Contents

# Contents

## Idea

A fuzzy sphere is a variant of an n-sphere in noncommutative geometry. Often the fuzzy 2-sphere is meant by default, but there are also fuzzy spheres of higher dimension.

## Definition

### Fuzzy 2-sphere

For $N \in \mathbb{N}$, $j \geq 2$, the fuzzy 2-sphere of $N$ bits is the formal dual to the associative algebra which is the sub-algebra in the matrix algebra $Mat_{j \times j}$ generated from the elements of the $N$-dimensional irreducible Lie algebra representation of su(2).

#### Algebra of functions

More in detail:

With $\mathfrak{su}(2)$ the Lie algebra of SU(2), write

$\sigma_i \in \mathfrak{su}(2) \,\,\, i \in \{1,2,3\}$

for a choice of linear basis such that the Lie bracket takes the form

(1)$[\sigma_i, \sigma_j] \;=\; i \underset{k}{\sum} \epsilon_{i j k} \sigma_k \,.$

Notice that the element

(2)$\underset{i}{\sum} \sigma_i \cdot \sigma_i \;\in\; U(\mathfrak{su}(2))$

is a Casimir element in the universal enveloping algebra.

Let then

$\rho_j \;\colon\; \mathfrak{su}(2) \longrightarrow Mat_{N \times N}$

be the $N$-dimensional irreducible Lie algebra representation of $\mathfrak{su}(2)$, and write

$X_i \;\coloneqq\; \tfrac {2} {\sqrt{N^2-1}} \rho_N(\sigma_i)$

for the matrices representing the generators $\sigma_i$ in this representation, suitably normalized.

Due to the normalization, the commutation relation (1) in this representation reads

(3)$[X_i, X_j] \;=\; i \tfrac {2} {\sqrt{N^2-1}} \underset{k}{\sum} \epsilon_{i j k} X_k$

and the image of the Casimir element (2) under this representation is the identity matrix

(4)\begin{aligned} R^2 & \coloneqq \underset{i}{\sum} X_i \cdot X_i \\ & = I \end{aligned}

Equation (3) shows that in the large N limit $N \to \infty$, the algebra generated by the $X_i$ becomes commutative, and (4) says that for any $N$, the algebra generated by the $X_i$ satisfies the same relation as the smooth algebra on generators $x_i$ restricted to the actual 2-sphere:

$C^\infty \big( S^2 \big) \;\simeq\; SmoothAlg\big( \{x_1, x_2,x_3\}\big) \big/ \Big( \underset{i}{\sum} x_i \cdot x_1 = 1 \Big)$

#### Integration

In this vein, one defines the fuzzy refinement of the integral of functions over the 2-sphere, against its canonical volume form, to be given by the matrix trace, normalized as follows

(5)$\array{ \langle X_1, X_2, X_3\rangle &\overset{\int_{S^2_N}}{\longrightarrow}& \mathbb{C} \\ M &\mapsto& 4 \pi \tfrac{1}{ \sqrt{N^2 -1 } } tr(M) }$

With this definition the volume of the fuzzy 2-sphere of $N$ bits comes out as

$vol(S^2_N) \;=\; \int_{S^2_N} I \;=\; \tfrac{4 \pi}{ \sqrt{N^2 - 1 } } \underset{N}{\underbrace{tr(I)}} \;=\; 4 \pi \tfrac{ N }{ \sqrt{N^2 -1} } \,.$

This indeed goes to the volume of the actual 2-sphere in the limit:

$\underset{N \to \infty}{\lim} vol(S^2_N) \;=\; 4 \pi \;=\; vol(S^2) \,.$

## Properties

### Shape observables as weight systems on chord diagrams

We discuss how the “shape observables” on the fuzzy 2-sphere (above) are given by single trace observables which are Lie algebra weight systems on chord diagrams (following Ramgoolam-Spence-Thomas 04, McNamara-Papageorgakis 05, see McNamara 06, Section 4 for review).

For more see at weight systems on chord diagrams in physics.

While in the commutative large N limit, all powers of the radius function are equal

$\underset{N\to \infty}{\lim} \int_{S^2_N} R^{2 k} \;=\; 4 \pi \,;$

for finite $N$ there is an ordering ambiguity: In fact, the number of functions on the fuzzy 2-sphere at finite $N$ that all go to the same function $R^{2k}$ in the large N limit grows rapidly with $k$.

At $k = 1$ there is the single radius observable (4)

$\int_{S^2_N} R^2 \;=\; \int_{S^2_N} \underset{i}{\sum} X_i \cdot X_i \;=\; 4 \pi \tfrac{ N }{ \sqrt{N^2 -1} }$

At $k = 2$ there are, under the integral (5), two radius observables:

1. $\int_{S^2_N} \underset{i,j}{\sum} X_i X_i X_j X_j$

2. $\int_{S^2_N} \underset{i,j}{\sum} X_i X_j X_j X_i$

(Here we are using that under the integral/trace, a cyclic permutation of the factors in the integrand does not change the result).

Similarly for higher $k$, where the number of possible orderings increases rapidly. The combinatorics that appears here is familiar in knot theory:

Every ordering of operators, up to cyclic permutation, in the single trace observable $Tr(R^2)^n$ is encoded in a chord diagram and the value of the corresponding single trace observable is the value of the su(2)-Lie algebra weight system on this chord diagram.

## References

On the fuzzy 2-sphere:

### Fuzzy 2-sphere

#### Observables via weight systems on chord diagrams

Relation of Dp-D(p+2)-brane bound states (hence Yang-Mills monopoles) to su(2)-Lie algebra weight systems on chord diagrams computing radii averages of fuzzy spheres:

### Fuzzy 3-sphere

The fuzzy 3-sphere was first discussed (in the context of D0-brane-systems) in

Discussion in the context of M2-M5-brane bound states/E-strings:

### Fuzzy 4-sphere

The fuzzy 4-sphere:

### Fuzzy 6-sphere and higher

The fuzzy 6-sphere and higher:

Last revised on December 1, 2019 at 16:55:20. See the history of this page for a list of all contributions to it.