[Paper Review] Fuzzy Orbifolds

Xavier Martín|arXiv (Cornell University)|May 7, 2004

Fuzzy Logic and Control Systems8 citations

TL;DR

This paper introduces fuzzy orbifolds as sub-algebras of the fuzzy sphere, providing the first fuzzified approximation of S²/Zₙ with singularities at the poles. One commutative case maps exactly onto a lattice, while others offer non-commutative fuzzy versions of orbifolds, generalizing fuzzification to singular spaces.

ABSTRACT

A family of fuzzy orbifolds are generated by looking at sub-algebras of the fuzzy sphere. One of them is actually commutative and can be mapped exactly onto a lattice. The others are fuzzy approximations of S^2/Z_N where Z_N is the cyclic group of rotations of angle 2pi/N and provides the first example of the ``fuzzification'' of a space with singularities (at the poles). This construction can easily be generalised to other fuzzy spaces.

Motivation & Objective

• To develop a fuzzification procedure for spaces with singularities, particularly orbifolds like S²/Zₙ.
• To construct fuzzy approximations of S²/Zₙ using sub-algebras of the fuzzy sphere.
• To identify a commutative sub-algebra that maps exactly onto a discrete lattice.
• To generalize the fuzzification method to other fuzzy spaces with singular structures.

Proposed method

• The construction begins with the fuzzy sphere algebra and identifies sub-algebras invariant under the action of the cyclic group Zₙ.
• The group Zₙ acts by rotations of angle 2π/N, inducing a quotient structure on the fuzzy sphere.
• A specific sub-algebra is selected that is commutative and corresponds to a lattice in the continuum limit.
• The resulting fuzzy orbifold approximates S²/Zₙ, preserving the singular structure at the poles.
• The method generalizes to other fuzzy spaces by applying similar group actions and sub-algebra reductions.
• The construction provides a non-commutative geometry framework for singular spaces via algebraic constraints.

Experimental results

Research questions

• RQ1 How can a fuzzy orbifold be constructed as a sub-algebra of the fuzzy sphere?
• RQ2 What is the role of the cyclic group Zₙ in defining the orbifold structure on the fuzzy sphere?
• RQ3 Can a commutative fuzzy orbifold be found that maps exactly onto a lattice?
• RQ4 How does fuzzification handle singularities, such as those at the poles of S²/Zₙ?
• RQ5 To what extent can this fuzzification method be generalized to other fuzzy spaces?

Key findings

• A commutative sub-algebra of the fuzzy sphere is identified that maps exactly onto a lattice in the continuum limit.
• The construction yields the first example of a fuzzy orbifold with singularities, specifically S²/Zₙ with poles as fixed points.
• The fuzzy orbifold provides a non-commutative approximation of S²/Zₙ, preserving its topological and group-theoretic structure.
• The method generalizes to other fuzzy spaces by applying similar group actions and sub-algebra reductions.
• The non-commutative fuzzy orbifolds retain the symmetry of the original Zₙ action while encoding singular geometry.