Yangian Symmetry for the Action of Planar N=4 Super Yang-Mills and N=6 Super Chern-Simons Theories

In this article we establish the notion of classical Yangian symmetry for planar N=4 supersymmetric Yang-Mills theory and for related planar gauge theories. After revisiting Yangian invariance for the equations of motion, we describe how the bi-local generators act on the action of the model such that the latter becomes exactly invariant. In particular, we elaborate on the relevance of the planar limit and how to act non-linearly with bi-local generators on the cyclic action.


Introduction
Integrability of planar N = 4 supersymmetric Yang-Mills (sYM) theory, its dual string theory on AdS 5 × S 5 and several other AdS/CFT dual pairs of models has proved to be an extremely powerful feature in computing various quantities at weak, strong and even at intermediate coupling strength; see [1] for reviews of the subject. However, despite the many successes related to planar integrability, a rigorous and unified notion of this feature is still missing. In particular, such a notion would be highly desirable towards finding a proof for planar integrability and thus towards firmly establishing a solid foundation for the results alluded to above.
In this regard, the worldsheet theory of AdS 5 ×S 5 strings at strong 't Hooft coupling is in rather good shape because there exist well-established notions for integrability in such two-dimensional field theories. Most importantly, Joe Polchinski with Bena and Roiban showed that the structures of integrability can be formulated in terms of a family of flat connections [2]. However, the non-ultra-locality of the algebra of worldsheet currents poses some difficulty, see [3], and the quantisation of these algebraic structures has several unresolved issues.
The exploration of integrable structures in planar N = 4 sYM took a different path: Various remarkable features were discovered in the investigation of particular observables. On a case-by-case basis, these discoveries led to the application, refinement and development of several integrability-based methods to compute these quantities much more efficiently than by ordinary field theory methods. Some of these features and methods directly matched known integrable structures: for example, the spectral problem of local operators at the leading one-loop order was related to an integrable spin chain [4], and the Bethe ansatz could be applied to the computation of the spectrum in various interesting limits.
Integrability of a physics model goes hand in hand with an enhancement of its ordinary symmetry algebra by a large amount of hidden symmetries. A unifying structure of the integrability-related features of planar N = 4 sYM turned out to be the Yangian algebra Y[psu(2, 2|4)] [5] based on the superconformal algebra psu(2, 2|4) of the model. The Yangian Y[g] is an infinite-dimensional quantum algebra based on a (finite-dimensional) Lie (super)algebra g [6]. Yangian algebras underlie large classes of integrable models where the Lie algebra g usually describes the ordinary symmetries of the model. The latter is spanned by the so-called level-zero generators J A , A = 1, . . . , dim g, and the Yangian algebra enhances it by the level-one generators J A , A = 1, . . . , dim g, which form a similar set as the J A . Importantly, the level-one generators act in a certain bi-local fashion and as non-local transformations they are not easily identified as symmetries and are therefore hidden from a standard analysis. Beyond these, the Yangian contains infinitely many other generators at higher levels which will not be of concern in this work.
The Yangian algebra has been observed and applied for various observables, but in most cases some caveats exist which prevent the action of the Yangian algebra from being considered a proper invariance: In the case of the spectral problem of local operators, the spin chain Hamiltonian almost commutes with the Yangian algebra [5]. However, the cyclic boundary conditions represent an obstacle here because the Yangian generators typically map cyclic physical states to non-cyclic states which have no proper meaning in the field theory context. Furthermore at higher loops, some issues related to gauge symmetry have been observed in [7] which obscure the closure of the algebra. The S-matrix describing scattering of magnon excitations of the spin chain was shown to have exact Yangian symmetry [8], even at finite coupling strength, but this symmetry is merely based on a psu(2|2)×psu(2|2) subalgebra of psu(2, 2|4), or, more precisely, an extension thereof. Also colour-ordered scattering amplitudes of ordinary particles of the four-dimensional gauge theory have been shown to display Yangian invariance [9] which can be understood as an interplay between ordinary and dual conformal symmetry [10]. Unfortunately, also here an exact and complete invariance is spoilt, this time by the presence of infrared divergences due to the scattering of massless particles. The divergences have been regularised in many different ways in order to arrive at suitable results, see e.g. [11], however, such regularisations obscure invariances, and thus make it much less clear in what sense the Yangian algebra can be regarded as a symmetry or how to prove a corresponding statement in general. Nevertheless, Yangian symmetry has proved useful in constructions of scattering amplitudes, see e.g. [12]. A similar picture arises from the study of null polyg-onal Wilson loops which are T-dual to scattering amplitudes [13]. The main difference w.r.t. scattering amplitudes is that the divergences arise from the cusps between light-like segments of the polygon are associated to the ultraviolet rather than the infrared regime, and are thus potentially easier to address. Yangian symmetry also plays a role in the construction of many other observables such as correlation functions and form factors of local operators. However, here the corresponding worldsheet topology of the observables is a clear obstruction for Yangian invariance. Yangian invariance without any of the above restrictions has only been observed for expectation values of smooth, non-intersecting Maldacena-Wilson loops with disc topology [14,15] (even though the analysis is limited to one loop so far).
Altogether, these results demonstrate that the Yangian algebra appears to be some kind of symmetry of planar N = 4 sYM, but it is not evident in what sense this statement can be made precise. They also show that the symmetry is restricted to finite objects with the worldsheet topology of a disc which are necessarily symmetric under ordinary superconformal symmetry. A relevant object with this set of features is the (unrenormalised, single-trace) action of N = 4 sYM itself. Showing invariance of the action could in fact be considered as a proof that the theory is symmetric under the Yangian algebra. Unfortunately, it was unclear how to show invariance largely because it was not understood precisely how to act with a Yangian generator on the action and how to make sense of the planar limit and, eventually, of quantum effects.
In our letter [16] we have laid the foundations for our work to establish the Yangian algebra Y[psu(2, 2|4)] as a symmetry of planar N = 4 sYM by proposing that it is a symmetry of the equations of motion in a strong sense. In this article, we provide a more detailed account of our construction and present explicitly in what precise way the action is classically invariant under the Yangian algebra. The fact that the model possesses a very non-trivial extended symmetry algebra and at the same time is apparently exactly integrable can hardly be a coincidence given the usual relationship between these two features. It should be evident that Yangian symmetry is a way to formally express the integrability of planar N = 4 sYM. Hence, we may declare integrability of a planar gauge theory model to be the presence of Yangian symmetry (or some other alike symmetry algebra). 1 To lend further credibility to our claim, we will treat N = 6 supersymmetric Chern-Simons theory, also known as ABJ(M) theory [17], as the second main example of a gauge field theory known to be integrable in the planar limit following the works [18]. Here, the Yangian Y[osp(6|4)] has been established as a symmetry of colour-ordered scattering amplitudes [19]. We will show that the action of this model is indeed invariant under a Yangian symmetry, and thus our proposed definition of integrability matches with the expectation. In order to convince ourselves of the non-triviality of Yangian symmetryafter all, it might in principle be a feature of a broad range of models -we will furthermore consider pure supersymmetric Yang-Mills theories with N < 4 supersymmetries. These sample field theories can be addressed straight-forwardly within our framework, and there is no indication that they become integrable in the planar limit. In line with this expectation, we will show that these models do not possess Yangian symmetry in the planar limit.
The present work is structured as follows: In Sec. 2 we introduce N = 4 sYM along with its superconformal symmetry and by doing that we outline the notation used in this work. We then review the results of our previous letter [16] on this subject in Sec. 3 with a much more detailed account of our constructions and further explanations and discussions. In Sec. 4 we convert these results to a statement on the invariance of the action which not only serves a definition of Yangian symmetry of the model but can also be viewed as an exact notion of integrability. In Sec. 5 and Sec. 6 we consider ABJ(M) theory and pure N < 4 sYM theory serving as two sample planar gauge theories for which we can test our proposal. Indeed we find that Yangian invariance of the action coincides with our expectations on planar integrability of these models. Finally, we conclude in Sec. 7 and we present a list of open issues regarding integrability within the AdS/CFT correspondence that can be addressed with our framework. The appendices contain a complete account of the superconformal algebra psu(2, 2|4) and its non-linear representation on the fields (App. A) as well as the enhancement to the Yangian algebra (App. B).

