Membranes on an orbifold

We provide an M theory interpretation of the recently discovered N=8 supersymmetric Chern-Simons theory with SO(4) gauge symmetry. The theory is argued to describe two membranes moving in the orbifold R8/Z2. At level k=1 and k=2, the classical moduli space M coincides with the infrared moduli space of SO(4) and SO(5) super Yang-Mills theory, respectively. For higher Chern-Simons level, the moduli space is a quotient of M. At a generic point in the moduli space, the massive spectrum is proportional to the area of the triangle formed by the two membranes and the orbifold fixed point.     

Vortices on surfaces with cylindrical ends

We consider Riemann surfaces obtained from nodal curves with infinite cylinders in the place of nodal and marked points, and study the space of finite energy vortices defined on these surfaces. To compactify the space of vortices, we need to consider stable vortices — these incorporate breaking of cylinders and sphere bubbling in the fibers. In this paper, we prove that the space of gauge equivalence classes of stable vortices representing a fixed equivariant homology class is compact and Hausdorff under the Gromov topology. We also show that this space is homeomorphic to the moduli space of quasimaps defined by Ciocan-Fontanine et al. (2014).     

Asymptotic Interactions of Critically Coupled Vortices

 At critical coupling, the interactions of Ginzburg-Landau vortices are determined by the metric on the moduli space of static solutions. Here, a formula for the asymptotic metric for two well separated vortices is obtained, which depends on a modified Bessel function. A straightforward extension gives the metric for N vortices. The asymptotic metric is also shown to follow from a physical model, where each vortex is treated as a point-like particle carrying a scalar charge and a magnetic dipole moment of the same magnitude. The geodesic motion of two well separated vortices is investigated, and the asymptotic dependence of the scattering angle on the impact parameter is determined. Formulae for the asymptotic Ricci and scalar curvatures of the N-vortex moduli space are also obtained. Received: 25 June 2002 / Accepted: 7 February 2003 Published online: 17 April 2003 Communicated by A. Kupiainen     

Universal reconnection of non-Abelian cosmic strings

We show that local and semilocal strings in Abelian and non-Abelian gauge theories with critical couplings always reconnect classically in collision, by using moduli space approximation. The moduli matrix formalism explicitly identifies a well-defined set of the vortex moduli parameters. Our analysis of generic geodesic motion in terms of those shows right-angle scattering in head-on collision of two vortices, which is known to give the reconnection of the strings.     

On vortex entities of 2D turbulence in wavenumber space

To connect vortices in physical space and scales in wavenumber space, spectral definitions for vortex size and momentum are introduced within the framework of a probabilistic method. At a late stage of 2D decaying turbulence, a simple solution is given for the vortex position and momentum probabilities. From the solution, an energy spectrum E(k) for self-similar vortices is constructed, which is in agreement with that observed in numerical simulations. (c) 1995 American Institute of Physics.     

Vortices as Degenerate Metrics

We note that the Bogomolny equation for abelian vortices is precisely the condition for invariance of the Hermitian–Einstein equation under a degenerate conformal transformation. This leads to a natural interpretation of vortices as degenerate Hermitian metrics that satisfy a certain curvature equation. Using this viewpoint, we rephrase standard results about vortices and make new observations. We note the existence of a conceptually simple, non-linear rule for superposing vortex solutions, and we describe the natural behaviour of the L ²-metric on the moduli space upon restriction to a class of submanifolds.     

Construction of non-Abelian walls and their complete moduli space

We present a systematic method to construct exactly all Bogomol'nyi-Prasad-Sommerfield multiwall solutions in supersymmetric U(N(C)) gauge theories in five dimensions with N(F) hypermultiplets in the fundamental representation for infinite gauge coupling. The moduli space of these non-Abelian walls is found to be the complex Grassmann manifold SU(N(F)) divided by SU(N(C))xSU(N(F)-N(C))xU(1) endowed with a deformed metric.     

The Group Property of Darboux Transformations from R2 to U(N)

It is proved that the basic Darboux transformations of harmonic maps from R² to U(N) generate an Abelian transformation group on the moduli space of harmonic maps.     

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. Received: 9 October 1998 / Revised version: 12 December 1998 / Published online: 18 June 1999     

Generalized Hitchin Systems and the Knizhnik–zamolodchikov–bernard Equation on Ellipic Curves

The Knizhnik–Zamolodchikov–Bernard (KZB) equation on an elliptic curve with a marked point is derived by classical Hamiltonian reduction and further quantization. We consider classical Hamiltonian systems on a cotangent bundle to the loop group L(GL(N, C)) extended by the shift operators, to be related to the elliptic module. After reduction, we obtain a Hamiltonian system on a cotangent bundle to the moduli of holomorphic principle bundles and an elliptic module. It is a particular example of generalized Hitchin systems (GHS) which are defined as Hamiltonian systems on cotangent bundles to the moduli of holomorphic bundles and to the moduli of curves. They are extensions of the Hitchin systems by the inclusion the moduli of curves. In contrast with the Hitchin systems, the algebra of integrals are noncommutative on GHS. We discuss the quantization procedure in our example. The quantization of the quadratic integral leads to the KZB equation. We present an explicit form of higher quantum Hitchin integrals which, upon reducing from GHS phase space to the Hitchin phase space, gives a particular example of the Beilinson–Drinfeld commutative algebra of differential operators on the moduli of holomorphic bundles.     

