Abstract
Extended supersymmetric σ-model is given, standing on the SO ( 2 N + 1 ) Lie algebra of fermion operators composed of annihilation–creation operators and pair operators. Canonical transformation, the extension of the SO ( 2 N ) Bogoliubov transformation to the SO ( 2 N + 1 ) group, is introduced. Embedding the SO ( 2 N + 1 ) group into an SO ( 2 N + 2 ) group and using SO ( 2 N + 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 ( 2 N + 2 ) U ( N + 1 ) . We construct a Killing potential which is just the extension of the Killing potential in the SO ( 2 N ) U ( N ) coset space given by van Holten et al. to that in the SO ( 2 N + 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 ( 2 N + 1 ) group parameters is obtained. We give bosonization of the SO ( 2 N + 2 ) Lie operators and vacuum functions in terms of the SO ( 2 N + 2 ) U ( N + 1 ) coset variables, a U ( 1 ) phase and the corresponding Kähler potential.
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