N = 4 sYM and superconformal symmetry
We start by introducing N = 4 sYM, the corresponding superconformal algebra and how the algebra acts as a symmetry of our model. In particular, we shall discuss the latter point at length by presenting several different notions of "symmetry". They will later serve as the foundation for our formulation of Yangian symmetry in this model.

N = 4 supersymmetric Yang-Mills theory
The fields of N = 4 sYM consist of a gauge potential A with associated field strength F , four Dirac fermions Ψ and six real scalars Φ. All matter fields transform in the adjoint representation of the gauge group which we assume to be U(N c ). We will thus express all (real) fields as N c × N c (hermitian) matrices, and sequences of fields correspond to matrix products.
As we will be mainly interested in aspects of the classical field theory model related to psu(2, 2|4) superconformal symmetry, it makes sense to use a notation where all vector indices are expressed as bi-spinors. In our convention, Latin indices a, b, . . . = 1, 2, 3, 4 denote su(4) internal spinors, and undotted / dotted Greek indices α, β, . . . = 1, 2 / α,β, . . . = 1, 2 correspond to left / right chiral sl(2, C) spacetime spinors. The coordinates x are given by a hermitian 2 × 2 matrix, and they carry the index structure x βα with the reality condition (x αγ ) † = x γα . The corresponding partial derivatives ∂α β are defined such that ∂α β x δγ = δγαδ δ β . We shall refrain from implicitly raising or lowering spinor indices. Instead we will explicitly contract indices with the help of the totally anti-symmetric symbols ε αγ and εαγ as well as ε abcd (all these symbols equal +1 when the upper or the lower indices are in proper order).
In this notation, the fields of N = 4 sYM all carry two spinor indices of various kinds Here, the bar denotes the hermitian conjugate of the unbarred field. 2 Importantly, the 2 All fields are spinor matrices, and hermitian conjugation is meant to act on the indices of the fields as well such that their order is reversed. Furthermore, hermitian conjugation raises or lowers internal su (4) indices and exchanges undotted with dotted sl(2, C) indices. Effectively we have (Φ ac ) † =Φ ca = −Φ ac , (Ψ a γ ) † =Ψγ a and (F αγ ) † =Fγα =Fαγ. In practice, we will not need these relations.
two scalar fields Φ andΦ are related by the reality condition The gauge-covariant derivative ∇ of some matter field Z and of the gauge potential A itself is defined as 3 Note that the latter definition of ∇A as the associated field strength F is merely a convenient notational assignment which will later allow us to write some expressions in a more uniform fashion. In the spinor notation, we can split this field strength into chiral and anti-chiral components F andF The Lagrangian is tightly constrained by supersymmetry and possesses two marginal couplings, the Yang-Mills coupling constant g YM as well as the topological angle θ; we shall not be interested in the latter, and the former can be expressed as an overall factor of the Lagrangian which we will also drop. The Lagrangian reads The equations of motion are obtained by varying the action with respect to the fields of the theory. To this end, we introduce the notationZ for the variational derivative of the action w.r.t. a generic field Z 4Z More concretely, the set of all variationsZ reads (2.7) With this notation, the equations of motion can be concisely written as {Z = 0}. Moreover, the Bianchi identity for the field strength relates the derivatives of the chiral components (with the opposite sign w.r.t.Ȃ)
In this article we mainly consider a gauge-covariant representation of the superconformal symmetries J ∈ psu(2, 2|4) on the fundamental fields Z ∈ {A, Φ, Ψ,Ψ } of our model. 5 The representation of the translation generators P on a generic field Z reads (2.9) Note that in our representation, shifts are generated by the covariant derivative ∇ rather than by a plain partial derivative ∂. 6 As usual, the gauge potential Z = A is a somewhat special case because it is not gauge-covariant in contradistinction to the matter fields. Notwithstanding, the above rule also explicitly applies to Z = A which is mapped to the (gauge-covariant) field strength ∇A according to the definition (2.3). Lorentz rotations of the fields are generated by the rule where L spin andL spin denote the spin contribution of the operator which acts non-trivially only on the spacetime indices of spinor fields Note that in particular there is no spin action on the gauge field even though it carries spacetime indices. Likewise, the dilatation generator D is represented on a generic field Z by a universal rule with the coefficients ∆ Φ = 1, ∆ Ψ = ∆Ψ = 3 / 2 and ∆ A = 0. Here, the case Z = A is special because ∆ A = 0 does not match the mass dimension 1 of the gauge potential. This curiosity is related to the fact that our representation also involves a gauge transformation, 5 Note that the representation on fields acts by replacing each fundamental field with a transformed field. It does not act on the coordinates, i.e. J·x = 0 for all generators J, and correspondingly it does not act on partial derivatives ∂ either. Consequently, it does not act on functions of the coordinates which are independent of the fundamental fields, such as, e.g., the parameter fields for gauge transformations (unless the latter are defined in terms of the fundamental fields). 6 The difference between a plain and a covariant representation amounts to a gauge transformation generated by the gauge field contracted to the Killing vector of the symmetry. However, this requires a full superspace formulation where all components (in particular the ones along the fermionic directions) of the gauge field and Killing vectors exist. Therefore a formulation in components must remain covariant to some extent.
as we shall discuss below. 7 The representation of the supercharges reads (2.13) We present the representation of the remaining superconformal generators explicitly in App. A. This representation of the superconformal algebra closes on-shell up to field-dependent gauge transformations. For example, it is easy to confirm that (2.14) where G[X] generates a gauge transformation with the gauge parameter field X as follows The appearance of a gauge transformation in the algebra is not surprising given the fact the we use covariant rather than partial derivatives to generate shifts: namely, one can view the difference between ∇ and ∂ as a (field-dependent) gauge transformation iG[A]. 8 9 The advantage of this representation is that covariant quantities remain manifestly covariant. The price to pay, however, is that the relations of the original algebra psu(2, 2|4) are not manifest -to recover them, the ideal generated by gauge transformations must be quotiented out. The related complications are minor for the superconformal algebra, but they introduce considerable difficulty in considering the algebraic relations for the Yangian algebra, as will be discussed in [20].

Notions of symmetry
In the following, we shall discuss three notions of symmetry of a model: invariance of the action as well as a strong and a weak version of invariance of the equations of motion. They will serve as a starting point for our discussion of Yangian symmetry of planar N = 4 sYM which has several complications that can be avoided by some of the notions. We will consider some generator J ∈ psu(2, 2|4) in the context of N = 4 sYM, but the following arguments will be rather general and apply to generic local symmetries of a field theory.
Invariance of the action. Typically, a symmetry J of a model is understood as an invariance of the action S J·S = 0. (2.16) This statement is strong because it implies important structures and relationships for a field theory. For instance, one can derive conserved currents and charges by means of Noether's theorem. Moreover, in a quantum field theory, the Ward-Takahashi identities imply a large set of relationships between various correlation functions. For the case of N = 4 sYM, invariance of the action (2.5) can be checked directly using the representations of J such as (2.9,2.12,2.13).
One can also perform one step of evaluating the symmetry representation on the action and write the above symmetry statement (2.16)  Here, the field index I refers to all dependencies of all fields including the gauge degrees of freedom as well as the full coordinate dependence. 10 As such it is evident that the statement must hold off shell for it becomes trivial when the equations of motionZ = 0 are imposed. In other words, a symmetry implies that (2.17) holds without making use of the equations of motion. Let us now reason in the opposite direction and ask ourselves the following question: Suppose we have a transformation J such that (2.20) holds. To what extent can we consider J a symmetry of the theory? When can we promote this transformation to a symmetry of the action of the theory? The following example shows that (2.20) can hardly be considered a sufficient condition for a symmetry. Weak invariance of the equations of motion is only a necessary condition. We propose that the strong version (2.19) is a sufficient condition for J to be a symmetry of the action.
Example. Let us now demonstrate that the strong invariance condition (2.19) indeed allows to differentiate which invariance of the equations of motion stems from a true symmetry of the action. To this end consider a simple example, the free complex scalar field φ defined by the following action (2.21) The equations of motion following from (2.21) are the wave equations The above equations of motion are weakly invariant as in (2.20) under a global complex rescaling of the fields

