2019 Fiscal Year Research-status Report
Localization formulae in Donaldson-Thomas theory of Calabi-Yau 4-folds
| Project/Area Number |
19K23397
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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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| Keywords | localization formulae / Donaldson-Thomas theory / Calabi-Yau 4-folds |
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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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| Causes of Carryover |
This is because of the outbreak of Coronavirus virus 2019, I have to cancel many visits, meetings and workshops from Feb. 2020 until today.
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