Gauge Theory on Twisted $\kappa$-Minkowski: Old Problems and Possible Solutions

We review the application of twist deformation formalism and the construction of noncommutative gauge theory on $\kappa$-Minkowski space-time. We compare two different types of twists: the Abelian and the Jordanian one. In each case we provide the twisted differential calculus and consider ${U}(1)$ gauge theory. Different methods of obtaining a gauge invariant action and related problems are thoroughly discussed.


Introduction
Noncommutative (NC) deformations of space-time, introduced either through effective models or as fundamental property of a theory, lead to models that can mimic some of the properties expected in a quantized theory of gravity. In particular, the combination of general relativity and quantum mechanics suggests that at the Plank scale the standard picture of space-time as a smooth manifold breaks down and should be replaced by some kind of fuzzy or foam-like space-time. This property can be naturally realized by NC deformations of space-time, with prominent examples being fuzzy spaces [1] and, more generally, matrix models [2]. Furthermore, it is expected that our understanding and implementation of symmetries should be modified when describing physics close to the Planck scale. Disentangling the geometry from matter content of a theory at this scale is non-trivial, and mixing of internal/gauge symmetries and geometry/diffeomorphisms occurs. In construction of a physical theory on NC space-time one implements this property by introducing noncommutative gauge transformations [3].
The construction of NC gauge theory is an important step in understanding the physics on NC space-time. Indeed, the main concept in quantum field theory underlying the success of the Standard model is the principle of local gauge symmetries. Therefore an important aspect of the construction of the gauge theory on NC spaces is a consistent implementation of local gauge symmetry in combination with non-locality introduced implicitly through a NC deformation of space-time. A key ingredient in such construction is the Seiberg-Witten map [4]. This mapping includes a set of non-local and non-linear field redefinitions relating commutative and NC gauge fields and parameters. Most importantly, it enables a consistent definition of NC gauge theories for arbitrary gauge groups.
In describing a NC space-time, we substitute the concept of manifold with an algebra of functions on manifold. NC deformations of such algebra correspond to NC deformations of space-time. There exist powerful methods for studying deformations of an algebra of functions if this algebra carries a representation of a Hopf algebra. In this case one first considers a deformation of the Hopf algebra, and then uses the Hopf algebra action in order to induce a deformation of the algebra on functions on manifold. Deformation via Drinfeld twists [5] is an example of such procedure. One of the main advantages of this formalism is the straightforward way to define a differential calculus, an important ingredient in the construction of a (gauge) field theory.
In this work our primary interest is to examine compatibility of the local gauge principle with the deformation of algebra of functions on a specific example of NC space-time, the κ-Minkowski space-time. This example of noncommutative space is the most analysed example of non-constant deformation with potentially interesting phenomenological consequences [6]. The (non-trivial) commutation relations of coordinates of the four-dimensional κ-Minkowski space-time are of the Liealgebra type [x 0 , x j ] = iκ −1 x j , where j denotes space directions and the zeroth component corresponds to the time direction. One of the important properties of this NC space-time is that there is a quantum group symmetry acting on it. It is a dimensionfull deformation of the global Poincaré group, the κ-Poincaré group [7]. The constant κ has dimension of energy and sets a deformation scale. The construction of field theory on this Lie algebraic type of noncommutative spacetime attracted a lot of interest, but was mainly concentrated on the scalar field theories [8][9][10][11][12]. The problem of constructing gauge field theory on κ-Minkowski space-time was addressed in [13].
All these results have shown that the construction of field theories on this space is plagued with ambiguities, mostly due to the lack of understanding of the symmetries of NC space. We proposed to resolve some of these ambiguities by using specific Abelian twist to introduce deformation in the algebra [14]. We constructed the action for the noncommutative U(1) gauge fields in a geometric way, as an integral of a maximal form. However, we could not maintain the κ-Poincaré symmetry; the corresponding symmetry of the twisted κ-Minkowski space was the twisted igl(1, 3) symmetry. It turned out that this is a generic situation; κ-Minkowski space-time obtained from κ-Poincaré algebra cannot be obtained by twisting the usual Poincaré algebra of symmetry [15]. Still, the extensions of Poincaré algebra are amenable to twist formulation, and in this work we concentrate on two particular examples. The twists with support in extensions of Poincaré algebra were also considered previously in [16][17][18].
In the first two sections we review the basics of the twist formalism and the Seiberg-Witten map; two main concepts we use in our analysis. Then, in Section 4 and Section 5 we introduce two different twists (Abelian and Jordanian). Commutation relations of coordinates that follow from those twists are commutation relations of the κ-Minkowski space-time; the twisted symmetry algebra is not the κ-Poincaré algebra. We construct the twisted differential calculus for both deformations. Finally, we discuss the U (1) gauge theory and problems related with it: integration and Hodge dual. We describe problems in details and give some possible solutions. The results concerning the Abelian twist were previously obtained in [14] while the results concerning the Jordanian twists are new.

