Symplectic 3-Algebras and D=3, N=4,5,6,8 superconformal Chern-Simons-Matter theories

M-theory is the underlying theory of five di fferent string theories and 11D supergravity theory. While strings (1+1D) are fundamental objects in string theory, M2-branes (1+2D) are fundamental objects in M-theory. According to the Gauge/Gravity duality, a gravity theory is equivalent to a gauge theory. Extended (N ? 4) superconformal Chern-Simonsmatter (CSM) theories in 3D are natural candidates of the dual gauge theories of multi M2-branes. In the last two years, the N = 4; 5; 6 CSM theories were constructed by using ordinary Lie 2-algebras, and the N = 8 theory was constructed by using 3-algebra. However, it remains unclear whether these theories can be constructed in a uni ed 3-algebra approach or not. It is also natural to ask whether there are new examples of the extended superconformal CSM theories. In this thesis, we propose to solve these two problems. We de fine a 3-algebra with structure constants being symmetric in the fi rst two indices. We also introduce an invariant antisymmetric tensor into this 3-algebra and call it a symplectic 3-algebra. The D = 3;N = 4; 5; 6; 8 CSM theories are constructed in terms of this unifi ed 3-algebraic structure, and some new examples of the N = 4 quiver gauge theories are derived as well. In particular, in order to realize the 3-algebra used to construct the N = 4 quiver gauge theories, we `fuse' two simple super Lie algebras into a single new super Lie algebra, by requiring that the even parts of these two simple super Lie algebras share one simple factor. We demonstrate how to construct this class of new super Lie algebras by presenting an explicit example. Finally, a quantization scheme for the 3-brackets is proposed.