Non-commutative deformation of Chern-Simons theory

The problem of the consistent definition of gauge theories living on the non-commutative (NC) spaces with a non-constant NC parameter Θ(x) is discussed. Working in the L∞ formalism we specify the undeformed theory, 3d abelian Chern-Simons, by setting the initial l1 brackets. The deformation is introduced by assigning the star commutator to the l2 bracket. For this initial set up we construct the corresponding L∞ structure which defines both the NC deformation of the abelian gauge transformations and the field equations covariant under these transformations. To compensate the violation of the Leibniz rule one needs the higher brackets which are proportional to the derivatives of Θ. Proceeding in the slowly varying field approximation when the star commutator is approximated by the Poisson bracket we derive the recurrence relations for the definition of these brackets for arbitrary Θ. For the particular case of su(2)-like NC space we obtain an explicit all orders formulas for both NC gauge transformations and NC deformation of Chern-Simons equation which is non-Lagrangian.


Introduction
In the standard approach to the definition of the gauge theory one needs the notion of the covariant derivative, D a = ∂ a − iA a , as a generalization of the partial derivative ∂ a which transforms covariantly, D a → e if (x) D a , under the gauge transformations δ f A a = ∂ a f . This notion is based on the Leibniz rule. The non-commutativity is a fundamental feature of the space-time which manifests itself at the very short distances. It can be introduced in the theory through the star product, where Θ ab (x) is the non-commutativity parameter depending on the specific physical model. In some cases, like the open string dynamics in the constant B-field [1], the non-commutativity parameter can be constant, however in general it is a function of coordinates. The coordinate dependence of Θ, in general, leads to the violation of the Leibniz rule, and makes impossible to follow the standard path for the formulation of NC gauge theory. Let us note that in some particular cases, like the NC gauge theory on D-branes in non-geometric backgrounds [2] the type of non-commutativity is compatible with the Leibniz rule, so the standard reasoning can be used for the definition of the NC field strength. At that, because of the non-geometry one has to shift the field strength tensor by a closed two-form on the D-brane worldvolume to construct the NC Yang-Mills action.
The problem with the violation of the Leibniz rule can be taken under control if, e.g., instead of the partial derivative ∂ a one takes the inner one defined through the star commutator, D a = i[ · , x a ] ⋆ , like it was done in the approach of covariant coordinates [3]. This however may lead to the problem with the correct commutative limit. Another possibility discussed in the literature consists in using the deformed Leibniz rule constructed with the help of the twist element of the Hopf algebra [4,5]. Here we mention that the twist element is known for the very few examples of NC spaces [6].
In the recent work [7] in collaboration with Ralph Blumenhagen, Ilka Brunner and Dieter Lüst we have formulated the L ∞ -bootstrap approach to the construction of non-commutative gauge theories. On the one hand, in the physical literature L ∞ structures were introduced for description of gauge theories [8], see also [9,10] for more details and recent references. On the classical level it contains all necessary information about the theory including the gauge symmetry, the field equations and the Noether identities. On the other hand, L ∞ algebras or the strong homotopy Lie algebras [11,12] is a natural framework for dealing with the deformation since the Jacobi identities are required to hold only up to the total derivative or the higher coherent homotopy. We note in particular that the proof of the key result in deformation quantization, the Formality Theorem, is based on the concept of L ∞ algebras [13].
The main idea of the L ∞ bootstrap approach consists in two steps. The first one is to represent the original undeformed gauge theory, like the Chern-Simons or the Yang-Mills, as well as the deformation introduced through the star commutator as a part of a new L ∞ algebra by specifying the initial brackets ℓ 1 , ℓ 2 , etc. Then solving the L ∞ relations (the higher Jacobi identities), J n = 0, one determines the missing brackets ℓ n and completes the L ∞ algebra which governs the NC deformation of the gauge transformations and the equations of motion. In [7] we found the expressions for the gauge transformations and the field equations up to the order O (Θ 2 ) in the non-commutativity parameter. However the calculations were extremely involved and it was not clear whether the procedure can be extended to the higher or potentially all orders in Θ.
The purpose of the current work is to develop the ideas proposed in [7] in part of the existence of the solution for the L ∞ bootstrap program, its construction and the explicit examples. The key observation we made is that in each given order n the consistency condition for the L ∞ bootstrap equations, J n = 0, is satisfied as a consequence of the previously solved relations, J m = 0, with m ≤ n. We use it to express the brackets ℓ n in terms of those which have been already found ℓ m , m ≤ n. Aiming to provide explicit calculations we work in the slowly varying field approximation when the higher derivative terms in the star commutator are discarded and it is approximated by the Poisson bracket, so we set, (1. 3) The construction of the algebra L gauge ∞ describing the NC deformation of the abelian gauge transformations was previously discussed in the proceedings of the Durham Symposium on Higher Structures in M-Theory [14]. We provide it in the Section 3 for the completeness. The essentially new results regarding the derivation of the algebra L f ull ∞ which also includes the equations of motion are contained in the Sections 4 and 5. In the Section 4 we discuss the NC deformation of the 3d abelian Chern-Simons theory for arbitrary Θ. The specific case of the rotation invariant NC space is analyzed in the Section 5. In this case the coordinates satisfy the su(2) algebra and thus, Θ ab (x) = 2 θ ε abc x c , with θ being the small parameter. An important algebraic relations involving the Levi-Civita tensor ε abc and arbitrary vector A e are given in the Appendix. These relations allowed us to find an explicit all order expressions for the NC gauge transformations satisfying the relation, 4) and the field equations, F a = 0, covariant under these gauge transformations, i.e., δ f F a = {F a , f } , and reproducing in the commutative limit, θ → 0, the standard Chern-Simons equations, ε abc ∂ b A c = 0.

