  In the first of the series of papers devoted to our project ``Holomorphic
Floer Theory" we discuss exponential integrals and related wall-crossing
structures. We emphasize two points of view on the subject: the one based on
the ideas of deformation quantization and the one based on the ideas of Floer
theory. Their equivalence is a corollary of our generalized Riemann-Hilbert
correspondence. In the case of exponential integrals this amounts to several
comparison isomorphisms between local and global versions of de Rham and Betti
cohomology. We develop the corresponding theories in particular generalizing
Morse-Novikov theory to the holomorphic case. We prove that arising
wall-crossing structures are analytic. As a corollary, perturbative expansions
of exponential integrals are resurgent. Based on a careful study of
finite-dimensional exponential integrals we propose a conjectural approach to
infinite-dimensional exponential integrals. We illustrate this approach in the
case of Feynman path integral with holomorphic Lagrangian boundary conditions
as well as in the case of the complexified Chern-Simons theory. We discuss the
arising perverse sheaf of infinite rank as well as analyticity of the
corresponding ``Chern-Simons wall-crossing structure". We develop a general
theory of quantum wave functions and show that in the case of Chern-Simons
theory it gives an alternative description of the Chern-Simons wall-crossing
structure based on the notion of generalized Nahm sum. We propose several
conjectures about analyticity and resurgence of the corresponding perturbative
series.
