Open membranes in a constant C-field background and noncommutative boundary strings

TL;DR

This paper investigates the noncommutativity of open membrane boundaries in a constant three-form field $C_{\mu\nu\rho}$ by adapting the Dirac-constraint approach used for open strings in a $B$-field. Using a bosonic membrane in a fixed background with boundary p-branes and a boundary two-form field to preserve gauge invariance, the authors derive the effective action and analyze the resulting boundary constraints in a controlled limit, employing a perturbative expansion in the field strength $C$. They compute the Dirac brackets to leading nontrivial orders, showing that boundary coordinates $X^i$ become noncommutative and that this noncommutativity has a loop-space character, with explicit expressions up to $\mathcal{O}(C^2)$. The findings indicate a richer, higher-dimensional version of noncommutative geometry on membrane boundaries, with potential implications for M-theory and matrix-model formulations, and they discuss the challenges of extending to more complete gauge choices and regularization schemes. Overall, the work reveals a concrete mechanism for membrane boundary noncommutativity in a constant $C$-field background and lays groundwork for further explorations in M-theory noncommutativity and boundary dynamics.

Abstract

We investigate the dynamics of open membrane boundaries in a constant C-field background. We follow the analysis for open strings in a B-field background, and take some approximations. We find that open membrane boundaries do show noncommutativity in this case by explicit calculations. Membrane boundaries are one dimensional strings, so we face a new type of noncommutativity, that is, noncommutative strings.

Figures (1)

• Figure 1: A membrane wrapped once around the compactification circle stretches between two fixed $p$-branes.