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On a complexification of the moduli space of Bohr - Sommerfeld lagrangian cycles
Symplectic Geometry · Submitted 30 Jul 2018
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim This implies that an appropriate section of the first map should define a complex structure on $T {\cal B}_S$; therefore it can be seen as a complexification of the moduli space ${\cal B}_S$.
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The abstract, sentence by sentence
In the previous papers we present a construction of the set ${\cal U}_{SBS}$ in the direct product ${\cal B}_S \times \mathbb{P} \Gamma (M, L)$ of the moduli space of Bohr - Sommerfeld lagrangian submanifolds of fixed topological type and the projectivized space of smooth sections of the prequantization bundle $L \to M$ over a given compact simply connected symplectic manifold $M$. Canonical projections $p: {\cal U}_{SBS} \to \mathbb{P} \Gamma (M, L)$ and $q: {\cal U}_{SBS} \to {\cal B}_S$ are studied in the present text: first, we show that the differential ${\rm d} p$ at a given point is an isomorphism, which implies that a natural complex structure can be defined on ${\cal U}_{SBS}$; second, the projection $q: {\cal U}_{SBS} \to {\cal B}_S$ splits as the combination ${\cal U}_{SBS} \to T {\cal B}_S \to {\cal B}_S$ such that the fibers of the first map are complex subsets in ${\cal U}_{SBS}$. This implies that an appropriate section of the first map should define a complex structure on $T {\cal B}_S$; therefore it can be seen as a complexification of the moduli space ${\cal B}_S$. The construction can be exploited in the Lagrangian approach to Geometric Quantization
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In the previous papers we present a construction of the set ${\cal U}_{SBS}$ in the direct product ${\cal B}_S \times \mathbb{P} \Gamma (M, L)$ of the moduli space of Bohr - Sommerfeld lagrangian submanifolds of fixed topological type and the projectivized space of smooth sections of the prequantization bundle $L \to M$ over a given compact simply connected symplectic manifold $M$. | background 0.95 | 0.42 | 0.06 | asserted 1.00 |
| s1 | Canonical projections $p: {\cal U}_{SBS} \to \mathbb{P} \Gamma (M, L)$ and $q: {\cal U}_{SBS} \to {\cal B}_S$ are studied in the present text: first, we show that the differential ${\rm d} p$ at a given point is an isomorphism, which implies that a natural complex structure can be defined on ${\cal U}_{SBS}$; second, the projection $q: {\cal U}_{SBS} \to {\cal B}_S$ splits as the combination ${\cal U}_{SBS} \to T {\cal B}_S \to {\cal B}_S$ such that the fibers of the first map are complex subsets in ${\cal U}_{SBS}$. | result 0.62 | 0.93 | 0.29 | asserted 1.00 |
| s2 | This implies that an appropriate section of the first map should define a complex structure on $T {\cal B}_S$; therefore it can be seen as a complexification of the moduli space ${\cal B}_S$. | interpretation 0.99 | 0.79 | 0.72 | asserted 0.52 |
| s3 | The construction can be exploited in the Lagrangian approach to Geometric Quantization | interpretation 0.72 | 0.31 | 0.10 | hedged possibility 0.88 |
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Original abstract
In the previous papers we present a construction of the set ${\cal U}_{SBS}$ in the direct product ${\cal B}_S \times \mathbb{P} \Gamma (M, L)$ of the moduli space of Bohr - Sommerfeld lagrangian submanifolds of fixed topological type and the projectivized space of smooth sections of the prequantization bundle $L \to M$ over a given compact simply connected symplectic manifold $M$. Canonical projections $p: {\cal U}_{SBS} \to \mathbb{P} \Gamma (M, L)$ and $q: {\cal U}_{SBS} \to {\cal B}_S$ are studied in the present text: first, we show that the differential ${\rm d} p$ at a given point is an isomorphism, which implies that a natural complex structure can be defined on ${\cal U}_{SBS}$; second, the projection $q: {\cal U}_{SBS} \to {\cal B}_S$ splits as the combination ${\cal U}_{SBS} \to T {\cal B}_S \to {\cal B}_S$ such that the fibers of the first map are complex subsets in ${\cal U}_{SBS}$. This implies that an appropriate section of the first map should define a complex structure on $T {\cal B}_S$; therefore it can be seen as a complexification of the moduli space ${\cal B}_S$. The construction can be exploited in the Lagrangian approach to Geometric Quantization
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