Noncommutative Grassmannian \(U(1)\) sigma-model and Bargmann-Fock space. (English) Zbl 1143.81019

Summary: We consider the Grassmannian version of the noncommutative \(U(1)\) sigma-model, which is given by the energy functional \(E(P) = \|[a, P]\|^2_{HS}\), 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.


81T10 Model quantum field theories
81T75 Noncommutative geometry methods in quantum field theory
14M15 Grassmannians, Schubert varieties, flag manifolds
51M35 Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations)