Max Karoubi, Mariano Suarez-Alvarez
We construct a braided structure on the algebra of Kähler differential forms of a commutative algebra twisted by an endomorphism. This generalises the construction done in M. Karoubi, Quantum Methods in Algebraic Topology, see http://www.math.jussieu.fr/~karoubi, for topological purposes.
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We construct the Atiyah-Hirzebruch spectral sequence (AHSS) for twisted differential generalized cohomology theories. This generalizes to the twisted setting the authors' corresponding earlier construction for differential cohomology theories, as well as to the differential setting the AHSS for twisted generalized cohomology theories, including that of twisted K-theory by Rosenberg and Atiyah-Segal. In describing twisted differential spectra we build on the work of Bunke-Nikolaus, but we find it useful for our purposes to take an approach that highlights direct analogies with classical bundles…
This is the first in a series of papers constructing geometric models of twisted differential K-theory. In this paper we construct a model of even twisted differential K-theory when the underlying topological twist represents a torsion class. By differential twists we will mean smooth U(1)-gerbes with connection, and we use twisted vector bundles with connection as cocycles. The model we construct satisfies the axioms of Kahle and Valentino, including functoriality, naturality of twists, and the hexagon diagram. This paper confirms a long-standing hypothetical idea that twisted vector bundles …
We use the geometry of the space of fields for gauged supersymmetric mechanics to construct the twisted differential equivariant K-theory of a manifold with an action by a finite group.
The main goal of the present paper is the construction of twisted generalized differential cohomology theories and the comprehensive statement of its basic functorial properties. Technically it combines the homotopy theoretic approach to (untwisted) generalized differential cohomology developed by Hopkins-Singer and later by the first author and D. Gepner with the oo-categorical treatement of twisted cohomology by Ando-Blumberg-Gepner. We introduce the notion of a differential twist for a given generalized cohomology theory and construct twisted differential cohomology groups (resp. spectra). …
In the background effective field theory of heterotic string theory, the Green-Schwarz anomaly cancellation mechanism plays a key role. Here we reinterpret it and its magnetic dual version in terms of differential twisted String- and differential twisted Fivebrane-structures that generalize the notion of Spin-structures and Spin-lifting gerbes and their differential refinement to smooth Spin-connections. We show that all these structures can be encoded in terms of nonabelian cohomology, twisted nonabelian cohomology, and differential twisted nonabelian cohomology, extending the differential ge…
Differential Models for the Anderson Dual to Twisted $\mathrm{Spin}^c$-Bordism and a Twisted Anomaly Map
通过构造三次扭曲 $\mathrm{Spin}^c$-配边及其 Anderson 对偶的微分模型，基于 Yamashita-Yonekura 框架，利用丛丛、丛模和 Dirac 算子的约化 eta 不变量，定义了从逆扭曲的微分扭曲 $K$-理论到扭曲 $\mathrm{Spin}^c$-配边的 Anderson 对偶的扭曲反常映射。
Twisted differential KO-theory
本文系统阐述了扭曲微分KO理论的构建方法，揭示了二度与一度扭曲的差异，提供了拓扑情况下AHSS中E₂和E₃页的微分明确识别，并通过几何、拓扑与物理中的例子展示其应用。
Ramond-Ramond fields and twisted differential K-theory
本文提出将RR场系统表述为扭曲微分K理论中的元素，并借助扭曲微分K理论的AHSS刻画其量子化，通过球面、环面和Calabi-Yau三维流形验证该框架。
The character map in (twisted differential) non-abelian cohomology
本文将K理论上的Chern特征推广至（扭曲、微分）非阿贝尔上同调，利用有理同伦论构建特征映射，统一并推广了Chern-Weil等同态，并在twistorial Cohomotopy理论中揭示了与M理论通量电荷量子化相关的拓扑关系。
We analyse in detail the language of partially non-abelian Deligne cohomology and of twisted differential K-theory, in order to describe the geometry of type II superstring backgrounds with D-branes. This description will also provide the opportunity to show some mathematical results of independent interest. In particular, we begin classifying the possible gauge theories on a D-brane or on a stack of D-branes using the intrinsic tool of long exact sequences. Afterwards, we recall how to construct two relevant models of differential twisted K-theory, paying particular attention to the dependenc…
Early in the history of higher homotopy algebra, it was realized that Massey products are homotopy invariants in a special sense, but it was the work of Tornike Kadeisvili that showed they were but a shadow of an A-infinity-structure on the homology of a differential graded algebra. Here we relate his work to that of Victor Gugenheim and K.T. (Chester) Chen. This paper is a personal tribute to Tornike and the Georgian school of homotopy theory as well as to Gugenheim and Chen, who unfortunately are not with us to appreciate this convergence.