Yangian symmetry of the equations of motion
In this section we will review and elaborate on our results of [16] on Yangian symmetry of the equations of motion of N = 4 sYM. The goal of that letter was to establish classical Yangian symmetry of planar N = 4 sYM theory and thus to establish a clear notion of integrability for this model which can later be carried over to quantum field theory. The Yangian algebra is an extension of psu(2, 2|4) superconformal symmetry, and the discussion of Yangian symmetry can follow along same lines as in the previous section. However, there are also several features that distinguish the Yangian algebra and its representation on fields from the more basic superconformal symmetry: • Yangian symmetry alias integrability ought to hold only in the planar limit; nonplanar terms are expected to violate the symmetry. However, there is no evident notion for the planar part of the action; the planar limit is largely related to the composition of Feynman diagrams. How should one realise the planar limit in the context of Yangian symmetry?
• The Yangian is a quantum algebra rather than a Lie algebra. As such there is a large number of presentations which are equivalent for physical purposes. Furthermore, most of the algebraic relations comprise many terms and thus require a lot of patience and attention to verify.
• Yangian representations are typically non-local. Non-local symmetries are harder to detect, and they may not be accessible with conventional tools of field theory.
In particular, Noether's theorem does not apply, at least not directly, and also the notion of quantum anomalies is obscure. In fact, non-locality does not refer to spacetime in our case, but rather to colour space. This again calls out for unconventional methods and ideas.
• A related issue is cyclicity violation. The action of N = 4 sYM is cyclic due to the trace in colour space. Yangian representations respect cyclicity only under very special conditions. We have to formulate these conditions and make sure that they are satisfied.
• Symmetry representations in interacting field theories are often non-linear in the fields. 11 It is largely unknown whether Yangian algebras possess such non-linear representations.
• A related issue is gauge symmetry. Gauge symmetry inevitably leads to interacting field theories with non-linear representations. The Yangian representation should be compatible with gauge symmetry such that the unphysical degrees of freedom do not interfere with the symmetry. Moreover, gauge symmetries have to be fixed in the process of quantisation such that the gauge field propagator can be defined. The process of gauge fixing can potentially violate Yangian symmetry.
Many of these features and in particular their interplay complicate the investigation of Yangian symmetry. We will therefore start with a thorough discussion of the algebra and more elementary considerations of symmetry before we move on to the two kinds of invariances of the equations of motion. The treatment of the equations of motion will be manifestly gauge covariant which provides some structural constraints on the novel terms that we shall encounter. Eventually, we will address Yangian invariance of the action in the next section where all of the complications come into play.

Algebra and representations
Given a semi-simple Lie algebra 12 g, the associated Yangian algebra Y[g] is an infinitedimensional quantum algebra generated by two sets of generators J A and J A with A = 1, . . . , dim(g) [21]. The so-called level-zero generators J A generate the universal enveloping algebra U[g] of the underlying Lie algebra g, and they obey the commutation relations with f AB C the structure constants of g. The level-one generators J A transform in the adjoint representation of g, and they obey the so-called Serre relation at level two Here, adjoint indices are raised and lowered with the Killing form k AB . 13 All the higherlevel generators are given as polynomials in the level-zero and level-one generators subject to the above commutation relations. The Yangian Y[g] is actually a Hopf algebra, and as such it possesses a coalgebra structure ∆ : Y → Y ⊗ Y defined by The coproduct is coassociative and compatible with the algebra product in a certain sense. These features allow one to use the coproduct for the definition of tensor product representations. The (n − 1)-fold iterated coproduct of J and J acting on n tensor factors reads Here, J A k indicates the action of the generator J A on the k-th factor or site of the tensor product (with a trivial action on the other sites). The above coproduct of J is how one would conventionally define a tensor product representation for the Lie algebra g. It acts on all sites in the same fashion, and therefore the ordering of the sites plays no role. The coproduct of a level-one generator J consists of a local part k J k analogous to the levelzero coproduct and of a bi-local contribution j<k J j ⊗ J k . This latter bi-local term does depend on the ordering of the sites of the tensor product; in other words, the coproduct is non-cocommutative.
As we have seen, the coproduct makes us deal with several generators of g at the same time requiring the use of adjoint indices A, B, . . . and structure constants f . A convenient abbreviation to avoid indices is Sweedler's notation: here one denotes by J (1) and J (2) the level-zero generators in the first and second factor, respectively, of the coproduct of some level-one generator J = J A with an implicit sum over all combinations 14 The above coproduct rule can thus be expressed more concisely as Our analysis of Yangian symmetry can furthermore be simplified by the fact that only a single level-one generator along with the level-zero generators is sufficient to generate the whole Yangian algebra; all other level-one generators follow from the adjoint property (3.2), and the higher-level ones from the Serre-relation (3.3). We are thus free to choose a particular generator for which the resulting expressions simplify as much as possible. Arguably, this is the level-one momentum P which is also known as the dual conformal generator. Its coproduct reads (including all appropriate signs due to fermionic terms) 15 where the anti-symmetric tensor product ∧ of any two objects X and Y is defined as Note that the action of P conveniently only needs the dilatation D next to the super Poincaré generators L,L, Q,Q and P. All of their representations are reasonably simple compared to the representations of the superconformal boosts S,S and K. In fact we will encounter further simplifications due to the choice of P later on. Expressions for the coproducts of the level-one generators J with J ∈ {Q,Q, R} as well as the level-one bonus symmetry B introduced in [22] can be found in App. B.

