Holomorphic Floer theory I: exponential integrals in finite and infinite dimensions

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.