Bootstrapping non-commutative gauge theories from L∞ algebras

Non-commutative gauge theories with a non-constant NC-parameter are investigated. As a novel approach, we propose that such theories should admit an underlying L∞ algebra, that governs not only the action of the symmetries but also the dynamics of the theory. Our approach is well motivated from string theory. We recall that such field theories arise in the context of branes in WZW models and briefly comment on its appearance for integrable deformations of AdS5 sigma models. For the SU(2) WZW model, we show that the earlier proposed matrix valued gauge theory on the fuzzy 2-sphere can be bootstrapped via an L∞ algebra. We then apply this approach to the construction of non-commutative Chern-Simons and Yang-Mills theories on flat and curved backgrounds with non-constant NC-structure. More concretely, up to the second order, we demonstrate how derivative and curvature corrections to the equations of motion can be bootstrapped in an algebraic way from the L∞ algebra. The appearance of a non-trivial A∞ algebra is discussed, as well.


Introduction
It is one of the rather appealing features of string theory that the effective theory on Dbranes in a two-form background is given by a non-commutative (NC) Yang-Mills theory [1]. For constant two-form flux this result can be explicitly derived by quantizing the open string and computing conformal field theory correlation functions [2]. In this case, the non-commutative theory is governed by the associative Moyal-Weyl star-product. From string theory it is known that there also exist consistent D-brane solutions of the string equations of motion that wrap curved submanifolds and carry a non-constant twoform flux, thus leading to a non-constant non-commutativity structure 1 Θ ij . Examples are branes in WZW models [3] or holographic duals of integrable deformations of AdS 5 sigma models [4]. In the latter case, the holographic dual gauge theory still lives on flat space and only receives a deformation in the non-commutativity structure. Therefore, one expect that one can formulate a non-commutative gauge theory also for such more general cases.

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Using techniques from conformal field theory, for the SU(2) WZW model it was shown that this theory is a non-commutative matrix valued gauge theory on the fuzzy 2-sphere. This theory is still associative, but in principle also this could be broken. Throughout this paper we will be agnostic about this point and admit also non-associative star-products.
There have been some attempts to provide a description of such gauge theories using the general Kontsevich [5] star-product [6][7][8][9] (for a recent application see [10]) and invoking techniques from Hopf-algebras [11,12]. For more information and literature on these attempts please consult the review [13]. However, these approaches were motivated rather mathematically while missing a clear physical guiding principle for their construction. It is the aim of this paper, to take such a physical principle from string theory and to analyze whether it works and gives reasonable results.
We note that more recently there have also been proposals for the appearance of noncommutative and non-associative structures in the closed string sector [14,15], in particular when one has a non-geometric flux background. Let us emphasize that in this paper we restrict to the open string case with D-branes.
In this paper we propose that the missing physically motivated guiding principle is the existence of an L ∞ (or A ∞ ) algebra. Before we investigate this idea in more detail, let us mention that these structures appeared for the first time in the context of string field theory [16]. Indeed, e.g. for bosonic closed string field theory, both the action of symmetries on the string field and their string equations of motion were governed by an L ∞ algebra. The latter can be considered as a generalization of a Lie algebra, where one allows field dependent gauge parameters. This weakens the closure constraint and motivates the introduction of in general infinitely many higher products satisfying generalized Jacobi identities. These are quadratic expressions involving for each n finitely many higher products. In particular, the usual Jacobi identity for the two-product (the commutator) can be violated by "derivative" terms, thus allowing a mild form of non-associativity. For this reason, in the mathematics literature such algebras have been called strong homotopy algebras [17].
In [18], the authors showed that L ∞ algebras do not only show up in string field theory, but also in much simpler field theories, like Chern-Simons (CS) and Yang-Mills (YM) theories. Here the structure is considerably truncated and only a finite number of higher products and relations were non-trivial. It is very tantalizing that again not only the action of the symmetry but also the dynamics of the whole gauge theory fit into such finite L ∞ algebras. The authors also proposed that every consistent gauge theory should be governed by such an underlying L ∞ algebra.
In [19,20], motivated by the AdS 3 -CFT 2 holographic duality, it was shown that Walgebras, describing infinitely many global symmetries in two-dimensional conformal field theories, also feature an underlying, highly non-trivial L ∞ structure. Here, it was the nonlinearity of the W-algebra that induced higher products and relations. Turning the logic around, if they were not already known, W-algebras could have been bootstrapped from the L ∞ algebra. Furthermore it was shown in [21] that the non-associative closed string R-flux algebra as well as the associated M-theory R-flux algebra of the seven octonions can be extended to a 2-term L ∞ algebra.

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Thus, so far there exist a couple of physical examples that could be rewritten in terms of L ∞ algebras. The motivation for this work is to advance the symmetry concept of L ∞ algebras and actually exploit it to determine the structure of the above mentioned NC gauge theories with general NC-structure. For this purpose we will follow a bootstrap approach, where we take some initial lower order products, like one-and two-products, and bootstrap the remaining higher products by invoking the L ∞ relations. The initial data are essentially the first term in the gauge variation and in the equations of motion, i.e. the one resulting from a kinetic term in the action. All of the rest follows. For a general NC-structure, we will see that all the other higher products receive derivative (∂Θ)-corrections. In other words, imposing the guiding principle of an underlying L ∞ algebra, we can algebraically bootstrap the derivative corrections to the action of the NC gauge symmetry onto the gauge fields and their equations of motion.
In this paper, we explicitly show how this procedure can be carried out in detail up to second order in Θ. For this purpose, in section 2 we review some facts about NC gauge theories and recall the mathematical definitions of L ∞ and A ∞ algebras. As a first application of L ∞ algebras, we show that NC-CS and NC-YM theories on the Moyal-Weyl plane fit into this scheme.
In section 3 we remind the reader of concrete string theory settings where NC gauge theories with non-constant NC-structure appeared. These are branes in WZW models and holographic duals of integrable deformations of AdS 5 sigma models. For the SU(2) WZW model, we recall that an NC matrix valued gauge theory has been derived via CFT techniques [22]. As a compelling first result, we show that this unconventional NC gauge theory on the fuzzy 2-sphere can be bootstrapped by imposing the existence of an L ∞ algebra.
In section 4, we apply the same technique to the more general class of NC gauge theory on flat and curved space with non-constant NC-structure. This is done in a perturbative approach in Θ. First, we point out the essential problem arising for non-constant Θ and argue that it receives a natural solution in the context of L ∞ algebras. We bootstrap the derivative corrections to the action of the NC gauge symmetry onto the gauge fields. Then, we extend the L ∞ algebra to also include the equations of motion of a NC-CS and a NC-YM theory on flat and curved space.
We show that up to second order in Θ one can even find an A ∞ algebra. As expected, the graded symmetrization of the obtained structure results in the corresponding L ∞ algebra. Since this involves a lengthy and tedious computation, we have delegated this part to an appendix -not because it is less important but for not too much disturbing the main flow of the paper.

