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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2410.09987 |
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| _version_ | 1866918099534479360 |
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| author | Langlais, Thibault |
| author_facet | Langlais, Thibault |
| contents | This paper is concerned with the geometry of the moduli space $\mathscr{M}$ of torsion-free $G_2$-structures on a compact $G_2$-manifold $M$, equipped with the volume-normalised $L^2$-metric $\mathscr{G}$. When $b^1(M) = 0$, this metric is known to be of Hessian type and to admit a global potential. Here we give a new description of the geometry of $\mathscr{M}$, based on the observation that there is a natural way to immerse the moduli space into a homogeneous space $\mathfrak{D}$ diffeomorphic to $GL(n+1)/ (\{\pm 1\} \times O(n))$, where $n = b^3(M) - 1$. We point out that the formal properties of this immersion $Φ: \mathscr{M} \rightarrow \mathfrak{D}$ are very similar to those of the period map defined on the moduli spaces of Calabi--Yau threefolds. With a view to understand the curvatures of $\mathscr{G}$, we also derive a new formula for the fourth derivative of the potential and relate it to the second fundamental form of $Φ(\mathscr{M}) \subset \mathfrak{D}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_09987 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Geometry and periods of $G_2$-moduli spaces Langlais, Thibault Differential Geometry High Energy Physics - Theory 53C26 (Primary), 53C29, 53A15, 58D27 (Secondary) This paper is concerned with the geometry of the moduli space $\mathscr{M}$ of torsion-free $G_2$-structures on a compact $G_2$-manifold $M$, equipped with the volume-normalised $L^2$-metric $\mathscr{G}$. When $b^1(M) = 0$, this metric is known to be of Hessian type and to admit a global potential. Here we give a new description of the geometry of $\mathscr{M}$, based on the observation that there is a natural way to immerse the moduli space into a homogeneous space $\mathfrak{D}$ diffeomorphic to $GL(n+1)/ (\{\pm 1\} \times O(n))$, where $n = b^3(M) - 1$. We point out that the formal properties of this immersion $Φ: \mathscr{M} \rightarrow \mathfrak{D}$ are very similar to those of the period map defined on the moduli spaces of Calabi--Yau threefolds. With a view to understand the curvatures of $\mathscr{G}$, we also derive a new formula for the fourth derivative of the potential and relate it to the second fundamental form of $Φ(\mathscr{M}) \subset \mathfrak{D}$. |
| title | Geometry and periods of $G_2$-moduli spaces |
| topic | Differential Geometry High Energy Physics - Theory 53C26 (Primary), 53C29, 53A15, 58D27 (Secondary) |
| url | https://arxiv.org/abs/2410.09987 |