Basic facts from L ∞ -algebras
For the convenience of the reader in this Section we will briefly review the basic facts form the theory of L ∞ -algebras and its relation to the gauge theories. We start with a formal definition. In fact, L ∞ -algebras are generalized Lie algebras where one has not only a two-bracket, that is the commutator, but more general multilinear n-brackets with n inputs ℓ n : defined on a graded vector space X = m X m , where m ∈ Z, denotes the grading of the corresponding subspace. Each element x ∈ X, has its own degree, meaning that if deg(x) = p, this element belongs to the subspace X p . The concept of the degree is essential for the definition of the products ℓ n . First, because these brackets are graded anti-symmetric according to, And second, because the result ℓ n (x 1 , . . . , x n ) ∈ X p , with 3) The set of higher brackets ℓ n define an L ∞ algebra, if they satisfy the infinitely many relations The permutations are restricted to the ones with from which one can deduce the scheme for the higher Jacobi idebtities J n . More precisely, denoting (−1) x i = (−1) deg(x i ) , the first two L ∞ relations read which means that that ℓ 1 is a nilpotent derivation with respect to the bracket ℓ 2 , and that in particular the Leibniz rule is satisfied. The full relation J 3 reads and means that the Jacobi identity for the ℓ 2 bracket holds up to ℓ 1 exact terms. For the future needs we will also provide here the complete form of the J 4 relation, The framework of L ∞ algebras is quite flexible and it has been suggested that every classical perturbative gauge theory (derived from string theory), including its dynamics, is organized by an underlying L ∞ structure [9]. To see this, let us assume that the field theory has a standard type gauge structure, meaning that the variations of the fields can be organized unambiguously into a sum of terms each of a definite power in the fields. First we choose only two non-trivial vector spaces as where physically X 0 corresponds to the space of gauge parameters or functions f , and X −1 contains the gauge fields A a . Note that in this case ℓ 1 (f ) ∈ X −1 and can be non-zero, while ℓ 1 (A) ∈ X −2 , which is empty by now, i.e., ℓ 1 (A) = 0, by the construction. In this case, the only allowed non-trivial higher bracket are the ones with one gauge parameter ℓ n+1 (f, A n ) ∈ X −1 , and two gauge parameters ℓ n+2 (f, g, A n ) ∈ X 0 . The graded symmetry in this case means, The gauge variations are defined in terms of the brackets ℓ n+1 (f, A n ) ∈ X −1 as follows, It was shown in [9,15,16], that the L ∞ relations with two gauge parameters, J n+2 (f, g, A n ) = 0, imply the off-shell closure of the symmetry variations where Here we stress that the closure relation allows for a field dependent gauge parameter. The Jacobi identity for gauge variaions is equivalent to the L ∞ relations with three gauge parameters J n+3 (f, g, h, A n ) = 0. Thus, we see that the action of gauge symmetries on the fundamental fields is governed by an L gauge ∞ algebra. We stress that in principle, the L ∞ algebra may have an infinite number of the brackets ℓ n , which however, are not arbitrary, since should satisfy L ∞ relations (2.4). As it was already mentioned in the introduction the idea of the L ∞ bootstrap approach consists in representing the original undeformed gauge theory together with a deformation as a part of a new L ∞ structure by setting initial brackets and solving L ∞ relations to determine the algebra L new ∞ , which corresponds to the consistent deformation of the original theory.

