Higher-dimensional invariants in $6D$ super Yang-Mills theory

We exploit the $6D, {\cal N}=(1,0)$ and ${\cal N}=(1,1)$ harmonic superspace approaches to construct the full set of the maximally supersymmetric on-shell invariants of the canonical dimension ${\bf d=12}$ in $6D, \,{\cal N}=(1,1)$ supersymmetric Yang-Mills (SYM) theory. Both single- and double-trace invariants are derived. Only four single-trace and two double-trace invariants prove to be independent. The invariants constructed can provide the possible counterterms of ${\cal N}=(1,1)$ SYM theory at four-loop order, where the first double-trace divergences are expected to appear. We explicitly exhibit the gauge sector of all invariants in terms of ${\cal N} = (1, 0)$ gauge superfields and find the absence of ${\cal N}=(1,1)$ supercompletion of the $F^6$ term in the abelian limit.


Introduction
Maximally supersymmetric theories in diverse dimensions are of particular interest in view of their close relation to superstrings and D-branes. This concerns, e.g., supersymmetric Yang-Mills (SYM) theories with sixteen supercharges existing in dimensions D ≤ 10. In particular, maximally supersymmetric 6D, N = (1, 1) SYM theory arises as a low-energy effective field theory on coincident D5-branes and is reduced to 4D, N = 4 SYM theory upon the appropriate compactification 6D → 4D. The study of quantum 6D, N = (1, 1) SYM theory attracts a heightened attention despite the fact that this theory is non-renormalizable by power-counting because of dimensionful coupling constant. The relevant full quantum effective action is expected to provide an ultra violet (UV) completion of this theory and to be identical, after summing up all loop contributions, to the full Born-Infeld-type world-volume action of D5-branes [1].
The quantum structure of the maximally extended gauge and supergravity theories is uncovered by studying their amplitudes, for which various powerful methods have been developed, see, e.g., refs. [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] (a more complete list of references can be found in the paper [15]). Using these methods, some surprising features of 6D, N = (1, 1) SYM theory in the quantum domain were observed. For instance, it is free of logarithmic divergences up to two loops, at three loops such divergences are seen only in the sector of planar (single-trace) diagrams, while the non-planar (double-trace) diagrams are finite.
The quantum (Wilsonian) effective action of 6D, N = (1, 1) SYM theory probably involves an infinite set of counterterms with the higher-order derivatives. Generically, they are not supersymmetric off shell and need not to be supersymmetric even on the shell of original equations of motion. Since every new term added to the original action can be looked upon as a deformation of the latter, the form of the "on-shell" supersymmetry transformations can vary from one loop order to another. It is the effective action with the whole set of counterterms that is expected to possess the properly realized deformed N = (1, 1) supersymmetry. As was argued in [28], up to the fifth order in loops it is legitimate to deal with the transformations in which the superfields involved are on the shell of the equations of motion corresponding to the original "microscopic" action. The deformation effects would manifest themselves only in the next orders. The corresponding canonical (scaling) dimensions of the component 2 6D spinor algebra and HSS We start by recalling generalities of 6D spinor algebra, as well as of N = (1, 0) and N = (1, 1) ordinary and harmonic superspaces (HSS). More details can be found in [17, 19-22, 26, 32]. We will basically follow the notation and conventions of [26].
The group Spin(5, 1) has two inequivalent irreducible spinor representations of complex dimension 4, that is (1, 0) and (0, 1) spinors. In contrast to 4D case, the complex conjugation does not interchange them, but produces an equivalent representation. In what follows, we will deal with the symplectic Majorana-Weyl spinors λ a i and ψ aA carrying extra SU (2) Sp(1) indices i = 1, 2; A = 1, 2 and satisfying the pseudoreality conditions, e.g., (2.5) In the spinor notation, 6D vectors are uniquely represented as Respectively, 6D Minkowski space can be parameterized by the coordinates x ab related to the standard 6D coordinates X M , M = 0, . . . , 5 as In what follows, we shall make use of the superspace approach in which 6D Minkowski space is extended by anticommuting Grassmann odd spinor coordinates θ a i and θ aA , with i = 1, 2; A = 1, 2 being doublet indices of some external automorphism (Rsymmetry) SU (2) groups.
