Integrable Symplectic Trilinear Interaction Terms for Matrix Membranes

TL;DR

This paper explores integrable cubic interaction terms in matrix membranes within 3 and 7 dimensions, transforming their equations into Nahm's form and providing explicit Lax pairs for the 3D case, with broader applicability to related algebraic structures.

Contribution

It introduces integrable symplectic trilinear interaction terms for matrix membranes and establishes their connection to Nahm's equations, including explicit Lax pairs for the 3D case.

Findings

Demonstrates integrability of certain membrane interaction equations

Provides explicit Lax pair for 3D case

Extends constructions to Moyal Bracket analogues

Abstract

Cubic interactions are considered in 3 and 7 space dimensions, respectively, for bosonic membranes in Poisson Bracket form. Their symmetries and vacuum configurations are discussed. 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. The constructions introduced also apply to commutator or Moyal Bracket analogues.