Equivariant Twisted Cartan Cohomology Theory
Debasis Sen
本文证明了Cartan关于局部系数等变单纯上同调的一个结果的等变形式。
In this note we prove an equivariant version of a result of Cartan for equivariant simplicial cohomology with local coefficients.
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Equivariant characteristic forms in the Cartan model and Borel equivariant cohomology
证明了紧李群的微分几何与等变特征类的拓扑构造的相容性。我们的分析激发了一个微分尺规作图法的等变特征类在非紧的情况下。这种相容性通常被假设并用于各种情况，但在文献中只有紧连通李群的证明，参见Raoul Bott和Loring Tu的工作。我们的证明适用并推广了Johan Dupont和Ezra Getzler的思想。
Equivariant cohomology and the Maurer-Cartan equation
设G是光滑作用在流形M上的紧致连通李群. Goresky-Kottwitz-MacPherson描述了M的等变上同调的小Cartan模型，拟同构于等变微分形式的标准Cartan复形。本文构造了一个从小Cartan模型到标准Cartan模型的显式上链映射，导出了一个上同调同构。该结构涉及一个显着的非齐次Maurer-Cartan方程的解决方案。这个解在Weil代数的海侵理论和主丛的上同调的Chevalley-Koszul理论中有进一步的应用。
Equivariant characteristic classes in the Cartan model
本文证明了紧连通李群作用的等变特征类的微分几何和拓扑公式的相容性。
Starting with a $\mathbb{C}^*$-valued cocycle on the global quotient orbifold $X // G$, we apply transgression techniques for 2-gerbes, as developed by Lupercio and Uribe, to construct a gerbe on the orbifold loop space $\mathcal{L}(X//G)$. This gives a loop based definition of twisted equivariant Tate K-theory for finite groups that conforms to the definition that Luecke provides for compact, connected Lie groups. We relate our construction to Ganter's work on Moonshine and Huan's quasi-elliptic cohomology.
Twisted equivariant quasi-elliptic cohomology and M-brane charge
本文提出了一种扭曲的准椭圆上同调理论，用于研究M膜电荷与D膜电荷在扭曲等价K理论中的关系。
On a theorem of Henri Cartan concerning the equivaraint cohomology
设G是紧李群，N是G的自由作用在光滑流形X上的闭正规子群.在标题中提到的嘉当定理假设了G-等变上同调X和X/N的G/N-等变上同调之间存在自然同构。本文利用J. Kalkman关于等变上同调的Cartan模型和Weil模型之间的显式同构，证明了：1）Cartan定理是Chern-Weil超进公式的简单推论; 2）在上链水平上显式地描述了这种同构。
Szczarba's twisted shuffle and equivariant path homology of directed graphs
本文研究了标记设定下路径同调的等变性质，通过扭曲洗牌定义了链同构，并应用于有向图的Borel构造。
Equivariant Simplicial Cohomology With Local Coefficients and its Classification
引入了具有离散群的单纯作用的单纯集的等变扭同调，并证明了对于由给定的等变局部系数诱导的适当扭函数，具有局部系数的Bredon-Illman上同调的单纯形式同构于等变扭同调.本文的主要目的是证明局部系数等变单纯上同调的一个分类定理。
A spectral sequence calculating the homology groups of some spaces of maps equivariant under compact group actions is described. For the main example, we calculate the rational homology groups of spaces of even and odd maps $S^m \to S^M$, $m<M$, or, which is the same, the stable homology groups of spaces of non-resultant homogeneous polynomial maps ${\mathbb R}^{m+1} \to {\mathbb R}^{M+1}$ of growing degrees. Also, we find the homology groups of spaces of ${\mathbb Z}_r$-equivariant maps of odd-dimensional spheres for any $r$. As a technical tool, we calculate the homology groups of configurat…
We consider twisted equivariant K--theory for actions of a compact Lie group $G$ on a space $X$ where all the isotropy subgroups are connected and of maximal rank. We show that the associated rational spectral sequence à la Segal has a simple $E_2$--term expressible as invariants under the Weyl group of $G$. Namely, if $T$ is a maximal torus of $G$, they are invariants of the $π_1(X^T)$-equivariant Bredon cohomology of the universal cover of $X^T$ with suitable coefficients. In the case of the inertia stack $ΛY$ this term can be expressed using the cohomology of $Y^T$ and algebraic invariants …
