On spherical harmonics for fuzzy spheres in diverse dimensions

TL;DR

This paper develops spherical harmonics for fuzzy spheres across various dimensions, linking finite matrix algebras with tensor representations of SO(n), and explores their applications in D-brane physics and non-associative algebra structures.

Contribution

It generalizes the construction of fuzzy spherical harmonics to higher and odd dimensions, connecting matrix algebras with SO(n) representations and analyzing their algebraic properties.

Findings

Finite matrix algebras correspond to tensor constructions of SO(n) irreducible representations.

Non-associativity and non-commutativity vanish in the large N limit, recovering classical geometry.

Heuristic models explain combinatorics of fuzzy four-spheres via giant fractional instantons.

Abstract

We construct spherical harmonics for fuzzy spheres of even and odd dimensions, generalizing the correspondence between finite matrix algebras and fuzzy two-spheres. The finite matrix algebras associated with the various fuzzy spheres have a natural basis which falls in correspondence with tensor constructions of irreducible representations of orthogonal groups SO(n). This basis is useful in describing fluctuations around various D-brane constructions of fuzzy spherical objects. The higher fuzzy spheres are non-associative algebras that appear as projections of associative algebras related to Matrices. The non-associativity (as well as the non-commutativity) disappears in the leading large $N$ limit, ensuring the correct classical limit. Some simple aspects of the combinatorics of the fuzzy four-sphere can be accounted by a heuristic picture of giant fractional instantons.