Preliminaries
For self-consistency, in this section we introduce some of the salient features of known NC gauge theories and the formal definitions of L ∞ and A ∞ algebras. In addition, we analyze the NC gauge theory based on the Moyal-Weyl star product with respect to an underlying L ∞ algebra.

Non-commutative gauge theories
First, let us recall that the conformal field theory of an open string ending on a D-brane supporting a non-trivial gauge flux F = B + 2πF features a non-commutative geometry. In this paper we choose α = 1. Indeed, by computing the disc level scattering amplitude of N -tachyons, certain relative phases appear which for constant gauge flux can be described by the Moyal-Weyl star-product The open string quantities governing the theory on the D-brane are related to the initial closed string variables g and F via G −1 + Θ = (g + F ) −1 , where the anti-symmetric bivector Θ ij is the one appearing in the star product (2.1). In the Seiberg-Witten limit the OPE exactly becomes the Moyal-Weyl star-product. This non-trivial product of functions leads to the non-commutative Moyal-Weyl plane with [x i , x j ] = i Θ ij . In [5] it has been shown that for every Poisson structure Θ ij that by definition satisfies one can define a corresponding associative star-product, which will also involve derivatives of the Poisson structure. The same product can also be considered for a quasi Poisson structure, but then leads to a non-associative star-product, which up to second order in Θ reads For the higher order expression see [23][24][25][26]. Often we will write this as that separates the derivative ∂Θ-corrections from the standard Moyal-Weyl terms. The associator for this product becomes In [1], for the Moyal-Weyl case with constant open string metric and NC-parameter, it was shown that the effective theory on a stack of N branes is given by a non-commutative gauge theory with gauge group U(N ). In the following we stick to the U(1) case. As in usual YM theory there is a gauge field A a (x) behaving under a gauge transformation as Using the Leibniz rule for the star-bracket [., .] and its associativity, 2 one can show that two gauge transformations close off-shell in the sense
Then, the vacuum equation of motion for the non-commutative U(1) Yang-Mills theory reads (2.10) In section 2.3 we will come back to these relations and study their implementation into an L ∞ algebra. Similarly one can also define a non-commutative Chern-Simons theory in threedimensions, whose equation of motion is For usual CS and YM-theory, it was explicitly shown in [18] that both their symmetries and their dynamics are governed in an algebraic way by the objects and relations of an L ∞ algebra. Before reviewing this, in the next section we give a brief introduction into the general notion of L ∞ and also of A ∞ algebras.

L ∞ and A ∞ algebras and gauge symmetries
Following [18], let us review the basis notion of L ∞ and A ∞ algebras and the generic relation of the first to the description of gauge symmetries and their dynamics.
Definition of L ∞ algebra. L ∞ algebras are generalized Lie algebras where one has not only a two-product, the commutator, but more general multilinear n-products with n inputs n : defined on a graded vector space X = n X n , where n denotes the grading. These products are graded anti-symmetric (2.14)

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The set of higher products n define an L ∞ algebra, if they satisfy the infinitely many relations The permutations are restricted to the ones with σ(1) < · · · < σ(i), σ(i + 1) < · · · < σ(n) , (2.16) revealing that 1 must be a nilpotent derivation with respect to 2 , i.e. that in particular the Leibniz rule is satisfied.
and means that the Jacobi identity for the 2 product is mildly violated by 1 exact expressions.
Definition of A ∞ algebras. The definition of an A ∞ algebras is very similar to the definition of an L ∞ algebra. While L ∞ algebras are generalized differential graded Lie algebras with a mild violation of the Jacobi identity, A ∞ algebras generalize algebras with a mild violation of associativity. One has higher products m n (x 1 , . . . , x n ) of degree n − 2, where x i are again elements of a graded vector space. The quadratic relations for the higher products are with the second product defined as

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The first three relations read where whenever an odd degree m n is exchanged with an odd degree x m one gets an extra minus sign. This will become clearer in appendix B where we will consider explicit examples. There we also need the next relation (2.23) Even though gauge theories arise for the open string and string field theory suggest that they are related by an A ∞ structure, [18] proposed that they also fit nicely into the structure of L ∞ algebras.
Gauge theories and L ∞ algebras. 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 [18]. For sure, the pure gauge algebra, called L gauge ∞ , of such theories satisfies the L ∞ identities. 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. Then we choose only two non-trivial vector spaces as (2.24) In this case, the only allowed non-trivial higher product are the ones with one and two gauge parameters n+1 (f, A n ) ∈ X −1 and n+2 (f, g, A n ) ∈ X 0 and the only non-trivial relations are J n+2 (f, g, A n ) ∈ X −1 and J n+3 (f, g, h, A n ) ∈ X 0 . Then, the gauge variations are expanded as This allows to read off the higher products n+1 (f, A n ) ∈ X −1 . It was shown in [18,27,28], that the off-shell closure of the symmetry variations

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are equivalent to the L ∞ relations with two and three gauge parameters. Here the closure relation allows for a field dependent gauge parameter which can be written in terms of L ∞ products as ) . (2.28) Thus, the action of gauge symmetries on the fundamental fields is governed by an L gauge ∞ algebra. However, this is not the end of the story, as string field theory suggests that also the dynamics of the theory, i.e. the equations of motion, are 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 many more higher products can be non-trivial and one has to check in a case by case study whether indeed the L full ∞ algebra closes. The higher products n (A n ) ∈ X −2 are special as they give the equation of motion that is expanded as Moreover, the structure admits that the closure condition (2.26) is only satisfied on-shell, i.e. there can be terms n+3 (f, g, F , A n ) ∈ X −1 on the right hand side. In case one has off-shell closure (like for the CS and YM theories considered in this paper) all these higher product are vanishing. Moreover, the gauge variation of F is given by reflecting that, as opposed to the gauge field A, it transforms covariantly. It was proposed that for writing down an action for these equations of motion one needs an inner product satisfying the cyclicity property for all A i ∈ X −1 . Then, the equations of motion follow from varying the action Now, as two examples, we analyze how U(1) NC Chern-Simons and Yang-Mills theories fit into the scheme of L ∞ algebras. In this section, we consider the case of the Moyal-Weyl star-product, i.e. the NC-parameter Θ is constant. In this case, the computation is very similar to the analysis of ordinary (non-abelian) CS and YM theories discussed in [18].
L ∞ structure of non-commutative CS. The vector spaces are still as in eq. (2.24) or as in eq. (2.29). Some of the relevant relations have already been given in section 2.1. From the gauge variation (2.6), we can read off 3 and from the off-shell closure condition (2.7) with all higher products vanishing, e.g. n+1 (f, A n ) = 0 for n ≥ 2. The equation of motion (2.11) motivates the choice for the non-vanishing products which fixes the last term to be Since all 3 are vanishing, the remaining four J 3 relations do only contain 2 2 -terms. As the star commutator satisfies the Jacobi identity, these are all satisfied. Let us also mention that the field strength can be expressed as

