Localization formulae in Donaldson-Thomas theory of Calabi-Yau 4-folds
| Project/Area Number |
19K23397
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| Research Category |
Grant-in-Aid for Research Activity Start-up
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| Allocation Type | Multi-year Fund |
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| Review Section |
0201:Algebra, geometry, analysis, applied mathematics,and related fields
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| Research Institution | The University of Tokyo |
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Principal Investigator |
Cao Yalong 東京大学, カブリ数物連携宇宙研究機構, 特任研究員 (80791459)
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| Project Period (FY) |
2019-08-30 – 2021-03-31
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| Project Status |
Granted (Fiscal Year 2019)
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| Budget Amount *help |
¥2,470,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥570,000)
Fiscal Year 2020: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2019: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
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| Keywords | localization formulae / Donaldson-Thomas theory / Calabi-Yau 4-folds |
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| Outline of Research at the Start |
In 2013-2015, the applicant with Leung and Borisov with Joyce made progress on the theory of Donaldson-Thomas invariants on Calabi-Yau 4-folds. Computations of such invariants are in general very difficult. The proposal aims to develop localization formulae to effectively compute such invariants when there are torus actions on the corresponding moduli spaces. In particular, we aim to compute DT4 invariants for: (1) moduli spaces of stable pairs, (2) Hilbert schemes.
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| Outline of Annual Research Achievements |
Under the support of my JSPS KAKENHI Grant Number JP19K23397, I have been making progress towards the research aims of the planned proposal. With my collaborators, we use localization formulae to compute Donaldson-Thomas invariants of Calabi-Yau 4-folds in several different settings. The output is that I wrote three research papers (with my collaborators) since September of 2019. To be specific, I wrote a paper "Curve counting via stable objects in derived categories of Calabi-Yau 4-folds" with Yukinobu Toda, where we defined the category of D0-D2-D8 bounded states and used it to give a wall-crossing explanation of the Gopakumar-Vafa type formula for stable pair invariants of Calabi-Yau 4-folds proposed by Maulik-Toda and myself. I wrote another paper with Yukinobu Toda, named "Gopakumar-Vafa type invariants on Calabi-Yau 4-folds via descendent insertions", where we gave a sheaf theoretic interpretation of Gopakumar-Vafa type invariants of Calabi-Yau 4-folds defined by Klemm-Pandharipande using Donaldson-Thomas invariants of one dimensional stable sheaves with descendent insertions. The third paper "Stable pair invariants of local Calabi-Yau 4-folds" is with Martijn Kool and Sergej Monavari from Utrecht University, the Netherlands, where we computed stable pair invariants by several methods, including using localization formulae to verify a conjecture of Maulik-Toda and myself.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
I have been fortunately studying Donaldson-Thomas type theory on Calabi-Yau 4-folds since my PhD period and I have a lot of experiences in this subject. The plan in the original proposal makes perfect sense. With the generous support from JSPS, we therefore make progress smoothly.
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| Strategy for Future Research Activity |
The plan will go exactly as we wrote in the original JSPS proposal. Currently I have two ongoing projects with several different people, we will apply localization formulae to compute Donaldson-Thomas invariants of Calabi-Yau 4-folds in those cases.
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Report
(1 results)
Research Products
(10 results)