Issues
The major complication in considering Yangian symmetry within an (interacting) field theory is that symmetry representations are often non-linear. By definition, the representation is still a linear map between observables; here, non-linearity refers to the fact that a single field can be mapped to a product of fields. In a gauge theory, the covariant derivative is a major source of non-linearity, see e.g. (2.9,2.12). Furthermore, the representation of supersymmetry (2.13) contains non-linear terms which are not due to covariant derivatives. The issue is that the concept of non-linear representations is in competition with the definition of tensor product representations via the coproduct: In a linear representation, each field corresponds to a single tensor factor. The representation acts on fields one-toone thus preserving the structure of the tensor product. For non-linear symmetries, the representation changes the number of fields and consequently the structure of the tensor product, see [23] for a discussion and construction of some aspects of non-linear representations. In fact, the representation does not split into sub-representations with a definite number of fields, but there is only the indecomposable representation on polynomials of the fields. For the local action of level-zero generators this complication is minor and one can still view the full representation as the sum of representations on component fields. Conversely, the construction of the bi-local action for the level-one generators is less evident, cf. some comments in [24]. Here, the action of the first constituent generator J (1) changes the tensor product on which the second constituent generator J (2) is supposed to act. This mainly refers to the precise definition of the bounds in the double sum in (3.5) given that n is not well-defined anymore. For example, it is conceivable that J (2) acts on the output of J (1) corresponding to an overlapping action. This leaves some ambiguity for the precise definition of a non-linear level-one representation. Unfortunately, this becomes a rather serious issue when taking gauge symmetry into account: for instance, the covariant derivative consists of a partial derivative and a gauge potential, and as such it relates terms of a different number of constituent fields. In that sense, the non-linear terms of the Yangian representations must be chosen delicately such that they will not violate gauge symmetry.
We will thus need representations which are analogous to the actions defined in (3.5). In particular, they should reduce to the ordinary coproduct rule (3.5) when restricting to linear terms. To that end, it will make sense to distinguish the local and bi-local contribution in the level-one representation, and write Here, the tensor product symbol ⊗ should be interpreted such that the first factor acts on a field which is to the left of the field on which the second factor acts. Formally, we define the tensor product acting on a sequence of fields by the rule (J (1) ⊗ J (2) )·(· · · Z I · · · Z J · · · ) := . . . + · · · (J (1) ·Z I ) · · · (J (2) ·Z J ) · · · + . . . , (3.11) where the dots on the r.h.s. represent further terms due to pairwise contractions with the omitted fields in the sequence. The local part J loc will just act on all fields homogeneously. Note that any overlapping contributions that could be part of some alternative definition of a bi-local action can and should be interpreted as local contributions.
The other main complication is that representations of the Yangian algebra tend to be incompatible with the boundary conditions imposed by the physical system at hand; consequently, the Yangian can merely be considered a symmetry of the bulk rather than of the system as a whole. Nevertheless, for planar N = 4 sYM our aim is to show exact Yangian invariance of the action. The colour trace within the action imposes periodic boundary conditions on the above tensor product. Moreover, the trace projects to states which are invariant under cyclic permutations; the physical information within the action is cyclic. However, the representations of the Yangian generically do not respect cyclicity.
To understand the violations of cyclicity, let us consider a cyclic state in a tensor product of n sites. The Yangian algebra acts on this state by the (linear) representation (3.5). To that end, we need to define an ordering of the sites within the tensor product. The adjacency relationship of the sites defines a local ordering, which is however globally inconsistent due to the periodic identification. In order to define an ordering, we need to choose a base point where to "cut open" the cycle. Let J (k,n+k−1) denote the action of J on the range of sites from k through n + k − 1 (modulo n), i.e. where we cut between sites k − 1 and k. We then have that Here J (j,k) denotes the representation of the level-zero generator on sites j through k. In a generic situation, the r.h.s. is not zero and therefore the resulting state is not cyclic. Likewise, the action of J will typically depend on the particular cyclic representative on which it acts. This means that there is no universal answer to the Yangian representation on cyclic states, and it makes no sense to ask whether or not any such state is Yangianinvariant. Gladly, the above difference can be rewritten as [9] J The first term on the r.h.s. contains the combination f A BC f BC D which is proportional to the dual Coxeter number of the underlying Lie algebra g. For the N = 4 superconformal algebra psu(2, 2|4) this number is zero, and therefore the term does not contribute. The second term is a product of two level-zero representations. The level-zero generator J C acts on the complete state, and annihilates it if is invariant. In other words, the level-one representation of a Yangian with vanishing dual Coxeter number respects cyclicity for those states which are invariant under the level-zero symmetry. In principle, this is an ideal starting point for considering Yangian invariance of the action of planar N = 4 sYM because these two prerequisites are satisfied.
Unfortunately, the argument about cyclicity is based on linear representations. For non-linear representations one may suspect something equivalent to hold, but it is unclear how to set up the representation precisely and how to show cyclicity in this case. The major challenge is thus to properly define non-linear representations of level-one generators and to learn how to work with them. In this situation, proving invariance of the action turns out to be a rather difficult task, and we shall at first consider invariance of the equations of motion in order to gain a better understanding of non-linear representations.

Weak invariance of the equations of motion
We start by considering the weak notion of a symmetry of the equations of motion (2.20): given a generator J and its action on the fields Z, it must leave the equation of motion For the Yangian level-one generators J, the non-trivial coproduct poses two problems: First, if we want to act with J biloc on the equations of motion, we have to fix an ordering prescription for each term that appears; second, we need to specify how J loc acts on a single field. Gladly, we can impose a "natural" ordering for the fields within the equations of motion for Yang-Mills theories with U(N c ) gauge group. All the fields in N = 4 sYM can be treated as N c × N c matrices and the (non-commutative) matrix product provides the ordering within a monomial of the fields. The equations of motion inherit this matrix structure supposing that all structure constants of the gauge group are written in terms of commutators of the fields. This is also where the large-N c limit comes into play: the possibility to write arbitrary (adjoint) combinations of the fields in terms of matrix polynomials requires a U(N c ) gauge group with correspondingly large N c . 16 Note that 't Hooft's (double) line notation for adjoint fields, which is prominently used in the discussion of the planar limit, directly depicts our ordering prescription.
For what concerns the single-field action J loc , we will fix it by first computing the bi-local action of J biloc = J (1) ⊗ J (2) on some equations of motion, which is completely determined by the conformal representation. We shall then require that the local terms eliminate all remaining terms J biloc + J loc ·Z ≈ 0. (3.15) The fact that suitable local terms can be found will be a first test for Yangian symmetry. Subsequently, we will consider the other equations of motion. Yangian symmetry will pass a more elaborate test if the same local terms compensate the remainders from the bi-local action for all equations of motion.
In the following we will focus on the level-one momentum generator J = P whose bi-local action is determined by the coproduct in (3.8) We will first compute the action of this bi-local term on the Dirac equation (2.7) We act on the latter with the coproduct (3.8) by means of the tensor product action (3.11) using the superconformal representation (2.9,2.12,2.13), to obtain 17 This expression does not vanish on-shell on its own. However, we can make an ansatz for a single-field action P·Z based on all terms with appropriate quantum numbers and symmetries, and add their contribution to the overall action of P. By choosing the coefficients of the ansatz as Pα β ·Φ cd := 0, we get that This shows that Pα β is a weak symmetry of the Dirac equation of N = 4 sYM. 16 For matrix products of N c fields or more, there are certain identities of polynomials related to antisymmetrisation which eventually introduce some ambiguity for the ordering of fields. Arguably this ambiguity does not apply to the equations of motion when N c > 3 (at least). Nevertheless, we will later want to establish Yangian symmetry for arbitrary correlators of the fields, in which case an arbitrarily large N c will be necessary. Of course there is the option to reverse-engineer the ordering rule such that the level-one generators are directly represented on fields contracted by structure constants (as long as N c is sufficiently large), however any such rule will be rather messy and cumbersome. 17 Note that all explicit x-dependence originating from the generators L,L and D cancels out exactly. This convenient feature is related to the fact that P commutes with P in the Yangian algebra.
One comment on the structure (3.18) of the single-field action is in order: It may appear unconventional as it is formulated in terms of anti-commutators in places where commutators are usually expected. However, this configuration is actually natural considering the action of a level-one Yangian generator J biloc on a commutator (3.20) This means that level-one generators typically map commutators to anti-commutators.
With the above single-field action of (3.18) it is now possible to show that all the equations of motion of N = 4 sYM are weakly invariant under P Moreover, we have verified explicitly that they are weakly invariant under the level-one generators J with J ∈ {Q,Q, R} as well as the level-one bonus symmetry B which extends the Yangian of psu(2, 2|4). This also fixes the single-field actions of these generators, and we present our results in App. B. Up to some issues w.r.t. the closure of the Yangian algebra onto gauge transformations, to be discussed in [20], we conclude that the Yangian of psu(2, 2|4) is a weak symmetry of the classical planar equations of motion of N = 4 sYM. As previously discussed, this represents a necessary condition for Yangian symmetry in planar N = 4 sYM. It is reassuring to see that it is met, and we can continue with the construction of a sufficient condition.

