S-duality in hyperkaehler Hodge theory

TL;DR

This paper surveys the Hodge theory of hyperkaehler quotients, exploring their geometric and topological properties in relation to S-duality in string theory, focusing on moduli spaces like instantons, monopoles, and Higgs bundles.

Contribution

It compiles and discusses recent results on the Hodge structures and harmonic forms of hyperkaehler moduli spaces motivated by string theory dualities.

Findings

Analysis of L^2 harmonic forms on hyperkaehler quotients

Results on Betti numbers and mixed Hodge structures

Connections between hyperkaehler geometry and S-duality

Abstract

Here we survey questions and results on the Hodge theory of hyperkaehler quotients, motivated by certain S-duality considerations in string theory. The problems include L^2 harmonic forms, Betti numbers and mixed Hodge structures on the moduli spaces of Yang-Mills instantons on ALE gravitational instantons, magnetic monopoles on R^3 and Higgs bundles on a Riemann surface. Several of these spaces and their hyperkaehler metrics were constructed by Nigel Hitchin and his collaborators.