Non-commutative deformation of the abelian gauge transformations
To define the undeformed model, the abelian gauge algebra, we set the bracket ℓ 1 (f ) = ∂ a f . The non-commutative deformation is introduced through the star commutator of functions which, from the consideration of anti-symmetry, should be assigned to the bracket ℓ 2 (f, g) = i[f, g] ⋆ . Just for the simplicity let us consider the limit of slowly varying, but not necessarily small gauge fields, i.e., we discard the higher derivatives terms in the star commutator and take, ℓ 2 (f, g) = −{f, g}, as a (quasi)-Poisson bracket defined in (1.3). This is a "self-consistent" approximation of non-commutativity since the main algebraic properties of the model are preserved. If we work with the NC deformations induced by the associative star product, the star commutator satisfies the Jacobi identity, so as the corresponding Poisson bracket.
Having non-vanishing brackets ℓ 1 (f ) and ℓ 2 (f, g), one has to check the L ∞ relation, J 2 (f, g) = 0, involving yet undetermined bracket ℓ 2 (f, A). It means that now the identity, J 2 (f, g) = 0, becomes an equation on ℓ 2 (f, A). Solving this equation one may proceed to the next L ∞ relation, J 3 (f, g, h) = 0, and define the next bracket ℓ 3 (f, g, A), etc. The procedure should be continued till no new bracket can be determined and all L ∞ relations are satisfied. Let us see how it works on practice.

Leading order contribution
The relation J 2 (f, g) = 0, reads, From which one finds Note that the solution is not unique, one may also set, e.g., The symmetry of s ij a (x) implies that this choice of the bracket ℓ ′ 2 (f, A) also satisfies the equation (3.1). However, the symmetric part s ij a (x) ∂ i f A j can be always "gauged away" by L ∞ -quasi-isomorphism, physically equivalent to Seiberg-Witten map [1], see [17] for more details.

Next to the leading order
Then we have to define the bracket ℓ 3 (f, g, A) from the identity J 3 (f, g, h) = 0, which reads, The first line is a Jacobiator, For associative non-commutative deformations we may just set, ℓ 3 (A, f, g) = 0, while in the non-associative case one needs non-vanishing bracket ℓ 3 (A, f, g) to satisfy it. We define The next step is the crucial for the whole construction. We have to analyze the relation J 3 (f, g, A) = 0, given by For simplicity, we replace it with J 3 (g, h, ℓ 1 (f )) = 0, written in the form We will follow the logic of [18] for the solution of the above algebraic equation. By construction, the equation (3.8) is antisymmetric with respect to the permutation of g and h. The graded symmetry of the ℓ 3 bracket, , implies the identity on the l.h.s. of (3.8): Which in turn requires the graded cyclicity of r.h.s. of the eq. (3.8), The latter is nothing but the consistency condition for the eq. (3.8).
It is remarkable that the consistency condition (3.9) follows from the previously satisfied L ∞ relations, namely J 2 (f, g) = 0, and J 3 (f, g, h) = 0. Indeed, taking the definition of G(f, g, h), one writes Using J 2 (f, g) = 0, we may push ℓ 1 out of the brackets and rewrite it as Which means that the consistency condition (3.9) holds true as a consequence of the previously satisfied L ∞ relations. Taking into account (3.9) one may easily check that the following expression (symmetrization in f and g of the r.h.s. of the eq. (3.8)): has required graded symmetry and solves one gets, At this point we would like to stress two main observations: • The consistency condition (graded cyclicity) (3.9) holds true as a consequence of L ∞ construction.
• Even in the associative case one needs higher brackets to compensate the violation of the standard Leibniz rule.

