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The Abelian Chern-Simons Gauge Field Theory in 2+1 dimensions and itsrelation with holomorphic Burgers' Hierarchy is considered. It is shown thatthe relation between complex potential and the complex gauge field as inincompressible and irrotational hydrodynamics, has meaning of the analyticCole-Hopf transformation, linearizing the Burgers Hierarchy in terms of theholomorphic Schr\"odinger Hierarchy. Then the motion of planar vortices inChern-Simons theory, appearing as pole singularities of the gauge field,corresponds to motion of zeroes of the hierarchy. Using boost transformationsof the complex Galilean group of the hierarchy, a rich set of exact solutions,describing integrable dynamics of planar vortices and vortex lattices in termsof the generalized Kampe de Feriet and Hermite polynomials is constructed. Theresults are applied to the holomorphic reduction of the Ishimori model and thecorresponding hierarchy, describing dynamics of magnetic vortices andcorresponding lattices in terms of complexified Calogero-Moser models.Corrections on two vortex dynamics from the Moyal space-time non-commutativityin terms of Airy functions are found.Comment: 15 pages, talk presented in Workshop `Nonlinear Physics IV: Theory  and Experiment`, 22-30 June 2006, Gallipoli, Ital

Abelian Chern-Simons vortices and holomorphic Burgers hierarchy