Advances in Theoretical and Mathematical Physics

Volume 22 (2018)

Number 3

SYZ transformation for coisotropic A-branes

Pages: 509 – 564

DOI: http://dx.doi.org/10.4310/ATMP.2018.v22.n3.a1

Authors

Kaileung Chan (Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong)

Naichung Conan Leung (Institute of Mathematical Sciences and Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong)

Yi Zhang (Institute of Mathematical Sciences and Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong)

Abstract

Kapustin and Orlov observed that natural boundary conditions in A-model are coisotropic A-branes, and also they need to be included for mirror symmetry.

In the SYZ conjecture, the transformation which takes a holomorphic bundle $E$ in $\check{X}$ to a Lagrangian A-brane in its mirror manifold $X$ uses the property that the restriction of $E$ to any Lagrangian torus fiber in $\check{X}$ is topologically trivial.

In the semiflat setting, without assuming that $E$ is fiberwise topologically trivial, we construct a SYZ transformation which takes holomorphic bundles in $\check{X}$ to coisotropic A-branes in $X$ and vice versa. The construction uses fiberwise Nahm transformations for twisted Dirac operators on tori.

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