Filippov-Nambu $n$-algebras relevant to physics

Abstract

Gauge symmetry based on Lie algebra has a rather long history and it successfully describes electromagnetism, weak and strong interactions in the nature. Recently the Filippov-Nambu 3-algebras have been in the focus of interest since they appear as gauge symmetries of new superconformal Chern-Simons non-Abelian theories in 2 + 1 dimensions with the maximum allowed number of N = 8 linear supersymmetries. These theories explore the low energy dynamics of the microscopic degrees of freedom of coincident M2 branes and constitute the boundary conformal field theories of the bulk AdS4 / S7 exact 11-dimensional supergravity backgrounds of supermembranes. These mysterious new symmetries, the Filippov-Nambu 3-algebras represent the implementation of non-associative algebras of coordinates of charged tensionless strings, the boundaries of open M2 branes in antisymmetric field magnetic backgrounds of M5 branes in the M2 -M5 system. A crucial input into this construction came from the study of the M2-M5 system in the Basu- Harvey's work where an equation describing the Bogomol'nyi-Prasad- Sommerfeld (BPS) bound state of multiple M2-branes ending on an M5 was formulated. The Filippov-Nambu 3-algebras are either operator or matrix representation of the classical Nambu symmetries of world volume preserving diffeomorphisms of M2 branes. Indeed at the classical level the supermembrane Lagrangian, in the covariant formulation, has the world volume preserving diffeomorphisms symmetry SDiff(M2+1). The Filippov-Nambu 3-algebras presumably correspond to the quantization of the rigid motions in this infinite dimensional group, which describe the low energy excitation spectrum of the M2 branes. It emphasizes the Filippov-Nambu n-algebras as the mathematical framework for describing symmetry properties of classical and quantum mechanical systems.