[Paper Review] The supermembrane with central charge as a bundle of D2-D0 branes

M.P. García del Moral, A. Restuccia|ArXiv.org|Oct 29, 2004

Methane Hydrates and Related Phenomena3 citations

TL;DR

This paper proposes that the D=11 supermembrane with non-zero central charge, arising from nontrivial winding (CSNW), is equivalent to a bundle of D2-D0 branes with flux-induced D0 charge. Using a noncommutative supersymmetric gauge theory on a Riemann surface of genus g≥1, it rigorously constructs the Hamiltonian's discrete spectrum and heat kernel via a Feynman formula, proving quantum consistency and linking the supermembrane to stable, recombined D2-D0 brane systems in M-theory compactifications.

ABSTRACT

We discuss the consistency of the D=11 supermembranes with non zero central charge arising from a nontrivial winding CSNW. The spectrum of its regularized Hamiltonian is discrete and its heat kernel in terms of a Feynman formula may be rigorously constructed. The $N o\infty$ limit is discussed. Since CSNW is equivalent to a noncommutative supersymmetric gauge theory on a general Riemann surface, its consistency provides a proof that all of them are well defined quantum theories. We interpret the supermembrane with central charge $n$, in the type IIA picture, as a bundle of D2 branes with $n$ units of D0 charge induced by a nonconstant magnetic flux.

Motivation & Objective

• To establish the quantum consistency of D=11 supermembranes with non-zero central charge arising from nontrivial winding (CSNW).
• To demonstrate that the regularized Hamiltonian of such supermembranes has a purely discrete spectrum.
• To construct the heat kernel of the Hamiltonian using a Feynman formula, ensuring mathematical rigor.
• To prove that the equivalence of CSNW to noncommutative supersymmetric gauge theories on Riemann surfaces implies the quantum consistency of all such theories.
• To provide a dual D-brane interpretation of the supermembrane as a bundle of D2 branes with n units of D0 charge induced by nonconstant magnetic flux.

Proposed method

• Formulate the D=11 supermembrane in the light-cone gauge with a potential term involving Poisson brackets of spatial coordinates: $ V(X) = \{X^M, X^N\}^2 $.
• Introduce a scalar density $ \sqrt{W(\sigma)} $ via partial gauge fixing to define the worldvolume metric.
• Show that the CSNW condition leads to a symplectic structure on the worldvolume, equivalent to a noncommutative supersymmetric gauge theory on a Riemann surface of genus $ g \geq 1 $.
• Apply the Darboux theorem to locally reduce the non-constant symplectic form to a constant antisymmetric tensor on open sets, enabling a local description as a conventional noncommutative gauge theory.
• Use the global structure of the Riemann surface to patch these local theories via symplectomorphisms, preserving area-preserving diffeomorphisms.
• Demonstrate that the D2-D0 brane system arises from the dual picture, where fluxes (not a constant B-field) induce D0 charge, and the recombination of intersecting branes leads to a stable bundle.

Experimental results

Research questions

• RQ1 Is the D=11 supermembrane with non-zero central charge quantum mechanically consistent, and does it possess a discrete spectrum?
• RQ2 Can the supermembrane with central charge be consistently described as a noncommutative supersymmetric gauge theory on a Riemann surface of genus $ g \geq 1 $?
• RQ3 How does the non-constant symplectic structure on the worldvolume affect the construction of the Hamiltonian and its heat kernel?
• RQ4 What is the physical interpretation of the supermembrane with central charge in terms of D-brane configurations in type IIA string theory?
• RQ5 Does the recombination of intersecting D2-D0 branes via flux-induced mass generation provide a stable, supersymmetric configuration?

Key findings

• The regularized Hamiltonian of the supermembrane with central charge has a purely discrete spectrum, resolving the issue of continuous spectrum in the standard supermembrane.
• The heat kernel of the Hamiltonian can be rigorously constructed using a Feynman formula, ensuring mathematical consistency of the quantum theory.
• The CSNW condition is equivalent to a noncommutative supersymmetric gauge theory on a Riemann surface of genus $ g \geq 1 $, and this equivalence proves the quantum consistency of all such noncommutative gauge theories.
• In the type IIA dual picture, the supermembrane with central charge $ n $ is interpreted as a bundle of D2 branes with $ n $ units of D0 charge, induced by a nonconstant magnetic flux.
• The D2-D0 brane system arises from the recombination of intersecting branes, where transverse scalar fields become massive via a worldvolume-induced Higgs mechanism.
• The stability of the system is guaranteed by the stability of the original supermembrane with central charge, and the configuration cannot exist in 10 dimensions due to the purely topological origin of the flux.