Strong invariance of the equations of motion
Next we would like to promote the above results to a strong invariance of the equations of motion. As we argued in Sec. 2.3, this would amount to an honest statement of Yangian symmetry in classical planar N = 4 sYM. To that end, we need an expression which predicts the exact form of the l.h.s. of J·Z ≈ 0 as a linear combination of theZ (which are zero on shell).
Let us start from the ordinary superconformal symmetry. As we have already shown, the invariance of the action under the generators J of psu(2, 2|4) implies an off-shell relationship for the equations of motion of (2.19) On the r.h.s. of this equality we have a sum of termsZ whose the coefficients are completely determined by the superconformal representation on the fields. Note that the sequence of fields for the term on the r.h.s. takes a particular form which is not completely evident from the above expression. It can be inferred from the colour structure which provides the correct adjacency information due to contraction of the field indices I and K. In particular, the contraction of indices I implies to take a colour trace. Let us explain it by means of an example: suppose J·Z I = Z L Z M Z N yields some cubic combination of fields, the term on the r.h.s. would take the form It is straight-forward to generalise this expression to any number of fields.
We can now ask ourselves if we can find an analogous expression for level-one Yangian generators: an off-shell relationship for the action of J onZ K , whose right-hand side is a sum ofZ I with coefficients completely determined by the action of the generators on the fields of the theory. With some inspiration from the expected structures related to level-one symmetry, we find that the answer is positive, and this novel relationship reads The anti-symmetric tensor product ∧ was defined in (3.9) and it acts in analogy to (3.11).
The insertion of the new fieldZ I , however, follows different rules based on the colour structure associated to the indices I, K in analogy to (3.23). For example, assume that J (2) ·Z I = Z L Z M Z N again is a cubic combination of fields. Then the bi-local term in (3.24) expands tȏ (3.25) Note that the colour trace implied by the contraction of I is cut open by the operator δ/δZ K , so thatZ I can appear in the middle of the resulting polynomial. This expression also generalises to any number of fields, and it can be non-zero only if J (2) ·Z I consists of (at least) two fields. In other words, the bi-local term in (3.24) only sees the non-linear contributions of J (2) .
First, we shall show that the above relationship holds for the Dirac equation in N = 4 sYM. The l.h.s. of (3.24) has been computed in (3.19). We need to show that it matches with the combination of terms on the r.h.s. of (3.24). The first term involves a variation of P·Z I byΨ . The only single-field action (3.18) containing a fieldΨ is the one for Z I = Ψ . Thus we can set Z I to Ψ f in the first term. The second term turns out to yield no contribution for a combination of reasons: First of all, J (2) ·Z I must yield an expression non-linear in the fields. One option is J (2) = Q,Q acting on Z I = Ψ,Ψ , which however never produces any terms containing Z K =Ψ . Another option is J (2) = L,L, D, but all non-linear terms have an explicit x-dependence which will eventually cancel against other terms. It remains to check J (2) = P. It must act on Z I =Ψ if the result is to contain Z K =Ψ . The only other field is Z J = A (within ∇Ψ ) which can be acted upon by J (1) = L,L, D. However, all these terms are explicitly x-dependent and as such they are needed to cancel against other terms. Hence the second term does not contribute, and we are left with This agrees precisely with (3.19). We have also verified that the relation (3.24) holds exactly for all equations of motion of N = 4 sYM and for the level-one generators J with J ∈ {P, Q,Q, R, B}, i.e. it correctly reproduces the r.h.s. of all terms in (3.21) and corresponding relations for the other generators.
Next, let us now explain the meaning of the two terms appearing on the r.h.s. in (3.24). The first of them is a direct counterpart of the one from the level-zero formula (3.22). The other term should be viewed a result of the non-trivial coproduct of the level-one generators (3.7). Interestingly, the action of the two constituent operators J (1) and J (2) is overlapping, as J (1) acts purely on the output of J (2) . We will thus call this contribution the overlapping term. Note that the overlapping term is a purely non-linear effect. 18 To conclude this discussion, we would like to emphasise that unlike its level-zero counterpart (3.22), the level-one formula (3.24) for invariance of the equations of motion has not been derived from first principles. We have merely verified that it holds exactly for all equations of motion of N = 4 sYM and for several level-one generators J. In other words, we have derived a strong form of invariance of the equations of motion for level-one generators. This invariance amounts to a set of non-trivial off-shell identities which hold in classical planar N = 4 sYM. Such identities are hard to get hold of, and independently of how we obtained them and of their precise form, they clearly indicate a (hidden) property of classical planar N = 4 sYM which is equivalent to a global symmetry.

Yangian invariance of the action
We would now like to address Yangian invariance of the action. Our starting point is the strong invariance of the equations of motion (3.24) which we claimed to be a valid statement of symmetry. In the following we will rearrange the terms in the relationship such that they take the form of a Yangian invariance of the action J·S = 0. (4.1) The goal is to find a precise formulation of this statement that holds for planar N = 4 sYM.

Notation
In order to perform the rearrangements, we will need a concise notation for the various terms that arise. First of all, we decompose all objects w.r.t. the number of fields that they contain. The action of N = 4 sYM has quadratic, cubic and quartic terms, and the superconformal representation has terms which preserve the number of fields as well as terms that increase the number of fields by one unit 19 We choose a particular level-one generator which minimises the non-linear terms, such as the level-one momentum J = P, see also App. A for further calculational simplifications. The relevant local and bi-local terms in J = J loc + J biloc then expand as follows 20 J loc = J loc, [1] , [0] + J (1) [0] + J [0] ⊗ J [1] . (4.3) Next we need to write out somewhat more explicitly how the representations act on the individual fields in the action. The action is a polynomial in the fields whose ordering matters. The contribution J [0] to the representation maps one field to one, and it is natural to denote the action on the field at site k by J [0],k . 21 For the higher contributions J [m] , m > 0, which map one field to m + 1 fields, the situation is not as evident, see [23] for discussions of this matter in a similar context. Here we make the choice that J [m],k acts on the field at site k and replaces it by an appropriate sequence of m + 1 fields. Consequently, the fields at sites 1, . . . , k − 1 are mapped to themselves, whereas the fields at sites k + 1, k + 2, . . . are shifted by m steps to sites k + m + 1, k + m + 2, . . ., e.g.
The change of length has a relevant implication on the commutation of two operators J and J . It is evident that two one-to-one operators acting on two different sites commute. Cyclicity. All the terms in the action are gauge-invariant by means of a trace. The trace establishes a neighbouring relationship between the last and the first sites, i.e. periodic boundary conditions. Moreover, the trace is cyclic, i.e. it is invariant under cyclic shifts. In order to deal with periodic boundary conditions, we introduce U as the operator that performs a cyclic shift by one site to the left; e.g., it shifts the second site to the first and the first site to the last. Cyclicity of a traced operator O is then expressed as Note that we will also work with polynomials O in the fields which are periodic but not cyclic. The cyclic shift operator shifts the insertion of an operator J according to the rule (4.7) 19 We define the expansion coefficients O [n] of traced polynomials O of the fields with an explicit symmetry factor of 1/n. Conversely, there is no symmetry factor for the expansion coefficients X [n] and J [n] of open polynomials X and operators J, respectively. Even though these different assignments may be confusing at times, they will avoid many combinatorial factors. 20 Even though there is in principle a bi-local term which adds a field at both insertion points, this term will not contribute in practice. 21 To avoid excessive clutter we may suppress the symbol '·' when an operator acts on a specific field.
It it worth emphasising that a term of the form U k J [m],1 with k ≤ m never appears on the r.h.s. of (4.7). Such a term cannot be written with the shift U residing to the right of J, the shift must remain to the left of J. This is because the term has a form which is not covered by J [m],k alone and which is different from (4.4). In this term, the output of J [m] extends past the last site of the polynomial and continues periodically to the first site or beyond. An example analogous to (4.4) could be written as where (J [m] Z) j denotes the j-th site of the output of J [m] on Z (with an implicit sum over all m-tuples of fields in the polynomial J [m] Z in similarity to Sweedler's notation).