Higher relations
Once the brackets ℓ 3 (f, g, A) and ℓ 3 (f, A, B) are determined we may proceed to the next L ∞ relation and find the brackets with four entries, ℓ 4 . First we analyze J 4 (f, g, h, A) = 0, which we rewrite in the form J 4 (f, g, h, ℓ 1 (k)) = 0. Taking into account (2.9) we write it explicitly as: The explicit form is given by Solution of the algebraic equations of the type (3.13) was given in [19]. By the construction F (f, g, h, k) is antisymmetric in first three arguments and the graded symmetry of ℓ 4 (ℓ 1 (f ), g, h, ℓ 1 (k)) implies the graded cyclicity (consistency condition) for F (f, g, h, k), which now reads: Again, the consistency condition (3.16) holds true as a consequence of the previous L ∞ relations, graded symmetry and multilinearity of the brackets ℓ n . As previously the solution of (3.13) is constructed by taking the corresponding symmetrization of the r.h.s.: Then, setting we conclude that To complete the picture in this order let us also consider the L ∞ relation: J 4 (f, g, A, B) = 0, which we replace with J 4 (f, g, ℓ 1 (h), ℓ 1 (k)) = 0, and write in the form of the equation: where By the construction, G(f, g, h, k) is antisymmetric in first two and symmetric in last two arguments, and as a consequence of the previous L ∞ relations it satisfies the graded cyclicity relation: Taking into account (3.19) one may check that the symmetrization in the last three arguments of the r.h.s. of the eq. (3 .18), has the required graded symmetry and satisfies the equation in question.

Recurrence relations
For the higher relations, J n+2 (g, h, A n ) = 0, we proceed in the similar way. First we substitute them by the equations J n+2 (g, h, ℓ 1 (f ) n ) = 0, which can be represented in the form where the right hand side, G(f 1 , . . . , f n , g, h), is defined in terms of the previously defined brackets ℓ m+2 (ℓ 1 (f ) m , ℓ 1 (g), h), with m < n. It is symmetric in the first n arguments and antisymmetric in the last two by the construction. The graded symmetry of ℓ n+2 (ℓ 1 (f ) n , ℓ 1 (g), h) implies the non-trivial consistency condition (since G(f 1 , . . . , f n , g, h) is symmetric in first n arguments, one needs to check the cyclicity relation with respect to the permutation of the last three slots), which follows from the previous L ∞ relations and can be proved by induction.
Following [18] the solution of the equation (3.21) can be constructed taking the symmetrization of the r.h.s. in the first n + 1 arguments, i.e., And finally we obtain the expression for ℓ n+2 (f, A n+1 ), substituting in the above expression all ℓ 1 (f ) with the corresponding fields A.
The identities with three gauge parameters J n+3 (f, g, h, A n ) = 0, n > 1, are substituted by the relations J n+3 (f, g, h, ℓ 1 (k) n ) = 0, written in the form: The r.h.s. F (f, g, h, k 1 , ..., k n ) is antisymmetric in first three arguments and symmetric in last n arguments, and also should satisfy the graded cyclicity relation, which as before follows from the previous L ∞ relations, graded symmetry and multi-linearity of the brackets ℓ n . The solution of (3.24) is constructed by taking the corresponding symmetrization of the r.h.s.: Again the expression for ℓ n+3 (f, g, A n+1 ) is obtained from (3.26) substituting all ℓ 1 (f ) by the fields A.

Non-commutative field dynamics and L ∞ structure
It is remarkable that the dynamics of the theory, i.e. the equations of motion, are also expected to fit into an extended L full ∞ algebra. For this purpose one extends the vector space to where X −2 also contains the equations of motion, i.e. F ∈ X −2 . Now more higher brackets, namely ℓ n (A n ) ∈ X −2 , ℓ n+2 (f, E, A n ) ∈ X −2 , and ℓ n+3 (f, g, E, A n ) ∈ X −1 , can be non-trivial and should satisfy the following identities The higher brackets ℓ n (A n ) are special since they define the equation of motion, Now the L ∞ structure admits that the closure condition (2.12) is only satisfied on-shell, i.e. there can be terms ℓ n+3 (f, g, F , A n ) ∈ X −1 on the right hand side. The gauge variation of F reads reflecting that, as opposed to the gauge field A, it transforms covariantly.
In this Section we discuss the consistent deformation of the field dynamics, i.e., the construction of L full ∞ algebra in the bootstrap approach. First we will make some general statements regarding the consistency condition and the solution of the equations (4.2). Then we will work out the non-commutative deformation of the abelian Chern-Simons theory. In this case we write the initial brackets as The brackets ℓ n+1 (f, A n ) and ℓ n+2 (f, g, A n ) defining the pure gauge algebra L gauge ∞ were determined in the Section 3. The rest brackets ℓ n (A n ), ℓ n+2 (f, E, A n ), and ℓ n+3 (f, g, E, A n ), should be found from the identities (4.2).

