Periodic Reporting for period 1 - AbQuantumSpec (Abelianisation of Connections, Quantum Curves, and Spectral Clusters)
Période du rapport: 2022-09-01 au 2024-08-31
The project focused on developing the theory of abelianisation of meromorphic connections. Meromorphic connections are a vast geometric generalisation of ordinary differential equations (ODEs) that encompasses many classical, widely studied, and very important equations in mathematical physics. Meromorphic connections encode highly transcendental analytic information which is often difficult to describe using existing mathematical frameworks. The project addressed the challenge of finding a geometric approach to systematically represent and study this data using the geometry of algebraic curves.
Major obstacles were encountered in the development of the relation of this theory with the exact WKB method. To overcome the challenges, several major old conjectures in the WKB method and exact perturbation theory were proved. These newly established theorems provide a bridge between the abelianisation of meromorphic connections and the semi-classical approximation in quantum theory.
In addition to the primary objectives, the project laid the foundations of a new geometric theory of resurgence and exact WKB method in quantum mechanics.