Noncommutative Grassmannian $U(1)$ Sigma-Model and Bargmann-Fock Space

Abstract

We consider the Grassmannian version of the noncommutative $U(1)$ sigma-model, which is given by the energy functional $E(P) = \|[a, P]\|_{HS}^2$, where $P$ is an orthogonal projection on a Hilbert space $H$ and the operator $a: H \to H$ is the standard annihilation operator. Using realization of $H$ as the Bargmann-Fock space, we describe all solutions with one-dimensional image and prove that the operator $[a, P]$ is densely defined on $H$ for some class of projections $P$ with infinite-dimensional image and kernel.