An analytic invariant of G_2 manifolds

Abstract

We prove that the moduli space of holonomy G_2-metrics on a closed 7-manifold is in general disconnected by presenting a number of explicit examples. We detect different connected components of the G_2-moduli space by defining an integer-valued analytic refinement of the nu-invariant, a Z/48-valued defect invariant of G_2-structures on a closed 7-manifold introduced by the first and third authors. The refined invariant is defined using eta invariants and Mathai-Quillen currents on the 7-manifold and we compute it for twisted connected sums à la Kovalev, Corti-Haskins-Nordström-Pacini and extra-twisted connected sums as constructed by the second and third authors. In particular, we find examples of G_2-holonomy metrics in different components of the moduli space where the associated G_2-structures are homotopic and other examples where they are not.

35 pages, 2 figures. v5: Added more explanation in section 4, minor other clarifications

References in corpus (4)

• Eta-Invariants and Determinant Lines
• G_2-manifolds and associative submanifolds via semi-Fano 3-folds
• New invariants of G_2-structures
• The classification of 2-connected 7-manifolds

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• Harmonic flow of geometric structures
• Eleven-Dimensional Supergravity in 4D, $N=1$ Superspace
• G2 instantons and the Seiberg-Witten monopoles
• Nu-invariants of extra-twisted connected sums
• Disconnecting the moduli space of G_2-metrics via U(4)-coboundary defects
• The asymptotic geometry of ${\mathrm{G}}_2$-monopoles
• On the incompleteness of $G_2$-moduli spaces along degenerating families of $G_2$-manifolds
• Geometry and periods of $G_2$-moduli spaces