Leading order contribution
The first non-trivial L ∞ relation is which we rewrite as The r.h.s. is given and can be calculated as while the two brackets in the l.h.s. should be determined. The bracket ℓ 2 (f, E) should be antisymmetric with respect to the permutation of its arguments, so we identify The rest of the eq. (4.7) can be written in the form where the coefficient functions P aijk 1 , Q aijk 1 and R aijkl 1 are given by (4.11) The solution of the equation (4.10) will be constructed following the logic of the previous section. There is a non-trivial consistency condition coming from the graded symmetry of the bracket ℓ 2 , which is satisfied as a consequence of the previously solved L ∞ relations. The relation J 2 (f, ℓ 1 (g)) = 0, can be written as ℓ 2 (ℓ 1 (f ), ℓ 1 (g)) = ℓ 1 (ℓ 2 (f, ℓ 1 (g))) . (4.12) The graded symmetry of ℓ 2 bracket, implies the consistency condition on the right hand side of (4.12), ℓ 2 (ℓ 1 (f ), ℓ 1 (g)) − ℓ 2 (ℓ 1 (g), ℓ 1 (f )) = ℓ 1 (ℓ 2 (f, ℓ 1 (g))) − ℓ 1 (ℓ 2 (g, ℓ 1 (f ))) = 0 . (4.14) The later however is automatically satisfied due to JI, J 2 (f, g) = 0, since In the specific case of the deformation of Chern-Simons theory, i.e., eq. (4.8) the relation (4.15) implies We stress that these relations can be checked explicitly taking into account (4.11), however they follow from the construction of L ∞ algebra. Using (4.17) the origynal equation (4.10) becomes implying the solution The explicit form of ℓ 2 (A, B) is given by 20) which is in the perfect agreement with our previous result [7].

Next to the leading order
At this order there appear higher brackets ℓ 3 . The expressions for ℓ 3 (A, f, g) and ℓ 3 (A, B, f ) were found in Sect. 3.1. Taking into account that now X −2 is non trivial, one may also have non-vanishing ℓ 3 (E, f, g) ∈ X −1 , ℓ 3 (E, A, f ) ∈ X −2 and ℓ 3 (A, B, C) ∈ X −2 . Let us start with ℓ 3 (E, f, g). Such a term contributes to the closure condition J 3 (f, g, A) = 0, which are however satisfied without it. Therefore, we can set ℓ 3 (E, f, g) = 0. Next we consider J 3 (E, f, g) = 0, i.e., from which one derives  By the construction the r.h.s., r 3 (A, B, f ), is symmetric with respect to the permutation of A and B. Before discussing the specific form of the r.h.s. for the deformation of the Chern-Simons theory let us prove the general formula: which is implied by the graded symmetry of ℓ 3 bracket, (4.25) First we write Using the graded symmetry and the previously satisfied L ∞ relations, J 2 (f, g) = 0, and, J 2 (A, f ) = 0, the r.h.s. of the above relation becomes ℓ 1 (A), g), f ) + ℓ 2 (ℓ 2 (ℓ 1 (A), f ), g) +ℓ 1 (ℓ 2 (ℓ 2 (f, A), g)) − ℓ 1 (ℓ 2 (ℓ 2 (g, A), f )) +ℓ 1 (ℓ 2 (ℓ 2 (g, f ), A)) − ℓ 2 (ℓ 2 (g, f ), ℓ 1 (A)) , (4.27) which in turn can be rearranged as a combination of two other previously satisfied L ∞ relations (4.28) Now let us discuss the solution of the eq. (4.23) for the non-commutative deformation of CS theory. The calculation of the r.h.s. is quite involved, but straightforward. We represent it as (4.30) The relation (4.24) becomes given by (4.30) correspondingly. They follow from the L ∞ relations, J 3 (g, f, A) = 0, J 3 (E, g, f ) = 0, etc., which were also used to obtain the eq. (4.24). The situation here is absolutely the same as in the previous Section for the construction of L gauge ∞ -algebra.
The solution of the L ∞ relations in each given order n imply the non-trivial consistency conditions, which in turn are satisfied due to the previously solved lower order L ∞ relations.
The following expression by construction is symmetric in all arguments and due to the relations (4.32) satisfies the equation (4.29). Rewriting the first two lines of (4.33) in the more compact form, the final answer is given by

(4.35)
It is written in this form to mutch with [7].

