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Fuzzy sphere is one of the simplest and most canonical examples of non-commutative geometry. Ordinarily, the functions defined on a sphere form a commuting algebra. A fuzzy sphere differs from an ordinary sphere because the algebra of functions on it is not commutative. It is generated by spherical harmonics whose spin l is at most equal to some j. The terms in the product of two spherical harmonics that involve spherical harmonics with spin exceeding j are simply omitted in the product. This truncation replaces an infinite-dimensional commutative algebra by a <math>j^2</math>-dimensional non-commutative algebra.

The simplest way to see this sphere is to realize this truncated algebra of functions as a matrix algebra on some finite dimensional vector space.
Take the three j-dimensional matrices <math>J_a,~ a=1,2,3</math> that form a basis for the j dimensional irreducible representation of the Lie algebra su(2). They satisfy the relations <math>[J_a,J_b]=i\epsilon_{abc}J_c</math>, where <math>\epsilon_{abc}</math> is the totally anti-commuting tensor with <math>\epsilon_{123}=1</math>, and generate via the matrix product the algebra <math>M_j</math> of j dimensional matrices. The value of the su(2) Casimir operator in this representation is

<math>J_1^2+J_2^2+J_3^2=\frac{1}{4}(j^2-1)I</math>

where I is the j-dimensional identity matrix.
Thus, if we define the ‘coordinates’
<math>x_a=kr^{-1}J_a</math>
where r is the radius of the sphere and k is a parameter, related to r and j by <math>4r^4=k^2(j^2-1)</math>, then the above equation concerning the Casimir operator can be rewritten as

<math>x_1^2+x_2^2+x_3^2=r^2</math>,

which is the usual relation for the coordinates on a sphere of radius r embedded in three dimensional space.

One can define an integral on this space, by

<math>\int_{S^2}fd\Omega:=2\pi k Tr(F)</math>

where F is the matrix corresponding to the function f.
For example, the integral of unity, which gives the surface of the sphere in the commutative case is here equal to

<math>2\pi k Tr(I)=2\pi k j =4\pi r^2\frac{j}{\sqrt{j^2-1}}</math>

which converges to the value of the surface of the sphere if one takes j to infinity.

See also

• Fuzzy torus

Notes

• John Madore, An introduction to Noncommutative Differential Geometry and its Physical Applications, London Mathematical Society Lecture Note Series. 257, Cambridge University Press 2002

References

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