标题： KR理论的傅里叶-穆凯变换 显示英文标题

标题： A Fourier-Mukai Transform For KR Theory

摘要： 在复K理论中，傅里叶-牟克伊变换是在环面及其对偶环面的K理论群之间的同构，它通过拉回、张量乘以庞加莱线丛和前推来定义。 当使用扭曲K理论时，傅里叶-牟克伊变换可以扩展到对偶环面族。 因此，傅里叶-牟克伊变换是$T$-对偶环面丛的扭曲K理论群之间的同构。 在本文中，我们将这些结果扩展到扭曲KR理论。 我们引入了具有实结构的环面丛的实$T$-对偶性的概念，并证明了实$T$-对偶的存在性。 然后，我们为实$T$-对偶环面丛定义了一个实傅里叶-牟克伊变换，并证明了它是同构。 最后，我们考虑了这些结果在实狄拉克算子族指标中的应用，这与实塞伯格-威滕理论有关。 显示英文摘要

摘要： 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 Poincar\'e 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.