Higher order relations
In the associative case the Jacobi identities of the type J n (A n−3 E, f, g) = 0, are satisfied automatically and we set ℓ n (A n−2 , E, f ) = 0. The missing brackets ℓ n (A n ) should be determined from the L ∞ relations J n (f, A n−1 ) = 0, which can be schematically represented as where the r.h.s. r n (A n−1 , f ) written in terms of the lower order brackets ℓ m , m < n, by the construction is symmetric in first n − 1 arguments. By the induction one may prove the following relation This relation is general, the specific form of undeformed theory, i.e., ℓ 1 (A) was not used to prove it. Before writing the eq. (4.36) for the Chern-Simons case let us first make some observations regarding the equations (4.10) and (4.29) describing the first and second order deformations of the CS theory correspondingly. In both cases the r.h.s. does not contain higher derivatives of fields A (i.e., second derivatives, ∂∂A, third derivatives, etc.). The later is in agreement with the slowly varying field approximation. At that, the order of the first derivative terms (∂A) is at the maximum second, e.g., R aijklm 2 ∂ i f ∂ j A k ∂ l A m , the terms of the form (∂A) 3 , etc., do not apear. The same form of the r.h.s. remains in the third order deformation which we do not write here explicitly. For the deformation of Chern-Simons theory in the n-th order we conjecture the following form of the r.h.s.: Using these relations one may show that

Lie-algebra like deformation
The main goal of this Section is to do some explicit calculations to illustrate the proposed ideas. We will work with the most simple and at the same time nontrivial situation taking the non-commutativity parameter Θ to be linear function of the coordinates and satisfying the Jacobi identity. Physically it corresponds, for exemple, to the Q-flux backgrounds in open string theory [20]. In this case (associative deformations) all higher brackets with two gauge parameters vanish, ℓ n+2 (f, g, A n ) = 0, for n > 0, so For non-associative deformations induced by the quasi-Poisson structures the non-vanishing brackets of the type ℓ n+2 (f, g, A n ) are required to compensate the violation of the associativity.

NC su(2)-like deformation
We choose the non-commutativity parameter Θ ij (x) = 2 θ ε ijk x k , which correspond to the rotationally invariant 3d NC space [21,22,23,24,25]. For the brevity of the calculations we will suppress the small parameter θ in this and the following subsections. However, we will restore θ in the Subsection 5.3 where we provide the summary of the main findings of this Section. The corresponding Poisson bracket is For the first two brackets with one gauge parameter one finds, Then, using the recurrence relations (5.64) we observe that the brackets ℓ n+3 (f, A n ) with the odd n vanish, while for even n they have the structure for some monomial function χ n (A 2 ). The combination of (5.3) and (5.4) in (2.11) results in the following ansatz for the gauge variation: where the function χ(A 2 ) should be determined from the closure condition (5.1).
Let us write After tedious but straightforward calculations we can rewrite the r.h.s. of (5.6) as That is, requiring that 2tχ ′ (t) + 1 + 3χ(t) + tχ 2 (t) = 0 , we will obtain zero in the r.h.s. of (5.6). The solution of (5.8) is Thus, we have obtained in (5.5), (5.42) an explicit form of the non-commutative su(2)-like deformation of the abelian gauge transformations in the slowly varying field approximation. Following the lines described in [26] this result can be generalized for the non-commutative deformations along any linear Poisson structure Θ ij (x).

