[Paper Review] D=10, N=IIB Supergravity: Lorentz-invariant actions and duality

Gianguido Dall’Agata, Kurt Lechner+1 more|arXiv (Cornell University)|Jun 17, 1998

Black Holes and Theoretical PhysicsPhysics and Astronomy73 citations

TL;DR

This paper presents a manifestly Lorentz-invariant and supersymmetric Lagrangian formulation for D=10, N=IIB supergravity using a novel method for chiral p-forms that introduces a single non-propagating auxiliary scalar. The approach enables duality-symmetric actions where the two-form and its Hodge dual six-form, as well as the zero-form and its dual eight-form, appear symmetrically, while preserving SL(2,R) duality invariance and correctly propagating the physical degrees of freedom.

ABSTRACT

We present a manifestly Lorentz invariant and supersymmetric component field action for $D = 10$, type $IIB$ supergravity, using a newly developed method for the construction of actions with chiral bosons, which implies only a single scalar non propagating auxiliary field. With the same method we construct also an action in which the complex two-form gauge potential and its Hodge-dual, a complex six-form gauge potential, appear in a symmetric way in compatibility with supersymmetry and Lorentz invariance. The duals of the two physical scalars of the theory turn out to be described by a $SL(2,R)$ triplet of eight-forms whose curvatures are constrained by a single linear relation. We present also a supersymmetric action in which the basic fields and their duals, six-form and eight-form potentials, appear in a symmetric way. All these actions are manifestly invariant under the global $SL(2,R)$-duality group of $D = 10$, $IIB$ supergravity and are equivalent to each other in that their dynamics corresponds to the well known equations of motion of $ D=10$, $IIB$ supergravity.

Motivation & Objective

• To resolve the longstanding challenge of constructing a manifestly Lorentz-invariant and supersymmetric Lagrangian for D=10, N=IIB supergravity, which contains a self-dual chiral four-form.
• To extend the novel method for chiral p-forms to include duality-symmetric formulations where gauge potentials and their Hodge duals appear on equal footing.
• To construct actions invariant under the global SL(2,R) duality group, ensuring consistency with the known equations of motion and physical degrees of freedom.
• To provide a framework compatible with quantum field theory on trivial topologies, enabling anomaly computations and effective action derivations.

Proposed method

• The paper employs a recently developed method for chiral p-forms in D=2 mod 4 dimensions, which introduces a single non-propagating auxiliary scalar field and two new bosonic symmetries to enforce self-duality.
• The auxiliary field is eliminated via one symmetry, while the second symmetry reduces the degrees of freedom of the p-form to those of a chiral boson, ensuring consistency with the self-duality condition.
• For duality-symmetric formulations, the method is adapted to treat both a two-form and its Hodge dual six-form simultaneously, with gauge-fixing conditions (7.4) and (7.5) preserving symmetry under A₂ ↔ A₆.
• The duals of the two physical scalars are described by a triplet of eight-form potentials whose curvatures satisfy an SL(2,R)-invariant linear constraint, with three nine-form curvatures transforming in the adjoint representation.
• The action is constructed to be manifestly invariant under SL(2,R), with curvatures related via Hodge duality and constrained by a single linear relation among the nine-form curvatures.
• The method allows for a canonical coupling to gravity and is shown to be consistent with supersymmetry and κ-symmetry, with the equations of motion correctly reproducing the known dynamics of IIB supergravity.

Experimental results

Research questions

• RQ1 How can a manifestly Lorentz-invariant and supersymmetric action be constructed for D=10, N=IIB supergravity, given the presence of a chiral four-form with self-dual field strength?
• RQ2 Can the duality between the two-form and its Hodge dual six-form be made manifest in a supersymmetric and Lorentz-invariant Lagrangian formulation?
• RQ3 How can the duals of the two physical scalars be consistently described using higher-rank form potentials while preserving SL(2,R) duality invariance?
• RQ4 What is the role of the auxiliary scalar field in the new method, and how does it ensure correct degrees of freedom without introducing propagating modes?
• RQ5 Can the functional integral over the duality-symmetric fields be consistently defined, and does it reproduce the expected effective interactions for electric and magnetic sources?

Key findings

• The proposed action is manifestly Lorentz-invariant and supersymmetric, with the chiral four-form handled via a single non-propagating auxiliary scalar field, avoiding the issues of previous approaches involving Lagrange multipliers.
• The method successfully constructs a duality-symmetric action where the two-form and six-form gauge potentials appear on equal footing, with equations of motion (7.6) and (7.7) confirming the correct 28 degrees of freedom for each.
• The duals of the two physical scalars are described by a triplet of eight-form potentials whose curvatures satisfy an SL(2,R)-invariant linear constraint, with two curvatures related by Hodge duality to the one-form curvatures of the scalars.
• The equations of motion derived from the action, including the residual constraints (7.7), correctly reproduce the dynamics of massless 2-form and 6-form fields in ten dimensions.
• The gauge-fixing conditions (7.4) and (7.5) are symmetric under A₂ ↔ A₆ and are shown to be suitable for functional integral quantization, yielding the expected effective interaction for electric and magnetic charges.
• The method is compatible with the computation of anomalies, such as the Lorentz anomaly, and has been successfully applied to derive the effective action of a 2D chiral boson, matching the index theorem prediction.