Fuzzy Riemann Surfaces

TL;DR

This paper introduces quantum analogues of compact Riemann surfaces called C-Algebras, defined via polynomial relations in non-commutative variables, and explores their finite-dimensional representations for quartic constraints.

Contribution

It defines C-Algebras as quantum analogues of Riemann surfaces and explicitly constructs finite-dimensional representations for certain quartic constraints.

Findings

C-Algebras generalize classical Riemann surfaces with a quantum parameter.

Finite-dimensional representations are explicitly constructed for quartic constraints.

Classical limits recover Poisson-bracket structures from polynomial constraints.

Abstract

We introduce C-Algebras (quantum analogues of compact Riemann surfaces), defined by polynomial relations in non-commutative variables and containing a real parameter that, when taken to zero, provides a classical non-linear, Poisson-bracket, obtainable from a single polynomial C(onstraint) function. For a continuous class of quartic constraints, we explicitly work out finite dimensional representations of the corresponding C-Algebras.