Holomorphic Floer theory I: exponential integrals in finite and infinite dimensions
Maxim Kontsevich, Yan Soibelman
TL;DR
The paper develops Holomorphic Floer Theory as a bridge between deformation quantization and Floer-theoretic approaches on complex symplectic manifolds, formalizing a global and local Riemann-Hilbert correspondence that relates de Rham and Betti data via exponential integrals and wall-crossing structures. It treats both finite- and infinite-dimensional exponential integrals, establishing twisted (De Rham and Betti) cohomologies, comparison isomorphisms, and analyticity/resurgence properties of wall-crossing data. A substantial portion is devoted to extending Morse-Novikov-type theories to holomorphic 1-forms, examining rationality and analyticity of WCS, and linking these to wheels of projective lines and non-archimedean perspectives. The 1st part lays a comprehensive framework and conjectures for infinite-dimensional counterparts (quantum wave functions, transport, Nahm sums, and Chern-Simons-like structures), with the aim of unifying RH-categorification, Fukaya categories, and holonomic DQ-modules under resurgence and analytic-wall-crossing regimes. Overall, the work proposes a programmatic pathway toward a fully-fledged holomorphic quantization theory and its implications for QFT-like path integrals, Fukaya categories, and non-perturbative invariants.
Abstract
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.