A Map between q-deformed and ordinary Gauge Theories

In complete analogy with Seiberg-Witten map defined in noncommutative geometry we introduce a new map between a q-deformed gauge theory and an ordinary gauge theory. The construction of this map is elaborated in order to fit the Hopf algebra structure.


Introduction
The concept of space-time as a space of commuting coordinates is perhaps too naive and must possibly be modified at the Planck scale. In their seminal paper Connes, Douglas and Schwarz [1] have found that noncommutative geometry [2] arises naturally in string theory. Quantum groups [3] provide another consistent mathematical framework to formulate physical theories on noncommutative spaces. They appeared first in the study of integrable systems [4] and are now applied to many branches of mathematics and physics. Although a tremendous amount of literature has been devoted to the study of quantum groups, we are still lacking a map which relates the quantum deformed gauge theories and the ordinary ones. In the present paper, we introduce such a map. This map is a quantum analog of the Seiberg-Witten map [5].
Let us first recall that Seiberg-Witten map was discovered in the contex of string theory where it emerged from 2D-σ-model regularized in different ways. Seiberg and Witten have schown that the noncommutativity depends on the choice of the regularization procedure: it appears in point-spliting regularization whereas it is not present in the Pauli Villars regularization. This observation led them to argue that there exist a map connecting the noncommutative gauge fields and gauge transformation parameter to the ordinary gauge field and gauge parameter. This map can be interpreted as an expansion of the noncommutative gauge field in θ. Along similar lines, we introduce a new map between the q-deformed and undeformed gauge theories. This map can be seen as an infinitesimal shift in the parameter q, and thus as an expansion of the deformed gauge field in q = e i η = 1 + iη + o η 2 .
This paper is organized as follows. In Sec. 2, we recall the Seiberg-Witten map. In Sec. 3, we present the SU q (2) quantum group techniques. In Sec. 4, we recall the Woronowicz differential calculus using the adjoint representation M a b of the group SU q (2). In Sec. 5, we consider an hybrid structure consisting of a noncommutative base space defined by a Moyal product and a q-deformed nonabelian gauge group. This structure allows us to define a map which relate the q-deformed noncommutative and the ordinary gauge fields. We close this paper by constructing a new map relating the q-deformed and ordinary gauge fields.
To ensure that an ordinary gauge transformation of A by λ is equivalent to a noncommutative gauge transformation of A by λ Seiberg and Witten have proposed the following relation They first worked the first order in θ and wrote Expanding (7) in powers of θ they found where they used the expansion The equation (9) is solved by The equations (11) are called the Seiberg Witten map.
3 The quantum group SU q (2) Let A be the associative unital C−algebra generated by the linear transformations M n m (n, m = 1, 2) the elements a, b, c, d satisfying the relations where q is a deformation parameter. The classical case is obtained by setting q equal to one.
The U q (2) is obtained by requiring that the unitary condition hold for this 2 × 2 quantum matrix: The 2 × 2 matrix belonging to U q (2) preserves the nondegenerate bilinear form [7] where and D q = ad − qbc is the quantum determinant. SU q (2) is obtained by taking the unimodularity condition D q = 1.
Let us take q = e i η ≃ 1 + iη. This gives The noncommutativity of the elements M n m is controlled by the braiding R becomes the permutation operator R nm kl = δ n l δ m k in the classical case q = 1.
The R matrix satisfy the Yang-Baxter equation The noncommutativity of the elements M n m is expressed as With the nondegenerate form B the R matrix has the form The first equation, in terms of η, gives:

