Abstract:
We consider the Abelian Chern–Simons gauge field theory in $2+1$
dimensions and its relation to the holomorphic Burgers hierarchy. We show
that the relation between the complex potential and the complex gauge field
as in incompressible and irrotational hydrodynamics has the meaning of
the analytic Cole–Hopf transformation, linearizing the Burgers hierarchy and
transforming it into the holomorphic Schrödinger hierarchy. The motion of
planar vortices in Chern–Simons theory, which appear as pole singularities
of the gauge field, then corresponds to the motion of zeros of the hierarchy.
We use boost transformations of the complex Galilei group of the hierarchy to
construct a rich set of exact solutions describing the integrable dynamics of
planar vortices and vortex lattices in terms of generalized Kampe de Feriet
and Hermite polynomials. We apply the results to the holomorphic reduction of
the Ishimori model and the corresponding hierarchy, describing the dynamics
of magnetic vortices and the corresponding lattices in terms of complexified
Calogero–Moser models. We find corrections (in terms of Airy
functions) to the two-vortex dynamics from the Moyal space–time
noncommutativity.
Keywords:Chern–Simons gauge theory, Burgers hierarchy, noncommutative vortex, Ishimori model, holomorphic equation, Kampe de Feriet polynomial.
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