Chern-Weil homomorphism in twisted equivariant cohomology
我们描述了扭曲等变上同调的Cartan和Weil模型以及两者之间的Cartan同态，并将Chern-Weil同态推广到扭曲等变上同调。阐明了为了建立上同调理论，扭曲等变上同调的系数必须取在$G$的对偶李代数上的完备多项式代数中.我们回顾了精确Courant代数胚的等变上同调和扭曲的等变上同调之间的关系，并展示了如何赋予Borel构造$M\times_G EG_k$的有限维近似的广义复结构，只要广义复流形$M$具有Hamilton $G$作用。
Twisted ambidexterity in equivariant homotopy theory
本文在参数化对称幺半∞-范畴中发展了扭曲双灵巧性概念，推广了Hopkins-Lurie双灵巧性与Wirthmüller同构，并证明了紧Lie群的genuine G-spectra是满足该性质的稳定参数化范畴的泛例，进而推广至非紧Lie群情形。
Twisted equivariant K-theory, groupoids and proper actions
本文定义了李群胚作用的扭等变K-理论。对于Bredon相容李群胚，这定义了具有等变稳定投射丛的有限CW-复形范畴上的周期上同调理论。示出了这些束的分类。我们还得到了一个完备化定理，并将这些结果应用于群的真作用。
For G a finite group and X a G-space on which a normal subgroup A acts trivially, we show that the G-equivariant K-theory of X decomposes as a direct sum of twisted equivariant K-theories of X parametrized by the orbits of the conjugation action of G on the irreducible representations of A. The twists are group 2-cocycles which encode the obstruction of lifting an irreducible representation of A to the subgroup of G which fixes the isomorphism class of the irreducible representation.
Equivariant Differential Cohomology
通过连接的曲率形式构造特征类是由德拉姆上循环细化积分上同调的一个动机-称为微分上同调。我们将讨论流形上的群作用的类似情况：由于Nicole Berline和Michèle Vergne在Cartan模型中定义了等变特征形式，激发了所有Cartan上循环对等变积分上同调的精化。有鉴于此，我们还将批判性地回顾以前的定义，特别是五味清则给出的定义。
D-branes in N=2 coset models and twisted equivariant K-theory
分析了Kazama-Suzuki陪集模型中D-膜的荷性。我们提供了相应的扭曲等变K理论的计算，并在格拉斯曼陪集的情况下，su（n+1）/u（n），比较这是从边界共形场论导出的电荷格。
Twisted topological Hochschild homology of $C_n$-equivariant spectra was introduced by Angeltveit, Blumberg, Gerhardt, Hill, Lawson, and Mandell, building on the work of Hill, Hopkins, and Ravenel on norms in equivariant homotopy theory. In this paper we introduce tools for computing twisted THH, which we apply to computations for Thom spectra, Eilenberg-MacLane spectra, and the real bordism spectrum $MU_{\mathbb{R}}$. In particular, we construct an equivariant version of the Bökstedt spectral sequence, the formulation of which requires further development of the Hochschild homology of Green f…
Power operations preserve Thom classes in twisted equivariant Real K-theory
本文为拓扑栈的扭曲KR-理论构造幂运算，证明其保持Thom类，并得出扭曲Atiyah-Bott-Shapiro定向与幂运算交换的结论。
Twisted equivariant K-theory with complex coefficients
利用等变Chern特征标的整体形式，我们用不动点数据描述了具有紧李群作用空间的复化扭曲等变K-理论。我们将其应用于紧李群通过共轭作用于自身的情况，并将结果与Verlinde代数和q=1时的Kac分子联系起来。Verlinde的公式也在这方面进行了讨论。
Twisted equivariant differential K-theory from gauged supersymmetric mechanics
利用规范超对称力学的场空间几何，构造了有限群作用流形的扭微分等变K-理论。
For $G$ a finite group, a normalized 2-cocycle $α\in Z^{2}\big(G,{\mathbb S}^{1}\big)$ and $X$ a $G$-space on which a normal subgroup $A$ acts trivially, we show that the $α$-twisted $G$-equivariant $K$-theory of $X$ decomposes as a direct sum of twisted equivariant $K$-theories of $X$ parametrized by the orbits of an action of $G$ on the set of irreducible $α$-projective representations of $A$. This generalizes the decomposition obtained in [Gómez J.M., Uribe B., Internat. J. Math. 28 (2017), 1750016, 23 pages, arXiv:1604.01656] for equivariant $K$-theory. We also explore some examples of thi…