The standard superspace approach is not too useful for gauge theories with extended supersymmetry. First, it does not allow to reveal the basic unconstrained objects of the theory and to construct the relevant off-shell superfield actions. Secondly, the standard superspace constraints on the basic gauge superfields frequently put the theory on mass shell and give no simple hints as to which portion of the total supersymmetry still admits an off-shell realization. Much more suggestive is the harmonic superspace (HSS) approach firstly discovered for 4D , N = 4 supersymmetry in [19,20]. Off-shell superfield formulations of both 6D, N = (1, 0) SYM theory and hypermultiplets become possible within 6D, N = (1, 0) HSS [21,22], a direct generalization of 4D, N = 2 HSS. On the other hand, in an analogous formulation of the maximally extended N = (1, 1) SYM theory in 6D HSS with the double set of harmonic and Grassmann variables [15,26] the total N = (1, 1) Poincaré supersymmetry admits a closure on the corresponding harmonic superfields only on shell. Hence, the maximally supersymmetric N = (1, 1) invariants can also be defined on shell (though they can still respect off-shell N = (1, 0) supersymmetry). Now we sketch the basic notions of N = (1, 0) HSS. The N = (1, 0) super Poincaré group in its bosonic sector involves the SU (2) automorphisms (R-symmetry) group, besides 6D Poincaré group. When defining the standard N = (1, 0) superspace, this SU (2) is factored out, so this superspace is parameterized by the coordinate set z := (x ab , θ c i ) .
It is convenient to define an analytic basis in this HSS, by passing to the coordinates where Now N = (1, 0) supersymmetry transformations prove to be closed also on the reduced set of coordinates ζ := (x (an)ab , θ +c , u ±i ) , (2.12) that parametrize a Grassmann-analytic subspace 3 . Since G-analytic superfields depend only on the half of odd variables, their component structure is simpler than that of general harmonic superfields. In order to define the G-analyticity condition, we introduce the spinor differential operators 13) which are related to the spinor derivatives in the central basis (2.9) by the standard relations, D ± a = D i a u ± i . Then an arbitrary G-analytic superfield Φ(ζ) can be defined as the one satisfying the Grassmann analyticity condition Like in 4D case [20], the existence of the analytic harmonic superspace with the half set of Grassmann coordinates and the opportunity to define "short" harmonic analytic superfields are the pivotal merits of the 6D, N = (1, 0) HSS approach. The presence of additional bosonic coordinates brings about new algebraic relations, those involving the harmonic derivatives, where (in the analytic basis) The commutation relations between harmonic and spinor derivatives can be easily established As was said earlier, an on-shell superfield description of 6D, N = (1, 1) SYM theory is achieved in the framework of bi-harmonic N = (1, 1) superspace. It is an extension of the superspace (2.9) by new right-handed Spin(5, 1) spinors θ aA , where on the doublet indices A = 1, 2 an additional R-symmetry SU (2) group acts, and by the corresponding extra set of harmonics u± A , u+ A u− A = 1, In this basis, both spinor derivatives D + a and D+ a are short, which reflects the existence of the closed analytic subspace with the double set of Grassmann and harmonic coordinates: Note that one can impose the Grassmann analyticity conditions either separately with respect to θ −a and θ− a , or simultaneously with respect to both these coordinates, without contradiction with the coordinate action of N = 4 Poincaré supersymmetry in the basis (2.18). The second option corresponds to the minimal-dimension "1/2" analytic subspace (2.20), while the first one to the existence of the additional "3/4" analytic subspacesζ I andζ II parametrized by the following sets of coordinateŝ (2.21) They are also closed under total N = (1, 1) supersymmetry. Whileζ involves eight Grassmann coordinates compared to sixteen such coordinates of the full N = (1, 1) HSS, both 3/4 analytic subspaces contain twelve Grassmann coordinates. The harmonic derivatives with respect to u± A in the analytic basis have the same structure as the N = (1, 0) ones defined earlier, modulo the proper altering of the position of the spinor indices. The same concerns their mutual commutators, their commutation relations with the hat-spinor derivatives, and the mutual anticommutators of the hat-spinor derivatives, taking into account the relation To complete this section, we define the integration measures for the full and analytic  (2.25) 3 N = (1, 0) supersymmetric Yang-Mills theory

