[Paper Review] Adiabatic Limit, Theta Function, and Geometric Quantization

Takahiko Yoshida|arXiv (Cornell University)|Apr 8, 2019

Geometry and complex manifolds32 references4 citations

TL;DR

This paper studies the adiabatic limit of Spin c Dirac operators on Lagrangian torus fibrations with prequantum line bundles, establishing orthogonal families of $L^2$-sections indexed by Bohr-Sommerfeld points. In the integrable case, these sections converge to delta-functions on Bohr-Sommerfeld fibers and relate to Jacobi theta functions; in the non-integrable case, the sections remain orthogonal and their Dirac operator norms vanish in the adiabatic limit.

ABSTRACT

Let $π\colon (M,ω) o B$ be a non-singular Lagrangian torus fibration on a complete base $B$ with prequantum line bundle $\bigl(L, abla^L\bigr) o (M,ω)$. Compactness on $M$ is not assumed. For a positive integer $N$ and a compatible almost complex structure $J$ on $(M,ω)$ invariant along the fiber of $π$, let $D$ be the associated Spin${}^c$ Dirac operator with coefficients in $L^{\otimes N}$. First, in the case where $J$ is integrable, under certain technical condition on $J$, we give a complete orthogonal system $\{ \vartheta_b\}_{b\in B_{ m BS}}$ of the space of holomorphic $L^2$-sections of $L^{\otimes N}$ indexed by the Bohr-Sommerfeld points $B_{ m BS}$ such that each $\vartheta_b$ 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 $\vartheta_b$ with Jacobi's theta functions when $(M,ω)$ is $T^{2n}$. Second, in the case where $J$ is not integrable, we give an orthogonal family $\big\{ { ilde \vartheta}_b\big\}_{b\in B_{ m BS}}$ of $L^2$-sections of $L^{\otimes N}$ indexed by $B_{ m BS}$ which has the same property as above, and show that each $D{ ilde \vartheta}_b$ converges to $0$ by the adiabatic(-type) limit with respect to the $L^2$-norm.

Motivation & Objective

• To understand the relationship between geometric quantization and real quantization via the adiabatic limit in non-compact Lagrangian torus fibrations.
• To construct orthogonal families of $L^2$-sections of $L^{ imes N}$ indexed by Bohr-Sommerfeld points in both integrable and non-integrable almost complex structures.
• To analyze the asymptotic behavior of these sections and their Dirac operator action under the adiabatic limit.
• To clarify the connection between the constructed sections and Jacobi theta functions when the manifold is a torus $T^{2n}$.

Proposed method

• Construct a compatible almost complex structure $J$ on $(M, ho)$ invariant along the fibers of the Lagrangian fibration $\pi: M \to B$.
• Define the Spin c Dirac operator $D$ with coefficients in $L^{igotimes N}$, and study its action on $L^2$-sections.
• For the integrable case, define sections $\vartheta_b$ using local holomorphic data and show they form a complete orthogonal system.
• For the non-integrable case, define $\widetilde{\vartheta}_b$ as an orthogonal family with similar asymptotic properties.
• Use the adiabatic-type limit to analyze the convergence of $\vartheta_b$ to delta-functions and $D\widetilde{\vartheta}_b \to 0$ in $L^2$-norm.
• Leverage integral affine structures and the lifting of monodromy actions to the prequantum line bundle to ensure consistency of the construction.

Experimental results

Research questions

• RQ1 How do the holomorphic $L^2$-sections of $L^{igotimes N}$ behave under the adiabatic limit in the integrable case?
• RQ2 What is the precise relation between the constructed sections $\vartheta_b$ and Jacobi theta functions when $M = T^{2n}$?
• RQ3 Can one construct an orthogonal family of $L^2$-sections in the non-integrable case that still exhibits the same adiabatic limit behavior?
• RQ4 Does the Dirac operator $D$ applied to the non-integrable sections $\widetilde{\vartheta}_b$ converge to zero in the $L^2$-norm under the adiabatic limit?
• RQ5 Under what conditions on the almost complex structure $J$ does the space of degree-zero harmonic spinors remain non-trivial?

Key findings

• In the integrable case, a complete orthogonal system $\{\vartheta_b\}_{b \in B_{BS}}$ of holomorphic $L^2$-sections of $L^{\otimes N}$ is constructed, indexed by Bohr-Sommerfeld points.
• Each $\vartheta_b$ converges to a delta-function section supported on the Bohr-Sommerfeld fiber $\pi^{-1}(b)$ in the adiabatic limit.
• When $M = T^{2n}$, the sections $\vartheta_b$ are explicitly related to Jacobi's theta functions.
• In the non-integrable case, an orthogonal family $\{\widetilde{\vartheta}_b\}_{b \in B_{BS}}$ of $L^2$-sections is constructed with the same asymptotic convergence to delta-functions.
• For the non-integrable case, $\|D\widetilde{\vartheta}_b\|_{L^2} \to 0$ as the adiabatic limit is taken.
• The construction relies on a technical condition ensuring the existence of non-trivial degree-zero harmonic spinors, which is satisfied under $\Gamma$-equivariance and integrability assumptions.