Low Energy Dynamics Of Vortices Research Articles

Equivariant Verlinde Formula from Fivebranes and Vortices

We study complex Chern-Simons theory on a Seifert manifold $M_3$ by embedding it into string theory. We show that complex Chern-Simons theory on $M_3$ is equivalent to a topologically twisted supersymmetric theory and its partition function can be naturally regularized by turning on a mass parameter. We find that the dimensional reduction of this theory to 2d gives the low energy dynamics of vortices in four-dimensional gauge theory, the fact apparently overlooked in the vortex literature. We also generalize the relations between 1) the Verlinde algebra, 2) quantum cohomology of the Grassmannian, 3) Chern-Simons theory on $\Sigma\times S^1$ and 4) index of a spin$^c$ Dirac operator on the moduli space of flat connections to a new set of relations between 1) the "equivariant Verlinde algebra" for a complex group, 2) the equivariant quantum K-theory of the vortex moduli space, 3) complex Chern-Simons theory on $\Sigma \times S^1$ and 4) the equivariant index of a spin$^c$ Dirac operator on the moduli space of Higgs bundles.

Continuous Preparation of a Fractional Chern Insulator.

We present evidence of a direct, continuous quantum phase transition between a Bose superfluid and the ν=1/2 fractional Chern insulator in a microscopic lattice model. In the process, we develop a detailed field theoretic description of this transition in terms of the low energy vortex dynamics. The theory explicitly accounts for the structure of lattice symmetries and predicts a Landau forbidden transition that is protected by inversion. That the transition is continuous enables the quasiadiabatic preparation of the fractional Chern insulator in nonequilibrium, quantum optical systems.

Low-energy vortex dynamics in the self-dual Chern–Simons–Higgs model

The relativistic Chern–Simons–Higgs theory finds application in anyonic superconductivity and contains topological vortices whose dynamics are poorly understood. The gauge fields are defined by a set of nonlinear constraint equations that can be accurately solved with effective Green’s functions, spectral methods, and a discretization scheme using lattice gauge techniques. Simulations show that low-energy two-vortex interactions are elastic with final scattering angles sensitive to vortex velocity; furthermore, vortex pairs form rotating breather states for certain impact parameters. In this study, a function that reproduces scattering angles in the adiabatic limit for nontangential collisions is presented. Simulation results are discussed in the context of analytical methods that extract vortex dynamics from low-energy effective Lagrangians, and a numerical method to calculate the effective Lagrangian is suggested. The numerical techniques used can be applied to the study of other Chern–Simon theories.

The dynamics of vortices on S2 near the Bradlow limit

The explicit solutions of the Bogomolny equations for N vortices on a sphere of radius R2>N are not known. In particular, this has prevented the use of the geodesic approximation to describe the low energy vortex dynamics. In this article we introduce an approximate general solution of the equations, valid for R2≳N, which has many properties of the true solutions, including the same moduli space CPN. Within the framework of the geodesic approximation, the metric on the moduli space is then computed to be proportional to the Fubini–Study metric, which leads to a complete description of the particle dynamics.

First and second order vortex dynamics

The low energy dynamics of vortices in selfdual Abelian Higgs theory is of second order in vortex velocity and characterized by the moduli space metric. When Chern-Simons term with small coefficient is added to the theory, we show that a term linear in vortex velocity appears and can be consistently added to the second order expression. We provides an additional check of the first and second order terms by studying the angular momentum in the field theory. We briefly explore other first order term due to small background electric charge density and also the harmonic potential well for vortices given by the moment of inertia.

Low-energy vortex dynamics in Abelian Higgs systems

The low-energy dynamics of the vortices of the Abelian Chern–Simons–Higgs system is investigated from the adiabatic approach. The difficulties involved in treating the field evolution as motion on the vortex moduli space in this system are shown. Another two generalized Abelian Higgs systems are discussed with respect to their vortex dynamics at the adiabatic limit. The method works well, and we find bound states in the first model and scattering at right angles in the second system.

Low energy dynamics of

We apply the adiabatic approximation to investigate the low energy dynamics of vortices in the parity invariant double self-dual Higgs model with only mutual Chern-Simons interaction. When distances between solitons are large they are particles subject to the mutual interaction. The dual formulation of the model is derived to explain the sign of the statistical interaction. When vortices of different types pass one through another they behave like charged particles in magnetic field. They can form a bound state due to the mutual magnetic trapping. Vortices of the same type exhibit no statistical interaction. Their short range interactions are analysed. Possible quantum effects due to the finite width of vortices are discussed.