Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2509.24284

A Fourier-Mukai Transform For KR Theory

David Baraglia
View original source ↗Read source document ↗Export metadata ↓

Abstract

In complex K-theory, the Fourier-Mukai transform is an isomorphism between K-theory groups of a torus and its dual torus which is defined by pullback, tensoring by the Poincaré line bundle and pushforward. The Fourier-Mukai transform extends to families of dual tori provided one works with twisted K-theory. The Fourier-Mukai transform is then an isomorphism between twisted K-theory groups of $T$-dual torus bundles. In this paper we prove an extension of these results to twisted KR-theory. We introduce a notion of Real $T$-duality for torus bundles with Real structures and prove the existence of Real $T$-duals. We then define a Real Fourier-Mukai transform for Real $T$-dual torus bundles and prove that it is an isomorphism. Lastly, we consider an application of these results to the families index of Real Dirac operators which is relevant to Real Seiberg-Witten theory.

Explore related subjects

math.KTmath-phmath.ATmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

David Baraglia. 2025-09-29. A Fourier-Mukai Transform For KR Theory. https://arxiv.org/abs/2509.24284

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

$E$-theory is compactly assembled

We show that the equivariant $E$-theory category $\mathrm{E}_{\mathrm{sep}}^{G}$ for separable $C^{*}$-algebras is a compactly assembled stable $\infty$-category. We derive this result as a consequence of the shape theory for $C^{*}$-algebras developed by Blackadar and Dardarlat and a new construction of $\mathrm{E}_{\mathrm{sep}}^{G}$. As an application we investigate a topological enrichment of the homotopy category of a compactly assembled $\infty$-category in general and argue that the results of Carrión and Schafhauser on the enrichment of the classical $E$-theory category can be derived by specialization.

math.KT↗
arXiv2026

Scissors automorphism groups I: Homological stability and K-theory

In any category with a reasonable notion of cover, each object has a group of scissors automorphisms. We prove that under mild conditions, the homology of this group is independent of the object, and can be expressed in terms of the scissors congruence K-theory spectrum defined by Zakharevich. We therefore obtain both a group-theoretic interpretation of Zakharevich's higher scissors congruence K-theory, as well as a method to compute the homology of scissors automorphism groups. In the classical case of scissors congruence of polytopes, this leads to calculations and structural results on the homology of the scissors automorphism group. In two sequels, we show that more generally our approach leads to homology calculations for various families of groups appearing in group theory and dynamics, recovering results of Szymik-Wahl, Li, and Tanner.

math.KT↗