Geometry of supersymmetric sigma models

Abstract

The geometrical structure of supersymmetric non-linear sigma models related to supersymmetric theories with spontaneous symmetry breaking is described. The target manifold for such models is a Kähler manifold M found by deforming the complex coset space G K so that only compact G isometries remain. The complex isotropy group K is determined by (although in general not identical to) the complex invariance group H of the vacuum of the fundamental theory, and may be larger than the complexification H of the unbroken global symmetry subgroup H of G. The true Goldstone bosons corresponding to the breaking G → H necessarily constitute only a subset of the massless scalar particles contained in the supersymmetric sigma model. By comparing the elegant description of the geometry of the manifold M in terms of the holomorphic coset represntative and vielbeins with an alternative formulation involving non-holomorphic quantities, it is shown how a conventional sigma model on the coset space G/H is embedded into the supersymmetric model on M. It is verified that the mutual interactions of the Goldstone bosons are entirely determined by the compact symmetries and are independent of the Kähler potential function K(ν †L(Φ) †L(Φ)ν) . In contrast, the interactions of the additional scalars, which are governed by the metric in the non-compact directions in M, depend explicitly on the function K.