The membrane as a perturbation around string-like configurations

TL;DR

This paper develops a perturbation theory for bosonic membranes around string-like solutions, showing that membrane dynamics can be transformed into a free string Hamiltonian plus boundary interactions, with implications for M-theory.

Contribution

It introduces a perturbative framework for membranes near string configurations and demonstrates how to transform membrane Hamiltonians into string-like forms with boundary terms.

Findings

First-order solutions of perturbative equations of motion.

Existence of canonical transformations to string-like Hamiltonians.

Membrane dynamics reduce to boundary theories on a world-sheet.

Abstract

The bosonic membrane in a partial gauge, where one space dimension is eliminated, is formulated as a perturbation theory around an exact free string-like solution. This perturbative regime corresponds to a situation where one of the world-volume space-like dimensions is much greater than the other, so that the membrane has the form of a narrow band or large hoop with string excitations being transverse to the widest dimension. The perturbative equations of motion are studied and solved to first order. Furthermore, it is shown for the open or semi-open cases and to any order in perturbation theory, that one may find canonical transformations that will transform the membrane Hamiltonian into a free string-like Hamiltonian and a boundary Hamiltonian. Thus the membrane dynamics in our perturbation scheme is essentially captured by an interacting boundary theory defined on a two-dimensional world-sheet. A possible implication of this to M-theory is discussed.