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Covariantly Constant Curvature Tensors and D=3, N=4, 5, 8 Chern-Simons Matter Theories

We construct some examples of D=3, N=4 GW theory and N=5 superconformal Chern-Simons matter theory by using the covariantly constant curvature of a quaternionic-Kahler manifold to construct the symplectic 3-algebra in the theories. Comparing with the previous theories, the N=4, 5 theories constructed in this way possess a local Sp(2n) symmetry and a diffeomorphism symmetry associated with the quaternionic-Kahler manifold. We also construct a generalized N=8 BLG theory by utilizing the dual curvature operator of a maximally symmetric space of dimension 4 to construct the Nambu 3-algebra. Comparing with the previous N=8 BLG theory, the theory has a diffeomorphism invariance and a local SO(4) invariance associated with the symmetric space.