Non-commutative Chern-Simons theory
The initial data in this case were specified in (4.5). The brackets ℓ 2 (A, A) and ℓ 3 (A 3 ) were calculated in (4.20) and (4.35) correspondingly. The resulting expression for the NC su(2)-like deformation of the abelian Chern-Simons equations of motion up to the order O(Θ 3 ) is given by: Let us emphasise that the contribution of the order O(A n ) to the e.o.m. corresponds to the order O(Θ n−1 ) of the bi-vector Θ. So, the correct commutative limit here is evident.
The L ∞ relations, J 4 (E, f, g, h) = 0, and J 4 (E, A, f, g) = 0, are satisfied automatically and we may set, ℓ 4 (E, A, B, f ) = 0. The same can be shown for higher brackets of the form ℓ n+2 (E, A n , f ). Thus, we conclude that the gauge variation of the field equation (4.4) in case of the the associative deformation should obey the equation The gauge variation of the field equation is proportional to the field equation itself, i.e., it is gauge invariant on-shell. Taking into account the form of the lower order brackets ℓ n (A n ) we are looking the solution of the equation (5.11) in the form with the initial condition P abc (0) = ε abc and R abc (0) = 1 2 ε abc . (5.13) Substituting (5.12) in (5.11) and introducing the notation one obtains in the l.h.s.: While the r.h.s. of (5.11) is just given by Taking into account that due to Jacobi identity, the eq. (5.11) becomes We set separately and

Definition of the P -term
Let us first discuss the eq. (5.19). Taking into account the explicit form ofδ f the eq. (5.19) reads From which we obtain two separate conditions Again the lower order brackets ℓ n (A n ) indicate the anzatz (5.24) The equation (5.23) implies the following relations on the coefficient functions: Our strategy is to substitute (5.24) in (5.22) and collect the coefficients at the different powers of fields A, modulo the A 2 . Starting with a quartic in A contribution, A a A b A c A e , then cubic in A structures, like ε abm A m A c A e , etc. up to the zero order in A terms like δ ab δ ce . Equating to zero these coefficients we will obtain the system of differential equations on the coefficient functions F, . . . , N.
At that, the key observation here is that not all these power in A structures are independent. There are algebraic relations involving the Levi-Civita tensors ε abc and vector fields A e described in the appendix. Using them we will reduce the number of different structures and thus the number of the equations on F , G, etc. These relations guarantee that the resulting system of differential equations is not overfull. The equation (5.22) does have the solution.
We start writing quartic in A term in the l.h.s. of (5.22): The cubic in the field A contribution is given by At this point for the first time we make use the algebraic relation from the appendix to reduce the number of structures. Namely employing the identity We stress that now it appeared the linear in A contribution coming from the cubic ones. We continue with the quadratic in the fields A terms in the l.h.s. of (5.22), Using the identity (7.5) from the appendix which we write here for the convenience of the reader, we rewrite (5.30) as (5.32) At this point it is convenient to invert the order. First we will analize the zero order in A contributions in the equation (5.22) and only then the linear in the fields A terms. Taking into account the first line of (5.32) the zero order in A terms in the l.h.s. of (5.22) are given by Here we remind that because of the algebraic identities from the appendix not all structures in the above expression are independent. Now using these identities and previously defend coefficients we will reduce the number of terms in (5.37). First, using (7.4) and (5.35) we get rid of the terms, −ε aem A m δ bc L + ε bem A m δ ac M , substituting them with, Then we utilize the identity (7.1) to convey the terms, We use that, H = −J = 0, from (5.34), and also notice that due to (5.35) and (5.36) the coefficients K, L and F satisfy the relation, We conclude that the linear in A contribution to the l.h.s. of the equation (5.22) given initially by (5.37) becomes, (5.38) Again we set to zero the coefficients in (5.38) and obtain the relations and The solution of the equation (5.39) with the initial condition, F (0) = 1, is The relation (5.41) defines the function G in terms of previously found ones χ and F . The equation (5.40) is satisfied as a consequence of the relation (5.41) and the differential equation (5.8). The same happens, for exemple, with the equation, (5.46) Since the coefficient function P abc (A) is already known, from the eq. (5.45) one finds, and U = 0 . Comparison to the lower order brackets As a consistency check let us calculate the first order contributions to the equations of motion. Since one finds, L(0) = M(0) = −1. Then N(0) = 2F (0) = 2, and S(0) = 1/2, so the first order contribution is given by which is in the perfect agreement with (5.10). Now, The term with the Poisson bracket is exactly the same as in (5.10), but the coefficients at the first two terms are different. However, adding to the (5.51) the algebraic identity (7.2) from the Appendix multiplied by the factor −2, we arrive exactly to the equation (5.10).

