A New Noncommutative Product on the Fuzzy Two-Sphere Corresponding to the Unitary Representation of SU(2) and the Seiberg-Witten Map

We obtain a new explicit expression for the noncommutative (star) product on the fuzzy two-sphere which yields a unitary representation. This is done by constructing a star product, $\star_{\lambda}$, for an arbitrary representation of SU(2) which depends on a continuous parameter $\lambda$ and searching for the values of $\lambda$ which give unitary representations. We will find two series of values: $\lambda = \lambda^{(A)}_j=1/(2j)$ and $\lambda=\lambda^{(B)}_j =-1/(2j+2)$, where j is the spin of the representation of SU(2). At $\lambda = \lambda^{(A)}_j$ the new star product $\star_{\lambda}$ has poles. To avoid the singularity the functions on the sphere must be spherical harmonics of order $\ell \leq 2j$ and then $\star_{\lambda}$ reduces to the star product $\star$ obtained by Preusnajder. The star product at $\lambda=\lambda^{(B)}_j$, to be denoted by $\bullet$, is new. In this case the functions on the fuzzy sphere do not need to be spherical harmonics of order $\ell \leq 2j$. Because in this case there is no cutoff on the order of spherical harmonics, the degrees of freedom of the gauge fields on the fuzzy sphere coincide with those on the commutative sphere. Therefore, although the field theory on the fuzzy sphere is a system with finite degrees of freedom, we can expect the existence of the Seiberg-Witten map between the noncommutative and commutative descriptions of the gauge theory on the sphere. We will derive the first few terms of the Seiberg-Witten map for the U(1) gauge theory on the fuzzy sphere by using power expansion around the commutative point $\lambda=0$.

= 1/(2j) and λ = λ the new star product ⋆ λ has poles. To avoid the singularity the functions on the sphere must be spherical harmonics of order ℓ ≤ 2j and then ⋆ λ reduces to the star product ⋆ obtained by Presnajder [10]. The star product at λ = λ (B) j , to be denoted by •, is new. In this case the functions on the fuzzy sphere do not need to be spherical harmonics of order ℓ ≤ 2j. The star product ⋆ λ has no singularity for negative values of λ and we can move from one representation λ = λ (B) j to another λ = λ (B) j ′ smoothly on the negative λ line. Because in this case there is no cutoff on the order of spherical harmonics, the degrees of freedom of the gauge fields on the fuzzy sphere coincide with those on the commutative sphere. Therefore, although the field theory on the fuzzy sphere is a system with finite degrees of freedom, we can expect the existence of the Seiberg-Witten map between the noncommutative and commutative descriptions of the gauge theory on the sphere. We will derive the first few terms of the Seiberg-Witten map for the U (1) gauge theory on the fuzzy sphere by using power expansion around the commutative point λ = 0.
Therefore in contrast to the ordinary spherical harmonics Y ℓm there is a truncation of the angular momentum.
Field theories on a fuzzy sphere have been studied by many people. [ [16]. In [10] an explicit formula for the star product on fuzzy sphere was constructed using the coherent state method [11].
Here J ab = x 2 δ ab − x a x b + ixǫ abc x c and x ≡ (x a ) 2 . The summation stops at m = 2j and f , g must be polynomials, i.e. spherical harmonics Y ℓm of order ℓ ≤ 2j. Extension to the fuzzy complex projective space CP N −1 was performed in [17]. There is also a star product in the integral form. In [12] by performing the stereographic projection of the sphere on to the plane and using generalized coherent states on the complex plane [13] another star product in the integral form was constructed. In [18] this product was also derived from the matrix model of [5].
In this paper we will derive the following expression for the star product on the fuzzy sphere.(sec.2) Here . Note that the summation extends to m = ∞. This product corresponds to an arbitrary representation of SU (2) including non-unitary ones. λ is a parameter introduced in (2) and this product gives the realization of (2). We will show the associativity of this product (5). For application to the field theories on the fuzzy two-sphere the values of λ must be selected by the condition of unitary representation.
We will show that there exist two values of λ for a single unitary representation j of SU (2), To truncate the summation the functions on the fuzzy sphere must be polynomials of order 2j, i.e.
The product (5) for λ = λ (B) j , which we will denote by •, is new and has interesting properties. The coefficient C m (λ (B) j ) = (−1) m (2j + 1)!/m!(2j + 1 + m)! is not singular and there is no restriction on the angular momentum of the functions, i.e. f (x) = ∞ ℓ=0 ℓ m=−ℓ a ℓm x ℓ Y ℓm . Furthermore, contrary to the case of the product (4) the spherical harmonics Y ℓm with the product • do not realize the algebra ofŶ ℓm explicitly! Especially, a product of polynomials of orders ℓ and ℓ ′ yields a polynomial of order ℓ + ℓ ′ . It turns out, however, that the integral of the star product of Y ℓm , Y * ℓ ′ m ′ corresponds to the trace of the product ofŶ ℓm ,Ŷ † ℓ ′ m ′ and the integral vanishes for ℓ = ℓ ′ or ℓ = ℓ ′ > 2j. Therefore just the combination of the star product and the integration gives a realization of the fuzzy sphere algebra.
For noncommutative theories in the flat space, the Seiberg-Witten map [19] between noncommutative and commutative descriptions has been investigated. [21][22] The Seiberg-Witten map is a transformation from the gauge fields in the commutative description to those in the noncommutative description in such a way that two field configurations in the gauge equivalence class in one description are mapped on to the two field configurations in the gauge equivalence class in the other description. A crucial problem in the study of the Seiberg-Witten map for theories on the fuzzy sphere is that while the angular momentum of the gauge fields in the noncommutative description is at most ℓ = 2j, that in the commutative description is arbitrary. Therefore it is difficult to establish a map from the commutative gauge fields to the noncommutative gauge fields. Furthermore the order of the polynomials will change when we move from λ = λ of the Seiberg-Witten map by using power expansion around λ = 0. We will give a brief summary in sec.5.

