Abstract
We review recent progress in formulating the worldvolume theory of M2-branes using the Nambu bracket. Although it is generally agreed that this formulation should be replaced by another using the superconformal Chern-Simons theory, we try to pursue a possible complementary role played by the Nambu bracket. Since the partition function of the superconformal Chern-Simons theory implies a hidden super gauge group, we study the supersymmetric generalization of the Nambu bracket. We explicitly construct the superunitary Nambu algebra. kiyoshig@sci.osaka-cu.ac.jp moriyama@sci.osaka-cu.ac.jp yanok@sci.osaka-cu.ac.jp
Highlights
As we have seen here, Nambu brackets have played an important role in understanding the M2-branes
Recent progress in understanding the multiple M2-branes starts from the Nambu algebra
From it we arrive at the BLG theory, and afterwards the ABJM theory
Summary
This section is devoted to a review of the three-algebra mainly from the algebraic viewpoint. We shall start with Lie algebra, which is defined as a linear space equipped with a binary operation [∗, ∗] : G × G → G, satisfying the following properties: 1See Refs. Antisymmetry, [X, Y ] = −[Y, X], bilinearity, [aX + bY, Z] = a[X, Z] + b[Y, Z], and Jacobi identity, [[X, Y ], Z] + [[Y, Z], X] + [[Z, X], Y ] = 0. If contracted with the Killing metric gZC, the structure constant f ABC = f ABZ gZC is completely antisymmetric in exchanging the three indices. The Nambu algebra is a natural generalization of the Lie algebra It is the linear space equipped with a ternary operation [∗, ∗, ∗] : G × G × G → G, which satisfies the fundamental identity, a natural generalization of (2.1),.
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