We formulate four-dimensional N=1 supersymmetric nonlinear sigma models on Hermitian symmetric spaces with higher derivative terms, free from the auxiliary field problem and the Ostrogradski's ghosts, as gauged linear sigma models. We then study Bogomol'nyi-Prasad-Sommerfield equations preserving 1/2 or 1/4 supersymmetries. We find that there are distinct branches, that we call canonical (F=0) and noncanonical (F≠0) branches, associated with solutions to auxiliary fields F in chiral multiplets. For the CPN model, we obtain a supersymmetric CPN Skyrme-Faddeev model in the canonical branch while in the noncanonical branch the Lagrangian consists of solely the CPN Skyrme-Faddeev term without a canonical kinetic term. These structures can be extended to the Grassmann manifold GM,N=SU(M)/[SU(M-N)×SU(N)×U(1)]. For other Hermitian symmetric spaces such as the quadric surface QN-2=SO(N)/[SO(N-2)×U(1)]), we impose F-term (holomorphic) constraints for embedding them into CPN-1 or Grassmann manifold. We find that these constraints are consistent in the canonical branch but yield additional constraints on the dynamical fields, thus reducing the target spaces in the noncanonical branch.
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