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Clearly, by setting all elements in X −2 to zero, one gets the sub-algebra L gauge ∞ . The latter is the same for NC-CS and NC-YM. Defining the inner product as one can integrate this to an action (2.34). Thus, we have seen that the U(1) NC-CS theory fits nicely into the L ∞ framework, where the highest appearing products are 2 .
L ∞ structure of non-commutative YM. A similar computation can also be done for NC-YM theory. The case of usual non-abelian YM-theory was first formulated in [29,30].
Here we follow the same path as in the more recent paper [18].
Since the action of a gauge transformation on the fields and its closure are the same as for NC-CS theory, the products 1 (f ) and 2 (f, A) from (2.35) and 2 (f, g) from (2.36) are still valid. The equations of motion (2.10) allow one to read-off the higher products

NC gauge theories arising in string theory
We just showed that both NC-CS and NC-YM on flat Minkowski space with constant NC-structure Θ fit into the scheme of L ∞ algebras. However, not all consistent D-branes (boundary states) in string theory are of this simple type, as there do also exist D-branes wrapping curved submanifolds and carrying a non-constant gauge flux on the brane worldvolume. Therefore, the question arises whether also the expected NC gauge theory on such JHEP05(2018)097 branes fit into the scheme of L ∞ algebras. In this case, the Kontsevich star product (2.3) indicates that one gets extra derivative terms ∂Θ.
Before we continue along these lines, in this section we want to remind the reader of a few stringy circumstances where non-commutativity with non-constant Θ does appear. These will be branes in exactly solvable WZW models for compact groups and recent advances related to integrable deformations of AdS 5 sigma models.

D-Branes in WZW models
In this section we review some of the features of D-branes 4 in WZW models relevant for us. WZW models are exactly solvable sigma models whose target spaces are group manifolds equipped with non-trivial NS-NS three-form fluxes. Their distinctive feature is that the corresponding two-dimensional conformal field theories are explicitly known and given by the unitary series of Kac-Moody algebras. As a consequence it was possible to construct boundary states in the CFT that turned out to correspond to certain branes wrapping conjugacy classes of the group manifold [3,31] carrying non-constant two-form flux. In this section, we will review the semi-classical description of these consistent branes.
Preliminaries. The starting point is the two-dimensional world-sheet action of a WZW model where h denotes the general element of a (simple) Lie-group G and Σ a three-manifold whose boundary is the closed string world-sheet. From the WZW sigma model action one can directly read off the metric and the NS-NS three-form flux Here the total derivative is with respect to the target space coordinates. Since this gives a CFT, the metric and the H-flux satisfy the string equations of motion for the metric and the B-field at any power in α , it only needs some additional input to also satisfy the dilaton equation of motion. This can be a linear dilaton ϕ(z) depending on an orthogonal direction (like it appears for the deep throat limit of the NS5-brane solution). The question which D-branes can be consistently introduced into these closed string backgrounds has been under intensive investigation. Here we just focus on the most simple JHEP05(2018)097 set of such branes. Since the WZW model describes a background with a non-trivial B-field, three issues arise.
First, one expects that the effective theory for the gauge field on the brane becomes non-commutative with the non-commutativity being controlled by an antisymmetric bivector Θ = Θ ij ∂ i ∧ ∂ j , which is part of the so-called open string fields, where the indices i, j are along the brane world-volume. Since the DBI action is established only for adiabatic field configurations, there will presumably be higher derivative corrections to this field equation. However, for constant F it includes all higher order α corrections. Third, due to the non-trivial H-flux in the bulk, its restriction on the brane has to satisfy dF = H, i.e. it is a total derivative of a globally defined two-form. Therefore, H must be trivial in the cohomology on the brane [H| D ] = 0. This is also called the Freed-Witten anomaly cancellation condition [32]. Note that this does not mean that H| D has to vanish identically on the brane.
Geometry of D-branes in WZW models. In this section we provide a set of branes for which the geometric semi-classical identification is known [33][34][35][36]. Our presentation follows the appendix of [36]. Indeed in the CFT there exist boundary states corresponding to branes wrapping conjugacy classes Here, for the element h one can always choose a representative from the Cartan torus M (χ) = exp(iχ · H). Then the position of the brane is labelled by χ and the coordinates along the D-brane can be parametrized by angular variables ψ according to This provides suitable coordinates for this brane configuration that admits a very explicit description of the geometry and the fluxes. Note that the (generic) dimension of these branes is d = dimG − rkG. Since these configurations do correspond to boundary states in the CFT, one expects that the open string equation of motion and the Freed-Witten anomaly condition are satisfied. Now one defines one-forms θ α = θ α i dψ i on the D-brane via
Then one can show that the metric (3.2) restricted to the brane world-volume can be expressed in the nice way where the sum is over all positive roots. This form neatly shows the separation of the dependence on the brane positions χ and the angular coordinates ψ along the brane. Moreover, one can choose a gauge so that the NS-NS two-form has legs only along the brane. Indeed, Choosing gives H = dB. Therefore, the restriction of the B-field onto the brane is also given by this expression, i.e. B| D = B. This by itself does not satisfy the open string equation motion (3.5), but has to be supplemented by a non-trivial gauge flux on the D-brane. This is also known quite explicitly as The quantization of the gauge flux fixes χ = 2π(λ + ρ)/k, where ρ = α>0 α/2 denotes the Weyl-vector and ρ an element from the weight-lattice. Thus, the total two-form flux on the brane is given by Now, one can explicitly compute similar expressions for the fields in the open string frame (3.4). For the metric we find the simple result and for the anti-symmetric bi-vector Note that in the last expression the dual one-vectorsθ α appear. For the dilaton in the open string frame one gets which does not depend on the coordinates along the brane. As a consequence, the open string equation of motion (3.5) is equivalent to which involves the Levi-Civita connection with respect to the open string metric G.

