Zero modes of the non-relativistic self-dual Chern-Simons vortices on the Toda backgrounds

The two-dimensional self-dual equations are the governing equations of the static zero-energy vortex solutions for the non-relativistic, non-Abelian Chern-Simons models. The zero modes of the non-relativistic vortices are examined by index calculation for the self-dual equations. The index for the self-dual equations is zero for non-Abelian groups, but a non-zero index is obtained by the Toda Ansatz which reduces the self-dual equations to the Toda equations. The number of zero modes for the non-relativistic Toda vortices is 2 Σα,βrKαβQβ which is twice the total number of isolated zeros of the vortex functions. For the affine Toda system, there are additional adjoint zero modes which give a zero index for the SU(N) group