Level zero
Equipped with this notation, we will first address superconformal symmetry of the action. This will provide us with some identities that we shall need later when we consider Yangian symmetry. 22 First, we decompose the statement J·S = 0 by the number of fields, and we obtain the following set of 4 statements Here it is necessary to use the cyclic shift operator U for one term to distribute the sequence of fields resulting from J [1] over the last and first site.
It is also useful to consider the strong invariance of the equations of motion (3.22). For any Z K specifying the equation of motion, it yields a relationship consisting of a 22 We would like to remind the reader of a general issue w.r.t. homogeneous non-linear representations acting on periodic objects: On the one hand, the periodic shift operator U classifies periodic objects according to the eigenvalue e 2πin/L where n = 0, . . . , L − 1. On the other hand, the non-linear representation changes the length L of the object. Now the spectra of U for different lengths L are largely distinct, the only eigenvalue which is present for all lengths is 1 corresponding to cyclic objects. Therefore non-linear representation can be homogeneous, i.e. commute with the shift operator U, only on the subspace of cyclic objects. In other words, the periodic object on which they act must be cyclic and the representation must be constructed such that the result is cyclic as well.
polynomial of fields. In order to handle the identities for all fields Z K at the same time, we prepend this field to the relationship polynomial and sum over all fields In this definition we explicitly do not take the colour trace such that the above field Z K will always reside at site 1 of the polynomial by construction. The expansion n Y  .12), respectively. In fact, the expression Y is cyclic; by rewriting the first term using (4.7) such that J [m] will always act on site 1, we can make cyclicity manifest Therefore, we lose no information by identifying terms by cyclic permutations, and we find a more concise statement To wrap this discussion up, we can rewrite the above definition (4.13) as As the operation Z K (δ/δZ K ) just counts the number of fields, it is not surprising that the symmetry variation of the action expands precisely to the coefficients Y [n] Therefore, the above transformations are a bit of a detour in this case, but they will help us find a corresponding expression J·S for the level-one Yangian generators J acting on the action S.

Level one
We would now like to construct a suitable level-one Yangian representation on the action, J·S, such that invariance amounts to J·S = 0. We will approach this construction by expanding in the number of fields, where the prefactors 1/n account for cyclic symmetry. We will start with the simplest cases at n = 2, 3 and later address arbitrary lengths. Before doing so, we derive some more identities which are needed to transform the expressions.
[l],k . (4.21) The leading term H [0] contains the combination f A BC f BC D which is proportional to the dual Coxeter number of the level-zero algebra. The dual Coxeter number for our superconformal algebra psu(2, 2|4) is zero, and consequently the linear term vanishes, H [0] = 0. The non-linear term H [1] is also zero for N = 4 sYM by explicit computation. One can relate this finding to the G-identity for N = 4 sYM found in [15]. Altogether, the constituent operators of the bi-local level-one generator commute (when summed over all pairs as implied by Sweedler's notation) Two fields. Next, we address level-one Yangian symmetry. The starting point is the strong invariance statement (3.24) of the equations of motion which we have already shown to hold. We will treat it analogously to the strong invariance of the equation of motion at level-zero (3.24): We prepend the field that selects the equation of motion to the relationship polynomial and denote the resulting expression by However, the relationship Y [2] = 0 at the level of two fields is empty, the first non-trivial relationship is at three fields.
What are the implications of Yangian symmetry at two fields? We use the fact that the combination Y [2] in (4.15) is exactly zero, 24 and apply another generator J to it to obtain a new relationship (4.25) The first term has the form H [0],1 S [2] and therefore it vanishes by (4.22). The second term is the bi-local part of the level-one representation on the quadratic part of the action. Furthermore, there is no linear contribution to the local part of the level-one representation. Altogether, (4.25) boils down to the statement [0] ·S [2] = 0. Three fields. The strong invariance relationship (3.24) at three fields reads Y [3] = 0 with [0],2 J (1) We would now like to reformulate this combination such that it looks more like the cubic term in the expansion of J·S. To that end, let us first address cyclicity of Y [3] . The first three terms are manifestly cyclic; the latter three are not. We therefore look at the violation of cyclicity Here we have made use of the algebraic identities (4.5) and (4.7) and the implicit antisymmetry in Sweedler's notation (3.6). All remaining terms could then be collected in the combinations Y (4.15) and H (4.21) which are zero as discussed above. Therefore the relationship Y [3] = 0 is effectively cyclic; all the non-cyclic contributions to the relationship are zero for lesser reasons than Yangian symmetry. It also allows us to compare modulo cyclic identifications (' ') without losing relevant information [0],2 J (1) [0],1 J (1) Let us now turn to the reformulation. Due to the effective cyclicity of Y [3] it is reasonable to define ( J·S) [3] to be proportional to it. To figure out the factor of proportionality, we will compare the terms in (4.29) to some canonical terms in ( J·S) [3] . The first term in (4.29) is equivalent to the local part of the level-one representation on S [2] J loc, [1] ·S [2] : 2 J [1],1 S [2] , (4.30) while the second term in (4.29) is equivalent to the bi-local part on S [3] J [0],2 S [3] . (4.31) The remaining two terms in (4.29) have a special form, one may interpret them as a nonlinear bi-local contribution where both legs overlap. Such terms are not provided by the usual coproduct rule for tensor product representations, but our non-linear representation is not exactly a tensor product, and hence it is conceivable to have them [0],1 J [1],1 S [2] . (4.32) The (insignificant) prefactors in this definition were chosen to agree with a desirable form of expression further below. Finally, we notice that Y [3] does not contain the standard bi-local contribution to the representation on S [2] . However, this term is zero modulo cyclic identifications J (1) Altogether we derive an invariance statement for the action at cubic order in the fields 1 3 ( J·S) [3] : 1 2 J loc, [1] ·S [2] Note that the prefactors of all terms agree with the symmetry factors 1/m of the S [m] which is acted upon. 25 We have verified explicitly that (4.34) holds for the level-one momentum J = P.
Non-linear level-one invariance. Now let us turn to a general number of fields n in order to understand the precise structure of the unconventional terms. The combination Y [n] of the strong invariance condition (3.24) can be written in our present notation as (4.37) The terms in the first line of the result represent the local terms of the level-one representation. The terms in the second line are clearly bi-local with two non-overlapping insertions. The terms in the third line take a similar form as the bi-local terms, but here the insertion of J (1) is within the range of J (2) and thus their actions overlap. 26 Some comments regarding the bi-local and overlapping terms are in order: Even through the prefactors for all terms in (4.37) are 1, those of the non-linear bi-local terms appear somewhat unnatural: when compared to their linear counterparts, they are off by a factor of (n − m)/n. On the one hand, this may be worrisome in view of gauge symmetry because such relative factors could easily upset the composition of covariant derivatives and thus spoil this essential symmetry. On the other hand, such factors can be compensated by the relative length of the objects and thus by the multiplicity of operator insertions. Moreover, the overlapping terms have no analog in the conventional level-one representation based on the coproduct rule.
How to make sense of this behaviour? Clearly, the novel non-linear representation of level-one symmetry on cyclic polynomials can have new features, which do not need to follow the coproduct rule strictly. What matters is that the above form follows by elementary transformations from the strong invariance relationship (3.24). The latter holds in planar N = 4 sYM for the level-one momentum J = P. It therefore makes sense to view Y [n] = 0 as given by (4.37) as the definition of the non-linear level-one invariance of the action. Whether or not it has the expected from, it certainly does describe a nontrivial relationship of planar N = 4 sYM. In any case, the structures in (4.37) clearly deserve further theoretical scrutiny.
Cyclic level-one representation. As the expression (4.37) is effectively cyclic, we may furthermore identify terms related by cyclic permutation. This yields a simpler expression Note that all bi-local terms could be combined in a uniform expression. Here the first and the last l + 1 terms labelled by k = 1, . . . , l + 1 and k = n − m + l + 1, . . . , n − m + 2l + 1 represent overlapping bi-local terms whereas the insertions do not overlap in the remaining middle range k = l + 2, . . . , n − m + l.
Since invariance of the action can and should be expressed modulo cyclicity of the trace, we may write the symmetry statement as J·S 0 with J·S : This form is a complete expression for level-one invariance of the action including all standard and non-standard terms. We have verified explicitly by computer algebra that it holds for planar N = 4 sYM for the level-one generators J with J ∈ {P, Q,Q, R, B}. Unfortunately, the calculations produce hundreds of intermediate terms which are subject to cyclic identifications and integrations by parts. Only the invariance under the levelone bonus symmetry B has a reasonably simple structure, and we shall show it explicitly below. As such we have not yet fully established that the extended symmetries generate a Yangian algebra; we would need to show that the adjoint property (3.2) as well as the Serre-relation (3.3) hold. A complication, to be addressed in [20], is that the algebra is mixed with gauge transformations of a novel kind. Independently of which algebraic relations the symmetry generators obey, they give rise to novel relations for planar N = 4 sYM. 27 Level-one bonus symmetry. In order to show invariance of the action under the levelone bonus symmetry B introduced in [22], see also [25], we first introduce the additional algebraic structures, see App. A and B for further details. The coproduct depends only on the odd level-zero generators Moreover, the representation of the superconformal boosts S andS are almost completely given in terms of the supersymmetriesQ and Q, respectively Both of the additional operators S andS act non-trivially only on a single type of field Based on these relationships, we can in fact show that almost all contributions to the invariance condition are trivially zero. First of all, by combining (4.40) and (4.42) one can observe that the x-dependent terms due toS∧Q and S∧Q mutually cancel irrespectively of what they act on. Therefore, the only non-zero contributions to the invariance condition can originate from the x-independent operators S andS as well as when S andS act on a partial derivative which subsequently removes the x-dependence (4.44) First we consider overlapping terms in the action of B: Overlapping terms require the first level-zero generator to act non-linearly. However all non-linear contributions from the odd generators act on fermions only and they produce two bosons without derivatives. When acting further on the result, all x-dependent contributions must cancel by the above arguments, and, as we have seen, bosons without derivatives cannot generate xindependent terms. Moreover, extra terms cannot be generated from the action of the first generator because S andS are purely linear and any partial derivatives acting on the original field can be pulled out from the calculation. Hence there are no overlapping terms. Analogously, the single-field action of B is zero.
Let us now consider the various terms in the action: The quadratic terms S [2] can only contribute via overlapping terms and therefore they are all trivially invariant on their own. Furthermore, all the quartic terms S [4] are purely bosonic and they do not involve partial derivatives. Since the operators S andS act on fermions only, and the absence of partial derivatives prevents the generation of further x-independent terms, also all terms in S [4] are invariant on their own. It remains to consider the cubic terms S [3] .
By elementary transformations we can summarise the (non-linear) action (4.38) on the cubic terms as Effectively, only x-independent terms can potentially contribute, and therefore we need to consider only two types of terms from this expression: The generators S andS can act on a derivative ∂Z in such a way that the derivative eliminates the x-dependence from the action of S andS. Alternatively, S andS can act on fermions and yield terms via the extra operators S andS. Curiously, the cubic terms in the action (2.5) split into two corresponding classes, namely terms with fermions and terms with derivatives Let us first consider the fermionic terms on the second line on which S andS can effectively act only via the extra operators S andS . By considering the types which can potentially be generated along with the parity-reversing nature of the level-one generators, we find terms of the types Now the former six terms are trivially zero because they all involve simultaneous symmetrisation and anti-symmetrisation of indices. Only the last term is non-zero. It can only be generated from the term {Ψ,Ψ }A in the action and the contributions read However, by explicit computation, the two terms cancel precisely. By similar arguments as above, the purely bosonic cubic terms of the action can yield terms of the kinds Therefore, both remaining contributions cancel, and altogether this proves that the bonus level-one Yangian generator B is a symmetry of planar N = 4 sYM.

