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The Fuzzy Sphere Star-Product and Spin Networks

Laurent Freidel, Kirill Krasnov

TL;DR

This work develops a spin-network framework for the fuzzy sphere to study its noncommutative product. By embedding $\\mathrm{SU}(2)$ representation theory into a graphical calculus, it derives a closed-form expression for the star_N product and its expansion in the noncommutativity parameter via $6j$-symbols, with associativity guaranteed by the Biedenharn-Elliott identity. It connects the fuzzy-sphere product to deformation quantization (Kirillov product) and shows the first-order correction reproduces the Poisson bracket on spherical harmonics, while the zeroth order is the commutative product. The approach provides a bridge between finite-dimensional NC geometry and deformation quantization, and suggests extensions to $q$-deformed fuzzy spheres.

Abstract

We analyze the expansion of the fuzzy sphere non-commutative product in powers of the non-commutativity parameter. To analyze this expansion we develop a graphical technique that uses spin networks. This technique is potentially interesting in its own right as introducing spin networks of Penrose into non-commutative geometry. Our analysis leads to a clarification of the link between the fuzzy sphere non-commutative product and the usual deformation quantization of the sphere in terms of the star-product.

The Fuzzy Sphere Star-Product and Spin Networks

TL;DR

This work develops a spin-network framework for the fuzzy sphere to study its noncommutative product. By embedding representation theory into a graphical calculus, it derives a closed-form expression for the star_N product and its expansion in the noncommutativity parameter via -symbols, with associativity guaranteed by the Biedenharn-Elliott identity. It connects the fuzzy-sphere product to deformation quantization (Kirillov product) and shows the first-order correction reproduces the Poisson bracket on spherical harmonics, while the zeroth order is the commutative product. The approach provides a bridge between finite-dimensional NC geometry and deformation quantization, and suggests extensions to -deformed fuzzy spheres.

Abstract

We analyze the expansion of the fuzzy sphere non-commutative product in powers of the non-commutativity parameter. To analyze this expansion we develop a graphical technique that uses spin networks. This technique is potentially interesting in its own right as introducing spin networks of Penrose into non-commutative geometry. Our analysis leads to a clarification of the link between the fuzzy sphere non-commutative product and the usual deformation quantization of the sphere in terms of the star-product.

Paper Structure

This paper contains 5 sections, 1 theorem, 97 equations.

Table of Contents

  1. Introduction
  2. Setup
  3. The $\star_N$-product
  4. Some formulas
  5. Poisson bracket of spherical harmonics

Key Result

Theorem 1

There exists a function $\Psi(l_1,l_2,l_3;\hbar)$, given by the analytic continuation of the function $\Psi_{(N)}(l_1,l_2,l_3)$ introduced in (Psi), which is analytic in $\hbar$ in the open disc of radius ${\rm min}(1/l_1,1/l_2,1/l_3)$ around zero, and For $N\geq l_{1},l_{2},l_{3}$, The product defined via: is an associative product. Moreover,

Theorems & Definitions (1)

  • Theorem 1