fuzzy sphere




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.


Fuzzy 2-sphere

For NN \in \mathbb{N}, N2N \geq 2, the fuzzy 2-sphere of NN bits is the formal dual to the associative algebra which is the sub-algebra in the matrix algebra Mat N×NMat_{N \times N} generated from the elements of the NN-dimensional complex irreducible Lie algebra representation of su(2).

We now say this more in detail:

Conventions and Normalizations

First to introduce relevant notation and to set out the proper choices of normalizations.

With 𝔰𝔲(2)\mathfrak{su}(2) the Lie algebra su(2) of SU(2), with complexification the special linear Lie algebra sl(2) (see here)

𝔰𝔲(2) 𝔰𝔩(2,) \mathfrak{su}(2) \otimes_{\mathbb{R}} \mathbb{C} \;\simeq\; \mathfrak{sl}(2,\mathbb{C})


(1)σ i𝔰𝔲(2) ,AAAi{1,2,3} \sigma_i \;\in\; \mathfrak{su}(2) \otimes_{\mathbb{R}} \mathbb{C} , \phantom{AAA} i \in \{1,2,3\}

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

(2)[σ i,σ j]=ikϵ ijkσ k. [\sigma_i, \sigma_j] \;=\; i \underset{k}{\sum} \epsilon_{i j k} \sigma_k \,.

(Here ϵ\epsilon is the Levi-Civita symbol, hence ϵ σ(1),σ(2)σ(3){±1}\epsilon_{\sigma(1), \sigma(2)\sigma(3)} \in \{\pm 1\} is the signature of the permutation σSym(3)\sigma \in Sym(3).)

Notice that the element

(3)iσ iσ iU(𝔰𝔲(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

(4)ρ:𝔰𝔲(2)Mat N×N \rho \;\colon\; \mathfrak{su}(2) \longrightarrow Mat_{N \times N}

denote the complex NN-dimensional complex irreducible Lie algebra representation of su(2) (hence of 𝔰𝔩(2,)\mathfrak{sl}(2,\mathbb{C})).

In the discussion of angular momentum in quantum mechanics the image

(5)iρ iρ iMat N×N \underset{i}{\sum} \rho_i \circ \rho_i \;\in\; Mat_{N \times N}

of the Casimir element (3) under the representation (4) is traditionally denoted J 2J^2, and the canonical linear basis for the N-dimensional representation ρ\rho (4) is then traditionally denoted {|j,m} m=j m=+j\{\vert j, m \rangle\}_{m = -j}^{m = +j} with

(6)J 2|j,m=j(j+1)AAAm{j,j+1,,j1,j}. J^2 \vert j,m\rangle \;=\; j(j+1) \phantom{AAA} m \in \{-j, -j+1, \cdots, j-1, j\} \,.

hence with

N(j)=2j+1j(N)=12(N1). N(j) = 2 j + 1 \;\; \Leftrightarrow \;\; j(N) = \tfrac{1}{2}(N - 1) \,.

This means that the eigenvalues of (5) as a function not of the angular momentum jj but of the dimension NN of the given irreducible representation are:

(7)j(N)(j(N)+1) =+(j(N)) 2+j(N) =+14(N1) 2+12(N1) =+14N 212N+14 =+12N12 =+14N 214 =+14(N 21) \begin{aligned} j(N)\big(j(N)+1\big) & = \phantom{+\;} (j(N))^2 + j(N) \\ & = \phantom{+\;} \tfrac{1}{4}(N-1)^2 + \tfrac{1}{2}(N-1) \\ & = \phantom{+\;} \tfrac{1}{4}N^2 - \tfrac{1}{2}N + \tfrac{1}{4} \\ & \phantom{=\;} + \tfrac{1}{2}N - \tfrac{1}{2} \\ & = \phantom{+\;} \tfrac{1}{4}N^2 - \tfrac{1}{4} \\ & = \phantom{+\;} \tfrac{1}{4} (N^2 - 1) \end{aligned}

Algebra of functions

Write now

(8)X i2N 21ρ N(σ i) X_i \;\coloneqq\; \tfrac {2} {\sqrt{N^2-1}} \rho_N(\sigma_i)

for the square matrices representing the generators σ i\sigma_i (1) in the NN-dimensional complex irrep (4), suitably normalized in view of (7).

