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

Abstract Let X be a hyperbolic Riemann surface. We study a convergent Wick–type star product $$\star _X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mo>⋆</mml:mo> <mml:mi>X</mml:mi> </mml:msub> </mml:math> on X which is induced by the canonical convergent star product $$\star _{{\mathbb {D}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mo>⋆</mml:mo> <mml:mi>D</mml:mi> </mml:msub> </mml:math> on the unit disk $${\mathbb {D}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> </mml:math> via Uniformization Theory. While by construction, the resulting Fréchet algebras $$(\mathcal {A}(X),\star _X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:msub> <mml:mo>⋆</mml:mo> <mml:mi>X</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> are strongly isomorphic for conformally equivalent Riemann surfaces, our work exhibits additional severe topological obstructions. In particular, we show that the Fréchet algebra $$(\mathcal {A}(X),\star _X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:msub> <mml:mo>⋆</mml:mo> <mml:mi>X</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> degenerates if and only if the connectivity of X is at least 3, and $$(\mathcal {A}(X),\star _X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:msub> <mml:mo>⋆</mml:mo> <mml:mi>X</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is noncommutative if and only if X is simply connected. We also explicitly determine the algebra $$\mathcal {A}_X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>X</mml:mi> </mml:msub> </mml:math> and the star product $$\star _X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mo>⋆</mml:mo> <mml:mi>X</mml:mi> </mml:msub> </mml:math> 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.