2019 Fiscal Year Annual Research Report
Singular topological field theory and classifying spaces of derived manifolds
| Project/Area Number |
19K14522
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| Research Institution | The University of Tokyo |
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Principal Investigator |
Macpherson Andrew 東京大学, カブリ数物連携宇宙研究機構, 特任研究員 (90800574)
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| Project Period (FY) |
2019-04-01 – 2020-03-31
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| Keywords | Virtual cycles / topological field theory / bordism / derived geometry / orientations |
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| Outline of Annual Research Achievements |
In my research plan I proposed a project whose objective was to assign cycles of integration in bordism theory to derived manifolds with tangential structures (such as orientations, spin structures, and so on), enhancing the theory of "virtual cycles" developed for use in Gromov-Witten theory and also Spivak's bordism theory of unoriented derived manifolds. Results in this direction would be a stepping stone to defining "enhanced" Gromov-Witten type invariants, with many conceivable applications.
I succeeded in proving the first main statement outlined in the application, that is, that families of derived manifolds with tangential structures are classified by a well-known object, a "Thom spectrum." This means that if a moduli space can be given the structure of a derived manifold --- something which is the case in many important examples --- it can be assigned a cycle in a bordism ring, which can be thought of as a more structured version of the notion of "counting points." Hence, this work is likely to have ramifications in Floer theory, symplectic field theory, and beyond.
I spoke on my results at an international conference in Osaka in November. Since the grant terminated early, I did not have time to write up and publish the arguments, so the paper remains in draft form. I continue to work on it and aim to publish before the grant would have terminated given its full length, next February.
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