Quantum differential surfaces of higher genera

• Published in: arXiv.org 2018-11
• Main Author: Bieliavsky, Pierre
• Format: Article
• Language: English
• Subjects: Asymptotic series; Deformation; Invariants; Riemann surfaces
• ISSN: 2331-8422

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 Fréchet algebra structure on \(C^\infty(\Sigma_\Gamma)\). The resulting algebra is pre - \(C^\ast\) and admits a continuous trace.