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

http://dx.doi.org/10.4153/CMB-2001-006-3
Canad. Math. Bull. 44(2001), 36-60

  Published:2001-03-01
 Printed: Mar 2001

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.