Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Quantum differential surfaces of higher genera   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.

35
The categories of this paper are: mtrl-th
9
The categories of this paper are: nlin
32
The categories of this paper are: atom-ph
22
The categories of this paper are: chao-dyn
21
The categories of this paper are: adap-org
30
The categories of this paper are: q-alg
11
The categories of this paper are: hep-lat
28
The categories of this paper are: alg-geom
25
The categories of this paper are: patt-sol
15
The categories of this paper are: nucl-ex
6
The categories of this paper are: astro-ph
13
The categories of this paper are: quant-ph
34
The categories of this paper are: chem-ph
27
The categories of this paper are: comp-gas
4
The categories of this paper are: cond-mat
12
The categories of this paper are: nucl-th