Holomorphic Floer Theory, Exponential Integrals and Resurgence

It is common in mathematics and physics to describe quantities as infinite series with respect to small parameters. On the other hand, these series can have global meanings in the regions where the parameters are not small. This phenomenon often appears in physics under the name of "duality". The PI will develop a theory encompassing a large class of examples associated with so-called exponential integrals whose associated global information gives rise to unexpected geometric structures. The results will reveal novel relations between the theory of exponential integrals and quantum physics. The PI will also integrate this research with educational efforts, developing new directions and projects for young researchers. Exponential integrals give a simple but deep special case of Holomorphic Floer Theory. The project will study exponential integrals with an emphasis on Hodge-theoretical aspects and their relation to the generalized Riemann-Hilbert correspondence introduced by the PI and Kontsevich. The PI will develop the de Rham and Betti cohomology theories associated with exponential integrals, including comparison isomorphisms. This is done in the framework of complex manifolds endowed with holomorphic functions or with closed one-forms. Some outcomes of the research will be a Holomorphic Morse-Novikov theory. Developing the notion of wall-crossing structures the PI will give a conceptual explanation of resurgent properties of perturbative expansions of exponential integrals in finite and infinite dimensions. This project is jointly funded by the Topology and Geometric Analysis program and the Established Program to Stimulate Competitive Research (EPSCoR). This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.