Supersymmetry and Hodge theory on Sasakian and Vaisman manifolds

TL;DR

This paper explores the geometric and algebraic structures of Sasakian and Vaisman manifolds, focusing on harmonic forms, Hodge theory, and supersymmetry algebra, providing new proofs and explicit computations.

Contribution

It introduces a Lie superalgebra for Sasakian manifolds and offers a coordinate-free proof of harmonic form decomposition results.

Findings

Construction of a Lie superalgebra for Sasakian manifolds

Coordinate-free proof of harmonic form decomposition

Explicit computation of supersymmetry algebra

Abstract

Sasakian manifolds are odd-dimensional counterpart to Kahler manifolds. They can be defined as contact manifolds equipped with an invariant Kahler structure on their symplectic cone. The quotient of this cone by the homothety action is a complex manifold called Vaisman. We study harmonic forms and Hodge decomposition on Vaisman and Sasakian manifolds. We construct a Lie superalgebra associated to a Sasakian manifold in the same way as the Kahler supersymmetry algebra is associated to a Kahler manifold. We use this construction to produce a self-contained, coordinate-free proof of the results by Tachibana, Kashiwada and Sato on the decomposition of harmonic forms and cohomology of Sasakian and Vaisman manifolds. In the last section, we compute the supersymmetry algebra of Sasakian manifolds explicitly.