Twist formalism
Within the twist formalism the NC deformations are introduced by twisting the underlying symmetry of the theory and then consistently applying the consequences of the deformation on the geometry of space-time itself. The underlying symmetries are described in the Hopf algebra language and the NC spaces (as deformed algebras of functions) are Hopf module algebras. The twisted symmetry does not have the usual dynamical significance and, in particular, there is no Noether procedure associated with it. We view this symmetry as a prescription that allows us to consistently apply deformation in the theory. The twist deformation equips the algebra of functions A with the twisted ⋆-product and can be represented by deformed, ⋆-commutators of noncommutative coordinates. Since we are interested in deformations of space-time symmetries, we concentrate on the Lie algebra of vector fields and its deformations; generalization to any Lie algebra is straightforward.
Vector fields provide an infinite dimensional Lie algebra, its universal enveloping algebra includes linear differential operators. Those act naturally on the algebra of functions (Hopf module algebra) on a manifold. We work in four dimensions and use the Lorentzian signature (mostly minus). Generalization to n dimensions is easily done. The Lie algebra of vector fields is denoted by Ξ and its elements are vector fields ξ, which can be written in the coordinate basis as: ξ = ξ µ ∂ ∂x µ = ξ µ ∂ µ . This algebra generates the diffeomorphism symmetry; one can also consider subgroups of Ξ like Poincaré algebra or conformal algebra as symmetry groups. The universal enveloping algebra of Ξ we denote by U Ξ. It can be equipped with the Hopf algebra structure: (2.1) The first line is the algebra relation: commutator of two vector fields is a vector field. In the second line the coproduct of the generator ξ is given; note that it is primitive. It encodes the Leibniz rule and specifies how the symmetry transformation acts on products of fields/representations. In the last line, the counit and the antipode maps are given.
A well defined way to deform the symmetry Hopf algebra is via twist. The twist F is an invertible element of U Ξ ⊗ U Ξ satisfying the following properties: 3. perturbative expansion where λ is a small deformation parameter 1 . The last property provides an expansion around the undeformed case in the limit λ → 0. We shall frequently use the notation (sum over α = 1, 2, ...∞ is understood) where, for each value of α,f α andf α are two distinct elements of U Ξ (and similarly f α and f α are in U Ξ). The twist acts on the symmetry Hopf algebra and gives the twisted symmetry (as deformed Hopf algebra) The algebra remains the same, while in general the comultiplication (coproduct) and antipode change. The whole deformation depends on formal parameters which control classical limit. Twisted (deformed) comultiplication leads to the deformed Leibniz rule for the symmetry transformations when acting on product of fields.
Now we use the twist to deform the commutative geometry of space-time (vector fields, 1-forms, exterior algebra of forms, tensor algebra). For convenience we consider one particular class of twists, the Abelian twists [19] F Here θ CD is a constant antisymmetric matrix, C, D = 1, . . . p, p ≤ 4 and X C = X µ C ∂ µ are commuting vector fields, [X C , X D ] = 0. This twist fulfils the requirements (2.2)-(2.4). We write all results for this particular twist, but we mention if and when they are valid in more general cases (more general twists).
Applying the inverse of the twist (2.7) to the usual point-wise multiplication of functions, µ(f ⊗ g) = f · g; f, g ∈ A, we obtain the ⋆-product of functions The action of the twist (f α andf α ) on the functions f and g is via the Lie derivative. This ⋆-product is noncommutative, associative and in the limit θ CD → 0 it reduces to the usual pointwise multiplication; the last property is guaranteed by (2.4) and the associativity is guaranteed by (2.2). In this way we obtain the noncommutative algebra of functions A F = (A, ⋆), i.e. the noncommutative space-time. The product between functions and 1-forms is defined as with an arbitrary 1-form ω. The action off α on forms is (again) given via the Lie derivative. We often use the Cartan's formula for the Lie derivative along the vector field ξ of an arbitrary form ω Here d is the exterior derivative and i ξ is the contraction along the vector field ξ.
Arbitrary forms form an exterior algebra with the wedge product. The ⋆-wedge product on two arbitrary forms ω and ω ′ is The usual (commutative) exterior derivative d : A → Ω satisfies: The first property if fulfilled because the usual exterior derivative commutes with the Lie derivative which enters in the definition of the ⋆-product. Therefore, we will use the usual exterior derivative as the noncommutative exterior derivative. Note that the last line of (2.12) gives a definition of the ∂ ⋆ µ derivatives. All the properties and definitions introduced so far are also valid for a more general twist. However with the definition of integral one has to be more careful. The usual integral is cyclic under the ⋆-exterior products of forms where d = deg(ω), d 1 + d 2 = 4 provided that S F (f α )f α = 1 holds [20]. One can check that this indeed holds for the Abelian twist (2.7).
Note that this approach to deformation of differential calculus, i.e. twisted approach differs from the bicovariant differential calculi formulation. More specifically, the covariance condition, i.e. L ⊲(f g) = L (1) ⊲ f L (2) ⊲ g , L ∈ U Ξ, f, g ∈ A is satisfied in both approaches (the NC spacetime is a Hopf module algebra). However the bicovariance condition (requiring that all dx µ , must be simultaneously left and right-invariant) is not satisfied in the twisted version. Moreover it has been shown that in the case of κ-Minkowski space-time the four-dimensional bicovariant differential calculi does not exist, but one can construct a five-dimensional one, which is bicovariant [21]. Alternative approaches to differential calculus on κ-Minkowski space-time were also considered in [22].

