Equivariant localization in factorization homology and applications in mathematical physics I: Foundations

Equivariant homology theory and twisted Yangian

We study the convolution algebra $H^{G\times \CC^{*}}_{*}(Z)$ of $G$-equivariant homology group on the Steinberg variety of type B/C and define an algebra $\widetilde{Y}$ that maps to $H^{G\times \CC^{*}}_{*}(Z)$. The Drinfeld new realization of the twisted Yangian associated to symmetric pairs is a quotient of $\widetilde{Y}$. We also study the $G$-equivariant case and prove that the twisted Yangian is the deformation of the twisted current algebra.

等变向量场的同调

Homology of equivariant vector fields

设K是紧李群。计算光滑K-流形X上等变向量场李代数的阿贝尔化。我们还计算了层保持光滑向量场的李代数在商X/K上的阿贝尔化。

Equivariant Tilting Modules, Pfaffian Varieties and Noncommutative Matrix Factorizations

We show that equivariant tilting modules over equivariant algebras induce equivalences of derived factorization categories. As an application, we show that the derived category of a noncommutative resolution of a linear section of a Pfaffian variety is equivalent to the derived factorization category of a noncommutative gauged Landau-Ginzburg model $(Λ,χ, w)^{\mathbb{G}_m}$, where $Λ$ is a noncommutative resolution of the quotient singularity $W/\operatorname{GSp}(Q)$ arising from a certain representation $W$ of the symplectic similitude group $\operatorname{GSp}(Q)$ of a symplectic vector spa…

Lectures on Factorization Homology, Infinity-Categories, and Topological Field Theories

These are notes from an informal mini-course on factorization homology, infinity-categories, and topological field theories. The target audience was imagined to be graduate students who are not homotopy theorists.

仿射格拉斯曼流形的等变 $K$-同调与 $K$-理论双重 $k$-Schur 函数

Equivariant $K$-homology of affine Grassmannian and $K$-theoretic double $k$-Schur functions

本文研究仿射格拉斯曼流形的环面等变 $K$-同调环，在 $A$ 型中引入 $K$-理论双重 $k$-Schur 函数作为 Schubert 基，并给出 $\mathrm{Gr}_{{SL}_n}$ 的等变 $K$-同调环的 Ginzburg-Peterson 型实现。

Equivariant Matrix Factorizations and Hamiltonian reduction

Let $X$ be a smooth scheme with an action of an algebraic group $G$. We establish an equivalence of two categories related to the corresponding moment map $μ: T^*X \to Lie(G)^*$ - the derived category of G-equivariant coherent sheaves on the derived fiber $μ^{-1}(0)$ and the derived category of $G$-equivariant matrix factorizations on $T^*X \times Lie(G)$ with potential given by $μ$.