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 – 2022-03-31
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| Project Status |
Granted (Fiscal Year 2020)
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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 |
During the support of the current JSPS Kakenhi funding, we have made several progress in computations of Donaldson-Thomas type invariants for Calabi-Yau 4-folds. For examples: (1) We used localization formulae to compute DT4 invariants for moduli spaces of one dimensional stable sheaves with descendent insertions and verified the conjectural relation with genus one GW invariants in several examples. (2) We used localization formulae to compute primary Pandharipande-Thomas stable pair invariants on local Calabi-Yau 4-folds and verified the conjectural formulae due to Cao-Maulik-Toda, (3) We used localization formulae to compute tautological invariants for moduli spaces of Le Potier stable pairs on local curves and verified our conjectural formula in many examples. (4) We found a K-theoretic localization formula and used it to compute K-theoretic Pandharipande-Thomas stable pair invariants and Hilbert scheme invariants on Calabi-Yau 4-folds. As a result of this computation, we found a K-theoretic DT/PT correspondence on toric Calabi-Yau 4-folds which generalises previous known DT/PT correspondence on toric Calabi-Yau manifolds of dimension three and four. The researches under the current JSPS support lead to the writing of six papers.
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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
We have made progress according to our original plan. The results we obtained had been written into 6 research papers (three are published in journals and three are under referee process).
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| Strategy for Future Research Activity |
We will keep on applying the powerful localization formulae to other moduli spaces in different examples. This will provide more verifications of our previous conjectures and possibly shed new light to directions we have not touched before.
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Report
(2 results)
Research Products
(22 results)