Segal's spectral sequence in twisted equivariant K-theory for proper and discrete actions
为了理解扭曲的G-等变的K理论，我们使用了Graeme Segal提出的谱序列。我们发现，这个谱序列的第二页是同构的一个版本的Bredon上同调与扭曲表示的局部系数。我们进一步解释了关于谱序列的三阶微分的一些现象，并且当扭曲来自离散扭曲的有穷级元素时，我们恢复了已知的结果。
Equivariant chain complexes, twisted homology and relative minimality of arrangements
证明了超平面排列A的补M（A）上的极小CW-结构X的等变链复形与X无关。当A是非球面排列的充分一般的线性截面时，我们给出了计算M（A）的扭曲同调的一种新方法。
Equivariant $\KK$-theory for generalised actions and Thom isomorphism in groupoid twisted $\K$-theory
利用实分次C ~*-代数和复分次C ~*-代数上的Morita等价，发展了局部紧广群作用的等变KK-理论.关于广义态射和博特周期性的功能性进行了讨论。本文引入局部紧群胚上实或复等变向量丛的Stiefel-Whitney类，建立扭群胚K-理论中的Thom同构定理.
Koszul duality and equivariant cohomology for tori
设T为环面。我们证明了Koszul对偶可以用来计算拓扑T-空间的等变上同调以及泛T-丛的拉回的上同调。新的特点是不再对空间作进一步的假设，而且系数环可以是任意的。这特别给出了一个任意系数的T-空间的等变上同调的Cartan型模型。我们的方法也适用于相交同调。
Equivariant Witt Complexes and Twisted Topological Hochschild Homology
本文研究扭曲拓扑霍奇同调（THH）的等变同伦群代数结构，定义等变Witt复形并证明其捕捉扭曲THH等变同伦的结构，为等变代数研究提供新贡献。
Twisted factorial Grothendieck polynomials and equivariant $K$-theory of weighted Grassmann orbifolds
本文研究加权格拉斯曼 orbifold 的等效K-理论中的Schubert类，引入扭曲阶乘Grothendieck多项式并证明其代表这些类，同时给出结构常数和限制公式。
This is an expository paper about a $\mathcal{C}_2$-equivariant analog of the bar construction. We will explain its construction and prove how it acts as a $σ$-fold delooping machine on twisted monoids, where $σ$ denotes the sign representation.
Let $G$ be a compact, connected, and simply-connected Lie group viewed as a $G$-space via the conjugation action. The Freed-Hopkins-Teleman Theorem (FHT) asserts a canonical link between the equivariant twisted $K$-homology of $G$ and its Verlinde algebra. In this paper we give a generalization of FHT in the presence of a Real structure of $G$. Along the way we develop preliminary materials necessary for this generalization, which are of independent interest in their own right. These include the definitions of Real Dixmier-Douady bundles, the Real third cohomology group which is shown to class…
We develop the basic theory of Maurer-Cartan simplicial sets associated to (shifted complete) $L_\infty$ algebras equipped with the action of a finite group. Our main result asserts that the inclusion of the fixed points of this equivariant simplicial set into the homotopy fixed points is a homotopy equivalence of Kan complexes, provided the $L_\infty$ algebra is concentrated in non-negative degrees. As an application, and under certain connectivity assumptions, we provide rational algebraic models of the fixed and homotopy fixed points of mapping spaces equipped with the action of a finite gr…
The $H \underline{\mathbb{F}}_2$-homology of $C_2$-equivariant Eilenberg-MacLane spaces
本文将 Ravenel-Wilson Hopf 环技术扩展至 $C_2$-等变同伦理论，计算了与常值 Mackey 函子关联的 $C_2$-等变 Eilenberg-MacLane 空间的 $RO(C_2)$-分次同调，并探讨了相关的扭曲 bar 谱序列。
For a compact Lie group acting on a smooth manifold, we define the differential cohomology of a certain quotient stack involving principal bundles with connection. This produces differential equivariant cohomology groups that map to the Cartan-Weil equivariant forms and to Borel's equivariant integral cohomology. We show the Chern-Weil homomorphism for equivariant vector bundles with connection naturally factors through differential equivariant cohomology.
