The term "Holomorphic Floer Theory" (HFT for short) was suggested in 2014

by Maxim Kontsevich and myself for the area of mathematics devoted

to the study of questions of ``Floer-theoretical nature"

in the framework of complex symplectic manifolds.

Morse theory of a holomorphic Morse function gives an example.

In my talk I am going to discuss an application of HFT to the study of exponential integrals. The corresponding

geometry is the one of a pair of holomorphic Lagrangian submanifolds of a complex sympectic manifold.

The key technical tool will be the notion of wall-crossing structure introduced jointly with Kontsevich in 2008 and revisited in 2013 and 2020.

Originally this notion was designed for the purposes of wall-crossing formulas in Donaldson-Thomas theory.

It was later discovered that it also appeared in the areas of mathematics and physics which did not have an obvious connection to the

DT-theory, e.g. in complex integrable systems.