Fuzzy sphere

Last updated October 15, 2023

In mathematics, the 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 $j^{2}$ -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 $J_{a},~a=1,2,3$ that form a basis for the j dimensional irreducible representation of the Lie algebra su(2) . They satisfy the relations $[J_{a},J_{b}]=i\epsilon _{abc}J_{c}$ , where $\epsilon _{abc}$ is the totally antisymmetric symbol with $\epsilon _{123}=1$ , and generate via the matrix product the algebra $M_{j}$ of j dimensional matrices. The value of the su(2) Casimir operator in this representation is

J_{1}^{2}+J_{2}^{2}+J_{3}^{2}={\frac {1}{4}}(j^{2}-1)I

where I is the j-dimensional identity matrix. Thus, if we define the 'coordinates' $x_{a}=kr^{-1}J_{a}$ where r is the radius of the sphere and k is a parameter, related to r and j by $4r^{4}=k^{2}(j^{2}-1)$ , then the above equation concerning the Casimir operator can be rewritten as

x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=r^{2}

,

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

{\displaystyle \int _{S^{2}}fd\Omega

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

2\pi k\,{\text{Tr}}(I)=2\pi kj=4\pi r^{2}{\frac {j}{\sqrt {j^{2}-1}}}

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

Notes

Jens Hoppe, "Membranes and Matrix Models", lectures presented during the summer school on ‘Quantum Field Theory – from a Hamiltonian Point of View’, August 2–9, 2000, arXiv : hep-th/0206192
John Madore, An introduction to Noncommutative Differential Geometry and its Physical Applications, London Mathematical Society Lecture Note Series. 257, Cambridge University Press 2002

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References

J. Hoppe, Quantum Theory of a Massless Relativistic Surface and a Two dimensional Bound State Problem. PhD thesis, Massachusetts Institute of Technology, 1982.

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