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Example: SU(2) WZW. Let us discuss the most familiar case of the SU(2) WZW model. In this case the target space is an S 3 with a non-constant H-flux through it. Due to the Freed-Witten anomaly condition, it is clear that there does not exist a D-brane wrapping the entire S 3 . However, a class of D-branes is given by the orbit O(D) := {k −1 Dk} with D denoting an element from the one-dimensional Cartan torus. Therefore, generically this describes a brane wrapping a two-dimensional submanifold of S 3 .
To apply the construction from the last section, we introduce a basis of correctly normalized generators of SU (2): Here σ i denote the Pauli matrices. The Cartan torus D(χ) = exp(iχH) is and the orthogonal directions to the D-brane can be parametrized by so that we write an element of SU(2) as M = N −1 D(χ)N . Evaluating (3.2), the metric on the SU(2) group manifold reads Thus, the semi-classical large radius limit corresponds to k → ∞.
Computing the total volume of the S 3 and comparing to other parametrizations of the SU(2) we can fix the ranges of the variables as 0 ≤ χ ≤ √ 2 π, 0 ≤ ϕ ≤ π 2 and 0 ≤ ψ ≤ 2π. Indeed we get √ G dχdϕdψ = 2π 2 R 3 . Next we evaluate (3.3) and obtain so that the flux integral 1 (2π) 2 S 3 H = −k is indeed quantized and the H-flux goes to zero for large k.
For the holomorphic one-forms (3.8) on the D-brane one obtains Inserting this into (3.9), we indeed find the metric (3.19) on S 3 restricted to the D-brane world-volume This metric describes an S 2 of radius r = R sin( χ √ 2 ). As described, for the B-field one can choose a gauge so that it has only legs on the D-brane. For SU(2) this simply reads The gauge flux quantization condition 1 2π F ∈ Z leads to χ = 2π k m √ 2 with 0 ≤ m ≤ k, which agrees with the formula below eq. (3.11) by observing that the weight lattice of SU (2) is λ = Z/ √ 2 and that its Weyl-vector reads ρ = 1/ √ 2. For the two choices m = 0, k, the co-dimension one D-brane degenerates to a point-like D-brane sitting at the north-or south-pole of the S 3 , respectively. Now, one can simply proceed by computing the globally defined two-form flux F = B + 2πF on the brane and the open string measure as Using the dual holomorphic one-vectorŝ and evaluating (3.14), we get for the antisymmetric bi-vector Θ ij on the brane Multiplying this with the measure (3.25) one realizes that the ϕ dependence drops out so that the open string equation of motion (3.5) is trivially satisfied on the D-brane.
Since the brane is two-dimensional, one trivially has Π ijk = 0, as well as ∇ k Θ ij = 0. In appendix B we also work out the SU(3) WZW case and show that there one has a codimension two D-brane supporting a non-vanishing Π ijk . Therefore, not all brane solutions of the leading order string equations of motion necessarily have Π ijk = 0 so that in our later analysis we will also admit a non-vanishing Π ijk .
In the semi-classical limit, the two-dimensional world-volume of the D-brane is expected to support a non-commutative (but still associative) gauge theory. Since the worldvolume is compact, for fixed but large k, there can only be a finite number of quantum cells so that the non-commutative gauge theory turned out not to be a field theory but rather a matrix theory. Using the operator product expansion of the corresponding vertex operators, this theory has been derived in [22] and, as we discuss next, provides the first non-trivial application of our L ∞ bootstrap program.

L ∞ algebra for the fuzzy 2-sphere
Let us first briefly review some relevant features of this construction of the NC gauge theory in the fuzzy sphere limit (for a little review see [37]) (3. 28) This means that one takes the zero-slope and the large radius limit.

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The rational boundary states are known explicitly and the open string excitations at lowest energy are given in terms of the ground states in the open string sector. As discussed above, branes wrapping the conjugacy classes S 2 ⊂ S 3 are labelled by an integer 0 ≤ m ≤ k. This integer determines a representation of the SU(2) current algebra. In this section, we use instead the half-integer representation labels j. The open string spectrum can be organized into the SU(2) k representations that appear in the fusion product of m with itself Here, one was working in the large k limit, such that no truncation appears in the fusion rules. It can now be observed that the representation (j ) contains 2j +1 ground states |Y j n labeled by j , n with n ∈ −j , . . . , j . Geometrically, one can think about them in terms of spherical harmonics on S 2 . These finitely many states are analogous to the infinitely many states exp(ikX) in the flat Moyal-Weyl case.
Note that the space of ground states is (2j + 1) 2 dimensional. As proposed in [31], these ground states can be identified with square matrices Mat(2j +1). SU(2) acts on these matrices in the adjoint representation. This representation is not irreducible; decomposing it into irreducible representations reproduces precisely the decomposition (3.29).
Furthermore, there is a product structure on the space of ground states arising from the (truncated) operator product expansion (OPE) between the corresponding vertex operators. Since their conformal weight h j = j(j + 1)/(k + 2) goes to zero in the k → ∞ limit, the boundary OPE becomes regular in this limit. As it turns out [22], the information about these OPEs is precisely encoded in the non-commutative matrix product f · g in Mat(2j + 1). This in particular allows to compute the correlation functions of arbitrary vertex operators in terms of traces over products of matrices. The upshot is that one deals with an associative matrix algebra. The matrix product plays the role of the Moyal-Weyl star-product, so that we define The above structure includes an action of angular momentum on the spherical harmonics. It is obtained from the OPE between the WZW currents and the vertex operators corresponding to the ground states. From this one obtains for any matrix A ∈ Mat(2j + 1) In the flat space limit, the operators L a can be thought of as derivatives L a → −i∂ a . However, these operators do not commute but satisfy where f abc are the totally antisymmetric SU(2) structure constants. The effective theory on N branes of type j wrapping this fuzzy 2-sphere was described as a gauge theory with the gauge potential A a ∈ Mat(N ) ⊗ Mat(2j + 1). Here Mat(N ) labels the Chan-Paton factors. This gauge field has to satisfy the physical state condition L a A a = 0.

