Grassmannian sigma models and nilpotent orbits

A sigma model of a manifold $X$ is a theory of harmonic maps from a Riemann surface to $X$, and, at the same time, an important example of (classical and quantum) field theory. Sigma models of complex homogeneous spaces have especially remarkable properties. For example, it turns out that Grassmannian (or, more generally, flag manifold) sigma models may be naturally formulated in terms of symplectic structures on nilpotent orbits of the respective complex groups. These orbits are quiver varieties, and, as such, carry natural variables, which are particularly useful for the quantum theory of sigma models. Apart from introducing the general circle of relevant ideas, I will discuss one-loop quantum corrections and, possibly, supersymmetric generalizations.