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Quantization of Bending Deformations of Polygons In $\mathbb{E}^3$, Hypergeometric Integrals and the Gassner Representation

 Printed: Mar 2001
  • Michael Kapovich
  • John J. Millson
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The Hamiltonian potentials of the bending deformations of $n$-gons in $\E^3$ studied in \cite{KM} and \cite{Kl} give rise to a Hamiltonian action of the Malcev Lie algebra $\p_n$ of the pure braid group $P_n$ on the moduli space $M_r$ of $n$-gon linkages with the side-lengths $r= (r_1,\dots, r_n)$ in $\E^3$. If $e\in M_r$ is a singular point we may linearize the vector fields in $\p_n$ at $e$. This linearization yields a flat connection $\nabla$ on the space $\C^n_*$ of $n$ distinct points on $\C$. We show that the monodromy of $\nabla$ is the dual of a quotient of a specialized reduced Gassner representation.
MSC Classifications: 53D30, 53D50 show english descriptions Symplectic structures of moduli spaces
Geometric quantization
53D30 - Symplectic structures of moduli spaces
53D50 - Geometric quantization

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