Conventional Floer theory studies real symplectic manifolds by means of algebraic and categorical structures associated with their Lagrangian submanifolds. The Floer complex of a pair of Lagrangian submanifolds is an example. Its upgrade to the Fukaya category of a symplectic manifold is another example.
In 2014 jointly with Maxim Kontsevich we started the project "Holomorphic Floer theory" (HFT) in which complex symplectic manifolds were the main object of study. This brings methods of symplectic topology into the realm of complex geometry without passing to the mirror dual manifold.
In this talk I plan to give an introduction to the project based mainly on our version of the Riemann-Hilbert correspondence.
The latter relates deformation quantization and the Fukaya category of the same complex symplectic manifold.
In particular it gives a tool for study some of the questions of "brane quantization" story in physics like e.g. the meaning of coisotropic branes, or how to associate a module over quantum algebra to a complex Lagrangian submanifold (maybe singular).
Another story in which HFT point of view is useful is the one of exponential integrals. HFT explains the geometric meaning of the Stokes phenomenon, wall-crossing formulas and resurgence in the framework of exponential integrals (maybe infinite-dimensional).