Woronowicz Differential Calculus
Now, we are going to consider the bicovariant bimodule [8] Γ over SU q (2). Let θ a be a left invariant basis of inv Γ, the linear subspace of all left-invariant elements of Γ i.e. ∆ L (θ a ) = I ⊗ θ a . In the q = 1 the left coaction ∆ L coincides with the pullback for 1-forms. There exists an adjoint representation M a b of the quantum group, defined by the right action on the left-invariant θ a : The adjoint representation is given in terms of the fundamental representation [9] as: where S (.) means antipode.
In the quantum case we have θ a M n m = M n m θ a in general, the bimodule structure of Γ being non-trivial for q = 1. There exist linear functionals f a b : F un (SU q (2)) → C for these left invariant basis such that where ∆ refers to coproduct. * is the convolution product of an element M n m ∈ A and a functional f a b [8].
Once we have the functionals f a b , we know how to commute elements of A through elements of Γ. These functionals are given by [10]: The representation with the lower index of θ a is defined by using the bilinear form B which defines the new functional f a b corresponding to the basis θ a We can also define the conjugate basis θ * a = (θ a ) * ≡ θ a . Then the linear functionals f We can easily find the transformation of the adjoint representation for the quantum group which acts on the generators M j i as the right coaction Ad R : As usual, in order to define the bicovariant differential calculus with the * −structure we have required that the * −operation is a bimodule antiautomorphism (Γ Ad ) * = Γ Ad . We found that the left invariant bases containing the adjoint representation are obtained by taking the tensor product θ i θ j ≡ θ j i of two fundamental modules. The bimodule generated by these bases is closed under the * −operation. We found the right coaction on the basis θ j We have also introduced the basis θ ij = θ i θ j The exterior derivative d is defined as where X = B ab θ ab = q −1/2 θ 12 − q 1/2 θ 21 is the singlet representation of θ ab and is both left and right co-invariant, N ∈ C is the normalization constant which we take purely imaginary N * = −N and χ ab are the quantum analog of left-invariant vector fields.
Using equation (36) Then the left invariant vector field is given by To construct the higher order differential calculus an exterior product, compatible with left and right actions of the quantum group was introduced. It can be defined by a bimodule automorphism Λ in Γ Ad ⊗ Γ Ad that generalizes the ordinary permutation operator: We found [10] Λ ef gh The external product is defined by The quantum commutators of the quantum Lie algebra generators χ ab are defined as where the convolution product of two functionals [8] is given by: The χ ab functionals close on the quantum Lie algebra where C abcd ef are the q-structure constants. They can also be expanded in terms of η as where C abcd ef is the classical matrix and c abcd ef is the quantum correction linear in η.
The quantum Killing metric is given by Let us recall that the quantum gauge theory on a quantum group SU q (2) is constructed in such a way that the gauge transformations fit the Hopf algebra structures [11]. Given a left F un (SU q (2))-comodule algebra V and a quantum algebra base X B , a quantum vector bundle can be defined. The matter fields ψ can be seen as sections: X B → V and V as fiber of E (X B , V, F un (SU q (2))) with a quantum structure group F un (SU q (2)).
where α = α ab χ ab : F un (SU q (2)) → X B and where the product (·) denotes the exterior product of two forms on the base X B .
We can write the last relation in terms of components as: 5 q−Deformed noncommutative Gauge Symmetry vs. Ordinary Gauge Symmetry Let us consider the quantum vector bundle E (X B , V, F un (SU q (2))) where the base space X B is the Moyal plane defined through the functions of operator valued coordinates x i satisfying (1) and where V is a left F un (SU q (2))-comodule algebra. We define the q-deformed noncommutative gauge transformations as where ⋆ is the Groenewold-Moyal star-product defined in (2), the convolution product * is defined in (46) and where the new product ⋄ is defined as The classical case is obtained setting q = 1 and θ = 0. Using first order expansion in θ and q = 1 + iη these relations give The map between q-deformed noncommutative gauge field and ordinary gauge field is given by: 6 q−Deformed Gauge Symmetry vs. Ordinary Gauge Symmetry We consider now a Manin plane x i = (x, y), defined by xy = qyx, as a base space X B of the quantum vector bundle. Instead of the Groenewold-Moyal star-product we use the Gerstenhaber [12] star product which is defined by: f ⋆ g = µ e i η x ∂x⊗y ∂y f ⊗ g where µ (f ⊗ g) = f g, q = e iη .
We can write this product as: The quantum Lie-algebra valued potential A is given by: Using the same method we find: and the new map is given by: We can also take a Jordanian plane as a base space of the Jordanian vector bundle [13,14]. These give a new map for the corresponding deformed structure.