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Using CFT techniques, the effective action was shown to be a sum of a Yang-Mills term and a Chern-Simons term. The two terms are separately invariant under the following gauge transformation where f is an arbitrary matrix in Mat(N ) ⊗ Mat(2j + 1). Note that the derivative operator L a only acts non-trivially on the degrees of freedom in Mat(2j + 1). The closure of two such gauge variations gives Introducing a field strength the action can be expressed as The resulting equation of motion can be written as (3.38) As a first application of our approach, we now show that the form of this NC gauge theory on the fuzzy sphere can be bootstrapped by invoking an L ∞ structure. The computation turns out to be similar to the Moyal-Weyl case but includes some corrections terms that can be traced back to the non-trivial commutator (3.32) of the derivatives. Let emphasize that we proceed not just by simply checking the L ∞ algebra but by bootstrapping the higher products via the L ∞ relations. Of course, one needs some initial information to get started.
As usual, we consider the graded vector space X 0 ⊕ X −1 ⊕ X −2 with now matrix valued gauge parameters in X 0 , gauge fields in X −1 and equations of motion in X −2 . From the gauge variation (3.33) and the closure condition (3.34) we read-off Then, imposing the L ∞ relation J 2 (f, g) = 0 fixes

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where, as indicated, the first two terms come from the variation of the YM action and the last term from the variation of the CS action. First, after using (3.32) we realize that so that only the combination of the two kinetic terms satisfies the relation J 1 (f ) = 1 ( 1 (f )) = 0. Therefore, if one were missing the contribution to the kinetic energy from the CS-term, one would be forced to introduce it by the nilpotency condition in the L ∞ algebra. Moreover, one can further simplify where the second term actually vanishes by the physical state condition L b A b = 0. Next, we consider the L ∞ relation J 2 (f, A) = 0. A straightforward computation reveals that this can be satisfied by defining Next, we observe that the 2 2 -terms in relations J 3 (f, g, h) = J 3 (f, g, A) = J 3 (f, g, E) = 0 involve only matrix commutators so that they can directly be satisfied by setting The only non-trivial relation is J 3 (f, A, B) = 0. However, one can check that the extra terms coming from the second line in (3.45) cancel against each other so that the computation is analogous to the Moyal-Weyl case presented in section 2.3. Thus, this relation fixes which is again consistent with the order O(A 3 ) term in the equation of motion. From the higher order relations only J 4 (f, A, B, C) = 0 is not trivially satisfied, but eventually vanishes by the Jacobi identity of the matrix commutator. Thus, after taking the initial data 1 (f ), 1 (A) and 2 (f, g) we have bootstrapped the remaining terms appearing in the gauge variations and the equations of motion by imposing the relations of an L ∞ algebra. In particular, we found the extra correction terms (∼f abc ) in the equations of motion. We consider this as first compelling evidence that the form of NC gauge theories (arising from string theory) is governed by an L ∞ structure. We will continue to elaborate on this idea in section 4.

Non-constant Θ via integrable deformations
As we have seen, branes in WZW models can lead to a non-constant Θ on a curved space. However, for compact group manifolds the effective NC-gauge theory in the large volume limit is rather a matrix model than a NC field theory based on a star-product with nonconstant Θ.
In this section, we recall that recently string theory examples have been identified that are supposed to give rise to NC-field theories on flat Minkowski space with non-constant Θ. These appeared in the context of integrable deformations of the AdS 5 sigma model [4]. Here, we do not want to review the whole construction and its refinements, as we are only interested in one of its aspects.
It was shown that from certain solutions to the classical Yang-Baxter equation one can extract a closed string metric g ij , Kalb-Ramond field B ij and dilaton φ (and R-R fields) that satisfies the string equations of motion and, for the deformation going to zero, gives back the AdS 5 geometry. This construction can be seen as a generalization of the early analysis [38,39] of the supergravity background dual to the Moyal-Weyl NC-gauge theory.
Expressing the deformed solution in the open string frame [40][41][42][43] revealed that the open string metric is still the one on AdS 5 , the open string dilaton is constant and the only change is the anti-symmetric bi-vector Θ. There are cases, where the latter restricted to the boundary R 1.3 ⊂ AdS 5 satisfies indeed the open string equation of motion (3.5). Therefore, one expects that the gravity theory in the bulk is dual to a NC-gauge theory on the flat boundary with non-constant Θ ij .
Here, we just present one example that we took from [42,43]. Taking one specific solution to the classical Yang-Baxter equation, in the open string frame gives a flat metric with Here η is the deformation parameter. One can readily check that ∂ i Θ ij = 0 is satisfied for all j = 0, . . . , 3 and that Π ijk = 0. Therefore, string theory admits solutions that are expected to give rise to NC gauge theories on flat (or curved) spaces. Here we just cite one comment from [40]: 'While likely technically involved, we believe it should in principle be possible to construct (supersymmetric) gauge theories on such non-commutative spaces using the methods developed in [1,6,8]'. Our proposal rather is that such theories can be bootstrapped via L ∞ algebras.

L ∞ structures for non-constant flux
As we have just seen, the question now arises whether one can deform the Moyal-Weyl case and formulate a consistent NC-gauge theory for non-constant Θ. Motivated also by the previous example of branes in the SU(2) WZW model, we investigate whether the L ∞ bootstrapping approach works and gives reasonable results. In this main section of the JHEP05(2018)097 paper, we investigate this novel approach for flat/curved space with a completely generic non-constant Θ ij (x) with even non-vanishing Π ijk . First we consider just the action of the gauge symmetry and its closure, i.e. we construct the corresponding L gauge ∞ algebra. In appendix B we explicitly show that this can be refined to an A gauge ∞ algebra. First, let us explain that for non-constant flux a serious problem appears, for which L ∞ offers a new solution.

An issue for non-constant Θ
In the previous computations in section 2.2 it was essential that the non-commutativity structure Θ ij was constant. For non-constant Θ ij , even at linear order one runs into the following issue concerning the Leibniz-rule. Consider the generic star-product between functions and apply ∂ a . One finds i.e. the derivative does not satisfy the usual Leibniz-rule with respect to the star-product. One way to resolve this issue is to generalize the co-product, leading to the general structure of a Hopf-algebra. For this purpose one recalls that for the usual product of functions where we have used the action H ⊗ A → A with δ a (f ) = ∂ a f . For consistency, the co-product should be co-associative, as well as admit a co-unit and an antipode, see e.g. [13] for details. Then, the Leibniz-rule can be abstractly written as The relation (4.2) can be written in an analogous manner by defining the new product

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between functions and the adjusted or deformed co-product ∆ (δ a ) for elements in H. Then (4.2) can still be written as Note, that the consistent definition of the deformed co-product satisfying the coassociativity condition (4.6) and the relation (4.9) is a highly nontrivial problem. The known solution is to use an invertible element F ∈ H ⊗ H, called a twist, with the help of which the original star product can be represented as Then the deformed co-product is given by However, only very few examples of star products originating from a twist are known. Another point which should be mentioned here is that the deformed co-product is still co-associative and that is why in the Hopf-algebra approach no higher products or brackets are needed to compensate the violation of the original Leibniz rule. This is the key difference with our proposal in this paper. Nevertheless, we leave for the future a better understanding of the relation between our approach and other previous approaches to the construction of non-commutative gauge theories.
In view of the proposal that generic (gauge) symmetries in string theory are related to L ∞ structures, let us have a second look at the violation of the naive Leibniz-rule (4.2). Recall that for NC-Yang-Mills theory we found 1 (f ) = ∂ a f . If we define 2 (f, g) = i[f, g] • = i(f • g − g • f ) ∈ X 0 then by anti-symmetrization the relation (4.2) is closely related to (4.12) From this point of view, the correction term ∂Θ only indicates that we should better not Note that in the L ∞ algebra, 2 (f, g) ∈ X 0 and 2 (f, A) ∈ X −1 are a priori different products. Thus, we can still satisfy the usual L ∞ Leibniz-rule by changing the action of the NC-gauge symmetry on the gauge fields