Syzygies in equivariant cohomology for non-abelian Lie groups
本文将Allday-Franz-Puppe关于环面上等变上同调合的工作推广到任意紧连通李群G.特别地，我们证明了对于紧致可定向G-流形X，Chang-Skjelbred序列的模拟是精确的当且仅当X的等变上同调是自反的，当且仅当X的等变Poincare配对是完美的。在此过程中，我们建立了由X的轨道滤子产生的等变上同调模是Cohen-Macaulay。我们允许奇异空间，并介绍了一个Cartan模型的等变上同调。我们还发展了一个G-流形的无穷小轨道类型数有限的判据。
Equivariant extensions of differential forms for non-compact Lie groups
考虑一个赋有李群作用的流形。利用Getzler等变De Rham复形研究了Cartan复形的上同调与等变上同调之间的关系，证明了Cartan复形的上同调位于收敛于等变上同调的谱序列的第二页的第0行.利用这个结果，我们推广了Witten关于紧李群上规范WZW作用量无反常等价于WZW项等变扩张存在的一个结果，推广到规范群是实系数特殊线性群的情形.
We prove that each exponential functor on the category of finite-dimensional complex inner product spaces and isomorphisms gives rise to an equivariant higher (ie. non-classical) twist of $K$-theory over $G=SU(n)$. This twist is represented by a Fell bundle $\mathcal{E} \to \mathcal{G}$, which reduces to the basic gerbe for the top exterior power functor. The groupoid $\mathcal{G}$ comes equipped with a $G$-action and an augmentation map $\mathcal{G} \to G$, that is an equivariant equivalence. The $C^*$-algebra $C^*(\mathcal{E})$ associated to $\mathcal{E}$ is stably isomorphic to the section …
Twisted K theory invariants
引入了三维流形上扭K理论类的一个不变量。然后将该不变量应用于基于SU（2）群的超对称Wess-Zumino-Witten模型的扭曲等变类。证明了由回路群LSU（2）的不同最高权表示所定义的类是不等价的。所得结果与Freed-Hopkins-Teleman将扭曲等变K理论识别为Verlinde代数是一致的。
We construct and analyse models of equivariant cohomology for differentiable stacks with Lie group actions extending classical results for smooth manifolds due to Borel, Cartan and Getzler. We also derive various spectral sequences for the equivariant cohomology of a differentiable stack generalising among others Bott's spectral sequence which converges to the cohomology of the classifying space of a Lie group.
