelian Chern-Simons Gauge Field Theory in 2+1 dimensions and its relation with holomorphic Burgers' Hierarchy is considered. It is shown that the relation between complex potential and the complex gauge field as in incompressible and irrotational hydrodynamics, has meaning of the analytic Cole-Hopf transformation, linearizing the Burgers Hierarchy in terms of the holomorphic Schr\"odinger Hierarchy. Then the motion of planar vortices in Chern-Simons theory, appearing as pole singularities of the gauge field, corresponds to motion of zeroes of the hierarchy. Using boost transformations of the complex Galilean group of the hierarchy, a rich set of exact solutions, describing integrable dynamics of planar vortices and vortex lattices in terms of the generalized Kampe de Feriet and Hermite polynomials is constructed. The results are applied to the holomorphic reduction of the Ishimori model and the corresponding hierarchy, describing dynamics of magnetic vortices and corresponding lattices in terms of complexified Calogero-Moser models. Corrections on two vortex dynamics from the Moyal space-time non-commutativity in terms of Airy functions are found.
展开▼
机译:考虑了2 + 1维的Abelian Chern-Simons规范场理论及其与全纯Burgers层次结构的关系。结果表明,在不可压缩和非旋转流体动力学中,复势与复规范场之间的关系具有解析Cole-Hopf变换的意义,根据全纯Schr“ odinger层级”将Burgers层级线性化。在Chern-Simons理论中,平面涡旋以标称场的极点奇点出现,对应于层级零的运动,利用层级复杂伽利略群的升压变换,获得了一组精确的解,描述了平面的可积动力学构造了广义Kampe de Feriet和Hermite多项式的涡旋和涡旋晶格,并将结果应用于Ishimori模型的全纯约简和相应的层次结构,根据复杂的Calogero-Moser描述了磁涡旋和相应晶格的动力学Moyal时空非交换性对两个涡旋动力学的修正找到关于Airy函数的ty。
展开▼