Anisotropic calibrations, Fueter maps and mirror symmetry

Abstract

Let $(M,g)$ be a Riemannian manifold. Choose a pair $(\alpha ,H)$, where $\alpha$ is a calibration and $H$ is a calibrated distribution. Using these data, we define a 1-parameter family of forms ${\alpha }_{\epsilon }$ and study its adiabatic limit as $\epsilon \to 0$. 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 endowed with an associative distribution. Here, one can also define the notion of Fueter maps. We prove that, in the case of isometric immersions, adiabatic calibrated submanifolds coincide with Fueter maps: this is a first-order analogue of the classical relationship between minimal submanifolds and harmonic maps. We provide explicit examples and prove local analytic existence theorems for adiabatic calibrated submanifolds. Applying mirror symmetry as described by the real Fourier-Mukai transform in the standard ``toy model'' situation, the picture is as follows: adiabatic limits correspond to large radius limits, calibrated (associative) submanifolds correspond to deformed Donaldson-Thomas connections, adiabatic calibrated submanifolds correspond to $G_2$-instantons.

v2: 67 pages. This version contains a new Section 9, emphasizing the relationship between our notion of adiabatic calibrated submanifolds and the classical notion of Fueter maps. We have correspondingly changed the title and adjusted some of the terminology throughout the paper. Sections which can be omitted upon first reading are now marked by (*)