PhD Studentship: Geometry and Difference Equations in String Theory

This project aims to explore the geometric and physical properties of discrete Schrödinger equations, with a particular focus on their connections to Calabi-Yau geometries and the quantum symmetries arising in String Theory and Quantum Field Theory. The investigation will also engage with concepts from algebraic geometry and, potentially, integrability, allowing the project to adopt either a predominantly mathematical or physical perspective based on the candidate’s interests and expertise. General context: Open Calabi-Yau threefolds (CY3s) are central object to probe nonperturbative aspects of string theory and its 11-dimensional uplift, M-theory. Quantum string theory effects associate these geometries to q-difference equations, a class of mathematical equations that, despite their significance, remain considerably less understood than their differential counterparts. The analytic properties of their solutions, geometrically described by their monodromy spaces, are a research topic at the forefront of both algebraic geometry and mathematical physics. These spaces hold profound physical importance, as they are expected to parametrise quantum field theories arising from M-theory reductions on open CY3s, as well as the structure of their extended operators.

The project: This project aims to advance a recently developed framework, introduced by the PI and a collaborator, for studying the geometric and analytic properties of q-difference equations. A central focus will be on understanding the monodromy and Stokes phenomena of their solutions through using the WKB approximation. As a cutting-edge area of research with numerous open questions, the project will initially build the necessary intuition by examining specific examples of discrete analogs of classical special functions associated with simple Calabi-Yau geometries. These will provide the foundations for further developments to be pursued in the later stages of the project.

References:

1. Del Monte, P. Longhi, The threefold way to quantum periods: WKB, TBA equations and q-Painlevé. SciPost Phys. 15 (2023) 3, 112.
2. Del Monte, BPS Spectra and Algebraic Solutions of Discrete Integrable Systems. Commun.Math.Phys. 405 (2024) 6, 147.
3. Del Monte, P. Longhi, Monodromies of Second Order q-difference Equations from the WKB Approximation. ArXiv: 2406.00175.

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