{"title":"环状 Calabi-Yau 四orbifold 上的瞬子计数和唐纳森-托马斯理论","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":"{\"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}","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}
Instanton counting and Donaldson–Thomas theory on toric Calabi–Yau four-orbifolds
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.
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.