Abelianisation of Connections, Quantum Curves, and Spectral Clusters

Project Information

AbQuantumSpec

Grant agreement ID: 101026083

DOI

10.3030/101026083

Project closed

EC signature date 18 March 2021

Start date 1 September 2022

End date 31 August 2024

Funded under

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Total cost

€ 224 933,76

EU contribution

€ 224 933,76

Coordinated by

THE UNIVERSITY OF BIRMINGHAM
United Kingdom

Periodic Reporting for period 1 - AbQuantumSpec (Abelianisation of Connections, Quantum Curves, and Spectral Clusters)

Reporting period: 2022-09-01 to 2024-08-31

Summary of the context and overall objectives of the project

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.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

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.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

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.