Twisted equivariant matter
我们展示了如何在量子力学的对称性的一般原则导致扭曲的概念群表示。这个框架概括了实/复/四元数表示的经典3重方式以及在凝聚态和核物理中出现的相应10重方式。我们建立了一个基础，讨论连续族的量子系统。这样做之后，量子系统的拓扑相可以被定义为带隙哈密顿算子的连续族的变形类。对于自由粒子有一个额外的代数结构的变形类自然导致扭曲等变K理论的概念。在系统的平移对称性的晶格，我们表明，有一个规范的扭曲的等变K理论的布里渊环面。给出了拓扑相的两个不变量的精确数学定义，这两个不变量在拓扑绝缘体的研究中起着重要的作用。扭曲等变K理论提供了比以前更精细的拓扑绝缘体分类。
Loop groups and twisted K-theory I
这是第一次在一系列的文件调查之间的关系扭曲等变K-理论的一个紧凑的李群G和“Verlinde环”的循环组。本文建立了扭等变K-群的基础，更一般地建立了群胚的扭K-理论。我们建立了足够的基本性质来进行有效的计算。利用Mayer-Vietoris谱序列计算了具有无挠的基本群的紧致连通李群G的挠等变K-群.我们把这种计算与循环群的表示论在与扭曲相关的水平上联系起来。
The ring structure for equivariant twisted K-theory
在较弱的条件下，证明了如果扭2-上圈是2-乘法的，则交叉模的等变扭K理论群有环结构.对于任何交叉模$N\to Γ$，我们也给出了一个明确的海侵映射$T_1: H^*(Γ;A) \to H^{*-1}((N\rtimes Γ;A)$的构造，并证明了该映象中的任何元素都是$\infty$ -乘法的.因此，我们证明了在一定的条件下，对于交叉模$N \to \gm$和任意$e \in \check{Z}^3(Γ;S^1)$，等变扭K理论群$K^*_{e,Γ}(N)$具有环结构.作为应用，我们证明了对于紧致连通单连通李群G，等变扭K理论群$K_{[c], G}^* (G)$具有正则环结构$K^{i+d}_{[c],G}(G)\otimes K^{j+d}_{[c],G}(G)\to K^{i+j+d}_{[c], G}(G)$，其中$d=dim G$和$[c]\in H^2(G\rtimes G;S^1)$ .
Extended manifolds and extended equivariant cohomology
我们定义了具有扩展的切丛的流形范畴，研究了它们的对称性，并考虑了这一范畴中李群作用的等变上同调的类似性。我们表明，当行动保持分裂的扩展切丛，我们的定义的扩展等变上同调同意扭曲等变德拉姆模型的Cartan，并在这种情况下，我们表明，有本地化的不动点集，à la Atiyah-Bott。
Periodic Cyclic Homology and Equivariant Gerbes
本文是我们利用非对易几何中的机械建立等变扭曲$K$ -理论的de Rham模型的第一步。设$G$是紧李群，$M$是$G$光滑作用于其上的紧流形.对于任何$α\in H^3_G (M, {\mathbb Z})$，我们引入了局部等变扭曲上同调$H^\bullet ({\barΩ}^\bullet (M, G, L)_g, d^α_{G^g})$的概念，索引为$g\in G$。本文证明了存在一类自然链映射族，其标号为$g\in G$，从而从$HP^G_\bullet ( C^\infty (M, α) )$导出一族态射到$H^\bullet ( {\barΩ}^\bullet (M, G, L)_g, d^α_{G^g})$，其中$C^\infty (M, α)$是由$α\in H^3_G (M,{\mathbb Z} )$定义的等变丛格构造的光滑代数.对等变扭曲$K$ -理论提出了Atiyah-Hirzebruch型定理的一个猜想.
