[Paper Review] Quantum differential surfaces of higher genera

Pierre Bieliavsky|arXiv (Cornell University)|Dec 18, 2017

Advanced Algebra and Geometry6 references3 citations

TL;DR

This paper constructs a family of $SL(2,\mathbb{R})$-invariant star products $\sharp_\theta$ on the Schwartz space $S(\mathbb{D})$ over the hyperbolic plane $\mathbb{D}$, deforming the pointwise product via the canonical Kähler form; it extends to compact Riemann surfaces $\Sigma_\Gamma = \Gamma\backslash\mathbb{D}$, yielding a Fréchet algebra structure on $C^\infty(\Sigma_\Gamma)$ that is pre-$C^*$ and admits a continuous trace.

ABSTRACT

We first construct a real family of $SL(2,\mathbb{R})$-invariant symbol composition product $\{\sharp_θ\}_{θ\in,\mathbb{R}}$ on the analogue of the Schwartz space $S(\mathbb{D})$ on the hyperbolic plane $\mathbb{D}\;:=\;SL(2,\mathbb{R})/SO(2)$. The value $θ=0$ 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 $Γ\subset SL(2,\mathbb{R})$ on $\mathbb{D}$ with associated Riemann surface $Σ_Γ\;:=\;Γ\backslash\mathbb{D}$. More precisely, the product $\sharp_θ$ extends from $S(\mathbb{D})$ to a smooth $SL(2,\mathbb{R})$- sub-module of $C^\infty(\mathbb{D})$ that contains the $Γ$-invariants $C^\infty(\mathbb{D})^Γ\simeq C^\infty(Σ_Γ)$ in $C^\infty(\mathbb{D})$. In particular, $\sharp_θ$ defines a Fréchet algebra structure on $C^\infty(Σ_Γ)$. The resulting algebra is pre - $C^\ast$ and admits a continuous trace.

Motivation & Objective

• To define a deformation quantization of smooth functions on hyperbolic surfaces of higher genus using $SL(2,\mathbb{R})$-invariant structures.
• To extend a symbol composition product $\sharp_\theta$ from the Schwartz space on $\mathbb{D}$ to the space of $\Gamma$-invariant smooth functions on $\mathbb{D}$, identifying with $C^\infty(\Sigma_\Gamma)$.
• To establish that the resulting product induces a Fréchet algebra structure on $C^\infty(\Sigma_\Gamma)$ that is pre-$C^*$ and admits a continuous trace.
• To provide a non-commutative geometric framework for higher-genus Riemann surfaces via quantum differential structures.

Proposed method

• Construct a real family of $SL(2,\mathbb{R})$-invariant symbol composition products $\sharp_\theta$ on $S(\mathbb{D})$, with $\theta = 0$ corresponding to the commutative pointwise product.
• Utilize the canonical $SL(2,\mathbb{R})$-invariant Kähler two-form on $\mathbb{D}$ to define the asymptotic deformation of the pointwise product in the $\theta \to 0$ limit.
• Extend the product $\sharp_\theta$ from $S(\mathbb{D})$ to a smooth $SL(2,\mathbb{R})$-submodule of $C^\infty(\mathbb{D})$ containing the $\Gamma$-invariant functions $C^\infty(\mathbb{D})^\Gamma \simeq C^\infty(\Sigma_\Gamma)$.
• Leverage the action of an arithmetic Fuchsian group $\Gamma \subset SL(2,\mathbb{R})$ to quotient $\mathbb{D}$ and obtain a compact Riemann surface $\Sigma_\Gamma$ of higher genus.
• Prove that $\sharp_\theta$ induces a Fréchet algebra structure on $C^\infty(\Sigma_\Gamma)$, with the product being continuous and compatible with the group action.
• Demonstrate that the resulting algebra is pre-$C^*$ and admits a continuous trace, ensuring compatibility with $C^*$-algebraic structures.

Experimental results

Research questions

• RQ1 How can a $SL(2,\mathbb{R})$-invariant deformation quantization be constructed on the hyperbolic plane $\mathbb{D}$ using a symbol composition product?
• RQ2 What is the asymptotic behavior of the product $\sharp_\theta$ as $\theta \to 0$, and how does it relate to the canonical Kähler form on $\mathbb{D}$?
• RQ3 Can the $\theta$-product on $S(\mathbb{D})$ be extended to the space of $\Gamma$-invariant smooth functions on $\mathbb{D}$ for an arithmetic Fuchsian group $\Gamma$?
• RQ4 Does the extended product $\sharp_\theta$ on $C^\infty(\Sigma_\Gamma)$ yield a Fréchet algebra with desirable analytic properties?
• RQ5 Is the resulting algebra on $C^\infty(\Sigma_\Gamma)$ pre-$C^*$ and does it admit a continuous trace?

Key findings

• The product $\sharp_\theta$ defines a deformation of the pointwise product on $S(\mathbb{D})$ that is asymptotically governed by the canonical $SL(2,\mathbb{R})$-invariant Kähler two-form on $\mathbb{D}$.
• The $\theta$-product extends from $S(\mathbb{D})$ to a smooth $SL(2,\mathbb{R})$-submodule of $C^\infty(\mathbb{D})$ containing the $\Gamma$-invariant functions $C^\infty(\mathbb{D})^\Gamma \simeq C^\infty(\Sigma_\Gamma)$.
• The restriction of $\sharp_\theta$ to $C^\infty(\Sigma_\Gamma)$ equips it with a Fréchet algebra structure compatible with the smooth topology.
• The resulting Fréchet algebra on $C^\infty(\Sigma_\Gamma)$ is pre-$C^*$, ensuring compatibility with $C^*$-algebraic frameworks.
• The algebra admits a continuous trace, which is a key analytic property for applications in noncommutative geometry and index theory.
• The construction provides a quantum differential structure on compact Riemann surfaces of higher genus via $SL(2,\mathbb{R})$-invariant deformation quantization.