We present a superspace formulation of the D ¼ 3, N ¼ 4, 5 superconformal Chern-Simons Matter theories, with matter supermultiplets valued in a symplectic 3-algebra. We first construct an N ¼ 1 superconformal action and then generalize a method used by Gaitto and Witten to enhance the supersymmetry from N ¼ 1 to N ¼ 5. By decomposing the N ¼ 5 supermultiplets and the symplectic 3-algebra properly and proposing a new superpotential term, we construct the N ¼ 4 superconformal Chern-Simons matter theories in terms of two sets of generators of a (quaternion) symplectic 3-algebra. The N ¼ 4 theories can also be derived by requiring that the supersymmetry transformations are closed on-shell. The relationship between the 3-algebras, Lie superalgebras, Lie algebras, and embedding tensors (proposed in [E. A. Bergshoeff, O. Hohm, D. Roest, H. Samtleben, and E. Sezgin, J. High Energy Phys. 09 (2008) 101.]) is also clarified. The general N ¼ 4, 5 superconformal Chern-Simons matter theories in terms of ordinary Lie algebras can be re-derived in our 3-algebra approach. All known N ¼ 4, 5 superconformal Chern-Simons matter theories can be recovered in the present superspace formulation for super-Lie algebra realization of symplectic 3-algebras.

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