NC gauge theory and the Seiberg-Witten map
In this section we describe how to construct a NC gauge theory on a deformed space-time obtained from an Abelian twist (2.7). To achieve our goal we use the enveloping algebra approach and the Seiberg-Witten map, as developed in [3]. Also, we specify the gauge group to be the SU (n) group; noncommutative fields we label with a "hat" and commutative without a "hat". Under the infinitesimal NC gauge transformations the NC gauge field 2Â =Â µ ⋆ dx µ transforms as (3.14) with the NC gauge parameterΛ α . We demand that the consistency condition is satisfied, i.e. transformations (3.14) have to close the algebra This will be the case provided that the gauge parameterΛ α is in the enveloping algebra of the su(n) algebra 3 . However, an enveloping algebra is infinite dimensional and the resulting theory seems to have infinitely many degrees of freedom. This problem is solved by the Seiberg-Witten map. The idea of the Seiberg-Witten map is that all noncommutative variables (gauge parameter, fields) can be expressed in terms of the corresponding commutative variables and their derivatives; then the NC gauge transformations are induced by the corresponding commutative gauge transformationŝ with the commutative gauge field A = A a T a and the commutative gauge parameter α = α a T a , with T a generators of su(n) algebra. In addition, we assume that we can expand all NC variables as power series in noncommutativity parameter θ CD introduced by the twist.
In the case of NC gauge parameter the expansion iŝ Inserting this expansion into (3.15) and expanding all ⋆-products gives a variational equation for the gauge parameterΛ α . This equation can be solved to all orders of the deformation parameter. The zeroth order solution is the commutative gauge parameter α. The recursive relation between the nth and the (n + 1)st order solution is given by [23,24] . . includes all possible terms of order n. We introduced the following notation:Â C = i X CÂ is a contraction of the 1-formÂ along the vector field X C and l D is a Lie derivative along the vector field X D .
2 Note that we can expand the noncommutative forms in the coordinate basis in two different ways ω = ωµ ⋆ dx µ = ωµdx µ . The difference will only be in the components of forms. Depending on the situation, we will use one or the other expansion, but we will be careful not to mix them. 3 Note that in (3.14) ⋆-commutators appear. These commutators do not close in the Lie algebra, namely having A = A a T a and B = B a T a leads to Only in the case of U (n) in the fundamental representation the anticommutator of generators is still in the corresponding Lie algebra.
Solving the equation (3.16) order by order in the NC parameter the NC gauge fieldÂ is expressed in terms of the commutative gauge field A. The recursive solution in this case is given bŷ Finally, the field-strength tensor is defined asF = dÂ − iÂ ∧ ⋆Â and transforms covariantly under infinitesimal NC gauge transformations, The recursive relation for the SW map solution is given bŷ Remark 1: The above SW map solutions are written in the language of forms and with the use of the recursive relations. One can also expand these relations in orders of the deformation parameter θ CD and write the solutions for the components. These will depend on the particular form of the twist as we will see later on. In Section 4 and Section 5, we discuss particular examples of the twisted κ-Minkowski. There we will write the component expansions and we will write them up to first order in the NC parameter.
Remark 2: The recursive solutions are valid for the Abelian twist. For a more general twist one has to solve the SW map order by order in parameter expansion.

