TOPICAL REVIEW: Lectures on nonlinear sigma-models in projective superspace

{\cal N}= 2 supersymmetry in four spacetime dimensions is intimately related to hyperkähler and quaternionic Kähler geometries. On one hand, the target spaces for rigid supersymmetric sigma-models are necessarily hyperkähler manifolds. On the other hand, when coupled to {\cal N}= 2 supergravity, the sigma-model target spaces must be quaternionic Kähler. It is known that such manifolds of restricted holonomy are difficult to generate explicitly. Projective superspace is a field-theoretic approach to construct general {\cal N}= 2 supersymmetric nonlinear sigma-models, and hence to generate new hyperkähler and quaternionic Kähler metrics. Intended for a mixed audience consisting of both physicists and mathematicians, these lectures provide a pedagogical introduction to the projective-superspace approach.

Invited lectures presented at the 30th Winter School Geometry and Physics, Srni, Czech Republic, 16-23 January 2010.