Deligne cohomology can be viewed as a differential refinement of integral cohomology, hence captures both topological and geometric information. On the other hand, it can be viewed as the simplest nontrivial version of a differential cohomology theory. While more involved differential cohomology theories have been explicitly twisted, the same has not been done to Deligne cohomology, although existence is known at a general abstract level. We work out what it means to twist Deligne cohomology, by taking degree one twists of both integral cohomology and de Rham cohomology. We present the main pr…
Symplectic Flatness and Twisted Primitive Cohomology
本文引入了辛流形上联络和纤维丛的辛平坦性概念，提出了基于$A_\infty$-代数的扭曲微分复形的平坦性条件，并证明辛平坦联络是Yang-Mills联络的特殊子类，同时研究了其上同调及消没定理。
Studying the topological aspects of M-branes in M-theory leads to various structures related to Wu classes. First we interpret Wu classes themselves as twisted classes and then define twisted notions of Wu structures. These generalize many known structures, including Pin^- structures, twisted Spin structures in the sense of Distler-Freed-Moore, Wu-twisted differential cocycles appearing in the work of Belov-Moore, as well as ones introduced by the author, such as twisted Membrane and twisted String^c structures. In addition, we introduce Wu^c structures, which generalize Pin^c structures, as w…
We combine rational homotopy theory and higher Lie theory to describe the WZW term in the M5-brane sigma model. We observe that this term admits a natural interpretation as a twisted 7-cocycle on super-Minkowski spacetime with coefficients in the rational 4-sphere. This exhibits the WZW term as an element in twisted cohomology, with the twist given by the cocycle of the M2-brane. We consider integration of this rational situation to differential cohomology and differential cohomotopy.
We demonstrate that twisted equivariant differential K-theory of transverse complex curves accommodates exotic charges of the form expected of codimension=2 defect branes, such as of D7-branes in IIB/F-theory on A-type orbifold singularities, but also of their dual 3-brane defects of class-S theories on M5-branes. These branes have been argued, within F-theory and the AGT correspondence, to carry special SL(2)-monodromy charges not seen for other branes, but none of these had previously been identified in the expected brane charge quantization law given by K-theory. Here we observe that it is …
We study the twisted cohomology groups of $A_\infty$-algebras defined by twisting elements and their behavior under morphisms and homotopies using the bar construction. We define higher Massey products on the cohomology groups of general $A_\infty$-algebras and establish the naturality under morphisms and their dependency on defining systems. The above constructions are also considered for $C_\infty$-algebras. We construct a spectral sequence converging to the twisted cohomology groups an show that the higher differentials are given by the $A_\infty$-algebraic Massey products.
Degree one twisting of Deligne cohomology, as a differential refinement of integral cohomology, was established in previous work. Here we consider higher degree twists. The Rham complex, hence de Rham cohomology, admits twists of any odd degree. However, in order to consider twists of integral cohomology we need a periodic version. Combining the periodic versions of both ingredients leads us to introduce a periodic form of Deligne cohomology. We demonstrate that this theory indeed admits a twist by a gerbe of any odd degree. We present the main properties of the new theory and illustrate its u…
Anyonic Topological Order in Twisted Equivariant Differential (TED) K-Theory
本文提出用扭曲等变微分K-理论分类相互作用二维半金属中的su(2)-任意子拓扑序，涵盖节点补集、构型空间及扭曲陈类，连接费米子与任意子量子。
Obstruction Theory for Bigraded Differential Algebras
本文发展了带扭曲系数的cbba的Hirsch扩张的障碍理论，并应用于极小cbba的结构定理、相对极小模型的构造与唯一性，以及自同构群和形式性的刻画。
We construct models of the differential $KO$-theory and the twisted differential $KO$-theory, by refining Karoubi's $KO$-theory [Kar78] in terms of gradations on Clifford modules. In order for this, we set up the generalized Clifford superconnection formalism which generalizes the Quillen's superconnection formalism [Qui85]. One of our models can be regarded as classifying "fermionic mass terms" in physics.
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