11 - Hyper-Kähler geometry in harmonic space

Summary

The aim of this chapter is to show that the concept of harmonic analyticity has deep implications not only in N = 2 (and N > 2) supersymmetry, but also in purely bosonic (N = 0) gauge theories. Namely, this concept allows one to obtain unconstrained geometric formulations of self-dual Yang–Mills theory and hyper-Kähler geometry. These formulations closely parallel those of N = 2 Yang–Mills theory and N = 2 supergravity in harmonic superspace. The basic objects are unconstrained potentials defined on an analytic subspace of the harmonic extension of the original space (in the general case it is ℝ⁴ⁿ × S²). They encode all the information about the quantities present in the conventional formulations, self-dual Yang–Mills connections in the first case and hyper-Kähler metrics in the second one. We show that the harmonic analytic potential of the most general hyper-Kähler manifold serves as the Lagrangian of the most general N = 2 sigma model (after identifying the coordinates of the analytic subspace of this manifold with the analytic superfields q⁺ describing N = 2 matter). This establishes a direct, one-to-one correspondence between N = 2 sigma models and hyper-Kähler manifolds.

Introduction

In the previous chapters we demonstrated the relevance of the harmonic variables for obtaining an adequate formulation of N = 2 supersymmetric theories.