Kappa-Minkowski from an Abelian twist
The main object of this review is the κ-Minkowski space-time. We will discuss two different ways of twisting that result in κ-Minkowski space-time. The starting point in both approaches is the symmetry algebra of the four dimensional Minkowski space-time, the Poincaré algebra iso(1, 3). It has 10 generators: 4 generators of translations P µ and 6 generators of Lorentz rotations M µν . The algebra relations are 4 with η µν = diag(+1, −1, −1, −1). The universal enveloping algebra of this algebra we label with U iso (1,3). Besides the algebra relations (4.21) U iso(1, 3) can be equipped with the additional structure It is the Hopf algebra we want to deform via twist. Unfortunately, we cannot choose a twist from U iso(1, 3) ⊗ U iso(1, 3) and obtain the κ-Minkowski space-time in the same time [15]. It follows from the fact that the κ-deformation of Poincaré algebra is characterized by classical r-matrix which satisfy inhomogeneous Yang-Baxter equation and one can not obtain the κ-Poincaré Hopf algebra and κ-Minkowski as its module from an internal twist. Therefore, in order to obtain the κ-Minkowski space-time by twisting, we have to enlarge the starting symmetry algebra.
In our first example we choose an Abelian twist given by with two commuting vector fields X 1 = ∂ 0 and X 2 = x j ∂ j and indices j = 1, 2, 3. The constant matrix θ CD is defined as This twist fulfils the conditions (2.2), (2.3) and (2.4) with the small deformation parameter λ = a. Detailed analysis of the consequences of this twist and the construction of the U (1) gauge theory was done in [14]. Therefore, we skip some details here and describe the main problems and results.
The vector field X 1 = ∂ 0 generates translations along x 0 and belongs to the Poincaré algebra iso(1, 3). However, the vector field X 2 = x j ∂ j belongs to the general linear algebra gl(1, 3). Therefore, we have to consider the inhomogeneous general linear algebra igl(1, 3) as our starting point for the symmetry analysis and the twist (4.23) then defines (4.24) The action of the twist (4.23) on the igl(1, 3) algebra follows from (2.6) and it has been analysed in detail in [25]. The most important results are: the algebra (4.24) remains the same; since X 2 = x j ∂ j does not commute with the generators P µ and L µν the comultiplication and the antipode change.
In this way we obtain the twisted igl(1, 3) Hopf algebra instead of the κ-Poincaré algebra found in [7].
The Lie algebra igl (1, 3) contains Poincaré algebra iso (1, 3) as a subalgebra where the Lorentz generators M µν are defined by: M µν = L µν − L νµ . The algebra igl (1, 3), as well as its classical subalgebras act on the algebra of functions A via first order differential operators, i.e. vector fields, defined by the natural representation L µν = x µ ∂ ν , P µ = ∂ µ . The inverse of the twist (4.23) defines the ⋆-product between functions (fields) on the κ-Minkowski space-time . This product is associative, noncommutative and hermitean f ⋆ g =ḡ ⋆f .
The usual complex conjugation we label with "bar". In the zeroth order (4.26) reduces to the usual point-wise multiplication. Calculating the commutation relations between the coordinates we obtain These are the commutation relations of the κ-Minkowski space-time with a = κ −1 .

Twisted differential calculus and integration
We have seen in Section 2 that the NC exterior derivative is the usual exterior derivative with the properties (2.12). We now discuss the specific properties due to the twist (4.23).
The basis 1-forms are dx µ . Knowing that the action of a vector field on a form is given via Lie derivative one can show that Since basis 1-forms anticommute the volume form remains undeformed The ⋆-derivatives follow from (2.12) and are given by Arbitrary forms ω 1 and ω 2 do not anticommute, the ∧ product is deformed to the ⋆-wedge product where d 1 and d 2 are the degrees of forms. However, under the integral forms anticommute This holds because the twist (4.23) fulfils the property S F f α f α = 1. The property (4.32) can be generalized to We say that the integral is cyclic. This property is very important for construction of NC gauge theories.

