Pure and Applied Mathematics Quarterly

Volume 17 (2021)

Number 5

The real Fourier–Mukai transform of Cayley cycles

Pages: 1861 – 1898

DOI: https://dx.doi.org/10.4310/PAMQ.2021.v17.n5.a7

Authors

Kotaro Kawai (Department of Mathematics, Faculty of Science, Gakushuin University, Toshima-ku, Tokyo, Japan)

Hikaru Yamamoto (Department of Mathematics, Faculty of Pure and Applied Science, University of Tsukuba, Ibaraki, Japan)

Abstract

The real Fourier–Mukai transform sends a section of a torus fibration to a connection over the total space of the dual torus fibration. By this method, Leung, Yau and Zaslow introduced deformed Hermitian Yang–Mills (dHYM) connections for Kähler manifolds and Lee and Leung introduced deformed Donaldson–Thomas (dDT) connections for $G_2$- and $\operatorname{Spin}(7)$-manifolds.

In this paper, we suggest an alternative definition of a dDT connection for a manifold with a $\operatorname{Spin}(7)$-structure which seems to be more appropriate by carefully computing the real Fourier–Mukai transform again. We also post some evidences showing that the definition we suggest is compatible with dDT connections for a $G_2$-manifold and dHYM connections of a Calabi–Yau 4-manifold.

Another importance of this paper is that it motivates our study in our other papers. That is, based on the computations in this paper, we develop the theories of deformations of dDT connections for a manifold with a $\operatorname{Spin}(7)$-structure and the “mirror” of the volume functional, which is called the Dirac–Born–Infeld (DBI) action in physics.

Keywords

mirror symmetry, deformed Donaldson–Thomas, special holonomy, calibrated submanifold

2010 Mathematics Subject Classification

Primary 53C07. Secondary 53C25, 53C38, 53D37.

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The first-named author is supported by JSPS KAKENHI Grant Number JP17K14181.

The second-named author is supported by JSPS KAKENHI Grant Number JP18K13415 and Osaka City University Advanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics).

Received 14 January 2021

Accepted 30 June 2021

Published 26 January 2022