Star Product for Arbitrary Representations
In [10] the star product (4) on the fuzzy two-sphere was obtained. This product corresponds to the spin j representation of SU (2). The star product (5) can be formally derived by rewriting the factorials in (4) by gamma functions and replacing 2j inside the gamma functions by 1/λ. 1 However, additionally, the upper limit of summation, 2j, must be replaced by ∞. Therefore the associativity of (5) is not a priori guaranteed. We will examine this problem in this section. This product corresponds to arbitrary representations of SU (2), including non-unitary ones. In the next section we will select the values of λ for unitary representations and find a new star product •.
Instead of beginning with (5) we will assume the following form for the star product ⋆ λ .
is a constant and will be determined by the requirement of the associativity. Here λ was introduced in (2) and works as an expansion parameter. We can easily show that this product realizes the algebra (2). J ab is a function defined by where ǫ abc is a completely anti-symmetric unit tensor (ǫ 123 = +1), and is a projector First of all we note the following identities of J ab . 1. 4.
Identities 1,2 guarantee that x is a constant with respect to the ⋆ λ product: i.e. for any functions f (x) of x and g(x a ) we obtain For example where f ′ (x) ≡ df (x)/dx. This is a necessary condition because ⋆ λ is the product on the sphere.
Let us check the associativity of (6). To the 1st order both (f ⋆ λ g) ⋆ λ h and f ⋆ λ (g ⋆ λ h) are given by and ⋆ λ is associative to this order. To the next order one can show that the difference The 1st term vanishes due to identity 3 and the other term vanishes if the condition is satisfied.
Therefore we have 3,3 = 1 6 , χ We can work out a similar analysis to higher orders. Study of a few more higher orders shows that the condition of the associativity leads to the recursion relation Here identities 1-5 must be used. The solution to (21) is not difficult to obtain.
Now the summation over n in (6) can be performed. For we immediately obtain and generally, we get a formula In summary we get the associative product ⋆ λ on the fuzzy two-sphere for arbitrary representation including non-unitary ones.
It is possible to show that with the integration on the sphere this star product satisfies the typical property of the trace of matrices.
where dΩ is the standard measure on the unit sphere. In the polar coordinate system it is given by dΩ = sin θdθdϕ. Therefore dΩ plays the role of the trace.