Shadows for bicategories, defined by Ponto, provide a useful framework that generalizes classical and topological Hochschild homology. In this paper, we define Hochschild-type invariants for monoids in a symmetric monoidal, simplicial model category $\mathsf V$, as well as for small $\mathsf V$-categories. We show that each of these constructions extends to a shadow on an appropriate bicategory, which implies in particular that they are Morita invariant. We also define a generalized theory of Hochschild homology twisted by an automorphism and show that it is Morita invariant. Hochschild homolo…
The Character Map in Twisted Equivariant Nonabelian Cohomology
本文提出了在扭曲等变非阿贝尔上同调理论中构造 Chern 特征映射类似物的一般方法，并以扭曲等变上同伦为示例，展望了其在推导 M5 膜上任意子量子态的应用。
Twisted actions on cohomologies and bimodules
本文引入单点等变上同调对 $L$-空间等变上同调的扭曲作用，构建双模结构并借此证明等变 Künneth 同构，进而应用于 Bott-Samelson 簇等变上同调的计算与双模态射的几何构造。
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 …
Twisted Orbifold K-Theory
我们用等变方法建立了轨道K-理论的基本性质。我们介绍的概念扭曲orbifold K-理论的存在下，离散扭转，并显示它可以明确计算的全球一致性。
Delocalized equivariant cohomology and resolution
一个精致的形式“民间定理”，一个光滑的行动，由一个紧凑的李群可以（规范）解决，通过迭代爆破，有独特的各向同性型建立了由作者在流形的背景下与角落;规范建设诱导纤维化的边界面上的决议，导致一个“等变的决议结构”。在这里，等变K-理论和Cartan模型的等变上同调跟踪下的决议程序，是离域等变上同调的鲍姆，Brylinski和麦克弗森。这导致了这些上同调理论中的每一个的解析模型，在相对对象的分辨率结构，因此，降阶模型作为平扭曲相对对象的分辨率的商。一个显式的等变陈省身字符，然后构建，基本上在非等变的情况下，在决议的商。
We introduce equivariant factorization homology, extending the axiomatic framework of Ayala-Francis to encompass multiplicative invariants of manifolds equipped with finite group actions. Examples of such equivariant factorization homology theories include Bredon equivariant homology and (twisted versions of) Hochschild homology. Our main result is that equivariant factorization homology satisfies an equivariant version of tensor excision, and is uniquely characterised by this property. We also discuss applications to representation theory, such as constructions of categorical braid group acti…
Equivariant cohomology and resolution
证明了在角点流形上紧李群的光滑作用可以通过迭代blow-up（正则）分解为具有唯一的各向同性型的“民间定理”.这个过程被证明是捕捉所有各向同性类型的同时分辨率在一个“分辨率塔”项目的决议，迭代边界纤维化的商。等变K-理论和Cartan模型的等变上同调跟踪下的决议程序是离域等变上同调的鲍姆，Brylinski和麦克弗森。这导致了这些上同调理论中的每一个的解析模型，在相对对象的分辨率塔，因此到降阶模型作为平扭曲相对对象的商的分辨率。其结果是一个明确的等变陈特征的构造，基本上在非等变的情况下，在决议的商。
Loop Groups and Twisted K-Theory II
这是第二次在一系列的文件调查之间的关系扭曲等变K-理论的一个紧凑的李群G和“Verlinde环”的循环组。我们引入与环群的正能量表示相关联的弗雷德霍姆算子的狄拉克族。它确定了一个从表示的同构类到扭K-理论的映射，证明了如果$G$与无扭基本群连通，则该映射是一个同构.我们还介绍了一个狄拉克家庭的有限维表示的紧李群，它是密切相关的Kirillov对应和等变托姆同构。在第三部分（数学。AT/0312155）中，我们将我们的主要定理的证明推广到任意紧李群G，并在各个方向提供补充。在第一部分（arXiv：0711.1906）中，我们发展了扭曲等变K理论，并进行了这里需要的一些计算。我们参考公告math.AT/0312155和math.AT/0206237以获得进一步的说明材料和动机。
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。
We prove that the parallel transport of a flat $n-1$-gerbe on any given target space gives rise to an $n$-dimensional extended homotopy quantum field theory. In case the target space is the classifying space of a finite group, we provide explicit formulae for this homotopy quantum field theory in terms of transgression. Moreover, we use the geometric theory of orbifolds to give a dimension-independent version of twisted and equivariant Dijkgraaf-Witten models. Finally, we introduce twisted equivariant Dijkgraaf-Witten theories giving us in the 3-2-1-dimensional case a new class of equivariant …
Given a finite group $G$ acting on a ring $R$, Merling constructed an equivariant algebraic $K$-theory $G$-spectrum, and work of Malkiewich and Merling, as well as work of Barwick, provides an interpretation of this construction as a spectral Mackey functor. This construction is powerful, but highly categorical; as a result the Mackey functors comprising the homotopy are not obvious from the construction and have therefore not yet been calculated. In this work, we provide a computation of the homotopy Mackey functors of equivariant algebraic $K$-theory in terms of a purely algebraic constructi…
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