Abelian Chern-Simons Vortices and Holomorphic Burgers' Hierarchy

TL;DR

This paper explores the connection between Abelian Chern-Simons vortices and holomorphic Burgers' hierarchy, revealing integrable vortex dynamics, exact solutions, and effects of non-commutativity in a unified mathematical framework.

Contribution

It establishes a novel link between Chern-Simons vortex dynamics and holomorphic Burgers' hierarchy, introducing new exact solutions and analyzing non-commutative corrections.

Findings

Exact solutions for vortex and vortex lattice dynamics.

Connection between gauge fields and Burgers' hierarchy via Cole-Hopf transformation.

Non-commutative corrections involving Airy functions.

Abstract

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.