Instanton counting and Donaldson–Thomas theory on toric Calabi–Yau four-orbifolds

IF 1 4区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL
Richard J. Szabo, Michelangelo Tirelli
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引用次数: 0

Abstract

We study rank $r$ cohomological Donaldson–Thomas theory on a toric Calabi–Yau orbifold of $\mathbb{C}^4$ by a finite abelian subgroup $\Gamma$ of $\mathsf{SU}(4)$, from the perspective of instanton counting in cohomological gauge theory on a noncommutative crepant resolution of the quotient singularity. We describe the moduli space of noncommutative instantons on $\mathbb{C}^4 / \Gamma$ and its generalized ADHM parametrization. Using toric localization, we compute the orbifold instanton partition function as a combinatorial series over $r$-vectors of $\Gamma$-coloured solid partitions. When the $\Gamma$-action fixes an affine line in $\mathbb{C}^4$, we exhibit the dimensional reduction to rank $r$ Donaldson–Thomas theory on the toric Kähler three-orbifold $\mathbb{C}^3 / \Gamma$. Based on this reduction and explicit calculations, we conjecture closed infinite product formulas, in terms of generalized MacMahon functions, for the instanton partition functions on the orbifolds $\mathbb{C}^2 / \mathbb{Z}_n \times \mathbb{C}^2$ and $\mathbb{C}^3 / (\mathbb{Z}_2 \times \mathbb{Z}_2) \times \mathbb{C}$, finding perfect agreement with new mathematical results of Cao, Kool and Monavari.
环状 Calabi-Yau 四orbifold 上的瞬子计数和唐纳森-托马斯理论
我们从共计量理论中的瞬子计数角度出发,研究了$\mathsf{SU}(4)$的有限无性子群$\Gamma$的$\mathbb{C}^4$的环状卡拉比-约轨道上的秩$r$共计量唐纳森-托马斯理论。我们描述了 $\mathbb{C}^4 / \Gamma$ 上的非交换瞬子模态空间及其广义 ADHM 参数化。利用环定位,我们计算了作为$r$-向量的$\Gamma$彩色实体分区的组合数列的轨道瞬子分区函数。当$\Gamma$作用固定了$\mathbb{C}^4$中的仿射线时,我们展示了环状凯勒三轨道$\mathbb{C}^3 / \Gamma$上秩为$r$的唐纳森-托马斯(Donaldson-Thomas)理论的维度还原。基于这种还原和显式计算,我们用广义麦克马洪函数来猜想封闭的无限乘积公式、和 $\mathbb{C}^3 / (\mathbb{Z}_2 \times \mathbb{Z}_2) \times \mathbb{C}$上的瞬子分区函数,发现与曹(Cao)、库尔(Kool)和莫纳瓦里(Monavari)的新数学结果完全一致。
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来源期刊
Advances in Theoretical and Mathematical Physics
Advances in Theoretical and Mathematical Physics 物理-物理:粒子与场物理
CiteScore
2.20
自引率
6.70%
发文量
0
审稿时长
>12 weeks
期刊介绍: Advances in Theoretical and Mathematical Physics is a bimonthly publication of the International Press, publishing papers on all areas in which theoretical physics and mathematics interact with each other.
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