Asian Journal of Mathematics

On the Quantization of Polygon Spaces

L. Charles

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Moduli spaces of polygons have been studied since the nineties for their topological and symplectic properties. Under generic assumptions, these are symplectic manifolds with natural global action-angle coordinates. This paper is concerned with the quantization of these manifolds and of their action coordinates. Applying the geometric quantization procedure, one is lead to consider invariant subspaces of a tensor product of irreducible representations of $SU(2)$. These quantum spaces admit natural sets of commuting observables. We prove that these operators form a semi- classical integrable system, in the sense that they are Toeplitz operators with principal symbol the square of the action coordinates. As a consequence, the quantum spaces admit bases whose vectors concentrate on the Lagrangian submanifolds of constant action. The coefficients of the change of basis matrices can be estimated in terms of geometric quantities. We recover this way the already known asymptotics of the classical $6j$-symbols.

Article information

Asian J. Math., Volume 14, Number 1 (2010), 109-152.

First available in Project Euclid: 8 October 2010

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Zentralblatt MATH identifier

Primary: 47L80: Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.) 53D30: Symplectic structures of moduli spaces 53D12: Lagrangian submanifolds; Maslov index 53D50: Geometric quantization 53D20: Momentum maps; symplectic reduction 81S10: Geometry and quantization, symplectic methods [See also 53D50] 81S30: Phase-space methods including Wigner distributions, etc. 81R12: Relations with integrable systems [See also 17Bxx, 37J35] 81Q20: Semiclassical techniques, including WKB and Maslov methods

Polygon space geometric quantization Toeplitz operators Lagrangian section symplectic reduction $6j$-symbol canonical base


Charles, L. On the Quantization of Polygon Spaces. Asian J. Math. 14 (2010), no. 1, 109--152.

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