Extended supersymmetric σ-model based on the SO(2N+1) Lie algebra of the fermion operators

Extended supersymmetric σ-model is given, standing on the SO(2N+1) Lie algebra of fermion operators composed of annihilation-creation operators and pair operators. Canonical transformation, the extension of the SO(2N) Bogoliubov transformation to the SO(2N+1) group, is introduced. Embedding the SO(2N+1) group into an SO(2N+2) group and using SO(2N+2)U(N+1) coset variables, we investigate a new aspect of the supersymmetric σ-model on the Kähler manifold of the symmetric space SO(2N+2)U(N+1). We construct a Killing potential which is just the extension of the Killing potential in the SO(2N)U(N) coset space given by van Holten et al. to that in the SO(2N+2)U(N+1) coset space. To our great surprise, the Killing potential is equivalent with the generalized density matrix. Its diagonal-block matrix is related to a reduced scalar potential with a Fayet-Ilipoulos term. The reduced scalar potential is optimized in order to see the behaviour of the vacuum expectation value of the σ-model fields and a proper solution for one of the SO(2N+1) group parameters is obtained. We give bosonization of the SO(2N+2) Lie operators and vacuum functions in terms of the SO(2N+2)U(N+1) coset variables, a U(1) phase and the corresponding Kähler potential.