We describe two constructions of hyperkahler manifolds, one based on a Legendre transform, and one on a symplectic 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.

No abstract

Ulf Lindstro ¨m was born in 1971 in Malmo ¨, Sweden. He studied Chemistry and Biology at Lund University to receive his B.Sc. degree in 1996. After earning a licentiate degree in Engineering at Lund Institute of Technology in 1998, he moved to Stockholm University to complete his doctoral thesis in 2000 under the supervision of Professor Peter Somfai. His Ph.D. work mainly concerned the development of new methods of preparing optically active nitrogen heterocycles. He then moved to California and Stanford University for postdoctoral research with Professor Eric T. Kool. There he was exposed to different aspects of nucleic acid chemistry, for example, the synthesis and molecular recognition of circular DNA. He joined the Department of Bioorganic Chemistry at Lund University as Assistant Professor in the spring of 2002. His research interest, in general, is molecular recognition and reactivity in water and, in particular, the development of chiral ligands for aqueous asymmetric synthesis.

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