Adiabatic Limit, Theta Function, and Geometric Quantization
Let π : (, ) → be a non-singular Lagrangian torus fibration on a complete base with prequantum line bundle (, ∇ᴸ) → (, ). Compactness on is not assumed. For a positive integer and a compatible almost complex structure on (, ) invariant along the fiber of π, let be the associated Spinᶜ Dirac op...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2024 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2024
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/212355 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Adiabatic Limit, Theta Function, and Geometric Quantization. Takahiko Yoshida. SIGMA 20 (2024), 065, 52 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862684624678813696 |
|---|---|
| author | Yoshida, Takahiko |
| author_facet | Yoshida, Takahiko |
| citation_txt | Adiabatic Limit, Theta Function, and Geometric Quantization. Takahiko Yoshida. SIGMA 20 (2024), 065, 52 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Let π : (, ) → be a non-singular Lagrangian torus fibration on a complete base with prequantum line bundle (, ∇ᴸ) → (, ). Compactness on is not assumed. For a positive integer and a compatible almost complex structure on (, ) invariant along the fiber of π, let be the associated Spinᶜ Dirac operator with coefficients in ⊗ᴺ. First, in the case where is integrable, under certain technical conditions on , we give a complete orthogonal system {ϑb}b ∈ BS of the space of holomorphic ²-sections of ⊗ᴺ indexed by the Bohr-Sommerfeld points BS such that each ϑb converges to a delta-function section supported on the corresponding Bohr-Sommerfeld fiber π⁻¹(b) by the adiabatic(-type) limit. We also explain the relation of ϑb with Jacobi's theta functions when (, ) is ²ⁿ. Second, in the case where is not integrable, we give an orthogonal family {ϑ~b}b ∈ BS of ²-sections of ⊗ᴺ indexed by BS which has the same property as above, and show that each ϑ~b converges to 0 by the adiabatic(-type) limit with respect to the ²-norm.
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| first_indexed | 2026-03-17T10:05:36Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-212355 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-17T10:05:36Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Yoshida, Takahiko 2026-02-05T09:56:19Z 2024 Adiabatic Limit, Theta Function, and Geometric Quantization. Takahiko Yoshida. SIGMA 20 (2024), 065, 52 pages 1815-0659 2020 Mathematics Subject Classification: 53D50; 58H15; 58J05 arXiv:1904.04076 https://nasplib.isofts.kiev.ua/handle/123456789/212355 https://doi.org/10.3842/SIGMA.2024.065 Let π : (, ) → be a non-singular Lagrangian torus fibration on a complete base with prequantum line bundle (, ∇ᴸ) → (, ). Compactness on is not assumed. For a positive integer and a compatible almost complex structure on (, ) invariant along the fiber of π, let be the associated Spinᶜ Dirac operator with coefficients in ⊗ᴺ. First, in the case where is integrable, under certain technical conditions on , we give a complete orthogonal system {ϑb}b ∈ BS of the space of holomorphic ²-sections of ⊗ᴺ indexed by the Bohr-Sommerfeld points BS such that each ϑb converges to a delta-function section supported on the corresponding Bohr-Sommerfeld fiber π⁻¹(b) by the adiabatic(-type) limit. We also explain the relation of ϑb with Jacobi's theta functions when (, ) is ²ⁿ. Second, in the case where is not integrable, we give an orthogonal family {ϑ~b}b ∈ BS of ²-sections of ⊗ᴺ indexed by BS which has the same property as above, and show that each ϑ~b converges to 0 by the adiabatic(-type) limit with respect to the ²-norm. The many part of this work were done while the author stayed at McMaster University. The author would like to thank the Department of Mathematics and Statistics, McMaster university and especially Megumi Harada for their hospitality. The author would also like to express our sincere gratitude to the referees who carefully read the manuscript and helped him improve it. This work is supported by Grant-in-Aid for Scientific Research (C) 15K04857 and 19K03479. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Adiabatic Limit, Theta Function, and Geometric Quantization Article published earlier |
| spellingShingle | Adiabatic Limit, Theta Function, and Geometric Quantization Yoshida, Takahiko |
| title | Adiabatic Limit, Theta Function, and Geometric Quantization |
| title_full | Adiabatic Limit, Theta Function, and Geometric Quantization |
| title_fullStr | Adiabatic Limit, Theta Function, and Geometric Quantization |
| title_full_unstemmed | Adiabatic Limit, Theta Function, and Geometric Quantization |
| title_short | Adiabatic Limit, Theta Function, and Geometric Quantization |
| title_sort | adiabatic limit, theta function, and geometric quantization |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212355 |
| work_keys_str_mv | AT yoshidatakahiko adiabaticlimitthetafunctionandgeometricquantization |