Integrable Symplectic Trilinear Interaction Terms for Matrix Membranes

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. Introduction A proposal for non-perturbative formulation of M-theory [1] has led to a revival of matrix membrane theory [2, 3]. Symmetry features of membranes and their connection to matrix models [3, 4, 5, 6, 7] have been established for quite some time. Effectively, infinite-N quantum mechanics matrix models presented as a restriction out of SU(∞) Yang-Mills theories amount to membranes, by virtue of the connection between SU(N) and area-preserving diffeomorphisms (Sdiff) generated by Poisson Brackets, in which “colour” algebra indices Fourier-transform to “membrane” sheet coordinates. The two are underlain and linked by Moyal Brackets. Below, we consider novel Poisson Bracket interactions for a bosonic membrane embedded in 3-space LIPB = 1 3 ǫX{X , X}, (1) which are restrictions of the Moyal Bracket generalization LIMB = 1 3 ǫX{{X , X}}, (2) which, in turn, also encompasses the plain matrix commutator term LIC = 1 3 ǫX[X , X] . (3) The structure of (1) may be recognized as that of the interaction term ǫijlφ ǫ∂μφ ∂νφ j of the 2dimensional SO(3) pseudodual chiral σ-model of Zakharov and Mikhailov [8]—this is a limit [9] of the 1 On research leave from the University of Durham, U.K. Curtright, Fairlie, & Zachos hepth@xxx/9704037 2 WZWN interaction term, where the integer WZWN coefficient goes to infinity while the coupling goes to zero, such that the product of the integer with the cube of the coupling is kept constant. The structure of (3) is also linked to what remains of the gauge theory instanton density,