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
{"title":"Instanton counting and Donaldson–Thomas theory on toric Calabi–Yau four-orbifolds","authors":"Richard J. Szabo, Michelangelo Tirelli","doi":"10.4310/atmp.2023.v27.n6.a2","DOIUrl":null,"url":null,"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.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":"78 1","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Theoretical and Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.4310/atmp.2023.v27.n6.a2","RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
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。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
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.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信