Lectures on nonlinear sigma-models in projective superspace

TL;DR

This paper provides a pedagogical introduction to the projective superspace approach for constructing N=2 supersymmetric nonlinear sigma-models, facilitating the generation of hyperkahler and quaternionic Kahler metrics.

Contribution

It offers a comprehensive, accessible overview of the projective superspace method for physicists and mathematicians to construct hyperkahler and quaternionic Kahler geometries.

Findings

Introduces the projective superspace formalism for N=2 sigma-models

Explains how to generate new hyperkahler and quaternionic Kahler metrics

Bridges physics and mathematics in supersymmetric geometry

Abstract

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