Null Sasaki eta-Einstein Structures in Five Manifolds

TL;DR

This paper classifies simply connected five-dimensional manifolds that admit null Sasaki eta-Einstein structures, extending previous results and describing their moduli space using lattice polarized K3 surfaces.

Contribution

It improves existing classification results for null Sasaki eta-Einstein structures on certain five-manifolds and determines their moduli space.

Findings

Manifolds diffeomorphic to #k(S^2×S^3) admit null Sasaki η-Einstein structures for k=3,...,21

Complete classification of simply connected null Sasaki η-Einstein structures on these manifolds

Description of the moduli space using lattice polarized K3 surface data

Abstract

We study null Sasakian structures in dimension five. First, based on a result due to Koll\'ar [Ko], we improve a result by Boyer, Galicki and Matzeu in [BGM] and prove that simply connected manifolds diffeomorphic to $# k(S^2\times S^3)$ admit null Sasaki $\eta$-Einstein structures if and only if $k\in \{3,..., 21\}$. After this, we determine the moduli space of simply connected null Sasaki $\eta$-Einstein structures. This is accomplished using information on the moduli of lattice polarized K3 surfaces.