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 describe two constructions of hyperkahler manifolds, one based on a Legendre transform, and one on a sympletic quotient. These constructions arose in the context of supersymmetric nonlinear σ-models, but can be described entirely geometrically. In this general setting, we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry, along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience.

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Hyperkahler Metrics and Supersymmetry

Scalar tensor duality and N = 1,2 non-linear σ-models

Self-interacting tensor multiplets in N = 2 superspace

Constrained local superfields

We present new constructions of hyperkahler metrics along with the three new classes ofN=4 supermultiplets that they derive from. Further, we provide a general setting for understanding the constructions and give a detailed description of the multiplets inN=2 andN=4 superspace.

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Ulf Lindström

Papers

Hyperkahler Metrics and Supersymmetry

Scalar tensor duality and N = 1,2 non-linear σ-models

Self-interacting tensor multiplets in N = 2 superspace

Constrained local superfields

New Hyperkahler Metrics and New Supermultiplets