Crepant resolutions and brane tilings II: Tilting bundles

On the Abuaf-Ueda Flop via Non-Commutative Crepant Resolutions

The Abuaf-Ueda flop is a 7-dimensional flop related to $G_2$ homogeneous spaces. The derived equivalence for this flop was first proved by Ueda using mutations of semi-orthogonal decompositions. In this article, we give an alternative proof for the derived equivalence using tilting bundles. Our proof also shows the existence of a non-commutative crepant resolution of the singularity appearing in the flopping contraction. We also give some results on moduli spaces of finite-length modules over this non-commutative crepant resolution.

膜镶嵌中的倾斜突变与箭图不变对偶性

Tilting Mutations and Quiver-Invariant Dualities in Brane Tilings

该研究针对H1121模型，证明箭图不变对偶性可通过膜镶嵌的倾斜突变实现，发现三类倾斜突变，给出相关对偶性实例并验证其对应同一Calabi-Yau 3-流形。

关于 Abuaf 翻转的导出等价：非交换crepant解消的变异与球面扭转

On derived equivalence for Abuaf flop: mutation of non-commutative crepant resolutions and spherical twists

本文研究Abuaf 5重翻转的导出等价，利用Toda-Uehara倾斜层和Iyama-Wemyss变异函子补充了Segal的结果，并观察到“翻转-翻转=扭转”和“多重变异=扭转”的结论。

通过倾转包证明类型$G_2^{\dagger}$的简单翻转的导出等价性

Derived equivalence for the simple flop of type $G_2^{\dagger}$ via tilting bundles

本文通过倾转包证明了类型$G_2^{\dagger}$简单翻转的局部模型的导出等价性，并给出了非交换crepant解析，该解析与翻转两侧导出等价。

Lefschetz decompositions and Categorical resolutions of singularities

Let $Y$ be a singular algebraic variety and let $\TY$ be a resolution of singularities of $Y$. Assume that the exceptional locus of $\TY$ over $Y$ is an irreducible divisor $\TZ$ in $\TY$. For every Lefschetz decomposition of $\TZ$ we construct a triangulated subcategory $\TD \subset \D^b(\TY)$ which gives a desingularization of $\D^b(Y)$. If the Lefschetz decomposition is generated by a vector bundle tilting over $Y$ then $\TD$ is a noncommutative resolution, and if the Lefschetz decomposition is rectangular, then $\TD$ is a crepant resolution.

吸引子不变量、膜砖与晶体

Attractor invariants, brane tilings and crystals

本文研究环面卡拉比-丘三维流形上的吸引子 DT 不变量，利用膜砖与带势箭图方法进行显式计算，预测了所有吸引子不变量的值，并验证其与熔融晶体计数的一致性。

Brane Tilings and Non-Commutative Geometry

We derive the quiver gauge theory on the world-volume of D3-branes transverse to an L(a,b,c) singularity by computing the endomorphism algebra of a tilting object first constructed by Van den Bergh. The quiver gauge theory can be concisely specified by an embedding of a graph into a face-centered cubic lattice. In this description, planar Seiberg dualities of the gauge theory act by changing the graph embedding. We use this description of Seiberg duality to show these quiver gauge theories possess periodic Seiberg dualities whose existence was expected from the AdS/CFT correspondence.

Segre积与遗传表示型Cohen-Macaulay环的非交换消解

Non-commutative resolutions for Segre products and Cohen-Macaulay rings of hereditary representation type

本文利用倾斜理论与丛倾斜理论研究遗传表示型Cohen-Macaulay环，证明Segre积上丛倾斜模或非交换crepant消解的存在性定理，构造多个Gorenstein环例子，并对一个4维非Gorenstein环给出刚性Cohen-Macaulay模的完全分类。

弱 del Pezzo 曲面由 $2$-倾斜丛的存在性刻画

Weak del Pezzo surfaces are characterized by the existence of $2$-tilting bundles

本文证明光滑真 $d$ 维簇上存在 $d$-倾斜丛时该簇必为弱 Fano 簇，并在二维情形证明反方向：代数闭域上的弱 del Pezzo 曲面存在 $2$-倾斜丛，从而给出 Daniel Chan 猜想在曲面情形的肯定回答。

膜编织中的图不变对偶性

Quiver-Invariant Dualities between Brane Tilings

研究共享相同真空模空间和 chiral 字段内容的4d N=1超对称规范理论对，通过膜编织图的单个翻转突变实现对偶性。

非交换crepant解析的三维孤立奇点上的稳定性条件

Stability conditions on noncommutative crepant resolutions of 3-dimensional isolated singularities

研究非交换crepant解析的三维孤立奇点的稳定性条件，通过构造突变锥并分析突变过程，证明突变锥的性质与最大修改模之间的联系，进而描述稳定性条件下的自同构群。

高表示无限代数与Picard数一或二的环面Fano栈

Higher representation infinite algebras and toric Fano stacks of Picard number one or two

本文证明了Picard数一或二的光滑环面Fano栈上由线丛构成的d-倾斜丛的存在性并进行了分类，其自同态代数给出了d-表示无限代数的自然例子，并建立了与dimer模型的联系。