发表机构
Dipartimento di Matematica, Informatica e Geoscienze, Università degli Studi di Trieste（的里雅斯特大学数学、信息学与地球科学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了ADE轨形[C^2/G]的C^*-等变上同调创消解猜想，通过McKay对应和特征标理论，将Bryan-Gholampour变换化为统一根系恒等式，并给出等变Chen-Ruan乘积的显式表示。
AI 中文摘要
设G是SL_2(C)的非平凡有限子群，p: X → C^2/G是极小消解。我们证明了ADE轨形[C^2/G]的C^*-等变上同调创消解猜想。更精确地说，在将量子参数特化到合适的单位根之后，我们证明了Bryan-Gholampour变换诱导了X的C^*-等变量子上同调与[C^2/G]的C^*-等变Chen-Ruan上同调之间的同构。我们的证明是直接的，基于McKay对应、ADE根系和特征标理论。遵循Bryan-Gholampour变量替换的特征标理论形式，我们使用McKay对应来对角化该变换，并将乘积的相容性归结为一个统一的根系恒等式。在A型中，这一归结允许离散傅里叶实现。在D型中，二元二面体群的循环子群则导致有限正弦变换，而例外情形E_6,E_7,E_8通过在相应分圆域上的精确矩阵计算来处理。作为预备结果，对于辛复向量空间V和Sp(V)的有限子群G，我们给出了[V/G]的C^*-等变Chen-Ruan乘积的显式表示，并将所得代数等同于群代数中心关于age滤过的Rees代数。特别地，等变Chen-Ruan代数在群代数中心与其伴随分次代数之间插值，后者恢复了通常的Chen-Ruan上同调。
英文摘要
Let G be a non-trivial finite subgroup of SL_2(C) and let p : X \to C^2/G be the minimal resolution. We prove the C^*-equivariant Cohomological Crepant Resolution Conjecture for the ADE orbifold [C^2/G]. More precisely, after specializing the quantum parameters at suitable roots of unity, we prove that the Bryan-Gholampour transformation induces an isomorphism between the C^*-equivariant quantum cohomology of X and the C^*-equivariant Chen-Ruan cohomology of [C^2/G]. Our proof is direct and is based on the McKay correspondence, ADE root systems, and character theory. Following the character-theoretic form of the Bryan-Gholampour change of variables, we use the McKay correspondence to diagonalize the transformation and to reduce the compatibility of the products to a uniform root-system identity. In type A, this reduction admits a discrete Fourier realization. In type D, the cyclic subgroup of the binary dihedral group leads instead to finite sine transforms, while the exceptional cases E_6,E_7,E_8 are treated by exact matrix computations over the corresponding cyclotomic fields. As a preliminary result, for a symplectic complex vector space V and a finite subgroup G of Sp(V), we give an explicit presentation of the C^*-equivariant Chen-Ruan product of [V/G] and identify the resulting algebra with the Rees algebra of the center of the group algebra with respect to the age filtration. In particular, the equivariant Chen-Ruan algebra interpolates between the center of the group algebra and its associated graded algebra, the latter recovering the ordinary Chen-Ruan cohomology.
Comments55 pages
