Strict Wick–Type Deformation Quantization on Riemann Surfaces: Rigidity and Obstructions

Let X be a hyperbolic Riemann surface. We study a convergent Wick–type star product X ⋆ X on X which is induced by the canonical convergent star product _D ⋆ D on the unit disk D D via Uniformization Theory. While by construction, the resulting Fréchet algebras (A(X), X) ( A ( X ) , ⋆ X ) are strongly isomorphic for conformally equivalent Riemann surfaces, our work exhibits additional severe topological obstructions. In particular, we show that the Fréchet algebra (A(X), X) ( A ( X ) , ⋆ X ) degenerates if and only if the connectivity of X is at least 3, and (A(X), X) ( A ( X ) , ⋆ X ) is noncommutative if and only if X is simply connected. We also explicitly determine the algebra AX A X and the star product X ⋆ X for the intermediate case of doubly connected Riemann surfaces X . As a perhaps surprising result, we deduce that two such Fréchet algebras are strongly isomorphic if and only if either both Riemann surfaces are conformally equivalent to an (not necessarily the same) annulus or both are conformally equivalent to a punctured disk.