The hyperbolic moduli space of flat connections and the isomorphism of symplectic multiplicity spaces

Abstract

Let $G$ be a simple complex Lie group, $\alg{g}$ be its Lie algebra, $K$ be a maximal compact form of $G$ and $\alg{k}$ be a Lie algebra of $K$. We denote by $X\rightarrow \overline{X}$ the anti-involution of $\alg{g}$ which singles out the compact form $\alg{k}$. Consider the space of flat $\alg{g}$-valued connections on a Riemann sphere with three holes which satisfy the additional condition $\overline{A(z)}=-A(\overline{z})$. We call the quotient of this space over the action of the gauge group $\overline{g(z)}=g^{-1}(\overline{z})$ a \emph{hyperbolic} moduli space of flat connections. We prove that the following three symplectic spaces are isomorphic: 1. The hyperbolic moduli space of flat connections. 2. The symplectic multiplicity space obtained as symplectic quotient of the triple product of co-adjoint orbits of $K$. 3. The Poisson-Lie multiplicity space equal to the Poisson quotient of the triple product of dressing orbits of $K$.