I will present various aspects (both physical and mathematical) of the theory of flag manifold sigma models. At the classical level these theories describe generalized harmonic maps of Riemann surfaces into flag manifolds. In recent years at least two interesting applications of such theories have been found: first of all, they describe continuum limits of spin chains, and the discovered relation allows to study the phase structure of such models; besides, it turns out that some of these models are integrable and, moreover, they open a "window" into a wide class of integrable sigma models, related to Nakajima varieties (from the physical viewpoint these models are generalized Gross–Neveu models). I plan to discuss both of these research directions, as well as the interrelations between them and especially the many open questions.

Financial support. The course is supported by the Simons Foundation and the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, Agreement no. 075-15-2019-1614).

Lecturer
Bykov Dmitrii Vladimirovich

Institutions
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center