Quantum differential surfaces of higher genera

Abstract

We first construct a real family of $SL(2,\mathbb{R})$-invariant symbol composition product $\{\sharp_\theta\}_{\theta\in,\mathbb{R}}$ on the analogue of the Schwartz space $S(\mathbb{D})$ on the hyperbolic plane $\mathbb{D}\;:=\;SL(2,\mathbb{R})/SO(2)$. The value $\theta=0$ consists in the pointwise commutative product of functions on $\mathbb{D}$. And admits an asymptotic expansion that deforms the pointwise product in the direction of the canonical $SL(2,\mathbb{R}) $-invariant Kahler two form on $\mathbb{D}$. We then extend this construction to any (non-homogeneous) compact surface by considering the left action of an arithmetic Fuschian group $\Gamma\subset SL(2,\mathbb{R})$ on $\mathbb{D}$ with associated Riemann surface $\Sigma_\Gamma\;:=\;\Gamma\backslash\mathbb{D}$. More precisely, the product $\sharp_\theta$ extends from $S(\mathbb{D})$ to a smooth $SL(2,\mathbb{R})$- sub-module of $C^\infty(\mathbb{D})$ that contains the $\Gamma$-invariants $C^\infty(\mathbb{D})^\Gamma\simeq C^\infty(\Sigma_\Gamma)$ in $C^\infty(\mathbb{D})$. In particular, $\sharp_\theta$ defines a Fr\'echet algebra structure on $C^\infty(\Sigma_\Gamma)$. The resulting algebra is pre - $C^\ast$ and admits a continuous trace.