U(1) gauge theory
In order to construct the NC U (1) gauge theory we now use the SW map solutions from Section 2. We expand the recursive relations up to the first order in the NC parameter a and use the particular form of the twist (4.23). Also, when writing the expanded solutions for the components of forms we have to use (4.28).
Expanding (3.18) we obtain the components of the gauge fieldÂ =Â µ ⋆ dx µ : with the commutative gauge field A µ The first order solution of the field-strength tensorF = 1 2F µν ⋆ dx µ ∧ ⋆ dx ν follows from (3.20) and is given by: Finally, in order to write a NC U (1) gauge invariant action we need a NC Hodge dual of the field-strength tensorF . We label it with * F ; it should have the following properties: with F ρσ = η ρµ η σν F µν . We use the flat metric η µν to raise and lower indices. The natural guess for the components of the NC Hodge dual ( * F ) µν = 1 2 ǫ µναβF αβ , (4.39) gives a 2-form * F = 1 2 ( * F ) µν ⋆ dx µ ∧ ⋆ dx ν which does not fulfil the property (4.37). Therefore, the action constructed using (4.39) is not gauge invariant. The construction of the Hodge dual on NC spaces turns out to be a problem in general. In some simple examples, like θ-constant deformation, the natural guess (4.39) works well, but for other more complicated deformations this is not the case. There are different ways of solving (or at least going around) this problem and we describe some of them next.

Discussion
In the following, we discuss three different ways of overcoming the problem of defining the NC Hodge dual.

Method 1
We introduce a two formẐ = 1 and demand that it fulfils the properties (4.37) and (4.38). Note thatẐ µν = ǫ µναβẐ αβ . Then, using the first property, we solve the SW map for this two form. Since it transforms in the adjoint representation, (4.37), the recursive relation is given bŷ .

(4.42)
Expanding this relation up to first order in the NC parameter a and using the twist (4.23) we obtainẐ The NC gauge invariant action can be written as The terms e −ia∂ 0Ẑ 0j and e −2ia∂ 0Ẑ ij come from ⋆-commuting basis 1-forms with the componentŝ Z µν . Inserting the SW map solutions (4.35), (4.36), (4.43) and (4.44) into (4.45) leads to This action is invariant under the commutative U (1); this is guaranteed by the SW map. One can further study coupling to the matter fields, equations of motion and their solutions. It seems that this method works fine. On the other hand, we had to introduce an additional fieldẐ as a replacement for the NC Hodge dual of the field-strength tensor. This field is an independent field, not a function ofF as in the case of the Hodge dual. After the expansion in the commutative fields, we see that there are no new degrees of freedom; SW map takes care of that. However, if one discusses the freedom of the SW map [26] one finds additional covariant terms that enter the action (4.46) with arbitrary coefficients. These terms could be fixed by imposing some additional physical requirements.

