Deformations of Asymptotically Conical $G_2$-Instantons

Abstract

We develop the deformation theory of instantons on asymptotically conical $G_2$-manifolds, where an asymptotic connection at infinity is fixed. A spinorial approach is adopted to relate the space of deformations to the kernel of a twisted Dirac operator on the $G_2$-manifold and to the eigenvalues of a twisted Dirac operator on the nearly Kähler link. This framework is then used to calculate the virtual dimension of the moduli spaces of $G_2$-instantons on which several known examples live. One such example considered is the $G_2$-instanton of Günaydin-Nicolai, which lives on $R^7$. As an application of the deformation theory, we show how knowledge of the virtual dimension of the moduli space allows us to prove that unobstructed connections in the moduli space are $G_2$-invariant. By classifying such connections we prove a uniqueness result for unobstructed $G_2$-instantons on the principal $G_2$-bundle over $R^7$.

arXiv comment66 Pages. Additional examples, on the Bryant-Salamon manifolds, have been added to this revision

References in corpus (8)

• Gauge Theory in higher dimensions, II
• Spinorial description of $\mathrm{SU}(3)$- and $G_2$-manifolds
• The Hermitian Laplace Operator on Nearly Kähler Manifolds
• Monopoles on the Bryant-Salamon $G_2$ Manifolds
• Instantons on the exceptional holonomy manifolds of Bryant and Salamon
• Deformation theory of G_2 conifolds
• Deformation Theory of Asymptotically Conical Coassociative 4-folds
• Asymptotically Conical Associative 3-folds

Cited by in corpus (3)

• Evidence for an Algebra of ${\mathbf{G}}_2$ Instantons
• Deformations of ${\mathrm{G}}_2$-instantons on nearly ${\mathrm{G}}_2$ manifolds
• Families of solutions of the heterotic G${}_2$ system