Matrix Membranes and Integrability

TL;DR

This paper reviews matrix membrane models, explores their symmetries and solutions, and demonstrates integrability in certain cases using explicit Lax pairs, with implications for related algebraic structures.

Contribution

It provides an explicit implementation of Euler's construction for solving Poisson Bracket dual Nahm equations and demonstrates integrability in specific membrane models.

Findings

Explicit Lax pair for 3D membrane equations

Symmetry analysis of cubic membrane interactions

Applicability to matrix commutator and Moyal Bracket analogs

Abstract

This is a pedagogical digest of results reported in Phys Lett B405 (1997) 37, and an explicit implementation of Euler's construction for the solution of the Poisson Bracket dual Nahm equation. But it does not cover 9 and 10-dimensional systems, and subsequent progress on them [hep-th/9707190]. Cubic interactions are considered in 3 and 7 space dimensions, respectively, for bosonic membranes in Poisson Bracket form. Their symmetries and vacuum configurations are explored. Their associated first order equations are transformed to Nahm's equations, and are hence seen to be integrable, for the 3-dimensional case, by virtue of the explicit Lax pair provided. Most constructions introduced also apply to matrix commutator or Moyal Bracket analogs.