Strict Wick-type deformation quantization on Riemann surfaces: Rigidity and Obstructions

TL;DR

This paper investigates a convergent Wick-type deformation quantization on hyperbolic Riemann surfaces, revealing topological obstructions to algebra isomorphisms and explicitly characterizing the algebraic structures for different surface connectivities.

Contribution

It introduces a canonical convergent star product on hyperbolic Riemann surfaces and identifies topological obstructions to algebra isomorphisms, providing explicit descriptions for doubly connected surfaces.

Findings

The algebra degenerates if the surface has connectivity at least 3.

The algebra is noncommutative only for simply connected surfaces.

Strong isomorphism classes correspond to annuli or punctured disks.

Abstract

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