Most mathematical models in theoretical physics cannot be solved exactly. “Integrable” models are those which, although certainly non-trivial, can in principle be completely solved by means of exact analytical techniques. Yet despite being rare, integrable models play an essential role in fundamental physics. Most notably, classical and quantum integrable field theories have been extensively studied over the past twenty years due to their importance in understanding the celebrated AdS/CFT (Anti de Sitter / Conformal Field Theory) correspondence.

In the context of quantum integrable field theories, a remarkable conjecture known as the ODE/IM correspondence began to emerge over two decades ago. It provides a fascinating description of the spectrum of the (infinite number of) Integrals of Motion (IM) of a quantum integrable field theory in terms of certain Ordinary Differential Equations (ODE). This mysterious conjecture has been observed in a growing list of examples of quantum integrable field theories. Mathematically, significant progress has been made in proving this conjecture in the particular example of quantum KdV (Korteweg-De Vries) theory, but an explanation of this correspondence for general quantum integrable field theories, in particular massive theories, remains elusive.

A promising avenue for understanding the ODE/IM correspondence comes from the observation that many classical integrable field theories can be seen as classical Gaudin models associated with affine Kac-Moody algebras. Indeed, it is known that the spectrum of the integrals of motion Gaudin models associated with finite-dimensional Lie algebras can be succinctly described in terms of certain ODEs. Conjecturally, the ODE/IM correspondence can then be seen as a generalisation of this story to the setting of affine Kac-Moody algebras.

The goal of the project will be to make progress towards obtaining a rigorous mathematical understanding of the ODE/IM correspondence. Specific initial tasks may include the study of certain quantum integrable field theories through the lens of Gaudin models associated with affine Kac-Moody algebras. Various dualities in quantum integrable field theories, such as strong/weak coupling dualities or boson/fermion dualities, will also be explored from this new perspective.

This three-year PhD will be held in the Department of Mathematics at the University of York, and supervised by Dr Benoit Vicedo.

You should be expecting to graduate with a First Class degree from a good university, and have a strong background in mathematics and theoretical physics. An existing or developing interest in integrable systems would be an advantage.

models” studentship, naming Dr Vicedo as your potential supervisors.

If you are interested in any other Maths Studentships at York, follow this link.