Deformations of Locally Conformal Spin(7) Instantons

TL;DR

This paper studies the deformation theory of instantons on locally conformal Spin(7) manifolds, revealing that the deformation space is governed by Levi-Civita geometry and establishing conditions for rigidity.

Contribution

It reformulates the linearized deformation equations using a family of Dirac operators, showing torsion terms cancel and simplifying the analysis to Levi-Civita geometry.

Findings

Deformation space governed by Levi-Civita geometry

Flat instanton on known manifolds is non-rigid

Established a rigidity criterion for LC Spin(7) instantons

Abstract

We explore the deformation theory of instantons on locally conformal (LC) $Spin(7)$ manifolds. These structures, characterized by a non-parallel fundamental 4-form $\Phi$ satisfying $d\Phi = \theta \wedge \Phi$, represent a significant, yet geometrically constrained, class of non-integrable $G$-structures. We analyze the infinitesimal deformation complex for $Spin(7)$-instantons in this setting. Our primary contribution is the reformulation of the linearized deformation equations -- comprising the linearized instanton condition and a gauge-fixing term -- using a $t$-parameter family of Dirac operators. We demonstrate that the $t$-dependent torsion terms arising from the Lee form $\theta$ cancel precisely. This unexpected simplification reveals that the deformation space $\mathcal{H}^1$ is governed entirely by the Levi-Civita geometry, effectively reducing the torsion-full problem to a more classical, torsion-free (Levi-Civita) setting. Using a Lichnerowicz-type rigidity theorem, we establish a general condition for an (LC) $Spin(7)$-instanton to be rigid (i.e., $\mathcal{H}^1 = \{0\}$). We apply this theory to the flat instanton ($A=0$) on known compact homogeneous (LC) $Spin(7)$ manifolds and conclude that the flat instanton on these spaces is non-rigid, thus possessing a non-trivial moduli space.