AbQuantumSpec · Abelianisation of Connections, Quantum Curves, and Spectral Clusters
Horizon 2020 — Marie Skłodowska-Curie Actions
- Duration
- 2022-09-01 → 2024-08-31
- EU contribution
- €224,934
- Participants
- 1
- Scheme
- MSCA-IF
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Results in brief
Abelianisation of Connections, Quantum Curves, and Spectral Clusters
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.
Data: CORDIS, © European Union
Project objective
This cross-disciplinary project lies at the interface of geometry, mathematical physics, perturbation theory, and integrable systems, combining techniques from algebraic topology, cluster algebras, ordinary differential equations, and asymptotic analysis. The main goal is to advance our understanding of the geometry of character varieties and their quantisation. This project -- carried out by Nikita Nikolaev under the supervision of Marta Mazzocco at the University of Birmingham -- is expected to result in a fundamental innovation in geometry and have important implications for quantum field theory. It will open a vast new scientific arena and will serve to establish Nikolaev amongst research leaders in this highly active research area. Character varieties are geometric spaces which are ubiquitous in mathematics and physics, where they capture important topological and quantum invariants. These spaces parameterise singular complex ordinary differential equations (such as the Airy and Bessel equations, or even time-independent Schrödinger equations), as well as their generalisations: meromorphic connections on vector bundles over a Riemann surface. However, character varieties are usually complicated singular spaces, so the valuable information they encode is not easy to access. This project will develop a new method to describe character varieties called abelianisation. Ideas behind abelianisation stem from the WKB method in quantum mechanics and have recently resurfaced in the context of quantum field theory and string theory. Abelianisation will allow to construct special coordinate systems on character varieties (called spectral coordinates) which naturally capture crucially important geometric structure of character varieties (most prominently the symplectic and cluster structures). In other words, spectral coordinates will be a geometric gadget to decrypt mathematical and physical information encrypted in character varieties.
Original text from CORDIS.
Participants
- THE UNIVERSITY OF BIRMINGHAM · BirminghamCoordinatorUnited Kingdom
Links
Data: CORDIS, © European Union