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By construction, this guarantees that the closure condition is indeed satisfied up to linear order in Θ. Proceeding in this way, the higher products of the L ∞ algebra will receive higher derivative corrections, leading to corrections to the action of the symmetries on the gauge fields and eventually to the equations of motion. This latter approach seems to be completely different from the resolution of the problem via twisted symmetries and Hopf-algebras and much closer to the structure of symmetries in string theory. In the remainder of this section, we will work this out in more detail and compute the corresponding derivative corrections to L gauge ∞ up to second order in Θ. At this order also the non-associativity enters.
The resolution of the Leibniz-rule was done at linear order in Θ and it is of course not clear whether the procedure can be consistently continued to higher orders in Θ. To get some confidence, starting with the Kontsevich star-product (2.3) at second order in Θ, in this section we construct the corresponding L gauge ∞ algebra. Thus, we have the vector space X 0 ⊕X −1 containing gauge parameters and gauge fields still choose Moreover, for 2 (f, g) we want to have the Kontsevich star-commutator Note that the even order terms in Θ drop out in the star-commutator. Therefore, the analysis of the Leibniz rule J 2 (f, g) = 0 from the previous section is still valid and we get Next, we have to impose the L ∞ relations J 3 (f, g, h) = 0 and J 3 (f, g, A) = 0. The first relation explicitly reads The first line is just the Jacobiator for the star-commutator and yields that we do not require to be vanishing. Taking into account the graded symmetry of the brackets we find

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Next, we have to analyze the relation J 3 (f, g, A) = 0, which is explicitly given by where we used 1 (A) = 0. The last two terms involve the three-product 3 (A, B, f ) that needs to be determined from this relation. For this purpose, we calculate that apparently satisfies the relation Taking into account the symmetry 3 (A, B, f ) = 3 (B, A, f ), (4.23) motivates to make the ansatz Inserting this into the L ∞ relation (4.19), using (4.25) and comparing coefficients we find Note that for β and γ we have 3 different equations for the two unknowns. Moreover, the relation (4.25) was used to solve for α, and is in fact the consistency condition for the existence of the solution of the equation (4.19), see [25] for more details. Thus, it is highly non-trivial that indeed the L ∞ relation J 3 (f, g, A) = 0 can be satisfied via (4.28) Note that, as opposed to 3 (A, f, g), the three-product 3 (A, B, f ) is non-vanishing in the associative case, either. Recall that our computation was exact only up to second order in Θ. Setting now all higher products to zero up to this order, all higher relations J n = 0 for JHEP05(2018)097 n ≥ 4 are automatically satisfied, as well. This is because 2 3 is already third order in Θ. Therefore, for non-constant Θ we have constructed a consistent L gauge ∞ algebra, for which derivative ∂Θ corrections induce non-vanishing higher products (even in the associative case). This is very compelling, as in the course of this computation there arose non-trivial consistency conditions that just happened to be satisfied.
We also analyzed whether the gauge structure features an underlying A ∞ algebra. Since the higher products are not any longer graded symmetric, for that purpose one has to determine many more individual higher products that are also constrained by more A ∞ relations. In this sense, an A ∞ structure can be considered as a refinement of an L ∞ structure. Since the computations turned out to be quite lengthy and involved, we delegated the presentation of the positive results into appendix B.
Let us proceed and extend the former L gauge ∞ algebra for the action of a gauge symmetry on gauge fields by a vector space X −2 that contains the equation of motion of a stardeformed 3D Chern-Simons theory and a star-deformed U(1) Yang-Mills theory.

NC Chern-Simons theory
Let us first consider a NC-CS theory. Since a metric does not appear neither in the Kontsevich star-product nor in the topological CS-theory, we suspect that the following considerations are valid irrespective of a metric. Let us first consider the appearing structure up to linear order in Θ. From the former discussion, we expect to find also here ∂Θ corrections.
As said, we have three vector spaces like with the following derivatives that clearly satisfy 1 ( 1 (f )) = 0. With we have already seen that the Leibniz-rule 1 ( 2 (f, g)) = . . . fixes Next, one has to check the Leibniz rule By making an ansatz for 2 (A, B) and 2 (f, E), using that the former is symmetric and fixing the coefficients one finds JHEP05(2018)097 Therefore, by requiring the consistency of an underlying L ∞ algebra for the case of a Θ deformed 3D Chern-Simons theory, we have extracted a ∂Θ correction to the equation of motion (4.36) We find it tantalizing that the algebraic structure of an L ∞ algebra allowed one to fix derivative corrections to the equations of motion of a non-commutative CS theory. Of course, so far this computation is only up to linear order in Θ, but we conjecture that it can be extended in a consistent way to higher orders. As a non-trivial check let us now consider the corrections at second order in Θ. This is the first instance where also the associators appear.
NC-CS at order O(Θ 2 ). The star commutator to this order remains unchanged, so do the previously defined structures for 2 . However, at this order there appear higher brackets 3 . The expressions for 3 (A, f, g) and 3 (A, B, f ) were found in section 3.1. Taking into account that now X −2 is non trivial, one may also have non-vanishing 3 (E, f, g) ∈ X −1 , Let us start with 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 3 (E, f, g) = 0. Next we consider J 3 (E, f, g) = 0 0 = 2 ( 2 (E, f ), g) + 2 ( 2 (g, E), f ) + 2 ( 2 (f, g), E) After a tedious computation we find a couple of cancellations and simplifications. First we obtain that two of the four terms do vanish The term with two derivatives using the relation (4.25) can be written in the convenient way Note that this is explicitly symmetric under the exchange of the gauge fields A and B. Moreover, the numerical prefactors are just right to directly read off a totally symmetric three product

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Next we come to the contribution with three derivatives. Here one has essentially two terms, one proportional to Π ijk and one that does not vanish in the associative case Again the relative coefficients are just right to define a totally symmetric three-product (4.45) Therefore invoking the L ∞ relations, in (4.43) and (4.45) we have determined a totally symmetric three-product 3 (A, B, C) = . Note that all the formerly determined higher products went into this computation and that all the prefactors just came out right to fit into the L ∞ structure. This procedure works both for the associative and the non-associative cases. Even in the associative case the three-product is non-vanishing and receives derivative corrections.
Moreover, all higher L ∞ relations are automatically satisfied at O(Θ 2 ). This is all very compelling and makes us believe that this procedure can be continued to higher orders in Θ. As a consequence, up to quadratic order, the equation of motion of the NC-CS gauge theory reads Let us emphasize that this is designed such that it transforms covariantly under an (∂Θ) corrected NC gauge transformations

