2019 Fiscal Year Final Research Report
Index theory for Dirac-type operators using Witten's deformation and its applications
| Project/Area Number |
15K04857
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Meiji University |
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Principal Investigator |
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| Project Period (FY) |
2015-04-01 – 2020-03-31
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| Keywords | 幾何学的量子化 / Lagrangeファイバー束 / Spin-c Dirac量子化 / 実量子化 / 断熱極限 |
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| Outline of Final Research Achievements |
In the geometric quantization by Kostant-Souriau, one temporary needs a geometric structure, polarization, to construct the quantum Hilbert space. It is the fundamental problem whether the obtained result depends on the choice of polarizations. A recent research reveals, on some examples, there exists a one-parameter family of Kahler polarization which converges to the real polarization.
Any symplectic manifold has many possibly non-integrable almost complex structures, and for them, one has a generalization of the Kahler quantization, called the Spin-c quantization. In this work, for a Lagrange fiber bundle on a complete base in the sense of affine geometry, one obtained an orthogonal family of sections of the prequantum line bundle indexed by Bohr-Sommerfeld points which satisfies, under the adiabatic limit: (i) the image of each section by the Spin-c Dirac operator converges to 0, (ii) each section converges to the delta-section supported on the corresponding Bohr-Sommerfeld fiber.
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| Free Research Field |
シンプレクティック幾何学
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| Academic Significance and Societal Importance of the Research Achievements |
シンプレクティック多様体の量子化には様々な方法が知られており，それらの間の関係性を調べることは基本的な問題である．これについて，Kahler偏極と実偏極の関係は最近の研究によって明らかになりつつあるが，概複素構造が可積分でない場合（つまり，Spin-c量子化）については，Hilbert空間の次元の一致以外のことは分かっていなかった．本研究では，先行研究を一般化し，概複素構造が可積分でない場合にも，Spin-c量子化の断熱極限として実偏極を用いた幾何学的量子化が現れることを示した点に学術的意義がある．
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