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We present a superspace formulation of the D=3, N=4,5 superconformalChern-Simons Matter theories, with matter supermultiplets valued in asymplectic 3-algebra. We first construct an N=1 superconformal action, and thengeneralize a method used by Gaitto and Witten to enhance the supersymmetry fromN=1 to N=5. By decomposing the N=5 supermultiplets and the symplectic 3-algebraproperly and proposing a new super-potential term, we construct the N=4superconformal Chern-Simons matter theories in terms of two sets of generatorsof a (quaternion) symplectic 3-algebra. The N=4 theories can also be derived byrequiring that the supersymmetry transformations are closed on-shell. Therelationship between the 3-algebras, Lie superalgebras, Lie algebras andembedding tensors (proposed in [E. A. Bergshoeff, O. Hohm, D. Roest, H.Samtleben, and E. Sezgin, J. High Energy Phys. 09 (2008) 101.]) is alsoclarified. The general N=4,5 superconformal Chern-Simons matter theories interms of ordinary Lie algebras can be rederived in our 3-algebra approach. Allknown N=4,5 superconformal Chern-Simons matter theories can be recovered in thepresent superspace formulation for super-Lie-algebra realization of symplectic3-algebras.

Superspace Formulation in a Three-Algebra Approach to D=3, N=4,5 Superconformal Chern-Simons Matter Theories