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Filippov-Nambu $n$-algebras relevant to physics
N. G. Pletnev Sobolev Institute of Mathematics, Novosibirsk, Russia
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
$\mathcal N=8$ 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.
Keywords:
Filippov $n$-algebra, Nambu bracket, supersymmetry, super $p$-branes.
UDC:
512.5
MSC: 13A99 Received July 8, 2009, published
October 16, 2009
Language: English