N = 6 supersymmetric Chern-Simons theory
Now that we have established an extended symmetry related to planar integrability of classical N = 4 sYM theory, we should consider further gauge theories where planar integrability has been observed. The main example of a supposedly integrable gauge theory in the planar limit which is substantially different from N = 4 sYM is ABJ(M) theory. This model is a superconformal field theory in three dimensions with an AdS/CFT dual string theory, it appears to be integrable in the planar limit and its observables display Yangian symmetry. ABJ(M) theory will serve as a crucial testing ground for our proposal: if our definition of integrability in the form of Yangian symmetry is correct, we should be able to show that its holds also for this theory. As for N = 4 sYM, we use a spinor-matrix notation x αβ to denote the coordinates of three-dimensional spacetime where α, β, . . . = 1, 2 denote sl(2, R) spacetime spinor indices. The coordinate matrices are symmetric x αβ = x βα and real. The corresponding partial derivatives ∂ αβ = ∂ βα are normalised by the rule ∂ αβ x γδ = δ γ α δ δ β + δ δ α δ γ β . Furthermore, Latin letters a, b, . . . = 1, 2, 3, 4 denote su(4) internal indices.

Action
For simplicity, we introduce two gauge fields A αβ andÃ αβ , one for each U(K) component of the gauge group. The fields of the theory can be represented as matrices: the field A αβ (Ã αβ ) is a hermitian M × M (N × N ) matrix, the fields Φ a and Ψ αb (Φ a and Ψ α b ) are complex M × N (N × M ) matrices; any sequence of fields is to be understood as the appropriate matrix product.
The gauge-covariant derivatives act on a covariant M × N field Z (N × M fieldZ) as and we define the field strengths as for N = 4 sYM The action of the theory is completely fixed by supersymmetry and its Lagrangian reads

Symmetries
As already stated, ABJ(M) theory is a superconformal theory; the algebra of superconformal transformations is 28 osp(6|4). The action of the Poincaré and dilatation generators reads where the conformal dimensions are ∆ Φ = 1 / 2 , ∆ Ψ = 1, ∆ A = 0, and where the spin operator L spin acts non-trivially only on the spinor fields Z α ∈ {Ψ αc ,Ψ α c }

Yangian invariance
In order to check the classical Yangian invariance of ABJ(M) theory, we can proceed the same way we did for N = 4 sYM: we start from the simplest equation of motion, we act on it with the bi-local part of a level-one Yangian generator, we fix the single-field action and then we check that the relationship of (3.24) is satisfied. The most convenient choice for the level-one generator of Y[osp(6|4)] is again P αβ . The bi-local part of the coproduct reads Furthermore, we choose the simplest of the equations of motion, the Dirac equationΨ = 0 withΨ We find that the Dirac equation is weakly invariant under the above level-one momentum generator with the following unique choice of single-field action P αβ ·Φ a = 0, P αβ ·Φ a = 0, More concretely, we find that the following equality holds Here,Ȃ andȂ denote the variations of the action w.r.t. the gauge fields, (5.11) and they vanish on-shell as doesΨ so that the Dirac equation is weakly invariant. Moreover, we have verified that the r.h.s. of (5.10) matches exactly the r.h.s. of (3.24), and we have performed this check for all the equations of motion of ABJ(M) theory. Therefore the equations of motion display Yangian symmetry in a strong sense. Finally, have shown by computer algebra that the ABJ(M) action is invariant in the sense of (4.39). Therefore classical planar ABJ(M) theory has Yangian symmetry.

Pure N < 4 sYM
In this section we would like to address a possible concern that our results discussed above, especially the formula (3.24), are a mere result of level-zero (superconformal) and gauge symmetries of the theories. If that was the case, our work would not provide any criterion for establishing Yangian symmetry of physical models. As we will now demonstrate, this is fortunately not the case and for formula (3.24) to hold increased symmetry is indeed required.
To this end let us consider a pure sYM theory, but keep N arbitrary. The action is again given by: The action (6.1) looks exactly like (2.5) of N = 4 sYM. The only difference is the range of indices a, b, . . . = 1, . . . , N , where N ≤ 4. As the reality condition (2.2) exists only for N = 4, the scalar potential is written here in a different form, valid also for N = 4. Moreover, for N = 1 the scalar fields do not exist at all, which here is implicitly taken care of by the anti-symmetry of their indices, Φ ab = −Φ ba = Φ 11 = 0. The symmetry algebra of (6.1) is psu(2, 2|N ) and for any N ≥ 0 its Yangian exists. What we want to demonstrate is that Y[psu(2, 2|N )] is a symmetry of the theory only for the special value N = 4. To that end it is already enough to show that for other values of N the weak invariance of the equations of motion does not hold, as its failure will assure that none of the (aforementioned) stronger criteria stand either. We thus can repeat the computation from Sec. 3.3 keeping N arbitrary. The action of all the generators carries over from Sec. 2 and 3 (see equations (2.13) and (3.18) for supersymmetry and local Yangian action respectively), of course up to the restricted range of indices mentioned above (e.g. the local action of P vanishes completely for N = 1). For general N ≥ 1 we act with P on the Dirac equation of motion 29 On shell we are left with a term of the form Even though we have the principal freedom to adjust the local action of P to the case N < 4, it is easy to see that this cannot remove the residual term: In order to cancel the term, one would need extra contributions of the form PA ∼F or PΨ ∼ ∇Ψ . However, these would merely produce commutators [F , Ψ ] rather than the desired anti-commutator {F , Ψ }.
Thus we see that the value N = 4 is special as it warrants the cancellation of the residual term and the on-shell invariance of the equations of motion of N = 4 sYM. More importantly, we observe that superconformal and gauge symmetries by themselves are not sufficient to warrant our results presented in the preceding sections, and indeed Yangian invariance of a theory is a non-trivial statement.

