A parabolic flow for the large volume heterotic $G_2$ system
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
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2025
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| _version_ | 1866908714654498816 |
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| author | Garcia-Fernandez, Mario Moreno, Andres J. Payne, Alec Streets, Jeffrey |
| author_facet | Garcia-Fernandez, Mario Moreno, Andres J. Payne, Alec Streets, Jeffrey |
| contents | We introduce a geometric flow of conformally coclosed $G_2$-structures, whose fixed points are large volume solutions of the heterotic $G_2$ system, with vanishing scalar torsion class $τ_0 = 0$. After conformal rescaling, it becomes a flow of coclosed $G_2$-structures, related to Grigorian's modified $G_2$ coflow, which is coupled to a flow for a dilaton function. Our main results establish fundamental short-time existence and Shi-type smoothing properties of this flow, as well as a classification of its fixed points. By a classical rigidity result in the string theory literature, the fixed points on a compact manifold correspond to torsion-free $G_2$-structures, that is, to metrics with holonomy contained in $G_2$. Thus, we establish in the affirmative a folklore question in the special holonomy community, about the existence of a well-posed flow for coclosed $G_2$-structures with fixed points given by torsion-free $G_2$-structures. The flow also satisfies a monotonicity formula for the $G_2$-dilaton functional (volume scale in string theory), which allows us to strengthen the rigidity result with an alternative proof. The monotonicity of the $G_2$-dilaton functional, combined with the Shi-type estimates, leads to a general result on the convergence of nonsingular solutions. A dimension reduction analysis reveals an interesting link with natural flows for $SU(3)$-structures, previously introduced in the literature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_14317 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A parabolic flow for the large volume heterotic $G_2$ system Garcia-Fernandez, Mario Moreno, Andres J. Payne, Alec Streets, Jeffrey Differential Geometry Analysis of PDEs 53E40, 53C10, 53C25, 53C29 We introduce a geometric flow of conformally coclosed $G_2$-structures, whose fixed points are large volume solutions of the heterotic $G_2$ system, with vanishing scalar torsion class $τ_0 = 0$. After conformal rescaling, it becomes a flow of coclosed $G_2$-structures, related to Grigorian's modified $G_2$ coflow, which is coupled to a flow for a dilaton function. Our main results establish fundamental short-time existence and Shi-type smoothing properties of this flow, as well as a classification of its fixed points. By a classical rigidity result in the string theory literature, the fixed points on a compact manifold correspond to torsion-free $G_2$-structures, that is, to metrics with holonomy contained in $G_2$. Thus, we establish in the affirmative a folklore question in the special holonomy community, about the existence of a well-posed flow for coclosed $G_2$-structures with fixed points given by torsion-free $G_2$-structures. The flow also satisfies a monotonicity formula for the $G_2$-dilaton functional (volume scale in string theory), which allows us to strengthen the rigidity result with an alternative proof. The monotonicity of the $G_2$-dilaton functional, combined with the Shi-type estimates, leads to a general result on the convergence of nonsingular solutions. A dimension reduction analysis reveals an interesting link with natural flows for $SU(3)$-structures, previously introduced in the literature. |
| title | A parabolic flow for the large volume heterotic $G_2$ system |
| topic | Differential Geometry Analysis of PDEs 53E40, 53C10, 53C25, 53C29 |
| url | https://arxiv.org/abs/2512.14317 |