Hochschild cohomology of quantized symplectic orbifolds and the Chen-Ruan cohomology

A Chen model for mapping spaces and the surface product

We develop a machinery of Chen iterated integrals for higher Hochschild complexes. These are complexes whose differentials are modeled on an arbitrary simplicial set much in the same way the ordinary Hochschild differential is modeled on the circle. We use these to give algebraic models for general mapping spaces and define and study the surface product operation on the homology of mapping spaces of surfaces of all genera into a manifold. This is an analogue of the loop product in string topology. As an application, we show this product is homotopy invariant. We prove Hochschild-Kostant-Rosenb…

Stringy K-theory and the Chern character

For a finite group G acting on a smooth projective variety X, we construct two new G-equivariant rings: first the stringy K-theory of X, and second the stringy cohomology of X. For a smooth Deligne-Mumford stack Y we also construct a new ring called the full orbifold K-theory of Y. For a global quotient Y=[X/G], the ring of G-invariants of the stringy K-theory of X is a subalgebra of the full orbifold K-theory of the the stack Y and is linearly isomorphic to the ``orbifold K-theory'' of Adem-Ruan (and hence Atiyah-Segal), but carries a different, ``quantum,'' product, which respects the natura…

开-闭 Hochschild 同调与相对圆盘映射空间

Open-Closed Hochschild Homology and the Relative Disk Mapping Space

本文建立 Chen 自由环路空间迭代积分模型的相对版本，利用开-闭同伦代数与开-闭 Hochschild 同调为相对圆盘映射空间构造模型，并在特定条件下证明其为拟同构，推广了 Chen 及 Getzler-Jones 的经典结果。

缠结的 Chen-Khovanov 谱

Chen-Khovanov spectra for tangles

本文指出 Khovanov 弧代数与缠结不变量的稳定同伦细化可诱导 Chen-Khovanov 和 Stroppel 平台代数及缠结不变量的细化，并研究其拓扑 Hochschild 同调。

Quantization, orbifold cohomology, and Cherednik algebras

We compute the Hochschild homology of the crossed product $\Bbb C[S_n]\ltimes A^{\otimes n}$ in terms of the Hochschild homology of the associative algebra $A$ (over $\Bbb C$). It allows us to compute the Hochschild (co)homology of $\Bbb C[W]\ltimes A^{\otimes n}$ where $A$ is the $q$-Weyl algebra or any its degeneration and $W$ is the Weyl group of type $A_{n-1}$ or $B_n$. For a deformation quantization $A_+$ of an affine symplectic variety $X$ we show that the Hochschild homology of $S^n A$, $A=A_+[\hbar^{-1}]$ is additively isomorphic to the Chen-Ruan orbifold cohomology of $S^nX$ with coef…

Trace Densities and Algebraic Index Theorems for Sheaves of Formal Cherednik Algebras

We show how a novel construction of the sheaf of Cherednik algebras on a quotient orbifold Y=X/G by virtue of formal geometry in author's prior work leads to results for the sheaf of Cherednik algebra which until recently were viewed as intractable. First, for every orbit type stratum in $X$, we define a trace density map for the Hochschild chain complex of the sheaf of Cherednik algebras, which generalises the standard Engeli-Felder's trace density construction for the sheaf of differential operators. Second, by means of the newly obtained trace density maps, we prove an isomorphism in the de…