2009-00-00 Spin Networks and SL(2,C) Character Varieties

Spin Networks and SL(2,C) Character Varieties (with Sean Lawton), Handbook of Teichmuller Theory Volume II (Chapter 16), 2009, Athanase Papadopoulos, ed., EMS Publishing House. arXiv: math.QA/0511271

Abstract: Denote the free group on 2 letters by $\mathcal{F}_2$ and the $\mathrm{SL}(2,\mathbb{C})$-representation variety of $\mathcal{F}_2$ by $\mathcal{R}=\mathrm{Hom}(\mathcal{F}_2,\mathrm{SL}(2,\mathbb{C}))$. The group $\mathrm{SL}(2,\mathbb{C})$ acts on $\mathcal{R}$ by conjugation. We construct an isomorphism between the coordinate ring $\mathbb{C}[\mathcal{F}_2,\mathrm{SL}(2,\mathbb{C})]$ and the ring of matrix coefficients, providing an additive basis of $\mathbb{C}[\mathcal{R}]^{\mathrm{SL}(2,\mathbb{C})}$ in terms of spin networks. Using a graphical calculus, we determine the symmetries and multiplicative structure of this basis. This gives a canonical description of the regular functions on the $\mathrm{SL}(2,\mathbb{C})$-character variety of $\mathcal{F}_2$ and a new proof of a classical result of Fricke, Klein, and Vogt.