(4.47)
Comments on an action. After we have successfully derived the equations of motion for the NC-CY theory up to order Θ 2 , one can ask whether these can be integrated to an action. Just to show what kind of issues appear, here we consider the simple case of the equations of motion up to linear order in Θ. Recall that the equation of motion is As mentioned in section 2.2, for defining an action one needs a inner product, which we choose to be the same one as for the Moyal-Weyl case For the action we also use the same form Varying now this action with respect to the gauge field one has to do a partial integration.
In doing this, we require

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which can be considered as the open string equation of motion for a flat brane or a natural topological generalization of the latter. Thus, after variation we can express the result as (4.52) Let us make two remarks: first, the first four terms are already the ones appearing in the equation of motion. Second, from the variation one gets extra terms, e.g. those multiplying δA i . However, all these terms explicitly contain a factor of ∂Θ and are proportional to Since the leading order equation of motion tells us that this combination vanishes, all the terms like ∂Θ(∂ k A l − ∂ l A k ) are already of order Θ 2 and can be safely dropped from the equation of motion at first order in Θ. Therefore, only after carefully working at linear order in Θ one gets the equation of motion (4.48) from the action (4.50). This means that the inner product itself does not satisfy the cyclicity property (2.33). It would be interesting to see whether such a reasoning also works up to second order in Θ or whether also the inner product receives some derivative ∂Θ corrections. We leave this involved study for future research.

NC Yang-Mills theory
So far we have considered NC deformations of the topological CS-theory. To really get into contact with the D-branes appearing as solutions of string theory, we finally consider non-commutative U(1) YM theory. Since in this case, the computations turn out to be more involved than in the CS case, we restrict ourselves in this section only to linear order in Θ and leave the generalization to higher orders for the future.
Since the equation of motion involves the metric, we first discuss the simplest case of a flat background with a non-constant NC-structure Θ and then generalize this to a curved background of metric g ij and non-constant Θ.
NC-YM on flat background. Recall from section 2.3 that for the Moyal-Weyl starproduct we have three vector spaces like with the following -products

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Note that at linear order in Θ the three-product 3 (A, B, C) is vanishing. As for the NC-CS theory, imposing the Leibniz-rule we expect to get a derivative correction (1) 2 (A, B). Indeed, going through the computation we arrive at the familiar expression (4.55) and the remaining term (4.56) Again, the relative coefficients are just right to be able to read off a symmetric 2 (A, B) As we see, the linear order corrections are much more involved than for NC-CS theory so that we stop here and leave higher order computations for future work. Here it is important to note that so far we did not encounter any obstacle for solving the L ∞ relations. Clearly, here we are entering new territory, as the usual star-product is constructed with respect to a Poisson structure only, without any mentioning of a metric or a connection. In this section, at least up to linear order in Θ, we investigate whether one can also bootstrap the first terms of a NC-YM theory on a curved manifold following our strategy of imposing the L ∞ algebra.
Looking at the usual abelian Yang-Mills theory on a Riemannian manifold, our starting point is that where the second definition follows from varying the action where indices are pulled up and down with the metric G ab . Moreover, for the star-product between two functions up to linear order we keep and define The Leibniz rule J 2 (f, g) = 0 can now be used to bootstrap the form of 2 (f, A), assuming that ∇ k Θ ij = 0

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where of course ∇ i f = ∂ i f . Thus, we realize that from this perspective it is more natural that indeed the covariant derivative appears when star-multiplying tensors. Since two covariant derivatives do not commute and give we must be prepared that there will arise curvature 5 corrections to the expression we encountered in the previous sections. Next, we impose the Leibniz rule J 2 (f, A) = 0 from which we are able to read-off 2 (A, B) and 2 (f, E). After reordering covariant derivatives and applying Bianchi-identities for the curvature, we finally arrive at and the more involved expression (4.66) Here, we have used . Note that this expression has the correct limit in the flat case and manifestly shows the curvature corrections. Since all 2 -product are at first order in Θ, all higher order relations are satisfied up to linear order and we have succeeded to bootstrap a NC-YM theory on a curved manifold up to linear order in a covariantly constant Θ. This could be continued to higher orders but we do not pursue this further here and just state that all the discussed examples exemplify that the string theory motivated L ∞ bootstrap program provides a promising novel approach to algebraically construct noncommutative gauge theories in regimes that were not completely accessible yet.

Conclusions
Motivated by its appearance in string theory and first successes when applied to the matrix valued NC gauge theory on the fuzzy 2-sphere, in the main part of this paper we have successfully carried out an L ∞ bootstrap program for determining higher derivative corrections to NC gauge theories for non-constant and in general non-associative NC-structure Θ. What is changed is both, the action of the gauge symmetry on the gauge fields and their equations of motion. An interesting open question is whether one can also find an action for them. We leave this non-trivial question for future work.
We believe that this approach is different from former attempts to solve this problem but is based on a string theoretic well motivated guiding principle. By successively applying or solving the L ∞ relations we managed to determine higher -products. Since we were pursuing a perturbative approach in Θ, the actual computations become more and more JHEP05(2018)097 involved. Note that at higher orders also the Kontsevich star-commutator receives derivative corrections that one needs to take into account. (For the A ∞ algebra from appendix B such corrections already appeared at second order and could consistently be handled.) But it is promising that, up to second order in Θ, we did not encounter any obstacle, even not in the non-associative case. To gain even more evidence for the self-consistency of this approach, one could try to implement this bootstrap algorithm and push the computer to iteratively produce higher and higher orders. Mathematically, one could also ask for a proof that our algorithm always works.
When considering NC-YM theory on a curved space with covariantly constant Θ, the L ∞ structure was telling us that one should better introduce a star-product that uses covariant derivatives when acting on tensors. It would be interesting to investigate whether our first order results can be extended to higher orders.
Of course, generally one could contemplate on other possible problems where such an L ∞ based bootstrap approach might be worthwhile to pursue.

