[Paper Review] The Sigma-Model Representation for the Duality-Symmetric D=11 Supergravity

A. J. Nurmagambetov|ArXiv.org|Dec 14, 2003

Black Holes and Theoretical Physics21 references3 citations

TL;DR

This paper extends the doubled field formalism of Cremmer, Julia, Lü, and Pope to D=11 supergravity by constructing a duality-symmetric action that incorporates fermions and dynamical brane sources. It formulates the theory using a sigma-model-like action with a twisted self-duality condition derived from a zero-curvature equation, ensuring the correct equations of motion for M2 and M5 branes coupled to supergravity.

ABSTRACT

In this paper which is based on the results obtained in collaboration with Igor Bandos and Dmitri Sorokin I present the extention of the "doubled fields" formalism by Cremmer, Julia, Lu and Pope to the supersymmetric case and lifting this construction onto the level of the proper duality-symmetric action for D=11 supergravity. Further extension of the doubled field formulation to include dynamical sources is also discussed.

Motivation & Objective

• To extend the doubled field formalism to include supersymmetry in D=11 supergravity.
• To construct a duality-invariant action that encodes the dynamics of gauge fields via a twisted self-duality condition.
• To incorporate dynamical sources for M2 and M5 branes into the doubled field framework.
• To generalize the construction to a sigma-model-like action that unifies the gravitational and gauge sectors in a duality-symmetric manner.
• To provide a foundation for a fully duality-symmetric formulation of M-theory via supergravity.

Proposed method

• Adapts the doubled field approach by Cremmer et al. to include fermions and supersymmetry in D=11 supergravity.
• Introduces a Cartan form extended by superalgebra-valued elements to encode gauge and gravitational fields.
• Implements a twisted self-duality condition $\ast\mathcal{G} = \mathcal{S}\mathcal{G}$ that reduces degrees of freedom while preserving duality symmetry.
• Derives the equations of motion from a zero-curvature condition $d\tilde{\mathcal{G}} + \tilde{\mathcal{G}} \wedge \tilde{\mathcal{G}} = 0$ with $\tilde{\mathcal{G}} = \mathcal{G} - \mathbf{g}$.
• Constructs a sigma-model-like action using $\mathcal{L}^{(11)}$ with terms for the 4-form, 7-form, and couplings to M2 and M5 brane sources.
• Uses distributional sources and regularization to model M2 and M5 branes via $J^{(3)}_e$ and $J^{(6)}_m$, ensuring consistency with classical supergravity.

Experimental results

Research questions

• RQ1 How can the doubled field formalism be extended to include fermions and supersymmetry in D=11 supergravity?
• RQ2 Can a duality-symmetric action be constructed such that the twisted self-duality condition arises as an equation of motion?
• RQ3 How can dynamical sources for M2 and M5 branes be consistently coupled within a doubled field theory framework?
• RQ4 What is the role of the sigma-model structure in encoding the full dynamics of D=11 supergravity with brane sources?
• RQ5 Can this construction be generalized to a unified, duality-invariant description of the full D=11 supergravity multiplet?

Key findings

• The paper constructs a duality-symmetric action for D=11 supergravity that includes fermions and dynamical brane sources via a sigma-model-like formulation.
• The twisted self-duality condition $\ast\hat{F}^{(4)} = \hat{F}^{(7)}$ and $\ast\hat{F}^{(7)} = \hat{F}^{(4)}$ is derived as an equation of motion from the zero-curvature condition.
• The action (28) correctly reproduces the second-order equations of motion $dF^{(4)} = 0$ and $dF^{(7)} = F^{(4)} \wedge F^{(4)}$ from the zero-curvature structure.
• The coupling to M2 and M5 branes is consistently implemented using distributional sources $J^{(3)}_e$ and $J^{(6)}_m$, with kinetic terms for the branes included explicitly.
• The extended Cartan form $\mathcal{G} = d\mathcal{A}\mathcal{A}^{-1} + \mathbf{g}$ allows the inclusion of gauge and gravitational fields in a unified duality-invariant way.
• The construction provides a template for a fully duality-symmetric formulation of M-theory, though the gravity sector remains outside the current formalism.