Topological Dynamics on Moduli Spaces

Let M be a Riemann surface of genus g with m boundary components (circles). Let {γ1, γ2, . . . , γm} ⊂ π1(M) be the elements in the fundamental group corresponding to these m boundary components. The space of SU(2)-gauge equivalence classes of SU(2)connections, YM2(SU(2)), is the well-known Yang-Mills two space of quantum field theory. Inside YM2(SU(2)) is the moduli space M(SU(2)) of flat SU(2)-connections. The moduli spaceM(SU(2)) has an interpretation that relates to the representation space Hom(π1(X),SU(2)) which is a real algebraic variety. The group SU(2) acts on Hom(π1(M),SU(2)) by conjugation, and the resulting quotient space is precisely M(SU(2)) = Hom(π1(M),SU(2))/ SU(2). Conceptually, the moduli spaceM(SU(2)) relates to the semi-classical limit of YM2(SU(2)). Assign each γi a conjugacy class Ci ⊂ SU(2) and let C = {C1, C2, . . . , Cm}. Definition 1.1. The relative character variety with respect to C is MC(SU(2)) = {[ρ] ∈M : ρ(γi) ∈ Ci, 1 ≤ i ≤ m}.