Conclusions and outlook
In this work we have elaborated upon the results of our previous letter [16] by spelling out in detail the different kinds of planar invariance conditions for the equations of motion of N = 4 supersymmetric Yang-Mills theory and ABJ(M) theory under a Yangian algebra. Subsequently, we have reformulated the strong version of this invariance as an invariance of the action and proved explicitly that this invariance holds indeed. This not only shows that these two classical gauge theory models possess Yangian symmetry in the planar limit, but it may also be viewed as a formal definition for their integrability.
A follow-up work [20] will address Yangian-invariance of field correlators at tree level: for a field theory with some set of symmetries, one should expect correlation functions of the fields to obey a corresponding set of Ward-Takahashi-identities reflecting these symmetries. An important aspect in the computation of correlators within a gauge theory is gauge fixing. It will be shown how Yangian symmetry can be made compatible with Faddeev-Popov gauge fixing and BRST symmetry and how to formulate Slavnov-Taylor identities for Yangian symmetry which properly take into account the effects of gauge fixing. To that end, the algebraic relations of the Yangian need to be studied in detail because they are intertwined with gauge transformations which are henceforth deformed by the gauge fixing procedure. A curious side effect is that this will amount to an extension of the gauge algebra by bi-local and non-local gauge symmetries.
Our works lay the foundations for many directions of further study. Most importantly, firm contact should be made with corresponding symmetry structures of the world-sheet theory for strings on AdS 5 × S 5 . By the AdS/CFT correspondence such structures should exist, but quantum effects on both sides, potentially including quantum anomalies, may play a significant role for a precise matching, see [26] for considerations on the string theory side. For example, the algebraic complications due to non-ultra-locality of the string theory sigma model [27] may well have a counterpart in the Yangian algebra for gauge theory.
Another obvious question is which other models beyond N = 4 sYM and ABJ(M) theory enjoy a Yangian (or related) symmetry in the planar limit? There are several deformations and orbifolds of N = 4 sYM which are apparently integrable [28] and which should therefore possess an extended symmetry. This has been shown explicitly for the beta-deformation in [29]. The simplistic fishnet theory [30] represents a particular contraction limit of an integrable deformation; however due to the non-vanishing Coxeter number of the level-zero symmetry, some adjustments to our treatment will be inevitable, see [31]. Furthermore, massive deformations of supersymmetric Chern-Simons theories [32] have an interesting non-conformal supersymmetry algebra [33] on which a Yangian algebra could in principle be established. One may also wonder whether and how the partial integrability of N < 4 sYM theories within certain sectors, see [34], can be formulated in analogy to our framework.
Yangian symmetry could also prove helpful in constructing gauge theory models which are predicted by the AdS/CFT correspondence but remain to be formulated. Such models include two-dimensional gauge theories as duals of the integrable string theories on AdS 3 × S 3 × S 3 × S 1 and AdS 3 × S 3 × T 4 , a q-deformation of N = 4 sYM as a dual of the eta/kappa-deformation of string theory on AdS 5 ×S 5 (where the Yangian symmetry would be deformed to a quantum affine group) as well as the infamous six-dimensional N = (2, 0) theory which is intrinsically non-perturbative.
Another direction for research is to firmly derive the applications of integrability to various observables from our framework of Yangian symmetry. For example, the Bethe equations for the spectrum of one-loop anomalous dimensions follow directly from the well-established Bethe ansatz framework for quantum integrable spin chains in connection to (conventional) quantum algebra. More interestingly, the methods developed for the spectrum at higher loops and at finite coupling should have a formal justification in Yangian symmetry of the model (potentially after implementing quantum corrections). Along the same lines, it will be useful to understand the origin of the extended psu(2|2) Yangian for the magnon scattering picture including the various master, boost and secret symmetries. Similarly, symmetries and techniques for other observables such as Wilson loops, scattering amplitudes, correlation functions of local operators and form factors might be traced back to Yangian symmetry.
Further open questions beyond generalisations and applications include: How to generalise Yangian symmetry to a superspace formulation of N = 4 sYM (either in a light-cone superspace [35] or in a full superspace [15])? Does the (somewhat non-local) Yangian symmetry have some associated Noether charges? What is the role of the Casimir operators (central elements) of the Yangian algebra? Can they be applied to a wider class of observables?
Finally, our framework for Yangian symmetry raises several questions of mathematical nature: First and foremost, does the Yangian algebra actually close, i.e. do the Serrerelations hold and how to formulate the closure precisely? How to construct the singlefield action of the level-one generators abstractly? Does it follow from some fundamental principles beyond the consistency requirements employed to derive it here? Last but not least, in what sense is the proposed action of the Yangian generators a representation, in particular when acting on cyclic states such as a the action functional.
Symmetry in Quantum Gauge Field Theory" of the German Research Foundation.

A N = 4 superconformal symmetry
In this appendix we summarise the N = 4 superconformal algebra psu(2, 2|4) and its representation on the fields.
Algebra. The supersymmetry algebra is spanned by the supersymmetry generators Q aγ ,Qα c and the momentum generator Pα γ . The special conformal generators are given by two fermionic generators S αc ,S aγ and the bosonic generator K αγ . Furthermore, the superconformal algebra includes the Lorentz and internal rotation generators L α γ ,Lαγ, R a c (whose trace over the indices vanishes) and the dilatation generator D. Finally, we will also need gauge transformations to discuss the gauge-covariant representations. These are generated by G[X] where the field X serves as the gauge parameter matrix.
Although we do not explicitly refer to real algebras, a suitable set of reality conditions for N = 4 sYM is given by as well as The below representations will be unitary w.r.t. these reality conditions. In the following we will list the most relevant algebra relations. The Lorentz and internal algebra relations take the form Here The relations involving the fermionic generators typically hold only on-shell, i.e. modulo the equations of motion.

B Level-one generators
In this appendix we will give explicit expressions for the coproducts as well as single-field actions of the level-one Yangian generators These are the level-one generators which commute with the ordinary momentum P (up to gauge artefacts) and hence their single-field action can be expected to have no explicit dependence on the position x.
Level-one supersymmetry. The level-one supersymmetries Q and Q have a nontrivial single-field action only on one of the fermionic fields: The single-field action on all the remaining fields Z of the theory is: The coproduct for Q is now: 6) and the corresponding expression for Q reads: Level-one internal rotations. The generator R a b has a trivial single-field action R a b ·Z = 0. (B.8) Heuristically, this is easy to understand: The generator R carries zero mass dimension so its action on a field Z should have the same mass dimension as Z, i.e. 1 for bosonic fields or 3 / 2 for fermionic fields. Furthermore, the action should have no explicit position dependence as argued above. Together, this implies that the result can only be a single field. However, the parity-inverting property of the level-one generators requires to have at least two fields in the result.
The following coproduct thus defines this symmetry completely: Level-one bonus symmetry. For the same reasons as for R, the level-one bonus symmetry B has a trivial single-field action B·Z = 0. (B.10) The coproduct reads: In fact we can now define a modified level-one internal rotation operator: R a b := R a b + 1 2 δ a b B, (B.12) whose trace is 2 B. This generator has a slightly simplified coproduct as compared to R: