A Fuzzy Three Sphere and Fuzzy Tori

TL;DR

This paper constructs fuzzy spheres and tori as finite-dimensional approximations of continuous spaces, enabling better numerical simulations by reducing unwanted states through modified Laplacians.

Contribution

It introduces a method to construct fuzzy spheres and tori by modifying Laplacians, effectively isolating low-energy states that approximate continuous geometries.

Findings

Fuzzy 3-sphere and circle constructed from complex projective spaces.

Modified Laplacians suppress unwanted states, improving approximation accuracy.

Fuzzy tori of any dimension can be realized, aiding numerical simulations.

Abstract

A fuzzy circle and a fuzzy 3-sphere are constructed as subspaces of fuzzy complex projective spaces, of complex dimension one and three, by modifying the Laplacians on the latter so as to give unwanted states large eigenvalues. This leaves only states corresponding to fuzzy spheres in the low energy spectrum (this allows the commutative algebra of functions on the continuous sphere to be approximated to any required degree of accuracy). The construction of a fuzzy circle opens the way to fuzzy tori of any dimension, thus circumventing the problem of power law corrections in possible numerical simulations on these spaces.