Trace Functionals on Non-commutative Deformations of Moduli Spaces of Flat Connections

Let G be a compact connected and simply connected Lie group, and Σ be a compact topological Riemann surface with a point p marked on it. One can associate to this data the moduli space of flat G connections on the punctured Riemann surface Σ denoted by MG = M[Σp]. This space decomposes into a union MG = ∪σM G(σ) of moduli spaces of flat connections with the holonomy around the point p fixed in some conjugacy class σ of G. For generic σ the space MG(σ) is a smooth real algebraic orbifold (manifold if G = SU(n)), and comes equipped with a canonically defined symplectic form, which induces an algebraic Poisson structure on it. In this paper, we study non-commutative deformations of the spaces of functions on MG and MG(σ). We bring together the algebraic approach initiated and developed in the works of Fock, Rosly [18], Alekseev et. al. [3], and Buffenoir, Roche [9], and the theory of formal deformations of symplectic manifolds, specifically the index theorem of Fedosov and Nest-Tsygan [12, 17, 27]. Our first result is a simple construction of a canonically defined noncommutative algebra Aq depending on a parameter q, from which F (M G), the algebraic functions on MG, may be recovered by setting q = 1. Substituting q = e2πi~ one obtains an algebra A~ over the formal power series C[[~]] which serves as a formal deformation of F (MG). The central object of the index theorem of Fedosov and Nest-Tsygan is a cyclic functional Tr : A~ → C[[~]], called the canonical trace , which plays the role of the index of an elliptic operator in this formal theory. The focus of our work is a conjectural lifting of the canonical trace, which takes values in formal power series, to a cyclic functional on Aq taking values in functions of q holomorphic in the unit disc. The two traces will be related by an asymptotic expansion at q = 1.