Anomaly-free supersymmetric $ \frac{{{\text{SO}}\left( {2N + 2} \right)}}{{{\text{U}}\left( {N + 1} \right)}} $ σ-model based on the SO(2N + 1) Lie algebra of the fermion operators

Abstract

The extended supersymmetric (SUSY) σ-model has been proposed on the bases of SO(2N + 1) Lie algebra spanned by fermion annihilation-creation operators and pair operators. The canonical transformation, extension of an SO(2N) Bogoliubov transformation to an 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 have investigated the SUSY σ-model on the Kähler manifold, the coset space SO(2N + 2)/U(N + 1). We have constructed the Killing potential, extension of the potential in the SO(2N)/U(N) coset space to that in the SO(2N + 2)/U(N + 1) coset space. That Killing potential is equivalent to the generalized density matrix whose diagonal-block part is related to a reduced scalar potential with a Fayet-Ilipoulos term. We rescale the Goldstone fields by a parameter f (inverse of mass m σ ). The f-deformed reduced scalar potential is optimized with respect to vacuum expectation value of the σ-model fields and a solution for one of the SO(2N + 1) group parameters has been obtained. The solution, however, is only a small part of all solutions obtained from anomaly-free SUSY coset models. To construct the coset models consistently, we must embed a coset coordinate in an anomaly-free spinor representation (rep) of SO(2N + 2) group and give corresponding Kähler and Killing potentials for an anomaly-free SO(2N + 2)/U(N + 1) model based on each positive chiral spinor rep. Using such mathematical manipulation we construct successfully the anomaly-free SO(2N + 2)/U(N + 1) SUSY σ-model and investigate new aspects of such a SUSY σ-model which have never been seen in the usual SUSY σ-model on the Kähler coset space SO(2N)/U(N). Then we reach a f-deformed reduced scalar potential in the anomaly-free SO(2N + 2)/U(N + 1) SUSY σ-model. It is minimized with respect to the vacuum expectation value of anomaly-free SUSY σ-model fields. Thus we find an interesting f-deformed solution very different from the previous solution for an anomaly-free SO(2 ⋅ 5 + 2)/(SU(5 + 1) × U(1)) SUSY σ-model.