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fuzzy sphere

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 Nβˆˆβ„•N \in \mathbb{N}, jβ‰₯2j \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 jΓ—jMat_{j \times j} generated from the elements of the NN-dimensional irreducible Lie algebra representation of su(2).

Algebra of functions

More in detail:

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

Οƒ iβˆˆπ”°π”²(2)i∈{1,2,3} \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)[Οƒ i,Οƒ j]=iβˆ‘kΟ΅ ijkΟƒ k. [\sigma_i, \sigma_j] \;=\; i \underset{k}{\sum} \epsilon_{i j k} \sigma_k \,.

Notice that the element

(2)βˆ‘iΟƒ iβ‹…Οƒ i∈U(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

ρ j:𝔰𝔲(2)⟢Mat NΓ—N \rho_j \;\colon\; \mathfrak{su}(2) \longrightarrow Mat_{N \times N}

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

X i≔2N 2βˆ’1ρ N(Οƒ i) X_i \;\coloneqq\; \tfrac {2} {\sqrt{N^2-1}} \rho_N(\sigma_i)

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

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

(3)[X i,X j]=i2N 2βˆ’1βˆ‘kΟ΅ ijkX k [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)R 2 β‰”βˆ‘iX iβ‹…X i =I \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 Nβ†’βˆžN \to \infty, the algebra generated by the X iX_i becomes commutative, and (4) 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:

C ∞(S 2)≃SmoothAlg({x 1,x 2,x 3})/(βˆ‘ix iβ‹…x 1=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_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)⟨X 1,X 2,X 3⟩ ⟢∫ S N 2 β„‚ M ↦ 4Ο€1N 2βˆ’1tr(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=4Ο€N 2βˆ’1tr(I)⏟N=4Ο€NN 2βˆ’1. 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:

limNβ†’βˆžvol(S N 2)=4Ο€=vol(S 2). \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 β€œshape 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

limNβ†’βˆžβˆ« S N 2R 2k=4Ο€; \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 (4)

∫ S N 2R 2=∫ S N 2βˆ‘iX iβ‹…X i=4Ο€NN 2βˆ’1 \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 (5), two radius observables:

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

  2. ∫ S N 2βˆ‘i,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.

References

On the fuzzy 2-sphere:

Fuzzy 2-sphere

General

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