Spinor Bilinears and Killing-Yano Forms in Generalized Geometry: Exploring the Relationship Between Generalized Killing Spinors and Generalized Killing-Yano Forms

Core Concepts

This research paper explores the connection between generalized Killing spinors and generalized Killing-Yano forms within the framework of generalized geometry, demonstrating that specific bilinear forms of generalized Killing spinors can be used to construct generalized Killing-Yano forms and generalized closed conformal Killing-Yano forms.

Abstract

• Bibliographic Information: A¸cık, Ö., Ertem, Ü., & Kelek¸ci, Ö. (2024). Spinor bilinears and Killing-Yano forms in generalized geometry. arXiv preprint arXiv:2411.00443.

• Research Objective: This paper investigates the properties of generalized spinor bilinears and their relationship to generalized Killing and twistor spinor equations within the framework of generalized geometry. The authors aim to define and construct generalized Killing-Yano (KY) forms using generalized spinors and analyze their connection to generalized Killing spinors.

• Methodology: The authors utilize the mathematical framework of generalized geometry, employing concepts like generalized tangent bundles, generalized metrics, Courant and Dorfman brackets, generalized connections, and spinor bilinears. They analyze the generalized Killing and twistor spinor equations, expressing them in terms of differential forms. By defining a difference operator between the generalized Lie derivative and generalized connection, they derive the generalized Killing equation in terms of the generalized connection. This allows them to define the generalized KY equation as an antisymmetric generalization of the generalized Killing equation.

• Key Findings: The paper reveals that solutions to the generalized Killing spinor equation correspond to solutions of the conformal Killing-Yano (CKY) equation with torsion. Additionally, the authors establish a connection between generalized Killing spinors and generalized KY forms, demonstrating that generalized KY forms and generalized closed conformal KY forms can be constructed from the bilinear forms of generalized Killing spinors.

• Main Conclusions: The study concludes that bilinear forms of generalized Killing spinors generally correspond to conformal KY forms. This finding has implications for the analysis of supergravity backgrounds, potentially allowing for refinements by extending Killing superalgebra structures to include higher-degree KY forms.

• Significance: This research contributes significantly to the field of generalized geometry by providing a novel construction of generalized KY forms using generalized Killing spinors. This connection deepens the understanding of the geometric structure of supergravity theories, which are central to modern theoretical physics.

• Limitations and Future Research: The paper primarily focuses on the mathematical framework and derivations within generalized geometry. Further research could explore the physical implications of these findings in the context of specific supergravity theories and their applications to string theory and cosmology. Investigating the role of generalized KY forms in the context of extended Killing superalgebra structures could also be a promising avenue for future research.

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Spinor bilinears and Killing-Yano forms in generalized geometry

by Özgü... at arxiv.org 11-04-2024

https://arxiv.org/pdf/2411.00443.pdf

Deeper Inquiries

How can the construction of generalized Killing-Yano forms using generalized Killing spinors be utilized to further our understanding of specific supergravity theories, such as 10-dimensional IIA and IIB theories?

The construction of generalized Killing-Yano (KY) forms from generalized Killing spinors offers a powerful tool for delving deeper into the structure of supergravity theories, particularly in 10-dimensional IIA and IIB contexts. Here's how:

Unveiling Hidden Symmetries: KY forms are intimately connected to hidden symmetries in supergravity backgrounds. By expressing generalized KY forms in terms of generalized Killing spinors, we gain a direct link between these symmetries and the supersymmetric structure of the theory. This can lead to the identification of new conserved charges and a more profound understanding of the dynamics of these theories.

Characterizing Supergravity Backgrounds: Generalized Killing spinors are crucial for classifying and characterizing supergravity solutions. The ability to construct generalized KY forms from these spinors provides additional geometric invariants that can further refine this classification. This is particularly relevant for understanding the moduli spaces of supergravity solutions and their potential applications to string theory and cosmology.

Exploring Dualities: Type IIA and IIB supergravity theories are famously related by dualities. The relationship between generalized Killing spinors and generalized KY forms could provide new insights into these dualities. For instance, it might be possible to relate specific KY forms in one theory to corresponding structures in the dual theory, shedding light on the underlying geometric connections.