Coherent vortex extraction in 3D turbulent flows using orthogonal wavelets

This Letter presents a wavelet technique for extracting coherent vortices from three-dimensional turbulent flows, which is applied to a homogeneous isotropic turbulent flow at resolution N = 256(3). The coherent flow is reconstructed from only 3%N wavelet coefficients that retain the vortex tubes, and 98.9% of the energy with the same k(-5/3) spectrum as the total flow. In contrast, the remaining 97%N wavelet coefficients correspond to the incoherent flow which is structureless, decorrelated, and whose effect can therefore be modeled statistically.     

Dynamics of Abelian Higgs vortices in the near Bogomolny regime

The aim of this paper is to give an analytical discussion of the dynamics of the Abelian Higgs multi-vortices whose existence was proved by Taubes ([JT82]). For a particular value of a parameter of the theory, , called the Higgs self-coupling constant, there is no force between two vortices and there exist static configurations corresponding to vortices centred at any set of points in the plane. This is known as the Bogomolny regime. We will develop some formal asymptotic expansions to describe the dynamics of these multi-vortices for close, but not equal to, this critical value. We shall then prove the validity of these asymptotic expansions. These expansions allow us to give a finite dimensional Hamiltonian system which describes the vortex dynamics. The configuration space of this system is the moduli space—the space of solutions of the static equations modulo gauge equivalence. The kinetic energy term in the Hamiltonian is obtained from the natural metric on the moduli space given by theL ² inner product of the tangent vectors. The potential energy gives the intervortex potential which is non-zero when is not given by its critical value. Thus the reduced equations for the evolution of the vortex parameters take the form of geodesics, with force terms to express the departure from the Bogomolny regime. The geodesics are geodesics on the moduli space with respect to the metric defined by theL ² inner product of the tangent vectors, in accordance with Manton's suggestion ([Man82]). This allows an understanding of the two main phenomenological issues—first of all there is the right angle scattering phenomenon, according to which two vortices passing through one another scatter through ninety degrees. Secondly there is the conjecture from numerical calculations that vortices repel for greater than the critical value, and attract for less than this value. The results of this paper allow a rigorous understanding of the right angle scattering phenomenon ([Sam92, Hit88]) and reduce the question of attraction or repulsion in the near Bogomolny regime to an understanding of the potential energy term in the Hamiltonian ([JR79]).     

A geometric foundation for a unified field theory

Generalizing the work of Einstein and Mayer, it is assumed that at each point of space-time there exists an N-dimensional linear vector space with N5. This space is decomposed into a four-dimensional tangent space and an (N - 4)-dimensional internal space. On the basis of geometric considerations, one arrives at a number of fields, the field equations being derived from a variational principle. Among the fields obtained there are the electromagnetic field, Yang-Mills gauge fields, and fields that can be interpreted as describing matter. As a simple example, the case N=6 is considered.     

Residue formulas for the largek asymptotics of Witten's invariants of Seifert manifolds. The case ofSU(2)

We derive the largek asymptotics of the surgery formula forSU(2) Witten's invariants of general Seifert manifolds. The contributions of connected components of the moduli space of flat connections are identified. The contributions of irreducible connections are presented in the residue form. This allows us to express them in terms of intersection numbers on their moduli spaces.Address after September 25: L. Rozansky, School of Mathematics, Institute for Advanced Study, Princeton, N.J. 08540, USA.     

The Hilbert series of the one instanton moduli space

The moduli space of k G-instantons on \( {\mathbb{R}^4} \) for a classical gauge group G is known to be given by the Higgs branch of a supersymmetric gauge theory that lives on Dp branes probing D(p + 4) branes in Type II theories. For p = 3, these (3 + 1) dimensional gauge theories have \( \mathcal{N} = 2 \) supersymmetry and can be represented by quiver diagrams. The F and D term equations coincide with the ADHM construction. The Hilbert series of the moduli spaces of one instanton for classical gauge groups is easy to compute and turns out to take a particularly simple form which is previously unknown. This allows for a G invariant character expansion and hence easily generalisable for exceptional gauge groups, where an ADHM construction is not known. The conjectures for exceptional groups are further checked using some new techniques like sewing relations in Hilbert Series. This is applied to Argyres-Seiberg dualities.