Adiabatic Limit, Theta Function, and Geometric Quantization

• Опубліковано в: : Symmetry, Integrability and Geometry: Methods and Applications
• Дата: 2024
• Автор: Yoshida, Takahiko
• Формат: Стаття
• Мова: Англійська
• Опубліковано: Інститут математики НАН України 2024
• Онлайн доступ: https://nasplib.isofts.kiev.ua/handle/123456789/212355
• 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

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.