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Instantons and Killing spinors

Derek Harland, Christoph Nölle

TL;DR

We analyze instantons on manifolds with real Killing spinors and their cones, showing that the instanton condition implies the Yang–Mills equation in this non-integrable-G-structure setting. A canonical tangent-bundle connection is constructed, proven to be an instanton, and extended to cones where a one-parameter family of instantons interpolates between Levi-Civita and canonical connections. The instanton construction is then lifted to heterotic supergravity, with explicit solutions that satisfy the gaugino, gravitino, and dilatino equations together with the Bianchi identity at leading order in α′; this yields new instantons on Euclidean spaces (including BPST, quaternionic, and octonionic cases) and a framework for Hermitian instantons on even dimensions. The results unify NK, nearly parallel G2, Sasaki–Einstein, and 3-Sasakian geometries, and provide explicit α′-corrected heterotic backgrounds, some of which extend smoothly over cone apexes for spheres.

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.

Instantons and Killing spinors

TL;DR

We analyze instantons on manifolds with real Killing spinors and their cones, showing that the instanton condition implies the Yang–Mills equation in this non-integrable-G-structure setting. A canonical tangent-bundle connection is constructed, proven to be an instanton, and extended to cones where a one-parameter family of instantons interpolates between Levi-Civita and canonical connections. The instanton construction is then lifted to heterotic supergravity, with explicit solutions that satisfy the gaugino, gravitino, and dilatino equations together with the Bianchi identity at leading order in α′; this yields new instantons on Euclidean spaces (including BPST, quaternionic, and octonionic cases) and a framework for Hermitian instantons on even dimensions. The results unify NK, nearly parallel G2, Sasaki–Einstein, and 3-Sasakian geometries, and provide explicit α′-corrected heterotic backgrounds, some of which extend smoothly over cone apexes for spheres.

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.

Paper Structure

This paper contains 39 sections, 3 theorems, 163 equations, 2 figures.

Table of Contents

  1. Introduction and summary
  2. Conventions.
  3. Instantons and the Yang-Mills equation
  4. Geometry of real Killing spinors
  5. Nearly parallel $\mathbf{G_2}$
  6. Canonical connection.
  7. $G_2$-instantons.
  8. Examples.
  9. Nearly Kähler
  10. Canonical connection.
  11. SU(3)-instantons.
  12. Examples.
  13. Sasaki-Einstein
  14. Canonical connection.
  15. SU($m$)-instantons.
  16. ...and 24 more sections

Key Result

Proposition 2.1

Suppose that $M$ is spin and carries a spinor $\epsilon$ solving equation Killingeq. If $A$ is gauge field on $M$ whose curvature form satisfies equation instcond1, then it solves the Yang-Mills equation.

Figures (2)

  • Figure 1: Instantons on the cone over Sasaki-Einstein manifolds, plotted in the $\psi,\chi$ plane. The dashed and dotted curves are instantons with $h=0$ and $m=1,2,4,8$. For $m=1$ we have $\psi =\chi$, and as $m$ increases the solutions get closer to the limiting solid curve $\psi^2=\chi$.
  • Figure 2: Sample solution of the Sasaki-Einstein supergravity equations with $m=2$.

Theorems & Definitions (6)

  • Proposition 2.1
  • proof
  • Proposition 3.1
  • proof : Proof of Prop. \ref{['prop:TorsionExSymmetry']}
  • Corollary 3.2
  • proof