Action principle
It was proposed in [9] that to define an action principle for these equations of motion one needs an inner product satisfying the cyclicity property A 0 , ℓ n (A 1 , . . . , A n ) = A 1 , ℓ n (A 0 , . . . , A n ) (5.53) for all A i ∈ X −1 . Then, the equations of motion follow from the action

(5.54)
For the field theoretical models on the NC su(2)-like space such a product coincides with the canonical Weyl-Moyal case, see [26], i.e.: (5.55) Taking this into account we observe that the term, in the equations of motion (5.12) simply cannot be reproduced from the variation of such an action, since In principle, using the identity (7.2) one may read off the expression (5.56) in terms of the other contributions to the equations of motion of the form it does not solve the problem with the Lagrangian description.
On the other hand, there is a known result about the rigidity of the Chern-Simons action [27], meaning essentially that up to the field redefinition any consistent deformation of the Chern-Simons action is proportional to the trivial one. Thus the absence of the action principle for the equation (5.12) means that possibly we obtained here some non-trivial deformation of the Chern-Simons theory.
As it was already mentioned in the introduction on the classical level L ∞ algebra encodes all necessary information about the gauge theory, see [10] for more details. In the Sections 2 and 4 we described how the gauge symmetry and the equations of motion fit into the L ∞ structure. The Noether identities are contained in the additional space X −3 which we didn't take into account in this research. While the existence of the action principle appears as an additional restriction on the field theoretical model. The example of the NC deformation of Chern-Simons theory shows that the model can be non-Lagrangian and admit the description within the L ∞ formalism. In this sense the formalism of L ∞ structures is broader then the action principle or the Batalin-Vilovisky formalism.

Summary of the results
Let us summarise the main results of the Section 5. Consider the three dimensional space endowed with the Poisson bracket, which corresponds in the slowly varying field approximation to the 3d rotation invariant NC space. At this point we restore the small parameter θ in the Poisson bracket (5.59).
In the subsection 5.1 we have shown that the gauge transformation of the gauge field A a given by In the commutative limit, θ → 0, the transformations (5.60) become an ordinary abelian gauge transformations, δ f A a = ∂ a f . That is why we call (5.60) as a non-commutative deformation of the abelian gauge transformation.
In the subsection 5.2 the problem of the consistent non-commutative deformation of the 3d abelian Chern-Somins equations, ε abc ∂ b A c = 0, was addressed. To construct this deformation we solve the equation The equations (5.64) are non-Lagrangian. The further physical properties and applications will be discussed elsewhere.

Conclusions
To construct the L ∞ structure with given initial terms one has to solve the L ∞ relations, J n = 0. The key observation we made in this work is that in each given order n the consistency condition of the equation, J n = 0, is satisfied as a consequence of the previously solved L ∞ relations, J m = 0, m ≤ n. Using this observation we were able to derive the recurrence relations for the construction of the L ∞ algebra describing the NC deformation of the abelian Chern-Simons theory in the slowly varying field approximation. Using these recurrence relations we made a conjecture regarding the form of the NC su(2)-like deformation of the gauge transformations and the corresponding field equations. The functional coefficients in the proposed anzatz were fixed from the closure conditions of the gauge algebra and the requirement of the gauge covariance of the equations of motion correspondingly.
We conclude that the problem formulated in the introduction regarding the existence of the solution to the L ∞ bootstrap programe has the positive answer. Moreover we were able to find an explicit exemple of such a solution. Thus we can see that L ∞ algebra is not only the correct mathematical framework to deal with the deformations but also is a powerful tool for the construction of these deformations.

Appendix: Important algebraic relations
Since we are in 3d, for any vector A e one may check that, ε abc A e − ε bce A a + ε cea A b − ε eab A c ≡ 0 . (7.1) Contracting the above identity with A e we arrive at, Taking the derivative of (7.2) with respect to A e one finds, Now using (7.1) in (7.3) we end up with ε abc A e − ε abm A m δ ce + ε acm A m δ be − ε bcm A m δ ae ≡ 0 . (7.4) One more identity we need is ε acm A m ε ben A n = δ ab δ ce − δ ae δ bc A 2 It can be obtained from (7.2) contracting it with ε cen and then renaming the indices.