Abstract
We investigate instantons on manifolds with Killing spinors and their cones. Examples of manifolds with Killing spinors include nearly Kaehler 6-manifolds, nearly parallel G_2-manifolds in dimension 7, Sasaki-Einstein manifolds, and 3-Sasakian manifolds. We construct a connection on the tangent bundle over these manifolds which solves the instanton equation, and also show that the instanton equation implies the Yang-Mills equation, despite the presence of torsion. We then construct instantons on the cones over these manifolds, and lift them to solutions of heterotic supergravity. Amongst our solutions are new instantons on even-dimensional Euclidean spaces, as well as the well-known BPST, quaternionic and octonionic instantons.
Highlights
Cone metric over a manifold with real Killing spinor does have special holonomy
The only manifolds with real Killing spinors are nearly parallel G2-manifolds in dimension 7, nearly Kahler manifolds in dimension 6, Sasaki-Einstein manifolds, and 3Sasakian manifolds
In doing so we construct a distinguished connection on the tangent bundle which solves the instanton equation, and which seems to be an analog of the LeviCivita connection in the geometry of real Killing spinors
Summary
Let E → M be a vector bundle over a Riemannian manifold (M, g) of dimension n, and A a connection on E with curvature. On manifolds of special holonomy the 4-form Q is both closed and coclosed, so the second term vanishes and we are left with the Yang-Mills equation ∇A ∧ ∗F = 0. The instanton equation does imply the Yang-Mills equation on a manifold with real Killing spinor, as the following proposition shows: Proposition 2.1. The instanton equation (2.2) implies that the second term vanishes, and since the action of 1-forms on spinors is invertible, we conclude that ∇A ∧ ∗F = 0 Note that this proposition applies to manifolds with parallel spinor as well as manifolds with real Killing spinor. On Kahler manifolds with holonomy group U(m) the most obvious instanton condition F ∈ u(m) does not automatically imply the Yang-Mills equation, because U(m) does not fix any spinor. Which imply in particular the full Yang-Mills equation for F
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