Monopole Condensates in Seiberg-Witten Theory

A product of two Riemann surfaces of genuses p_1 and p_2 solves the Seiberg-Witten monopole equations for a constant Weyl spinor that represents a monopole condensate. Self-dual electromagnetic fields require p_1=p_2=p and provide a solution of the euclidean Einstein-Maxwell-Dirac equations with p-1 magnetic vortices in one surface and the same number of electric vortices in the other. The monopole condensate plays the role of cosmological constant. The virtual dimension of the moduli space is zero, showing that for given p_1 and p_2, the solutions are unique.Comment: 10 page

Singular solutions to the Seiberg-Witten and Freund equations on flat space from an iterative method

Although it is well known that the Seiberg-Witten equations do not admit nontrivial $L^2$ solutions in flat space, singular solutions to them have been previously exhibited -- either in $R^3$ or in the dimensionally reduced spaces $R^2$ and $R^1$ -- which have physical interest. In this work, we employ an extension of the Hopf fibration to obtain an iterative procedure to generate particular singular solutions to the Seiberg-Witten and Freund equations on flat space. Examples of solutions obtained by such method are presented and briefly discussed.Comment: 7 pages, minor changes. To appear in J. Math. Phy

Liouville Vortex And φ4\varphi^{4} Kink Solutions Of The Seiberg--Witten Equations

The Seiberg--Witten equations, when dimensionally reduced to \bf R^{2}\mit, naturally yield the Liouville equation, whose solutions are parametrized by an arbitrary analytic function $g(z)$. The magnetic flux $\Phi$ is the integral of a singular Kaehler form involving $g(z)$; for an appropriate choice of $g(z)$ , $N$ coaxial or separated vortex configurations with $\Phi=\frac{2\pi N}{e}$ are obtained when the integral is regularized. The regularized connection in the \bf R^{1}\mit case coincides with the kink solution of $\varphi^{4}$ theory.Comment: 14 pages, Late

Geometry of 2d spacetime and quantization of particle dynamics

We analyze classical and quantum dynamics of a particle in 2d spacetimes with constant curvature which are locally isometric but globally different. We show that global symmetries of spacetime specify the symmetries of physical phase-space and the corresponding quantum theory. To quantize the systems we parametrize the physical phase-space by canonical coordinates. Canonical quantization leads to unitary irreducible representations of $SO_\uparrow (2.1)$ group.Comment: 12 pages, LaTeX2e, submitted for publicatio

Is my ODE a Painleve equation in disguise?

Painleve equations belong to the class y'' + a_1 {y'}^3 + 3 a_2 {y'}^2 + 3 a_3 y' + a_4 = 0, where a_i=a_i(x,y). This class of equations is invariant under the general point transformation x=Phi(X,Y), y=Psi(X,Y) and it is therefore very difficult to find out whether two equations in this class are related. We describe R. Liouville's theory of invariants that can be used to construct invariant characteristic expressions (syzygies), and in particular present such a characterization for Painleve equations I-IV.Comment: 8 pages. Based on talks presented at NEEDS 2000, Gokova, Turkey, 29 June - 7 July, 2000, and at the AMS-HKMS joint meeting 13-16 December, 2000. Submitted to J. Nonlin. Math. Phy

The Binet-Legendre Metric in Finsler Geometry

For every Finsler metric $F$ we associate a Riemannian metric $g_F$ (called the Binet-Legendre metric). The transformation $F \mapsto g_F$ is $C^0$-stable and has good smoothness properties, in contrast to previous constructions. The Riemannian metric $g_F$ also behaves nicely under conformal or bilipshitz deformation of the Finsler metric $F$. These properties makes it a powerful tool in Finsler geometry and we illustrate that by solving a number of named Finslerian geometric problems. We also generalize and give new and shorter proofs of a number of known results. In particular we answer a question of M. Matsumoto about local conformal mapping between two Minkowski spaces, we describe all possible conformal self maps and all self similarities on a Finsler manifold. We also classify all compact conformally flat Finsler manifolds, solve a conjecture of S. Deng and Z. Hou on the Berwaldian character of locally symmetric Finsler spaces, and extend the classic result of H.C. Wang about the maximal dimension of the isometry groups of Finsler manifolds to manifolds of all dimensions. Most proofs in this paper go along the following scheme: using the correspondence $F \mapsto g_F$we reduce the Finslerian problem to a similar problem for the Binet-Legendre metric, which is easier and is already solved in most cases we consider. The solution of the Riemannian problem provides us with the additional information that helps to solve the initial Finslerian problem. Our methods apply even in the absence of the strong convexity assumption usually assumed in Finsler geometry. The smoothness hypothesis can also be replaced by that of partial smoothness, a notion we introduce in the paper. Our results apply therefore to a vast class of Finsler metrics not usually considered in the Finsler literature.Comment: 33 pages, 5 figures. This version is slightly reduced fron versions 1 and 2. The paper has been published in Geometry & Topolog

