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Complex Geometry Seminar
Complex Geometry Seminar
Anisotropic calibrations, adiabatic limits and mirror symmetry
Anisotropic calibrations, adiabatic limits and mirror symmetry
Organizers
Speaker
Time
Wednesday, May 20, 2026 11:00 AM - 12:00 PM
Venue
A3-4-301
Online
Zoom 293 812 9202
(BIMSA)
Abstract
This seminar will provide an overview of recent work, joint with K. Kawai, on the following topics.
Let $(M,g)$ be a Riemannian manifold. Choose a pair $(\alpha,H)$ where $\alpha$ is a calibration and $H$ is a calibrated distribution. Using this data we define a 1-parameter family of forms $\alpha_\epsilon$ and study its adiabatic limit as $\epsilon\rightarrow 0$. We show that (i) the limit is a calibration in a generalized sense, (ii) 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 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, $\alpha$-calibrated (associative) submanifolds correspond to deformed Donaldson--Thomas connections, adiabatic calibrated submanifolds correspond to $G_2$-instantons.
Let $(M,g)$ be a Riemannian manifold. Choose a pair $(\alpha,H)$ where $\alpha$ is a calibration and $H$ is a calibrated distribution. Using this data we define a 1-parameter family of forms $\alpha_\epsilon$ and study its adiabatic limit as $\epsilon\rightarrow 0$. We show that (i) the limit is a calibration in a generalized sense, (ii) 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 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, $\alpha$-calibrated (associative) submanifolds correspond to deformed Donaldson--Thomas connections, adiabatic calibrated submanifolds correspond to $G_2$-instantons.