Quantization of bending deformations of polygons in double-struck E sign3, hypergeometric integrals and the Gassner representation

Abstract

The Hamiltonian potentials of the bending deformations of n-gons in double-struck E sign3 studied in [KM] and [Kly] give rise to a Hamiltonian action of the Malcev Lie algebra Pn of the pure braid group Pn on the moduli space Mr of n-gon linkages with the side-lengths r = (r1 , . . . , rn) in double-struck E sign3. If e ∈ Mr is a singular point we may linearize the vector fields in Pn at e. This linearization yields a flat connection ▽ on the space ℂn* of n distinct points on ℂ. We show that the monodromy of ▽ is the dual of a quotient of a specialized reduced Gassner representation.