Deformations of Locally Conformal Spin(7) Instantons
Eyup Yalcinkaya
TL;DR
This work addresses the deformation theory of Spin(7)-instantons on locally conformal Spin(7) manifolds, where the fundamental 4-form satisfies $d\Phi = \theta \wedge \Phi$ with Lee form $\theta$. By recasting the linearized deformation equations as a $t$-parameter Dirac problem $D_{t,A} = \mathcal{D}_{A,LC} + t \mathcal{T}_{\theta}$ and exploiting a cancellation of torsion terms, the authors prove that the infinitesimal deformation space $\mathcal{H}^1$ is independent of $t$ and isomorphic to $\ker(\mathcal{D}_{A,LC} + 3 \cdot \mathrm{id})$, i.e. governed by Levi-Civita geometry. They establish a Lichnerowicz-type rigidity criterion $\lambda_{\mathcal{L}} > 9 - \frac{1}{4}\mathrm{Scal}_g$ for rigidity and apply it to flat instantons on compact homogeneous LC Spin(7) manifolds, finding non-rigidity due to their comparatively small scalar curvature. For the flat connections on $M=SU(3)$ and $M=Sp(2)/T^2$, the computed scalar curvatures are well below the threshold, confirming non-trivial local moduli and motivating further study of the global moduli space structure. Overall, the work reduces a torsionful deformation problem to a classical Levi-Civita setting and provides concrete non-rigidity results that illuminate the local geometry of 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 $Φ$ satisfying $dΦ= θ\wedge Φ$, 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 $θ$ 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.