Classical Mechanics on Grassmannian and Disc

Abstract

In these notes, we will discuss from a purely geometric point of view classical mechanics on certain type of Grassmannians and discs. We will briefly discuss a superversion which in some sense combines these two models, and corresponds to the large-$N_c$ limit of $SU(N_c)$ gauge theory with fermionic and bosonic matter fields, both in the fundamental representation, in $1 + 1$ dimensions [12]. This result is a natural extension of ideas in [16]. There it has been shown that the large-$N_c$ phase space of $1+1$ dimensional QCD is given by an infinite dimensional Grassmannian. The complex scalar field version of this theory is worked out in [18] and it is shown that the phase space is an infinite dimensional disc.