An analytic invariant of G_2 manifolds

TL;DR

This work shows that the moduli space of holonomy $G_2$-metrics on a closed 7-manifold can be disconnected by constructing explicit extra-twisted and twisted connected sums and by refining the $\nu$-invariant to an integer-valued $\bar{\nu}$. The authors formulate $\bar{\nu}$ analytically via eta-invariants and a Mathai-Quillen current, prove its key topological and geometric properties, and prove its local constancy on the $G_2$-metric moduli space. They develop a refined gluing framework for both the odd-signature and spin Dirac operators, including a deformation-based approach and precise Maslov-angle computations, to compute $\bar{\nu}$ for a broad class of constructions. The main contributions include explicit calculations of $\bar{\nu}$ for extra-twisted connected sums with gluing angles $\vartheta = \pi/4$ and $\vartheta = \pi/6$, yielding examples where two $G_2$-holonomy metrics lie in different components of the moduli space even when their underlying $G_2$-structures are homotopic. Together, these results demonstrate how analytic invariants can distinguish connected components of $G_2$-moduli spaces and provide a framework for systematically producing such distinct components in higher-dimensional holonomy geometries.

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.

Figures (2)

• Figure 1: angle pi/4
• Figure 2: angle pi/6

Theorems & Definitions (67)

• Theorem 1
• Corollary 2
• Corollary 3
• Theorem 4
• Theorem 5
• Definition 1.1: Crowley--Nordström CrN
• Theorem 1.2
• proof
• Lemma 1.3
• proof