Method 2
If we look closely at the problem of definition of the Hodge dual, we see that the problem arises because the basis 1-forms do not ⋆-commute with functions (4.29). But we are free to choose a different basis, because we write the action in a basis independent form. Instead of working in the coordinate basis, we now redo calculations in the natural basis (frame in the sense of Madore [27]). This basis is defined as a basis in which basis 1-forms θ a ⋆-commute with functions, f ⋆ θ a = θ a ⋆ f . Its particular form will in general depend on the choice of the twist. For the twist (4.23) it is given by [11] x µ = (t = x 0 , x, y, z), dx µ = (dt, dx, dy, dz), ∂ µ = (∂ t , ∂ x , ∂ y , ∂ z ) ↓ x a = (t, r, θ, ϕ), θ a = (dt, dr r , dθ, dϕ), e a = (∂ t , r∂ r , ∂ θ , ∂ ϕ ). (4.47) The twist (4.23) is then rewritten as with X 1 = ∂ t = e 0 and X 2 = r∂ r = e 1 . The ⋆-product in this basis is Note that the new basis is not flat, the metric is given by g ab = diag(1, −r 2 , −r 2 , −r 2 sin 2 θ). Since the metric does not depend on t the ⋆-inverse is the same as the usual inverse g ab ⋆g ac = g ab g ac = δ c a . The volume element is The Hodge dual * F we define generalizing the usual expression for the Hodge dual in curved space given by * In order to have a NC gauge invariant action it is necessary that * F transforms covariantly under NC gauge transformations. To ensure this we have to covariantize the expression √ −gg ae g bf . We HereĜ aebf is the quantity that under NC gauge transformations transforms covariantly (4.53) and in the limit a → 0 reduces to √ −gg ae g bf . The SW map solution forĜ aebf up to first order in a is given byĜ aebf = √ −gg ae g bf − aA 0 e 1 ( √ −gg ae g bf ). (4.54) The SW map solutions for A and F in the new basis arê Finally, we construct and expand the NC U (1) gauge invariant action and obtain with d 4 x = r 2 sin θdtdrdθdϕ. Note that the action (4.57) has the same form as the expanded action for the NC gauge field in the case of θ-constant deformation [3]. This is the consequence of the particular choice of basis, the natural basis, in which the twist looks like the Moyal-Weyl twist. One can do a coordinate transformation and write back the action (4.57) in the coordinate basis. The result (as expected) is (4.46). Concerning the degrees of freedom, the situation is the same as in Method 1. There we introduced the fieldẐ, while in Method 2 we introduced the fieldĜ aebf . Therefore, the SW map freedom will contribute here as well and (we expect) with the same number of terms, since bothẐ andĜ transform covariantly.

Method 3
In [28] yet another approach is discussed. The action for the commutative U (1) gauge theory coupled to gravity can be defined as 5 Here we have three fields: the 2-form field-strength tensor F = 1 2 F µν dx µ ∧ dx ν , the 1-form vierbein V a = V a µ dx µ and the auxiliary scalar field f ab . Indices a, b, · · · = 0, 1, 2, 3 are flat Lorentz indices and are raised and lowered with the metric η ab = diag(1, −1, −1, −1). Since we work in the flat space the vierbeins reduce to the coordinate basis 1-forms V a = δ a µ dx µ . Under the commutative U (1) gauge transformation all three fields are invariant and therefore the action (4.58) is also invariant.
The equation of motion for the field f ab identifies f ab = 1 2 F ab . Inserting this into the action (4.58) and using the flat space form of vierbeins gives We see that by introducing the auxiliary field f ab one can avoid the explicit use of the Hodge dual in the action. All this should now be generalized to the NC spaces.
We define the NC U (1) gauge field action as with noncommutative fieldsf ab ,F andV a . All these fields transform covariantly, that is Then one can solve the SW map for these fields and expand the action, calculate the equations of motion and finally evaluate the on-shell action. Solution for the field-strength tensor we already have; it is given by (3.20). The solutions for the other fields are easy to find, these fields are in the adjoint representation and the form of the SW map solution will be the same as (4.42). We will not do the explicit calculations here, the steps are similar to the steps in Method 1 and Method 2.
We just comment that this method indeed enables avoiding the introduction of the Hodge dual, by introducing one additional fieldf ab . In this way, it resembles to first two methods and the freedom in the SW map will again contribute in the similar way.
Finally, to conclude this analysis: There are different ways to solve (or at least to go around) the problem of the definition of NC Hodge dual. What seems to be common for all of them is the introduction of an additional NC field in the adjoint representation. These fields do not change the number of degrees of freedom due to the SW map, but they can introduce additional covariant terms in the expanded actions provided one discusses the freedom of the SW map. Then these additional terms can be used to render some nice properties of the theory, like renormalizability [29].

