Noncommutative Grassmannian $U(1)$ sigma model and a Bargmann–Fock space

We consider a Grassmannian version of the noncommutative $U(1)$ sigma model specified by the energy functional $E(P)=\bigl\|[a,P]\bigr\|_{\mathrm{HS}}^2$, where $P$ is an orthogonal projection operator in a Hilbert space $H$ and $a\colon H\to H$ is the standard annihilation operator. With $H$ realized as a Bargmann–Fock space, we describe all solutions with a one-dimensional range and prove that the operator $[a,P]$ is densely defined in $H$ for a certain class of projection operators $P$ with infinite-dimensional ranges and kernels.