[Paper Review] 11d Electric-Magnetic Duality and the Dbrane Spectrum

Shyamoli Chaudhuri|ArXiv.org|Sep 2, 2004

Black Holes and Theoretical Physics33 references3 citations

TL;DR

This paper resolves a puzzle in 10d type II string theory by showing that the D-brane spectrum, when extended to include pointlike sources of the zero-form field strength, completes electric-magnetic duality in eleven dimensions. Using a gedanken worldsheet calculation in strongly coupled M-theory, the authors demonstrate that the missing D(-2)- and D(-3)-branes—Poincaré duals of the D8- and D9-branes—are realized as sources for the Romans mass term, confirming that the full duality structure is 11-dimensional rather than 10-dimensional.

ABSTRACT

We consider the gedanken calculation of the pair correlation function of spatially-separated macroscopic string solitons in strongly coupled type IIA string/M theory, with the macroscopic strings wrapping the eleventh dimension. The supergravity limit of this correlation function with well-separated, pointlike macroscopic strings corresponds to having also taken the IIA string coupling constant to zero. Thus, the pointlike limit of the gedanken correlation function can be given a precise worldsheet description in the 10D weakly-coupled type IIA string theory, analysed by us in hep-th/0007056 [Nucl. Phys. B591 (2000) 243]. The requisite type IIA string amplitude is the supersymmetric extension of the worldsheet formulation of an off-shell closed string tree propagator in bosonic string theory, a 1986 analysis due to Cohen, Moore, Nelson, and Polchinski. We point out that the evidence for pointlike sources of the zero-form field strength provided by our worldsheet results clarifies that the electric-magnetic duality in the Dirichlet-brane spectrum of type II string theories is eleven-dimensional.

Motivation & Objective

• To resolve the apparent conflict between Poincaré-Hodge duality and the observed D-brane spectrum in 10d type II string theories.
• To demonstrate that the missing D(-2)- and D(-3)-branes—dual to D8- and D9-branes—are realized as pointlike sources of the zero-form field strength in 11d supergravity.
• To show that the full D-brane spectrum, including these sources, is consistent with 11-dimensional electric-magnetic duality.
• To provide worldsheet evidence for the existence of pointlike sources of the Romans mass term in 10d type IIA supergravity via a gedanken correlation function in M-theory.

Proposed method

• Perform a gedanken calculation of the pair correlation function of spatially separated macroscopic string solitons in strongly coupled type IIA string/M-theory, with strings wrapping the 11th dimension.
• Take the supergravity limit of this correlation function, which corresponds to the weakly-coupled type IIA string theory limit with vanishing string coupling.
• Use the supersymmetric extension of the Cohen-Moore-Nelson-Polchinski 1986 worldsheet formulation of the off-shell closed string tree propagator to describe the amplitude.
• Analyze the worldsheet path integral for a twisted Neumann scalar on a torus with magnetic flux, using contour integration and zeta function regularization to compute the functional determinant.
• Derive the functional determinant of the Laplacian on a twisted complex scalar, expressing it in terms of Jacobi theta functions and Dedekind eta functions.
• Combine results to show that the worldsheet amplitude supports the existence of pointlike sources for the zero-form field strength, identified as the D(-2)-brane.

Experimental results

Research questions

• RQ1 Why does the D-brane spectrum in 10d type II string theory appear to violate Poincaré-Hodge duality, missing the D(-2)- and D(-3)-branes?
• RQ2 How can the Dirac quantization condition for electric and magnetic charges be consistently satisfied across all D-brane charges in the spectrum?
• RQ3 What is the worldsheet origin of the zero-form field strength in Romans' massive 10d type IIA supergravity?
• RQ4 How does the inclusion of pointlike sources for the zero-form field strength restore 11-dimensional electric-magnetic duality?
• RQ5 What is the precise worldsheet description of the D-brane tension and charge in the context of strongly coupled M-theory?

Key findings

• The D-brane spectrum in type II string theories is completed by including pointlike sources for the zero-form field strength, corresponding to a D(-2)-brane, which resolves the duality puzzle.
• The worldsheet analysis of the closed string tree-level amplitude in weakly-coupled type IIA string theory provides direct evidence for the existence of these pointlike sources.
• The functional determinant of the Laplacian on a twisted scalar on a torus with magnetic flux is computed using zeta function regularization and contour integration, yielding a result expressed in terms of Jacobi theta functions.
• The resulting amplitude contains a factor proportional to the Dedekind eta function and a theta function with argument related to the magnetic flux, confirming the presence of a source for the zero-form field strength.
• The D(-2)-brane is identified as the Poincaré dual of the D8-brane and the source for the Romans mass term in 10d type IIA supergravity.
• The full D-brane spectrum, including D(-1) through D9-branes and the D(-2)-brane, is shown to be consistent with 11-dimensional electric-magnetic duality.