[Paper Review] IIB PP-Waves with Extra Supersymmetries

Makoto Sakaguchi|ArXiv.org|Jun 2, 2003

Black Holes and Theoretical Physics32 references3 citations

TL;DR

This paper establishes a uniqueness theorem for IIB pp-wave backgrounds with extra supersymmetries, proving that any such background admitting at least one non-harmonic extra Killing spinor must reduce to a form with $ H = A_{mn}(x^{-})x^m x^n $ and $ \xi = \xi(x^{-}) $, modulo coordinate transformations. It identifies new IIB pp-wave solutions with 18, 20, 24, and 32 supersymmetries by analyzing Killing spinor equations using Cartan matrices related to $ SU(4) $ and $ Spin(7) $ holonomy structures.

ABSTRACT

We examine Killing spinor equations of the general IIB pp-wave backgrounds, which contain a scalar H(x^m,x^-) in the metric and a self-dual four-form ξ(x^m,x^-) in the self-dual five-form flux. Considering non-harmonic extra Killing spinors, we find that if the backgrounds admit at least one extra Killing spinor in addition to 16 standard Killing spinors, backgrounds can be reduced to the form with H=A_{mn}(x^-)x^mx^n and ξ(x^-), modulo coordinate transformations. We examine further the cases in which the extra Killing spinors are characterized by a set of Cartan matrices. Solving Killing spinor equations, we find IIB pp-wave backgrounds which admit 18, 20, 24 and 32 Killing spinors.

Motivation & Objective

• To establish a uniqueness theorem for IIB pp-wave backgrounds that admit at least one non-harmonic extra Killing spinor beyond the standard 16.
• To classify IIB pp-wave solutions with enhanced supersymmetry (18, 20, 24, 32) by analyzing the structure of extra Killing spinors.
• To explore the role of Cartan matrices in characterizing extra Killing spinors and their relation to special holonomy manifolds ($ SU(4) $, $ Spin(7) $).
• To determine whether such backgrounds automatically satisfy the supergravity equations of motion, suggesting algebraic richness in their supersymmetry structure.
• To extend the uniqueness principle observed in eleven-dimensional supergravity to IIB pp-waves, potentially simplifying classification of flux backgrounds.

Proposed method

• Derive the Killing spinor equations for general IIB pp-wave backgrounds with a self-dual five-form flux $ F = dx^{-} \wedge \xi $, where $ \xi $ is a self-dual four-form on $ \mathbb{E}^8 $.
• Apply the condition of non-harmonic extra Killing spinors to constrain the dependence of $ H(x^m, x^{-}) $ and $ \xi(x^m, x^{-}) $, proving they reduce to $ H = A_{mn}(x^{-})x^m x^n $ and $ \xi = \xi(x^{-}) $ modulo coordinate transformations.
• Use mutually commuting projectors to characterize extra Killing spinors, linking them to geometric structures such as the Kähler form of a Calabi-Yau four-fold ($ SU(4) $ holonomy) and the Cayley four-form of a $ Spin(7) $ manifold.
• Solve the resulting Killing spinor equations under these projector conditions, identifying specific forms of $ A_{mn} $ and $ \xi $ that yield enhanced supersymmetry.
• Construct explicit solutions for 18, 20, 24, and 32 supersymmetries by solving the system with parameters $ \mu_i $, and verify consistency with the supergravity equation $ \triangle H = -\frac{32}{4!} \xi_{klmn} \xi^{klmn} $.
• Demonstrate that the 32-supersymmetric solution is related to the maximally supersymmetric pp-wave by a coordinate transformation, confirming consistency with known results.

Experimental results

Research questions

• RQ1 Under what conditions does the existence of a single non-harmonic extra Killing spinor restrict the form of the pp-wave metric and flux to $ H = A_{mn}(x^{-})x^m x^n $ and $ \xi = \xi(x^{-}) $?
• RQ2 How do the geometric structures of $ SU(4) $ and $ Spin(7) $ holonomy manifolds manifest in the classification of extra Killing spinors for IIB pp-waves?
• RQ3 What are the explicit forms of IIB pp-wave backgrounds that admit 18, 20, 24, and 32 supersymmetries, and how are they related to the underlying algebraic structure of the Killing spinor equations?
• RQ4 Can the supergravity equations of motion be automatically satisfied by backgrounds with extra supersymmetries, suggesting deeper algebraic constraints?
• RQ5 Is there a generalization of the uniqueness theorem from eleven-dimensional supergravity to IIB pp-waves, and what does it imply for the classification of flux backgrounds?

Key findings

• Any IIB pp-wave background admitting at least one non-harmonic extra Killing spinor must be reducible to $ H = A_{mn}(x^{-})x^m x^n $ and $ \xi = \xi(x^{-}) $, modulo coordinate transformations, establishing a uniqueness theorem.
• Solutions with 18, 20, 24, and 32 supersymmetries are explicitly constructed by solving the Killing spinor equations under Cartan matrix conditions related to $ SU(4) $ and $ Spin(7) $ holonomy.
• The 32-supersymmetric solution is shown to be related by a coordinate transformation to the maximally supersymmetric IIB pp-wave background, confirming consistency with known results.
• The 24-supersymmetric solution arises from a specific combination of parameters in the $ \xi $-operator, corresponding to a $ Spin(7) $-invariant four-form structure.
• Two distinct three-parameter families of 20-supersymmetric solutions are found, each associated with different combinations of $ \mu_i $, corresponding to $ SU(4) $ and $ Spin(7) $ structures.
• The 18-supersymmetric solution is derived from a seven-parameter family of $ \xi $-operators, with $ A_m $ and $ \xi $ expressed in terms of symmetric combinations of $ \mu_i $, indicating a richer algebraic structure.