Fuzzy Toric Geometries

TL;DR

This paper introduces a method to construct fuzzy approximations of projective toric varieties, enabling the quantization of their geometric structures and expanding the class of fuzzy spaces available.

Contribution

It presents a novel construction of fuzzy spaces for toric varieties using their embedding into complex projective spaces, applicable to a wide range of subvarieties.

Findings

Explicit examples of fuzzy weighted projective spaces

Construction of fuzzy Hirzebruch and del Pezzo surfaces

Potential to create fuzzy Calabi-Yau manifolds

Abstract

We describe a construction of fuzzy spaces which approximate projective toric varieties. The construction uses the canonical embedding of such varieties into a complex projective space: The algebra of fuzzy functions on a toric variety is obtained by a restriction of the fuzzy algebra of functions on the complex projective space appearing in the embedding. We give several explicit examples for this construction; in particular, we present fuzzy weighted projective spaces as well as fuzzy Hirzebruch and del Pezzo surfaces. As our construction is actually suited for arbitrary subvarieties of complex projective spaces, one can easily obtain large classes of fuzzy Calabi-Yau manifolds and we comment on fuzzy K3 surfaces and fuzzy quintic three-folds. Besides enlarging the number of available fuzzy spaces significantly, we show that the fuzzification of a projective toric variety amounts to a quantization of its toric base.