A Semi classical branes in SU(3) WZW
As we have seen, the SU(2) group manifold is not rich enough to non-trivially check whether on-shell brane configurations exist with non-vanishing Π ijk . The reason was that the boundary states describe at most two-dimensional branes on which H restricts trivially. Thus, we now consider the eight-dimensional SU(3) WZW model which admit sixdimensional branes. The non-vanishing Betti numbers of this group manifold are This manifold can be considered as a S 3 fibration over a five-dimensional base M 5 . Clearly, in order to satisfy the Freed-Witten anomaly condition, the D-brane world-volume should better not contain the S 3 . From these topological considerations this example seems to be rich enough to provide a non-trivial example with Π ijk = 0. To explicitly describe the SU(3) group manifold we introduce the matrices  ψ 3 ). Thus, the six coordinates φ i along the brane are φ ∈ {ϕ 1 , ψ 1 , ϕ 2 , ψ 2 , ϕ 3 , ψ 3 } Moreover, in this normalization the positive roots are given by For the metric on SU(3) one obtain where the second term is the metric restricted to the D-brane. The metric components g ij (φ) are partially long expressions in terms of the coordinates along the brane. Therefore, we just list a few components to convince the reader that the expressions are indeed very explicit At generic positions χ 1,2 this gives a smooth metric on the D6-brane. However, at the three boundaries α i · χ = 0 the metric degenerates to a four-dimensional metric. Therefore, at these positions one gets D4-branes. At the intersection of two such lines the whole metric degenerates thus yielding D0-branes. Therefore, the position moduli space has the form displayed in figure 1. The determinant of the metric has a simple form Integrating this over the domain Next, utilizing (3.10) one computes the B-field and its total derivative to get the Hflux. To find a compact expression for the sechs-bein along the D-brane world-volume it is useful to define the structure "constants" Now, using one can express the H-flux in a very compelling way as Thus, in contrast to the former SU(2) case, the restriction of H onto the D6-brane is not vanishing. However, the restriction onto the D4 and D0 branes at boundary of the moduli space vanishes. Due to i (−1) i α i · χ = 0 these linear terms in χ do not contribute to the three-form flux H| D on the brane. Taking into account (3.11), this is nothing else than the manifestation of the fact that a pure gauge flux F = dA satisfies dF = 0. Therefore, in accord with the Freed-Witten anomaly, the restriction of the bulk H-flux onto the brane JHEP05(2018)097 can be expressed as H| D = dF . Being non-vanishing, there is a good chance to finally also get a non-vanishing Π ijk .
The quantization of the gauge flux F only admits a finite number of allowed D-branes. These are parametrized by 16) with m 1 , m 2 ∈ Z such that they lie inside the domain in figure 1. For k = 1 one only gets the three D0-branes but for k = 3, for the first time, also a D6 is allowed.
Next one can derive an explicit expression of the flux F and compute Similarly, from (3.14) one can get an expression for the anti-symmetric bi-vector Θ and explicitly check that the equation of motion (3.5) is indeed satisfied. Starting with the general expression (3.14), one can check in more detail that the underlying reason for this result are the relations (no sum over α in F αα α ) leading to a cancellation already for each term in the sum over the positive roots in (3.14). Thus, as expected, the highly curved, fluxed D6-brane is a consistent solution of the string equations of motion. Similarly to the H-flux one can write the non-vanishing antisymmetric three-vector Π = [Θ, Θ] S in the very compelling form This explicitly shows that the SU(3) WZW model admits a six-dimensional brane that carries a non-trivial Π. That means that Θ is not a Poisson structure and the related JHEP05(2018)097 Next we analyze the A ∞ relation A 3 (f, g, h) = m 2 (m 2 (f, g), h) − m 2 (f, m 2 (g, h)) + m 1 (m 3 (f, g, h)) + m 3 (m 1 (f ), g, h) + m 3 (f, m 1 (g), h) + m 3 (f, g, m 1 (h)) = 0 . (B.8) For the associator one finds Since m 1 (m 3 (f, g, h)) ∈ X 1 vanishes one can solve (B.8) by with α − β + γ = 2/3. This guarantees that in the associative case, all these three-products vanish. It is straightforward to confirm the relation to the corresponding L ∞ three-product

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and where besides α, γ the κ i , λ i and x 1 , x 2 , y 2 are still free parameters. However, when computing the graded symmetrization of these m-products, all these parameters precisely cancel and one gets the corresponding 3 -product Finally, one has to check the A ∞ relation A 4 (2.23). There are only two possible sets of a priori non-trivial relations with entries A 4 (f, g, h, A) and A 4 (f, g, A, B) and permutations thereof. The A 4 (f, g, h, A) relations are all satisfied up to order O(Θ 2 ) so that we choose a vanishing four-product m 4 (f, g, A, B) ∈ X 0 . The A 4 (f, g, A, B) relations are also all satisfied in the associative case, but in the non-associative case, one needs to introduce non-trivial four-products m 4 (f, A, B, C) ∈ X −1 that are proportional toΠ ijk . As before, we make a general ansatz and similarly for m 4 (A, f, B, C), m 4 (A, B, f, C) and m 4 (A, B, C, f ). One realizes that there appear consistency conditions for the existence of a solution, that are however satisfied once the relations that we encountered before are satisfied. After all, the four parameters in m 4 (f, A, B, C) remain as free parameters with the other three four-products given as

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and We note that via graded symmetrization the corresponding 4 -product is vanishing, being consistent with our findings in section 4.
Having now a non-trivial m 4 -product one also has to worry about the relation A 5 that for m 5 = 0 contains the order O(Θ 2 ) term (B.20) We have checked that all ten relations of the type A 5 (f, g, A, B, C) are satisfied. All higher relations are trivially satisfied up to orderΘ 2 . Let us summarize our findings: we have explicitly constructed the A ∞ algebra up to order O(Θ 2 ) that underlies the non-commutative gauge theory governed by a non-constant and in general even non-associative star-product. By constructing the higher products in a step-by-step procedure, we encountered many derivative ∂Θ-corrections that make the whole algebra and relations highly non-trivial. At each step, we observed that the A ∞ relation under question led to some consistency conditions that were automatically satisfied once the lower A ∞ relations were already satisfied. This is very encouraging and makes us believe that the whole procedure continues also to higher orders in Θ. Up to the level that we were considering, we found that in the associative case, a non-constant Θ induces non-vanishing higher products of type m 2 (f, g) , m 2 (A, g) m 3 (A, B, g) . (B.21) In the non-associative case a further three and a four-product had to be introduced In the conclusion of this section we stress that both, for the consistency of the proposed construction of A ∞ and for the correct relation to the L ∞ , the product m 2 (f, g) was taken to be the Kontsevich star product f • g. Up to second order in Θ, 2 (f, g) = i[f, g] • , coincides with the "classical" (quasi)-Poisson bracket, −{f, g}, which is not the case of the JHEP05(2018)097 product m 2 . It contains "quantum" information in the sense of deformation quantization corrections. A separate question is whether there exists a "classical" L ∞ algebra, where the two-product is always simply 2 (f, g) = {f, g} or whether for consistency one should necessarily take 2 (f, g) as a star commutator, i.e. construct "quantum" L ∞ .
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