Quantization of bending deformations of polygons in \(\mathbb{E}^3\), hypergeometric integrals and the Gassner representation (Q2739250)

In [\textit{M. Kapovich} and \textit{J. J. Millson}, J. Differ. Geom. 44, No. 3, 479-513 (1996; Zbl 0889.58017) and \textit{A. Klyachko}, Algebraic geometry appl., Yaroslavl, 1992, 67-84 (1994; Zbl 0820.51016)], certain Hamiltonian flows on the moduli space \(M_r\) of \(n\)-gon linkages in \(\mathbb{E}^3\) were studied. There, flows were interpreted geometrically and called bending deformations of polygons. Klyachko pointed out that the Hamiltonian potentials of the bending deformations give rise to a Hamiltonian action of \(\mathcal{P}_n\), the Malcev Lie algebra of the pure braid group \(P_n\) on \(M_r\). It is a remarkable fact, that a representation \(\rho:\mathcal{P}_n\to \text{End}(V)\) gives rise to a flat connection \(\nabla\) in the vector bundle \(\mathbb{C}^n_*\times V\) over \(\mathbb{C}^n_*\), the space of distinct points in \(\mathbb{C}\). Accordingly, the monodromy representation of \(\nabla\) yields a representation \(\widehat{\rho}:P_n\to \text{Aut}(V)\). NEWLINENEWLINENEWLINEIn this paper the representation \(\widehat{\rho}_{\varepsilon,r}:P_n\to \text{Aut}(T_{\varepsilon,r})\) associated to a degenerate \(n\)-gon \(P\) (i.e. an \(n\)-gon which is contained in a line \(L\)) is computed. Here \(T_{\varepsilon,r}=T_P(M_r)\), \(\varepsilon=(\varepsilon_1,\dots ,\varepsilon_n)\), \(\varepsilon_i=\pm 1\), and \(r=(r_1,\dots ,r_n)\), \(r_i\in \mathbb{R}_{+}\) are defined as follows. Fix an orientation on \(L\). The number \(r_i\) is the length of the \(i\)-th edge of \(P\). Define \(\varepsilon_i\) to be +1 if the \(i\)-th edge is positively oriented and \(\varepsilon_i=-1\) otherwise. NEWLINENEWLINENEWLINETheorem 1. There is a \(\mathcal{P}_n\)-invariant almost complex structure \(J^{\varepsilon}\) on \(T_{\varepsilon,r}\) such that there is an isomorphism of \(\mathcal{P}_n\)-modules \(T_{\varepsilon,r}^{1,0}\cong W_{\lambda}\) for \(\lambda=(\sqrt{-1}\varepsilon_1r_1,\dots ,\sqrt{-1}\varepsilon_nr_n)\). NEWLINENEWLINENEWLINELet \(\gamma_j\), \(1\leq j\leq n\), be the free generators of the free group \(\mathbb{F}_n\). Define the character \(\chi:\mathbb{F}_n\to \mathbb{C}^*\) by \(\chi(\gamma_j)=e^{2\pi i \lambda_j}\). Let \(\mathbb{C}_{\chi^{-1}}\) be the 1-dimensional module in which the free group \(\mathbb{F}_n\) acts by \(\chi^{-1}\). The pure braid group \(P_n\) acts by automorphisms on \(\mathbb{F}_n\) so that the character \(\chi\) is fixed. Thus we have the associated action of \(P_n\) on \(H_1(\mathbb{F}_n,\mathbb{C}_{\chi^{-1}})\). Let \(\Gamma_n=\pi_1(\mathbb{CP}^1-\{z_1,\dots ,z_n\})\). Hence \(\Gamma_n\) is the quotient of \(\mathbb{F}_n\) by the normal subgroup generated by \(\gamma_1\dots\gamma_n\). Since \(\chi(\gamma_1\dots\gamma_n)=1\), the character \(\chi\) is induced by a character of \(\Gamma_n\). The group \(P_n\) fixes \(\gamma_1\dots\gamma_n\) and consequently acts on \(\Gamma_n\) and on \(H_1(\Gamma_n,\mathbb{C}_{\chi^{-1}})\). NEWLINENEWLINENEWLINETheorem 2. The monodromy representation of \(\nabla\) is equivalent to the representation of \(P_n\) on \(H_1(\Gamma_n,\mathbb{C}_{\chi^{-1}})\). NEWLINENEWLINENEWLINELet \(\mathcal{L}\) be the \(\mathbb{C}\)-algebra of Laurent polynomials on \(t_1,\dots ,t_n\). NEWLINENEWLINENEWLINETheorem 3. The monodromy representation of \(\nabla\) is dual to the quotient of the reduced Gassner representation \(Z^1(\Gamma_n,\mathcal{L})\) specialized at \(t_j=e^{-2\pi \varepsilon_j r_j}\), where we quotient by the 1-dimensional subspace \(B^1(\Gamma_n,\mathbb{C}_{\chi})\) fixed by \(P_n\).