Microwave assisted organic synthesis-a review

Phases of N = 2 theories in two dimensions

In this paper we shall study a special class of solutions of the self-dual Yang-Mills equations. The original self-duality equations which arose in mathematical physics were defined on Euclidean 4-space. The physically relevant solutions were the ones with finite action-the so-called 'instantons'. The same equations may be dimensionally reduced to Euclidean 3-space by imposing invariance under translation in one direction. These equations also have physical relevance-the solutions which have finite action in three dimensions are the 'magnetic monopoles'. If we take the reduction process one step further and consider solutions which are invariant under two translations, we obtain a set of equations in the plane. Here, however, there is no clear physical meaning and, indeed, attempts to find finite action solutions have failed. Nevertheless, these are the equations we shall consider. Despite the lack of interesting solutions in R2, the equations have the important property-conformal invariance-which allows them to be defined on manifolds modelled on R2 by conformal maps, namely Riemann surfaces. We shall consider here solutions of the self-duality equations defined on a compact Riemann surface. There are in fact solutions, as we shall show, and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study. It brings together in a harmonious way the subjects of Riemannian geometry, topology, algebraic geometry, and symplectic geometry. Illuminating all these facets of the same object accounts for the length of this paper. The self-duality equations are equations from gauge theory; geometrically they are defined in terms of connections on principal bundles. While the group of the principal bundle may be chosen arbitrarily for the equations to make sense, we restrict attention here to the simplest case of SU(2) or SO(3). There are two reasons for this. The first, and most obvious, is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general Lie group of higher rank. The second reason is that solutions for SU(2) have an intimate relationship with the internal structure of the Riemann surface. As a consequence of results we shall prove about solutions to the self-duality equations, we learn something about the moduli space of complex structures on the surface itself, namely Teichmuller space.

The self-duality equations on a riemann surface

Superstring theoryをめぐって(放談室)

Type iib superstrings, bps monopoles, and three-dimensional gauge dynamics

We study dualities in off-shell 4D N = 2 supersymmetric sigma-models, using the projective superspace approach. These include (i) duality between the real O(2n) and polar multiplets; and (ii) polar-polar duality. We demonstrate that the dual of any superconformal sigma-model is superconformal. Since N = 2 superconformal sigma-models (for which target spaces are hyperkahler cones) formulated in terms of polar multiplets are naturally associated with Kahler cones (which are target spaces for N = 1 superconformal sigma-models), polar-polar duality generates a transformation between different Kahler cones. In the non-superconformal case, we study implications of polar-polar duality for the sigma-model formulation in terms of N = 1 chiral superfields. In particular, we find the relation between the original hyperkahler potential and its dual. As an application of polar-polar duality, we study self-dual models.

New supersymmetric σ-model duality

We consider the axially symmetric coupled system of gravitation, electromagnetism and a dilaton field. Reducing from four to three dimensions, the system is described by gravity coupled to a non-linear $\sigma$-model. We find the target space isometries and use them to generate new solutions. It seems that it is only possible to generate rotating solutions from non-rotating ones for the special cases when the dilaton coupling parameter $a=0, \pm \sqrt{3}$. For those particular values, the target space symmetry is enlarged.

Constructing new rotating dilaton black holes.

We introduce a phase space with spinorial momenta, corresponding to fermionic derivatives, for a 2d supersymmetric (1, 1) sigma model. We show that there is a generalisation of the covariant De Donder-Weyl Hamiltonian formulation on this phase space with canonical equations equivalent to the Lagrangian formulation, find the corresponding multisymplectic form and Hamiltonian multivectors. The covariance of the formulation makes it possible to see how additional non-manifest supersymmetries arise in analogy to those of the Lagrangian formulation. We then observe that an intermediate phase space Lagrangian defined on the sum of the tangent and cotanget spaces is a first order Lagrangian for the sigma model and derive additional supersymmetries for this.

Ulf Lindström

Papers

New supersymmetric σ-model duality

Constructing new rotating dilaton black holes.

Covariant Hamiltonians, sigma models and supersymmetry

Rigid strings from field theory

From the superparticle path integral to superfield theory