Anisotropic calibrations, adiabatic limits and mirror symmetry
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| Format: | Preprint |
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2026
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| author | Kawai, Kotaro Pacini, Tommaso |
| author_facet | Kawai, Kotaro Pacini, Tommaso |
| contents | Let $(M,g)$ be a Riemannian manifold. Choose a pair $(α,H)$ where $α$ is a calibration and $H$ is a calibrated distribution. Using this data we define a 1-parameter family of forms $α_\varepsilon$ and study its adiabatic limit as $\varepsilon\rightarrow 0$. We show that (i) the limit is a calibration in a generalized sense, (ii) under the usual closedness assumptions, the adiabatic calibrated submanifolds are anisotropic minimal in the classical sense defined in the calculus of variations/PDE theory.
We apply this construction to $G_2$-manifolds. In this case the adiabatic calibrated condition is equivalent to a Fueter-type equation. We provide explicit examples and prove local analytic existence theorems for the adiabatic calibrated submanifolds. Applying mirror symmetry as described by the real Fourier-Mukai transform, the general picture is as follows: adiabatic limits correspond to large radius limits, $α$-calibrated (associative) submanifolds correspond to deformed Donaldson-Thomas connections, adiabatic calibrated submanifolds correspond to $G_2$-instantons. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_21161 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Anisotropic calibrations, adiabatic limits and mirror symmetry Kawai, Kotaro Pacini, Tommaso Differential Geometry Let $(M,g)$ be a Riemannian manifold. Choose a pair $(α,H)$ where $α$ is a calibration and $H$ is a calibrated distribution. Using this data we define a 1-parameter family of forms $α_\varepsilon$ and study its adiabatic limit as $\varepsilon\rightarrow 0$. We show that (i) the limit is a calibration in a generalized sense, (ii) under the usual closedness assumptions, the adiabatic calibrated submanifolds are anisotropic minimal in the classical sense defined in the calculus of variations/PDE theory. We apply this construction to $G_2$-manifolds. In this case the adiabatic calibrated condition is equivalent to a Fueter-type equation. We provide explicit examples and prove local analytic existence theorems for the adiabatic calibrated submanifolds. Applying mirror symmetry as described by the real Fourier-Mukai transform, the general picture is as follows: adiabatic limits correspond to large radius limits, $α$-calibrated (associative) submanifolds correspond to deformed Donaldson-Thomas connections, adiabatic calibrated submanifolds correspond to $G_2$-instantons. |
| title | Anisotropic calibrations, adiabatic limits and mirror symmetry |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2605.21161 |