Bryant-Salamon ${\mathrm{G}}_2$ manifolds and coassociative fibrations

Abstract

Bryant-Salamon constructed three 1-parameter families of complete manifolds with holonomy ${\mathrm{G}}_2$ which are asymptotically conical to a holonomy ${\mathrm{G}}_2$ cone. For each of these families, including their asymptotic cone, we construct a fibration by asymptotically conical and conically singular coassociative 4-folds. We show that these fibrations are natural generalizations of the following three well-known coassociative fibrations on ${\mathbb{R}}^7$: the trivial fibration by 4-planes, the product of the standard Lefschetz fibration of ${\mathbb{C}}^3$ with a line, and the Harvey-Lawson coassociative fibration. In particular, we describe coassociative fibrations of the bundle of anti-self-dual 2-forms over the 4-sphere ${\mathcal{S}}^4$, and the cone on $\mathbb{C}{\mathbb{P}}^3$, whose smooth fibres are $T^{\ast }{\mathcal{S}}^2$, and whose singular fibres are ${\mathbb{R}}^4/\{\pm 1\}$. We relate these fibrations to hypersymplectic geometry, Donaldson's work on Kovalev-Lefschetz fibrations, harmonic 1-forms and the Joyce--Karigiannis construction of holonomy ${\mathrm{G}}_2$ manifolds, and we construct vanishing cycles and associative "thimbles" for these fibrations.

76 pages, 7 tables, 11 colour figures. Version 2: Discussion of multimoment maps clarified. Other minor revisions and improvements made as per suggestions of the referee. Final version, to appear in Journal of Geometry and Physics. Version 3: Corrected three spelling errors, updated publication info of reference [18], and added funding information to acknowledgements

References in corpus (2)

• 2-Ruled Calibrated 4-folds in R^7 and R^8
• Mukai duality on adiabatic coassociative fibrations

Cited by in corpus (5)

• $G_2$-Manifolds from 4d N=1 Theories, Part I: Domain Walls
• Cayley fibrations in the Bryant-Salamon $Spin(7)$ manifold
• Deformed $G_2$-instantons on ${\mathbb{R}}^4\times S^3$
• Deformations of calibrated subbundles in noncompact manifolds of special holonomy via twisting by special sections
• Cohomogeneity one solitons for the isometric flow of $G_2$-structures