Filippov-Nambu $n$-algebras relevant to physics

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 $\mathcal N=8$ linear supersymmetries. These theories explore the low energy dynamics of the microscopic degrees of freedom of coincident $\mathrm M2$ branes and constitute the boundary conformal field theories of the bulk $AdS_4\times S_7$ 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 $\mathrm M5$ branes in the $\mathrm M2$-$\mathrm M5$ system. A crucial input into this construction came from the study of the $\mathrm M2$-$\mathrm M5$ system in the Basu–Harvey's work where an equation describing the Bogomol'nyi–Prasad–Sommerfield (BPS) bound state of multiple $\mathrm M2$-branes ending on an $\mathrm M5$ was formulated. The Filippov–Nambu $3$-algebras are either operator or matrix representation of the classical Nambu symmetries of world volume preserving diffeomorphisms of $\mathrm M2$ branes. Indeed at the classical level the supermembrane Lagrangian, in the covariant formulation, has the world volume preserving diffeomorphisms symmetry $SDiff(M_{2+1})$. 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 $\mathrm M2$ branes. It emphasizes the Filippov–Nambu $n$-algebras as the mathematical framework for describing symmetry properties of classical and quantum mechanical systems.