PropositionStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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Period-lattice monodromy obstructs global action–angle coordinates

Statement

This item assumes ${A C}_{ω}$ , namely countable choice. In the propagated dependency chain, that assumption is required through Liouville–Arnold action–angle theorem; after those interfaces are fixed, the remaining local or finite argument makes no additional countable-family choice.

On the regular base of a compact Lagrangian torus fibration, local bases of the period lattice differ by matrices in $G L (n, Z)$ . Parallel transport therefore defines a monodromy representation $π_{1} (B) \to G L (n, Z)$ . Nontrivial monodromy obstructs global action–angle coordinates.

Facts & Assumptions

Given: ${A C}_{ω}$ , a regular compact connected torus fibration covered by the local action–angle charts of Liouville–Arnold.

[A1]

${A C}_{ω}$ is countable choice and is required here through Liouville–Arnold action–angle theorem; after those supplied interfaces are fixed, the remaining local or finite calculation makes no additional countable-family choice.

[F1]

Each local chart chooses a $Z$ -basis of the stabilizer lattice. Liouville–Arnold action–angle theorem.

Proof

technique · direct

1.1

On an overlap, two ordered period bases generate the same rank- $n$ lattice. Each is therefore an integer linear combination of the other, and the two change matrices are inverse integer matrices; hence the transition lies in $G L (n, Z)$ . Products of these transitions around loops give the monodromy representation.

A1F1givenalgebra

2.1

A single global angle system would label the same $n$ fundamental period loops on every fibre. Those labels would be a global basis of the lattice local system, so transport around every loop would return the basis unchanged. Therefore nonidentity monodromy rules out global action–angle coordinates. Trivial monodromy is only necessary: a global Lagrangian-section obstruction may remain.

F1step 1.1∎

Depends on

Used by

Dependency tree · two levels

17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Eckhard Meinrenken, Symplectic Geometry (standard reference, not scraped)