Einstein manifolds of dimension five with small first eigenvalue of the Dirac operator

Rigid supersymmetry on 5-dimensional Riemannian manifolds and contact geometry

“…Prior to [25], manifolds with real Killing spinor are better known in low dimensions. For instance, 4-dimensional complete manifolds with real Killing spinor were shown to be isometric to the 4-sphere [26]. In 5-dimension, simply-connected manifolds with real Killing spinors were shown to be round S 5 or Sasaki-Einstein manifolds, with solutions coming down from covariantly constant spinors on their Calabi-Yau cone.…”

“…Using the 2nd and 3rd equation, one can solve (partially) the auxiliary fields in terms of the bilinears (field V is decomposed as V = V H + κ ∧ η): 26) where Ω ± are self-dual (+) and anti-self-dual (−) 2-forms respectively, satisfying extra condition…”

Calabi-Yau cones from contact reduction

“…Let H be a solvable 3-dimensional Lie group. Then there exists a semidirect product H ⋉ R 4 admitting a left-invariant contact SU(3)-structure whose associated spinor is generalized Killing if and only if H has structure equations (11)(12)(13)(14).…”

“…Next we prove the "if" part of Theorem 5, by giving explicit examples for each solvable Lie group (11)(12)(13)(14).…”

“…Einstein-Sasaki manifolds can be characterized in terms of real Killing spinors [14]; infinite families of explicit examples of Einstein-Sasaki manifolds have been constructed in [16,10]. These examples are in fact toric contact manifolds (see [21]).…”