## Journal of Geometry and Symmetry in Physics

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

Aleksandr Komlov

#### 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.

#### Article information

Source
J. Geom. Symmetry Phys., Volume 10 (2007), 41-50.

Dates
First available in Project Euclid: 20 May 2017

https://projecteuclid.org/euclid.jgsp/1495245622

Digital Object Identifier
doi:10.7546/jgsp-10-2007-41-50

Mathematical Reviews number (MathSciNet)
MR2380049

Zentralblatt MATH identifier
1143.81019

#### Citation

Komlov, Aleksandr. Noncommutative Grassmannian $U(1)$ Sigma-Model and Bargmann-Fock Space. J. Geom. Symmetry Phys. 10 (2007), 41--50. doi:10.7546/jgsp-10-2007-41-50. https://projecteuclid.org/euclid.jgsp/1495245622