[Paper Review] Special Bohr - Sommerfeld geometry

Nikolai Andreevich Tyurin|arXiv (Cornell University)|Aug 27, 2015

Geometric and Algebraic Topology5 references3 citations

TL;DR

This paper introduces special Bohr-Sommerfeld (SBS) Lagrangian submanifolds—Lagrangian cycles satisfying both the Bohr-Sommerfeld condition and a new speciality condition tied to a holomorphic section—leading to a finite-dimensional moduli space. For a quadric threefold with line bundle O(1,1), the moduli space is isomorphic to ℂℙ³∖Q′, with a natural compactification to ℂℙ³, offering a new geometric framework for Mirror Symmetry via quantization-compatible structures.

ABSTRACT

We present a new approach to special lagrangian geometry which works for Bohr - Sommerfeld lagrangian submanifolds of symplectic manifolds with integer symplectic forms. This leads to construction of finite dimensional moduli spaces of SBS lagrangian cycles over algebraic varieties.

Motivation & Objective

• To resolve the infiniteness of Lagrangian moduli spaces in Mirror Symmetry by introducing a new finiteness condition.
• To unify Bohr-Sommerfeld and special Lagrangian geometry via a new speciality condition on prequantized Lagrangians.
• To construct finite-dimensional moduli spaces of Lagrangian submanifolds that are compatible with geometric quantization and mirror duality.
• To provide a geometric setting where quantum line bundles and connections can be naturally defined on moduli spaces, enabling further quantization.

Proposed method

• Define SBS Lagrangians as Bohr-Sommerfeld submanifolds where the restriction of a fixed section has constant argument relative to the covariantly constant section.
• Use prequantization data: a Hermitian line bundle L with curvature 2πiω to define the Bohr-Sommerfeld condition via covariantly constant sections.
• Construct the incidence cycle in ℙ(Γ(M,L)) × ℬ_S, where ℬ_S is the moduli space of Bohr-Sommerfeld Lagrangians of fixed topological type.
• Equip the moduli space of SBS cycles with a tensor product line bundle E = p₁*𝒪(1) ⊗ p₂*L, endowed with a hermitian connection A derived from curvature forms and U(1)-invariant horizontal lifts.
• Use a compatible complex structure and Riemannian metric to define a canonical horizontal distribution on the line bundle ℒ over ℬ_S, enabling a well-defined connection A_I.
• Investigate the curvature of the total connection 𝔸 on E to determine whether it is proportional to the Kähler form, a key condition for geometric quantization.

Experimental results

Research questions

• RQ1 Can a new finiteness condition be imposed on Lagrangian submanifolds to stabilize their moduli space in symplectic geometry?
• RQ2 How does the interplay between the Bohr-Sommerfeld condition and a section-based speciality condition constrain the geometry of Lagrangian cycles?
• RQ3 What is the structure of the moduli space of special Bohr-Sommerfeld Lagrangians for specific algebraic varieties like the quadric threefold?
• RQ4 Can the moduli space of SBS Lagrangians be naturally compactified, and what is its geometric structure?
• RQ5 Does the induced connection on the moduli space have curvature proportional to the Kähler form, enabling a full geometric quantization framework?

Key findings

• For the quadric threefold Q with O(1,1) and S ≅ S² of class (1,-1), the moduli space ℳ_SBS is isomorphic to ℂℙ³∖Q′, where Q′ is the projectively dual quadric.
• The moduli space ℳ_SBS admits a natural compactification isomorphic to ℂℙ³, indicating a complete geometric structure.
• The incidence cycle 𝒰_SBS is embedded in ℙ(Γ(M,L)) × ℬ_S, parameterizing pairs (s, S) where s is a section and S is an SBS Lagrangian.
• A natural line bundle ℒ over ℬ_S is constructed via direct image of the prequantum bundle along the incidence cycle, with a U(1)-invariant connection A_I.
• The total connection 𝔸 on the tensor product bundle E over ℳ_SBS combines curvature from 𝒪(1) and A_I, and its curvature is under investigation for proportionality to the Kähler form.
• The framework suggests a path toward geometric quantization on moduli spaces, provided the curvature of 𝔸 is proportional to the Kähler form, a condition currently under active investigation.