Binary Black Hole Mergers from Planet-like Migrations

If supermassive black holes (BHs) are generically present in galaxy centers, and if galaxies are built up through hierarchical merging, BH binaries are at least temporary features of most galactic bulges. Observations suggest, however, that binary BHs are rare, pointing towards a binary lifetime far shorter than the Hubble time. We show that, regardless of the detailed mechanism, all stellar-dynamical processes are insufficient to reduce significantly the orbital separation once orbital velocities in the binary exceed the virial velocity of the system. We propose that a massive gas disk surrounding a BH binary can effect its merger rapidly, in a scenario analogous to the orbital decay of super-jovian planets due to a proto-planetary disk. As in the case of planets, gas accretion onto the secondary (here a supermassive BH) is integrally connected with its inward migration. Such accretion would give rise to quasar activity. BH binary mergers could therefore be responsible for many or most quasars.Comment: 8 pages, submitted to ApJ Letter

Coadjoint orbits of the Virasoro algebra and the global Liouville equation

The classification of the coadjoint orbits of the Virasoro algebra is reviewed and is then applied to analyze the so-called global Liouville equation. The review is self-contained, elementary and is tailor-made for the application. It is well-known that the Liouville equation for a smooth, real field $\phi$ under periodic boundary condition is a reduction of the SL(2,R) WZNW model on the cylinder, where the WZNW field g in SL(2,R) is restricted to be Gauss decomposable. If one drops this restriction, the Hamiltonian reduction yields, for the field $Q=\kappa g_{22}$ where $\kappa\neq 0$ is a constant, what we call the global Liouville equation. Corresponding to the winding number of the SL(2,R) WZNW model there is a topological invariant in the reduced theory, given by the number of zeros of Q over a period. By the substitution $Q=\pm\exp(- \phi/2)$, the Liouville theory for a smooth $\phi$ is recovered in the trivial topological sector. The nontrivial topological sectors can be viewed as singular sectors of the Liouville theory that contain blowing-up solutions in terms of $\phi$. Since the global Liouville equation is conformally invariant, its solutions can be described by explicitly listing those solutions for which the stress-energy tensor belongs to a set of representatives of the Virasoro coadjoint orbits chosen by convention. This direct method permits to study the `coadjoint orbit content' of the topological sectors as well as the behaviour of the energy in the sectors. The analysis confirms that the trivial topological sector contains special orbits with hyperbolic monodromy and shows that the energy is bounded from below in this sector only.Comment: Plain TEX, 48 pages, final version to appear in IJMP

Onsager-Manning-Oosawa condensation phenomenon and the effect of salt

Making use of results pertaining to Painleve III type equations, we revisit the celebrated Onsager-Manning-Oosawa condensation phenomenon for charged stiff linear polymers, in the mean-field approximation with salt. We obtain analytically the associated critical line charge density, and show that it is severely affected by finite salt effects, whereas previous results focused on the no salt limit. In addition, we obtain explicit expressions for the condensate thickness and the electric potential. The case of asymmetric electrolytes is also briefly addressed.Comment: to appear in Phys. Rev. Let

Families of exact solutions of a 2D gravity model minimally coupled to electrodynamics

Three families of exact solutions for 2-dimensional gravity minimally coupled to electrodynamics are obtained in the context of ${\cal R}=T$ theory. It is shown, by supersymmetric formalism of quantum mechanics, that the quantum dynamics of a neutral bosonic particle on static backgrounds with both varying curvature and electric field is exactly solvable.Comment: 13 pages, LaTeX, to be published in JM