Extended supersymmetric σ-model based on the Lie algebra of the fermion operators

Abstract

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, vacuum functions and differential forms for their bosons expressed in terms of the SO(2N +2) U (N +1) coset variables, a U (1) phase and the corresponding Kähler potential. * A preliminary version of this work has been presented by S. Nishiyama at the YITP workshop YITP-W-07-05 on String Theory and Quantum Field Theory held at Kinki University

Mean-field theory based on the Jacobi hsp ≔ semidirect sum hN⋊sp(2N,R)C algebra of boson operators

“…As already shown for a fermion system [38,39], it is also easily verified that the result of (7.9) satisfies the gradient of the real function M coset space obtained by van Holten et al [41] and also by the present authors et. al.…”

“…On the other hand, we also have obtained several years ago the matrixtype Riccati-Hartree-Bogoliubov equation on the compact symmetric coset space SO(2N ) U (N ) for a fermion system accompanying with a general two-body interaction [37]. Along the direction taken by the present authors et al [38,39], here we also use the matrix elements of U and U † as the coordinates on the manifold Sp(2N+2) instead of the manifold SO(2N+2). Even in the case of Sp(2N+2), it turns out that a real line element can be defined by a hermitian metric tensor on the manifold.…”

“…3) The condition that a certain manifold is a Kähler manifold is that its complex structure is covariantly constant relative to the Riemann connection and that it has vanishing torsions: As in the case of SO(2N +2) [38,39], the hermitian metric tensor G pq rs can be locally given through a real scalar function, the Kähler potential, which takes the well-known form 5) and the explicit expression for the components of the metric tensor is given as…”

“…Notice that the above function does not determine the Kähler potential K(Q B † , Q B ) uniquely since the metric tensor G pqrs is invariant under transformations of the Kähler potential, [38,39], we consider an Sp(2N+2) infinitesimal left transformation of an Sp(2N+2) matrix G to G ′ , G ′ = (1 2N+2 +δG)G, by using the first equation of (B.3):…”

Remarks on the mean-field theory based on the SO(2N + 1) Lie algebra of the fermion operators

“…Following a fundamental prescription prepared in the fermion SO(2N+1) many-body theory, the SO(2N) σ-model has been extended by an algebraic manner to an SO(2N +1) σ-model by the present authors et al [17,18]. Through the minimization of a scalar potential, we have first determined a symbolical group-parameter z 2 specifying the property of the SO(2N +1) group.…”