Abstract:
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.

Abstract:
We study the motion of vortices in the planar Ginzburg-Landau model with Schrodinger-Chern-Simons dynamics. We compare the moduli space approximation with the results of numerical simulations of the full field theory and find that there is agreement if the coupling constant is very close to the critical value separating Type I from Type II superconductors. However, there are significant qualitative differences even for modest deviations from the critically coupled regime. Radiation effects produce forces which are of the same order of magnitude as the intervortex force and therefore have a significant impact on vortex motion. We conclude that the moduli space approximation does not provide a good description of the dynamics in this regime.

Abstract:
Gauge invariant conservation laws for the linear and angular momenta are studied in a certain 2+1 dimensional first order dynamical model of vortices in superconductivity. In analogy with fluid vortices it is possible to express the linear and angular momenta as low moments of vorticity. The conservation laws are compared with those obtained in the moduli space approximation for vortex dynamics.

Abstract:
A non-dissipative model for vortex motion in thin superconductors is considered. The Lagrangian is a Galilean invariant version of the Ginzburg--Landau model for time-dependent fields, with kinetic terms linear in the first time derivatives of the fields. It is shown how, for certain values of the coupling constants, the field dynamics can be reduced to first order differential equations for the vortex positions. Two vortices circle around one another at constant speed and separation in this model.

Abstract:
The moduli space of solutions to the vortex equations on a Riemann surface are well known to have a symplectic (in fact K\"{a}hler) structure. We show this symplectic structure explictly and proceed to show a family of symplectic (in fact, K\"{a}hler) structures $\Omega_{\Psi_0}$ on the moduli space, parametrised by $\Psi_0$, a section of a line bundle on the Riemann surface. Next we show that corresponding to these there is a family of prequantum line bundles ${\mathcal P}_{\Psi_0} $on the moduli space whose curvature is proportional to the symplectic forms $\Omega_{\Psi_0}$.

Abstract:
We give upper and lower bounds for the order of the top Chern class of the Hodge bundle on the moduli space of principally polarized abelian varieties.

Abstract:
Vortex lattices in the high temperature superconductors undergo a first order phase transition which has thus far been regarded as melting from a solid to a liquid. We point out an alternative possibility of a two step process in which there is a first order transition from an ordinary vortex lattice to a soft vortex solid followed by another first order melting transition from the soft vortex solid to a vortex liquid. We focus on the first step. This premelting transition is induced by vacancy and interstitial vortex lines. We obtain good agreement with the experimental transition temperature versus field, latent heat, and magnetization jumps for YBCO and BSCCO.

Abstract:
In this paper, we initiate our investigation of log canonical models for the moduli space of curves with the boundary divisor $\a \d$ as we decrease $\a$ from 1 to 0. We prove that for the first critical value $\a = 9/11$, the log canonical model is isomorphic to the moduli space of pseudostable curves, which have nodes and cusps as singularities. We also show that $\a = 7/10$ is the next critical value, i.e., the log canonical model stays the same in the interval $(7/10, 9/11]$. In the appendix, we develop a theory of log canonical models of stacks that explains how these can be expressed in terms of the coarse moduli space.

Abstract:
Hysteresis in cycling through first-order phase transitions in vortex matter, akin to the well-studied phenomenon of supercooling of water, has been discussed in literature. Hysteresis can be seen while varying either temperature T or magnetic field H (and thus the density of vortices). Our recent work on phase transitions with two control variables shows that the observable region of metastability of the supercooled phase would depend on the path followed in H-T space, and will be larger when T is lowered at constant H compared to the case when H is lowered at constant T. We discuss the effect of isothermal field variations on metastable supercooled states produced by field-cooling. This path dependence is not a priori applicable to metastability caused by reduced diffusivity or hindered kinetics.

Abstract:
We present a systematic study of the topology of the vortex solid phase in superconducting Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8}$ samples with low doses of columnar defects. A new state of vortex matter imposed by the presence of geometrical contours associated with the random distribution of columns is found. The results show that the first order liquid-solid transition in this vortex matter does not require a structural symmetry change.