Membrane and Noncommutativity

TL;DR

This paper investigates the dynamics of open membranes with boundary conditions attached to p-branes, revealing noncommutative properties of boundary string coordinates through algebraic analysis in the low-energy limit.

Contribution

It provides a detailed analysis of boundary conditions and constraints for open membranes, deriving the exact noncommutative algebra of boundary string coordinates without relying solely on primary constraints.

Findings

Noncommutative algebra of boundary string coordinates is derived.

Boundary conditions influence the membrane's effective dynamics.

Low-energy approximation reduces membrane to an open string with noncommutative features.

Abstract

We analyse the dynamics of an open membrane, both for the free case and when it is coupled to a background three-form, whose boundary is attached to $p$-branes. The role of boundary conditions and constraints in the Nambu-Goto and Polyakov formulations is studied. The low-energy approximation that effectively reduces the membrane to an open string is examined in detail. Noncommutative features of the boundary string coordinates, where the cylindrical membrane is attached to the D$p$-branes, are revealed by algebraic consistency arguments and not by treating boundary conditions as primary constraints, as is usually done. The exact form of the noncommutative algebra is obtained in the low-energy limit.