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This paper presents a derived version of Laumon's homogeneous Fourier transform, extending its application from vector bundles to derived vector bundles, and explores the resulting duality between derived and stacky phenomena, particularly the persistence of the involutivity property in this new context.

Abstract

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arxiv.org

Khan, A. A. (2024). The derived homogeneous Fourier transform. arXiv:2311.13270v2 [math.AG].

This paper aims to extend Laumon's homogeneous Fourier transform from traditional vector bundles to the realm of derived vector bundles. The study investigates the properties and implications of this extended transform, focusing on the duality between derived and stacky phenomena that emerges in this context.

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by Adeel A. Kha... at **arxiv.org** 10-10-2024

Deeper Inquiries

The concept of the derived homogeneous Fourier transform, as presented in the context of derived vector bundles, exhibits a rich interplay between algebraic and geometric structures. This suggests several promising avenues for generalization and application:
1. Beyond Vector Bundles:
Derived Coherent Sheaves: A natural extension is to consider the derived category of coherent sheaves on a derived stack. This would involve generalizing the notion of the evaluation map (1.1) and the kernel (1.2) to this setting. The challenge lies in defining a suitable duality for derived coherent sheaves that mirrors the role of vector bundle duality in the current construction.
Principal Bundles: One could explore a generalization to principal G-bundles for a more general algebraic group G. This would require developing a suitable notion of "Fourier duality" in the context of G-equivariant sheaves.
Derived Loop Spaces: The homogeneous Fourier transform can be viewed as a kind of "linearization" procedure. This suggests a connection with derived loop spaces, which are naturally equipped with an S^1-action. Exploring this link could lead to new insights into both areas.
2. Applications:
Derived Symplectic Geometry: The derived homogeneous Fourier transform, particularly its connection to the Fourier-Sato transform mentioned in the context, has the potential to provide new tools for studying derived symplectic geometry and its applications to mirror symmetry.
Geometric Representation Theory: The appearance of Borel-Moore homology and the generalization of Kashiwara's Fourier isomorphism (Example 1.34) hint at deeper connections with geometric representation theory. The derived homogeneous Fourier transform could potentially be used to study representations of algebraic groups on derived categories of sheaves.
Motivic Homotopy Theory: The fact that the framework accommodates motivic settings (as mentioned in the "Conventions and notation" section) opens up possibilities for applications in motivic homotopy theory. The derived homogeneous Fourier transform could provide new tools for studying motivic spectra and their associated invariants.

Yes, alternative approaches to defining a derived homogeneous Fourier transform could offer fresh perspectives and unveil new properties:
1. Six Functor Formalism Perspective:
Alternative Kernels: Instead of directly generalizing Laumon's kernel, one could explore different kernels on the product E∨ ×S E. This could lead to variants of the Fourier transform with modified properties, potentially highlighting different aspects of the underlying geometry.
Categorical Approach: One could attempt a more abstract categorical construction of the derived Fourier transform, perhaps leveraging the language of ∞-categories and their functoriality. This might provide a more conceptual understanding of the transform and its properties.
2. Derived Algebraic Geometry Techniques:
Deformation Theory: One could study the derived homogeneous Fourier transform using deformation theory, considering families of derived vector bundles and analyzing how the transform behaves under deformations. This might reveal connections with moduli spaces of sheaves and their derived structures.
Derived Intersection Theory: The derived homogeneous Fourier transform naturally involves intersections, as seen in the definition of the kernel. Employing tools from derived intersection theory could provide a more refined understanding of the transform and its interaction with derived Chern classes and other invariants.
3. Connections to Other Transforms:
Fourier-Mukai Transforms: Exploring the relationship between the derived homogeneous Fourier transform and more general Fourier-Mukai transforms on derived stacks could lead to fruitful interactions. This might involve developing a suitable notion of a "kernel" for Fourier-Mukai transforms in the derived setting.
Quantum Field Theory: Drawing inspiration from quantum field theory, where Fourier transforms play a fundamental role, one could seek alternative definitions of the derived homogeneous Fourier transform motivated by path integrals or other QFT techniques.

The duality between derived and stacky phenomena, as illuminated by the derived homogeneous Fourier transform, hints at a profound shift in our understanding of space and geometry:
1. Space as a Spectrum:
Beyond Points: The traditional notion of space as a collection of points is being challenged. Derived algebraic geometry suggests that spaces have a richer structure, encoded in their derived categories of sheaves. The derived homogeneous Fourier transform, by bridging derived and stacky phenomena, further blurs the lines between points and more general objects like line bundles.
Geometry from Algebra: The duality suggests a deep connection between the algebraic structure of derived categories and the geometric properties of spaces. This aligns with the broader theme in modern mathematics and physics of understanding geometry through algebraic invariants.
2. Implications for Physics:
Quantum Geometry: In quantum field theory, the distinction between particles and fields becomes blurred. The derived homogeneous Fourier transform, with its ability to exchange derived and stacky features, might provide a mathematical framework for describing such quantum geometries where the classical distinction between points and extended objects breaks down.
Mirror Symmetry and Duality: The duality between derived and stacky phenomena resonates with the concept of mirror symmetry in string theory, where different geometric spaces give rise to the same physics. The derived homogeneous Fourier transform could potentially provide a mathematical language for describing and studying such dualities.
3. New Foundations for Geometry:
Homotopical Perspective: Derived algebraic geometry emphasizes the importance of homotopy theory in understanding geometry. The derived homogeneous Fourier transform, by connecting derived and stacky structures, further reinforces this perspective, suggesting that the fundamental building blocks of space and geometry might be homotopical in nature.
Categorification: The use of derived categories and stacks points towards a "categorification" of geometry, where geometric objects are replaced by categories and geometric constructions are lifted to functors between categories. The derived homogeneous Fourier transform exemplifies this trend, providing a concrete example of how categorification can reveal hidden structures and dualities.

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