Kappa-Minkowski from a Jordanian twist
The twisted symmetry of the κ-Minkowski space-time constructed in Section 4 is the twisted igl(1, 3). However, we would like to stay as close as possible to the Poincaré symmetry, that is we do not want to enlarge the symmetry algebra too much. Therefore, in this section we discuss twisting of the Poincaré-Weyl algebra denoted by iwso (1, 3). This algebra has 11 generators: 10 of the Poincaré algebra and the dilatation generator J. The algebra is given by (4.21) and additional commutators: On the space of functions (scalar fields) the (anti-hermitean) generators are given by M µν = x µ ∂ ν − x ν ∂ µ , P µ = ∂ µ and J = −x µ ∂ µ . The universal enveloping algebra of this algebra is U iwso (1, 3); it becomes a Hopf algebra with the structure (2.1).
In order to have a hermitean ⋆-product we work with the symmetrized version of Jordanian twist related with the Borel subalgebra of iwso(1, 3) given by dilatation J and momenta P 0 generators: [J, P 0 ] = P 0 . The inverse of such symmetrized Jordanian twist is given by [31]: where the following notation is used: Under the action of the twist the algebra relations do not change. However, the coalgebra sector is deformed. We give here the deformed coproduct for momenta generators only It will be used to calculate the coproduct for the new derivatives ∂ ⋆ µ in the next subsection. For the rest of deformed coproducts we refer the reader to the Appendix. Once again, the twisted Hopf algebra is not the κ-Poincaré algebra from [7]. It is the twisted ]. The inverse of the twist defines the ⋆-product between functions and in a compact form can be written in the following way: For the future use we rewrite F −1 order by order, using expansion in the deformation parameter a: The ⋆-product is then The commutation relations between coordinates are the κ-Minkowski commutation relations: Note that the star product for κ-Minkowski space-time up to the first order is the same when coming from the Jordanian twist (also the non-symmetric one, see e.g. [25]) and from the Abelian twist (4.26).

Twisted differential calculus
The usual exterior derivative is the ⋆-exterior derivative. As in Section 4, we use the coordinate basis; the basis 1-forms are dx µ . As the action of a vector field on a form is given via Lie derivative, we obtain Using these relations one can show that the basis 1-forms anticommute Note that the relations (5.71) are valid to all orders in a.
One can rewrite the usual exterior derivative of a function using the ⋆-product as where the new derivatives ∂ ⋆ µ are defined by this equation. We obtain the following relation, again valid to all orders in a: However, the coproduct for the new derivatives ∂ ⋆ µ can only be calculated order by order, using (5.64) and the expansion in a. Up to second order we obtain: Unfortunately, the twist (6.89) does not fulfil S F f α f α = 1 and the integral will not be cyclic. This is a problem when one wants to discuss NC gauge theories and use the variational principle. We postpone the discussion until next subsection.

U(1) gauge theory
Having defined differential calculus we are ready to formulate NC gauge theory on κ-Minkowski space-time possessing the twisted Weyl-Poincaré symmetry. We use the SW map solutions from Section 3 and expand everything up to the first order in the deformation parameter a. Strictly speaking, formulae from the Section 3 are only valid in the case of Abelian twist deformation and will not be valid in the case of Jordanian deformation (5.63). But since the first order of the ⋆-product is the same for both twists we can use these solutions up to first order 7 .
Let us write the first order expansions of the SW map solutions. The NC gauge parameterΛ α up to first order in the NC parameter a is given bŷ The NC gauge fieldÂ =Â µ ⋆ dx µ we calculate from (3.18) The components of the field-strength tensorF = 1 2F µν ⋆ dx µ ∧ ⋆ dx ν follow from (3.20) Note that this solution is different then (4.35) and (4.36). This is a consequence of the Jordanian deformation and the difference in the differential calculus, compare (4.29) and (5.71). Next step is the construction of gauge invariant action. Here we face the following problems: 1. In order to write a NC action for the gauge fields, we need a NC generalization of the Hodge dual of the field-strength tensorF . The problem is the same as in Section 4.

The integral is not cyclic.
In Section 4 we saw that there are three ways to "step around" the problem 1. In the case of Jordanian deformation, due to the non-cyclicity of the integral, problems 1 and 2 interfere and cannot be analyzed separately. To solve this we will use a modification of the first method in Section 4. The other methods we comment in the next subsection.
We modify the integral by introducing a measure function µ(x) in the following way where µ(x) satisfies P 0 µ = 0 and Jµ = −µ. This measure function is a-independent and does not vanish in the limit a → 0. One possible solution in four dimensions is given by A more precise mathematical description (and justification) of adding the measure function (changing the volume element) can be found in [34].
Next, we need a Hodge dual to write a gauge invariant action for the NC gauge field. In addition, since the measure function µ does not vanish in the commutative limit, we have to find a way to cancel it from the zeroth order of the equations of motion. Having these two requirements in mind, we construct the following action where µ(x) is defined by (5.78) and (5.79) andŶ is a 2-form which satisfieŝ The action (5.80) is gauge invariant and the good commutative limit is ensured by (5.83). Expanding the action (5.80) we obtain The SW map solution forŶ can be found solving perturbatively (5.82) and up to first order in a is given byŶ Inserting the SW map solutions (5.77) and (5.85) into the action (5.84) we obtain It is obvious that the obtained action is invariant under the commutative U (1) gauge transformations, that is guaranteed by the SW map. The result is different than the one obtained in Section 4. We obtained the same terms as in (4.46), however the relative coefficient between the second and the third term is different compared with (4.46). This difference can be attributed to the properties of Jordanian twist, i.e., to the different twisted symmetry than in the previous case.

