Dynamics of Cayley forms

Krasnov, Kirill

Abstract

The most natural first-order PDE's to be imposed on a Cayley 4-form in eight dimensions is the condition that it is closed. As is well-known, this implies integrability of the Spin(7)-structure defined by the Cayley form, as well as Ricci-flatness of the associated metric. We address the question as to what the most natural second-order in derivatives set of conditions is. We start at the linearised level, and construct the most general diffeomorphism-invariant second order in derivatives Lagrangian that is quadratic in the perturbations of the Cayley form. We find that there is a two-parameter family of such Lagrangians. We then describe a non-linear completion of the linear story. We parametrise the intrinsic torsion of a Spin(7)-structure by a 3-form, and show that this 3-form is completely determined by the exterior derivative of the Cayley form. The space of 3-forms splits into two Spin(7) irreducible components, and so there is a two-parameter family of diffeomorphism-invariant Lagrangians that are quadratic in the torsion, matching the linearised story. We then describe a first-order in derivatives version of the action functional, which depends on the Cayley 4-form and auxiliary 3-form as independent variables. There is a unique functional whose Euler-Lagrange equation for the auxiliary 3-form states that it is equal to the torsion 3-form. For any member of our family of theories, the Euler-Lagrange equations are written only using the operator of exterior differentiation of forms, and do not require the knowledge of the metric-compatible Levi-Civita connection. Geometrically, there is a preferred member in the family of Lagrangians, and we propose that its Euler-Lagrange equations are the most natural second-order equations to be satisfied by Cayley forms. Our construction also leads to a natural geometric flow in the space of Cayley forms, defined as the gradient flow of our action functional.

Citation

Krasnov, K. (2025). Dynamics of Cayley forms. Pure and Applied Mathematics Quarterly, 21(5), 1959-2003. https://doi.org/10.4310/pamq.250421122254

• Journal Article Type: Article
• Acceptance Date: Apr 14, 2025
• Online Publication Date: Jun 24, 2025
• Publication Date: 2025
• Deposit Date: Jun 27, 2025
• Publicly Available Date: Jun 30, 2025
• Journal: Pure and Applied Mathematics Quarterly
• Print ISSN: 1558-8599
• Electronic ISSN: 1558-8602
• Publisher: International Press
• Peer Reviewed: Peer Reviewed
• Volume: 21
• Issue: 5
• Pages: 1959-2003
• DOI: https://doi.org/10.4310/pamq.250421122254
• Public URL: https://nottingham-repository.worktribe.com/output/50716689
• Publisher URL: https://link.intlpress.com/journals/journalList?id=1804417850613493761