Moduli Spaces of Abelian Vortices on Kahler Manifolds

We consider the self-dual vortex equations on a positive line bundle L --> M over a compact Kaehler manifold of arbitrary dimension. When M is simply connected, the moduli space of vortex solutions is a projective space. When M is an abelian variety, the moduli space is the projectivization of the Fourier-Mukai transform of L. We extend this description of the moduli space to the abelian GLSM, i.e. to vortex equations with a torus gauge group acting linearly on a complex vector space. After establishing the Hitchin-Kobayashi correspondence appropriate for the general abelian GLSM, we give explicit descriptions of the vortex moduli space in the case where the manifold M is simply connected or is an abelian variety. In these examples we compute the Kaehler class of the natural L^2-metric on the moduli space. In the simplest cases we compute the volume and total scalar curvature of the muduli space. Finally, we note that for abelian GLSM the vortex moduli space is a compactification of the space of holomorphic maps from M to toric targets, just as in the usual case of M being a Riemann surface. This leads to various natural conjectures, for instance explicit formulae for the volume of the space of maps CP^m --> CP^n.Comment: v2: 48 pages; significant changes; description of the vortex moduli spaces of the GLSM extended to allow general values of the parameters, beyond the generic values of v

Vortex equations in abelian gauged sigma-models

We consider nonlinear gauged sigma-models with Kahler domain and target. For a special choice of potential these models admit Bogomolny (or self-duality) equations -- the so-called vortex equations. We find the moduli space and energy spectrum of the solutions of these equations when the gauge group is a torus T^n, the domain is compact, and the target is C^n or CP^n. We also obtain a large family of solutions when the target is a compact Kahler toric manifold.Comment: v2: 60 pages, more details than in CMP versio

Non-transitive maps in phase synchronization

Concepts from the Ergodic Theory are used to describe the existence of non-transitive maps in attractors of phase synchronous chaotic systems. It is shown that for a class of phase-coherent systems, e.g. the sinusoidally forced Chua's circuit and two coupled R{\"o}ssler oscillators, phase synchronization implies that such maps exist. These ideas are also extended to other coupled chaotic systems. In addition, a phase for a chaotic attractor is defined from the tangent vector of the flow. Finally, it is discussed how these maps can be used to real time detection of phase synchronization in experimental systems