Simplifying Calculations: In practical terms, expressing generalized KY forms in terms of generalized Killing spinors can significantly simplify calculations within these supergravity theories. This is because the spinorial formalism often provides a more compact and efficient way to handle the complicated tensorial structures that arise in supergravity.
In essence, this construction acts as a bridge between the supersymmetry encoded in Killing spinors and the hidden symmetries represented by KY forms. This bridge has the potential to uncover new features and deepen our understanding of the geometric structure of 10-dimensional IIA and IIB supergravity theories.

Could there be alternative geometric frameworks beyond generalized geometry where the relationship between Killing spinors and Killing-Yano forms manifests differently or reveals new insights?

Yes, exploring the relationship between Killing spinors and Killing-Yano (KY) forms within alternative geometric frameworks beyond generalized geometry holds exciting possibilities for uncovering novel insights. Here are a few promising avenues:

Exceptional Generalized Geometry: This framework extends generalized geometry by incorporating exceptional Lie groups, allowing for the description of supergravity theories with more intricate gauge groups. Investigating the interplay between Killing spinors and KY forms in this setting could reveal new connections between supersymmetry, hidden symmetries, and the underlying exceptional structures.

Double Field Theory: This approach aims to realize T-duality as a geometric symmetry by doubling the spacetime coordinates. Studying Killing spinors and KY forms within double field theory could provide a more unified perspective on these objects and their transformation properties under T-duality.

Noncommutative Geometry: This framework generalizes classical geometry by allowing for non-commuting coordinates. Exploring the relationship between Killing spinors and KY forms in this context could lead to new insights into the interplay between supersymmetry, hidden symmetries, and non-commutativity, potentially with implications for quantum gravity.

Higher Structures: Moving beyond conventional differential geometry, exploring higher geometric structures like gerbes and stacks could provide a richer framework for understanding the relationship between Killing spinors and KY forms. This could lead to new geometric invariants and a deeper understanding of the topological aspects of these objects.
Each of these frameworks offers a unique perspective on geometry and its relation to physics. By investigating the interplay between Killing spinors and KY forms within these alternative settings, we can potentially uncover hidden connections, novel geometric structures, and a more profound understanding of the fundamental symmetries of nature.

What are the implications of this research for the development of novel mathematical tools and techniques in areas beyond theoretical physics, such as differential geometry and topology?

The research on generalized Killing-Yano (KY) forms and their connection to generalized Killing spinors has the potential to stimulate the development of novel mathematical tools and techniques with applications extending far beyond theoretical physics, particularly in the realms of differential geometry and topology. Here are some key implications:

New Geometric Invariants: The construction of generalized KY forms provides a new set of geometric invariants that can be used to characterize and classify manifolds. These invariants could potentially capture subtle topological and geometric properties that are not easily accessible through traditional methods.

Generalized Spinor Calculus: The use of generalized spinors in this context motivates further development of a generalized spinor calculus. This would involve extending concepts like spinor bundles, Dirac operators, and spinor representations to the setting of generalized geometry and beyond. Such a calculus would be a powerful tool for studying geometric structures and could find applications in areas like index theory and geometric analysis.

Connections to G-structures: Generalized KY forms are naturally associated with G-structures, which are reductions of the structure group of the tangent bundle. This research could lead to a deeper understanding of the relationship between G-structures, spinors, and special geometric structures. This could have implications for the study of Riemannian holonomy groups, special Kähler manifolds, and other areas of differential geometry.

Topological Field Theories: The relationship between generalized KY forms and generalized Killing spinors could provide new insights into the construction and classification of topological field theories. These theories are intimately connected to the geometry and topology of manifolds, and new geometric tools could lead to the discovery of new examples and dualities.

Applications to Geometric Analysis: The techniques developed in this research could also find applications in geometric analysis, particularly in the study of elliptic operators on manifolds. The interplay between spinors, differential forms, and geometric structures is central to this field, and new insights from generalized geometry could lead to progress on problems related to spectral geometry, heat kernel asymptotics, and other areas.
In summary, the exploration of generalized KY forms and their connection to generalized Killing spinors not only deepens our understanding of fundamental physics but also opens up new avenues of research in pure mathematics. The development of novel mathematical tools and techniques inspired by this research has the potential to significantly advance our understanding of geometry, topology, and their applications in various branches of mathematics and physics.