Unitary representation of the Fuzzy Sphere Algebra and New Product •
For application to field theory on the fuzzy two-sphere we must pick up unitary representations. We will determine the allowed values of λ. From the product (27) we can derive the fuzzy sphere algebra.
This equation has two series of solutions.
Let us first consider the case (A). The coefficient is given by When m ≥ 2j, this coefficient is infinite. Therefore the functions f (x), g(x) must be polynomials of order at most 2j. This restriction corresponds to the size (2j + 1) × (2j + 1) of the matrices in the spin j representation. The summation must now be restricted to 0 ≤ m ≤ 2j. The product (27) with this coefficient agrees with the product (4) obtained in [10]. If we do not restrict the order of f (x), g(x), the product (4) does not satisfy the associativity. The product reduces to the commutative one in the j → ∞ limit, where the matrix size becomes infinite.
Let us turn to the case (B). The coefficient does not diverge for any m and there is no restriction on the functions f , g on the fuzzy sphere. We will denote the corresponding product by •.
This also reduces to the ordinary commutative product for j → ∞.
Let us investigate the case j = 1/2 in detail. For the product (4) with λ = λ (A) 1/2 , functions must be a linear combination of 1 and x a and we get the multiplication rule This is the algebra of the Pauli matrices σ a and the 2 by 2 unit matrix I 2 .
x a x b does not appear on the RHS of x a ⋆ x b and the angular momentum is truncated. For the product (37) we get Therefore the algebra of (Pauli) matrices is not realized. When we integrate these products over the sphere, however, we obtain These correspond to T rσ a σ b , T rσ a σ b σ c . Therefore the combination of the star product and the integration realizes the matrix multiplication rule.
We also note that for the ℓ = 2 spherical harmonics, f (x) = α ab x a x b , g(x) = β ab x a x b (α ab and β ab are symmetric, traceless constant tensors), we obtain For j ≥ 1 and −1/4 ≤ λ < 0 this does not vanish. However, when j = 1/2 and λ = −1/3, this integral vanishes. As this example shows the space of functions is actually finite dimensional.
In the case (B) the angular momentum is not cut off at some finite value and the relation of the algebra of functions to that ofŶ ℓm is not clear. Only after integration the relation of the algebra of functions to that of matrices is manifest. However, this product does realize the fuzzy sphere algebra. It is known that a star product is not unique. Given a star product ⋆ and a differential operatorD, a new product defined by gives another star product. Clearly, the relation between ⋆ and • is not of this type. We need more understanding of the relation between ⋆ and •.

Seiberg-Witten Map on the Fuzzy Sphere
In this section we will construct the Seiberg-Witten map [19] for the gauge theory on the two-sphere by using perturbation theory in λ up to order O(λ 2 ). After expansion we will j . So we will use • for the star product in what follows. We will consider only the U (1) gauge theory for simplicity.
On the fuzzy sphere the gauge transformation is defined [5] by Here L a is the angular momentum operator L a = −iǫ abc x b ∂ c and satisfies the relation The strength of the gauge field is defined [5] bŷ and transforms under (43) by the formula δF ab = i[F ab ,Λ] • .
In commutative gauge theory the gauge field A a transforms as δA a = −iL a Λ and a field strength is invariant.
We will use the cohomological approach to the Seiberg-Witten map. [20][21] We introduce the ghost field C andĈ. In the commutative description the BRST transformation is defined by sC = iC 2 = 0, where s is the BRST operator. Analogously, in the noncommutative description we have We will expandĈ andÂ a in formal power series in λ whose coefficients are local polynomials in C and A a .Ĉ The fieldsĈ,Â a reduce to C, A a at λ = 0. We will substitute the expansion (48) into (47) and obtain eqs for C (i) , A (i) . We get The solution to the 1st eq of (49) is given by Similarly a can be obtained. Only the results are presented here.
In this way one can compute the Seiberg-Witten map for the gauge theory on the two-sphere order by order in perturbation theory in powers of λ ( = λ (B) j ).

Summary
In this paper the star product ⋆ λ for the field theories on the fuzzy sphere corresponding to an arbitrary representation of SU (2) including non-unitary representation is constructed and the associativity of this product is proved. By imposing the condition of unitary representation we obtained a new noncommutative product • for λ = λ An investigation such as that of the correspondence of the Dirac-Born-Infeld actions in the noncommutative and commutative descriptions will be reported elsewhere.

Acknowledgments
The work of R. N. is supported in part by Grant-in-Aid (No.13135201) from the Ministry of Education, Science, Sports and Culture of Japan (Priority Area of Research (B)(2)).

Note Added
(1) We can prove the following formula for the inner product of ordinary spherical harmonics.
Especially, for λ = λ The inner product is positive definite for ℓ = 0, 1, . . . , 2j and the norm vanishes for ℓ > 2j. This shows that our new star product • with λ = λ (B) j defines a unitary representation. This also completes the observation made around eq (41) concerning the finite dimensionality of the functional space.
The proof of eq(52) will be presented elsewhere.
(2) Generally, the solution to (49) is not unique and has ambiguity of s-exact terms. We also found a simpler solution.