(It is here that we need the assumption N2N \geq 2, hence excluding the complex 1-dimensional trivial irrep.)

Due to the normalization, the commutation relation (2) in this representation reads

(9)[X i,X j]=i2N 21kϵ ijkX k [X_i, X_j] \;=\; i \tfrac {2} {\sqrt{N^2-1}} \underset{k}{\sum} \epsilon_{i j k} X_k

and, by (7), the image of the Casimir element (3) under this representation is the identity matrix:

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

Equation (9) shows that in the large N limit NN \to \infty, the algebra generated by the X iX_i becomes commutative, and (10) says that for any NN, the algebra generated by the X iX_i satisfies the same relation as the smooth algebra on generators x ix_i restricted to the actual 2-sphere of unit radius:

C (S 2)SmoothAlg({x 1,x 2,x 3})/(ix ix i=1) 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_i = 1 \Big)


With the above, the volume density of the fuzzy 2-sphere scales with 1N 21\tfrac{1}{\sqrt{N^2-1}}

4πi3!ϵ ijkX iX jX k =4πi3!12ϵ ijk[X i,X j]X k =4πi3!iN 21ϵ ijkϵ ijlX lX k =4πi3!2iN 21X kX k =4πi3!2iN 21I =4π31N 21I \begin{aligned} \tfrac{4\pi i}{3!} \epsilon^{i j k } X_i \cdot X_j \cdot X_k & = \tfrac{4\pi i}{3!} \tfrac{1}{2} \epsilon^{i j k } [X_i, X_j] \cdot X_k \\ & = \tfrac{4\pi i}{3!} \tfrac {i} {\sqrt{N^2-1}} \epsilon^{i j k } \epsilon_{i j l} X^l \cdot X_k \\ & = \tfrac{4\pi i}{3!} \tfrac {2i} {\sqrt{N^2-1}} X^k \cdot X_k \\ & = \tfrac{4\pi i}{3!} \tfrac {2i} {\sqrt{N^2-1}} \, I \\ & = - \tfrac{4\pi}{3} \tfrac {1} {\sqrt{N^2-1}} \, I \end{aligned}

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

(11)X 1,X 2,X 3 S N 2 M 4π1N 21tr(M) \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 NN bits comes out as

vol(S N 2)= S N 2I=4πN 21tr(I)N=4πNN 21. 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 NN \to \infty:

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


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

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

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

At k=1k = 1 there is the single radius observable (10)

S N 2R 2= S N 2iX iX i=4πNN 21 \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=2k = 2 there are, under the integral (11), two radius observables:

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

  2. S N 2i,jX iX jX jX i\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 kk, 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) nTr(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.

from Sati-Schreiber 19c

Appearance in D-brane geometry

The fuzzy spheres appear in D-brane geometry:

  1. the fuzzy funnels of Dp-D(p+2)-brane intersections have fuzzy 2-sphere slices

  2. the fuzzy funnels of Dp-D(p+4)-brane intersections have fuzzy 4-sphere slices

  3. the supersymmetric classical solutions of the BMN matrix model are precisely fuzzy 2-sphere configurations (BMN 02 (5.4)).



Review in the context of D-brane geometry, matrix models of string theory/M-theory (BFSS matrix model, BMN matrix model, IKKT matrix model):

Fuzzy 2-sphere

On the fuzzy 2-sphere:


The fuzzy 2-sphere was introduced in:

Discussion of the spectral Riemannian geometry of the fuzzy 2-sphere:

In the context of the IKKT matrix model:

See also:

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 January 2, 2021 at 09:45:02. See the history of this page for a list of all contributions to it.