Speaker
Description
The ODE/IM correspondence identifies spectral data of ordinary differential equations with the structures of quantum integrable models. These lectures present the correspondence from a modern viewpoint, based on the concept of opers.
We begin with the classical prototype: the anharmonic oscillator ODE. We will work through the derivation of special entire functions, known as spectral determinants, and of the many functional relations that these functions satisfy. In order to get there, we will discuss distinguished solutions, the Stokes phenomenon, Wronskian identities, and related analytic structures. This first example will serve to illustrate the central mechanism in a reasonably concrete way: the spectral determinants of a linear differential problem coincide, as functions, with the eigenvalues of T- and Q-operators of an integrable quantum field theory, in this case a family of conformal field theories.
Next, we will reformulate this prototype in an increasingly structural way, naturally arriving at the concept of opers — roughly, distinguished gauge-equivalence classes of connections. This will allow us to formulate the ODE/IM correspondence in a more general framework, applicable to several different families of integrable field theories.
Finally, we will place this picture in the contemporary research landscape, discussing its relation with Gaudin models, the extension to excited states and off-critical theories, and recent developments involving q-deformations.
Given the breadth of the subject and the limited lecture time, the lectures will emphasise the main ideas, constructions and conceptual links, rather than technical details. A set of detailed lecture notes will be provided, and students are encouraged to use them as a companion text both during and after the course.