Discussion
Modifying the first method of Section 4, we managed to construct the gauge invariant action in the case of the Jordanian twist. However, the origin of the measure function µ is not very clear. We can speculate that in higher dimensions the twist (5.63) looks simpler and when doing the dimensional reduction to four dimensions the twist obtains its Jordanian form and the measure function appears in the integral. So far we were not able to find a proper higher dimensional theory that could give all this.
There were two other methods discussed in Section 4. Let us comment briefly how they do (not) apply here. It is obvious that they have to be modified due to the non-cyclicity of the integral.
One can find a natural basis in the case of Jordanian deformation and it is given by The vector fields entering the twist (5.63) are rewritten as X 1 = ∂ 0 = 1 r e 0 and X 2 = −x µ ∂ µ = − t r e 0 − e 1 . Then one can check that The basis 1-forms θ a are frame 1-forms, they ⋆-commute with functions: The construction of the Hodge dual is done following the same steps as in Section 4. Only when writing the integral, one has to be careful and add the measure function. Of course, the a → 0 limit of Hodge dual has to be modified to cancel the measure function, similar to (5.83). This basis simplifies calculations but does not lead to a big improvement. The measure function in this basis is µ(x) = r.
The method of an auxiliary field f ab introduced in [28] again does not improve a lot. In the sense of the ambiguities of the SW map all three methods are the same as the methods discussed in Section before. All this suggests that the question of NC gauge theory on a Jordanian κ-Minkowski needs to be understood better.

Conclusions
In this paper we demonstrated how one could construct NC gauge theory consistent with deformation of algebra of functions on κ-Minkowski space-time. Two key ingredients were twist formalism and the Seiberg-Witten map. We used Drinfeld twist to deform the Hopf algebra of symmetry generators and then used the Hopf algebra action to induce deformation of geometry. This procedure provides differential calculus needed for the construction of a field theory. As a next step, we defined NC gauge transformations. The Seiberg-Witten map insures that these NC gauge transformations are actually induced by the corresponding commutative gauge transformation. Expanding in the deformation parameter led to effective models which could be seen as a possible non-local and non-linear extension of classical electrodynamics. Moreover, we showed that different underlying symmetries, igl(1, 3) and iwso(1, 3), led to two different deformations of the standard theory.
We also described in details the obstacles encountered in our analyses and offered some possible solutions. The failure of the Jordanian twist to provide cyclic integral could be understood as an indication that the underlying symmetry iwso(1, 3) and its twisted version are not compatible with the flat metric. The measure we introduced, seemingly ad hoc, might be seen as a consistency requirement. The obstruction we have encountered in the construction of the Hodge dual fieldstrength tensor is a manifestation of the fact that the introduction of a NC geometrical structure prevents decoupling of diffeomorphisms and gauge symmetries. The Hodge dual field-strength tensor includes both gauge and metric degrees of freedom. Consistent NC deformation of both gauge and geometry imposes that the metric degrees of freedom should transform covariantly under the gauge transformation. This brings in mind the ideas of generalized geometry, a framework in which one organizes and extends the gauge transformations and the diffeomorphisms within O(d, d) group. Applying these ideas in the present context would imply extending our analyses to quasi-Hopf algebras [35,36] and/or Hopf algebroids [37].
to the third order are the following: (P k ⊗ P 0 J + P 0 J ⊗ P k + P k P 0 ⊗ J + J ⊗ P k P 0 ) Here we introduced the following notation for Poincaré algebra generators M i = 1 2 ǫ ijk M jk for rotations and N i = M 0i for boosts. The twisted coproduct for momenta written in a compact form in Section 5 (5.64) can be also expanded: Twisted antipodes are the following: To complete the definition of U F iwso (1,3) [[a]] we mention that counits stay undeformed : ǫ (M i ) = ǫ (N k ) = ǫ (P µ ) = ǫ (J) = 0.