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# 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

#### 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.