[Paper Review] Filippov-Nambu $n$-algebras relevant to physics

Н. Г. Плетнев|arXiv (Cornell University)|Dec 13, 2010

Black Holes and Theoretical Physics30 references4 citations

TL;DR

This paper proposes Filippov-Nambu $n$-algebras as the mathematical framework for describing gauge symmetries in 2+1-dimensional superconformal Chern-Simons theories with N=8 supersymmetry, which arise as low-energy effective theories of multiple M2-branes ending on M5-branes. The key contribution is establishing these non-associative algebras as the quantized realization of worldvolume-preserving diffeomorphisms (SDiff(M2+1)) of M2-branes, linking them to the BPS bound state structure in the M2-M5 brane system via Basu-Harvey-type equations.

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.

Motivation & Objective

• To establish a mathematical framework for gauge symmetries in M-theory compactifications involving M2 and M5 branes.
• To connect the Filippov-Nambu 3-algebras to the low-energy dynamics of multiple M2-branes ending on M5-branes.
• To interpret these algebras as the quantized version of worldvolume-preserving diffeomorphisms (SDiff(M2+1)) of M2-branes.
• To clarify the role of non-associative algebras in describing the coordinates of tensionless strings at the boundary of open M2-branes in M-theory.

Proposed method

• Utilizes the Basu-Harvey equation to describe the BPS bound state of multiple M2-branes ending on an M5-brane.
• Applies the classical supermembrane Lagrangian in covariant formulation to identify the worldvolume-preserving diffeomorphism symmetry SDiff(M2+1).
• Identifies Filippov-Nambu 3-algebras as operator or matrix representations of the classical Nambu symmetries associated with SDiff(M2+1).
• Analyzes the quantization of rigid motions within the infinite-dimensional SDiff(M2+1) group as the origin of Filippov-Nambu $n$-algebras.
• Establishes a correspondence between the algebraic structure of Filippov-Nambu $n$-algebras and the dynamics of open M2-brane boundaries in M5-brane backgrounds.
• Relies on the AdS4/S7 duality and 11-dimensional supergravity backgrounds to contextualize the physical realization of these algebras.

Experimental results

Research questions

• RQ1 How do Filippov-Nambu 3-algebras emerge as gauge symmetries in 2+1-dimensional Chern-Simons theories with N=8 supersymmetry?
• RQ2 What is the geometric and algebraic origin of these algebras in the context of M2-brane worldvolume symmetries?
• RQ3 How are the Filippov-Nambu $n$-algebras related to the worldvolume-preserving diffeomorphisms SDiff(M2+1) of M2-branes?
• RQ4 In what way do these algebras describe the low-energy spectrum of M2-branes ending on M5-branes?
• RQ5 How does the Basu-Harvey equation encode the structure of these non-associative algebras in the M2-M5 brane system?

Key findings

• Filippov-Nambu 3-algebras are identified as the gauge symmetry algebra of 2+1-dimensional superconformal Chern-Simons theories with maximal N=8 supersymmetry.
• These algebras arise as the quantized version of the worldvolume-preserving diffeomorphisms (SDiff(M2+1)) of M2-branes.
• The algebras provide a mathematical realization of non-associative coordinates for tensionless strings at the boundary of open M2-branes in an M5-brane background.
• The construction is rooted in the Basu-Harvey equation, which describes the BPS bound state of multiple M2-branes ending on an M5-brane.
• The Filippov-Nambu $n$-algebras are shown to be essential for describing the low-energy dynamics of M2-branes in the context of AdS4/S7 supergravity and M-theory dualities.
• The framework establishes a direct link between non-associative algebraic structures and the quantum symmetry of M-theory brane systems.