Supersymmetry and the cohomology of (hyper)Kaehler manifolds

TL;DR

This paper demonstrates how supersymmetry underpins the Lie algebra actions on the cohomology of compact Kaehler and hyperKaehler manifolds, linking physical symmetries to geometric structures.

Contribution

It reveals the origin of Lie algebra actions on cohomology from supersymmetry principles, connecting physics and complex geometry in a novel way.

Findings

Lie algebra actions derived from supersymmetry

Fundamental identities in Hodge-Lefschetz theory explained via supersymmetry

Supersymmetric sigma models underpin geometric cohomology structures

Abstract

The cohomology of a compact Kaehler (resp. hyperKaehler) manifold admits the action of the Lie algebra so(2,1) (resp. so(4,1)). In this paper we show, following an idea of Witten, how this action follows from supersymmetry, in particular from the symmetries of certain supersymmetric sigma models. In addition, many of the fundamental identities in Hodge-Lefschetz theory are also naturally derived from supersymmetry.