An analytic invariant of G_2 manifolds
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2015
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| _version_ | 1866916606736596992 |
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| author | Crowley, Diarmuid Goette, Sebastian Nordström, Johannes |
| author_facet | Crowley, Diarmuid Goette, Sebastian Nordström, Johannes |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1505_02734 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | An analytic invariant of G_2 manifolds Crowley, Diarmuid Goette, Sebastian Nordström, Johannes Geometric Topology Differential Geometry 57R20 (Primary) 53C29, 58J28 (Secondary) 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. |
| title | An analytic invariant of G_2 manifolds |
| topic | Geometric Topology Differential Geometry 57R20 (Primary) 53C29, 58J28 (Secondary) |
| url | https://arxiv.org/abs/1505.02734 |