Vacuum configurations for superstrings

Constraints on Supersymmetry Breaking

Strings in background fields

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

A symmetry of the string background field equations

A complete geometrical classification of supersymmetric σ-models is given. Extended supersymmetry requires covariantly constant complex structures, and Kahler and hyperkahler manifolds play a special role. As an application of the classification, it is shown that a particular class of these models is on-shell ultraviolet finite to all orders in perturbation theory.

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Luis Alvarez-Gaumé

Papers

Geometrical structure and ultraviolet finiteness in the supersymmetric σ-model

The background field method and the ultraviolet structure of the supersymmetric nonlinear σ-model

Three-loop finiteness in Ricci-flat supersymmetric non-linear σ-models