Kinematics
Now we turn to supersymmetric Yang-Mills (SYM) theory in 6D, N = (1, 0) HSS [21,22,26,32] 4 . First, we introduce the gauge superfield V ++ that covariantizes the harmonic derivative The connection V ++ is a G-analytic superfield that transforms as δV ++ = ∇ ++ Λ under gauge transformations with a G-analytic gauge parameter Λ(ζ). The second harmonic derivative D −− is covariantized by a non-analytic connection V −− , This connection is uniquely determined from the zero-curvature ("harmonic flatness") condition, as a series over the connections V ++ given at different harmonic "points": where 1/(u + u + 1 ), . . . are harmonic distributions defined in [20]. It is straightforward to and imposing the commutation relations we can express the spinor and vector connections in terms of V −− : The covariant spinor superfield strength W +a is defined as For later use, we also define The spinor superfield strengths obey the following off-shell relations (as well as some other ones derived through acting by ∇ −− on eqs. (3.11), (3.12)). An important Bianchi identity stemming from (3.8) and (3.10) is which, in particular, implies

Dynamics
The superfield action of 6D, N = (1, 0) SYM theory can be written in the form [22] , where f is a coupling constant with the dimension of inverse mass, so that the relevant component Lagrangian density has the canonical ("engineering") dimension d = 4. The action (3.17) is invariant under the supergauge transformations of V ++ and yields the equation of motion Both these properties can be derived by employing the variation formula Using the representation eq. (3.18) can be rewritten in terms of the spinor superfield strength as Due to the gauge freedom with the analytic super-parameter Λ, one can choose the Wess-Zumino gauge for V ++ where Ψ +3 d := ε abcd θ +a θ +b θ +c , Ψ +4 := ε abcd θ +a θ +b θ +c θ +d . Along with V ++ , an important ingredient of the harmonic superspace formulation of 6D, N = (1, 1) SYM theory is the analytic hypermultiplet superfield q +A (ζ) which also takes values in the adjoint representation of the gauge group and satisfies the reality condition Its action reads The sum of two actions, reveals a hidden N = (0, 1) supersymmetry [26], where A a is the corresponding Grassmann parameter 6 . So the total action describes N = (1, 1) SYM theory. The equation of motion for the hypermultiplet is (3.28) The equation of motion for V ++ following from the action S (1,1) undergoes the evident modification as compared to (3.18): On the mass shell, the previously listed Bianchi identities are simplified and some new relations of this kind can be derived. In the limit of vanishing hypermultiplet superfield (this limit will be employed in Sect. 7), the equations of motion (3.21), combined with the relation (3.14), imply The Bianchi identity (3.16) yields 31) or, in an equivalent form, The last two identities, together with (3.15), give rise to the equation Eqs. (3.31) and (3.33) are none other than the covariantized 6D Dirac and Klein-Fock-Gordon equations for the N = (1, 0) vector multiplet. In the abelian limit they are reduced to ∂ ab W ±b = 0 , ∂ ab ∂ ab W ±c = 0 .

N = (1, 1) SYM in bi-HSS formulation
When 6D, N = (1, 1) SYM theory is formulated in terms of N = (1, 1) superfields ( [17], [23] and [15]), the constraints of the theory imply equations of motions, i.e. this formulation supplies an example of the on-shell description. The original N = (1, 1) SYM constraints, before introducing the harmonic coordinates, are written in terms of the gauge-covariantized spinor derivatives, where A i a andÂ aA are the spinor connections and the convention ∇ ab = 1 2 ε abcd ∇ cd is assumed. The constraints read [17,23] Through the Bianchi identities, eqs. (4.2) and (4.3) imply The constraints (4.2) and (4.3) define 6D, N = (1, 1) SYM supersymmetric theory in the standard N = (1, 1) superspace parametrized by the coordinates (x ab , θ a i ,θ Aa ). As was already mentioned, they imply the equations of motion for the superfields involved and so put the theory on shell.
The standard steps of the harmonic interpretation of the above constraints can be summarized as: (i) contracting the SU (2) indices of the spinor derivatives and superfield strengths with the harmonics u ± i , u± A , thereby passing to the harmonic projections of these derivatives; (ii) replacing the property of linearity in harmonics by the property of vanishing of some commutation relations with the harmonic derivatives; (iii) passing to the analytic basis and frame, so as to make "short" (i.e. having no connection in this frame) as much spinor derivatives as possible. After these steps the constraints are rewritten as the integrability conditions for some sort of Grassmann analyticity, while the underlying gauge multiplet proves to be carried by the harmonic gauge connections covariantizing flat harmonic derivatives.
This general scheme, being applied to the case at hand, yields the following equivalent set of the constraints [26] where we have chosen the "hat-analytic" basis, with a short hat-spinor derivative, ∇+ a = D+ a = ∂/∂θ− a . All other derivatives have non-trivial gauge connections The constraints (4.5) are covariant under the gauge transformations with the 3/4 hatanalytic parameterΛ: (4.7) Eqs. (4.5) impose non-trivial on-shell conditions on the gauge connections. The notable property of the constraints in the form (4.5) is that the derivative ∇ + a cannot be chosen "flat" simultaneously with ∇+ a because of their non-vanishing anticommutator.
The full set of the relations solving (4.5) in terms of N = (1, 0) superfields was found in [26]. Below we shall briefly summarize them.
• An important result is that, after imposing the appropriate gauge fixing and due to the vanishing of the commutator between the hatted and unhatted covariant harmonic derivatives, the connections V ++ and V −− prove to coincide with their N = (1, 0) prototypes, • For the N = (1, 1) SYM superfield strength φ ++ the following solution was obtained where Ψ+ 3d := ε abcd θ+ a θ+ b θ+ c , Recall that D ab = 1 2 ε abcd D cd , D cd = ∂ cd + A cd and the vector N = (1, 0) connection A cd was defined in (3.7).
• The final expression for the harmonic connection V++ is as follows As a consequence of the second flatness condition, • The constraints (4.5) accompanied by the suitable gauge choice fix the form of the spinor connection A + a entering ∇ + a in (4.6) as • The constraints also imply the equations of motion (3.29), (3.28) for the superfields V ++ and q +A . Using these equations of motion together with the explicit formulas quoted above (as well as some remote consequences of them collected in Appendix A of [26]), one can check the important properties which, in fact, directly follow from the constraints (4.5) as the Bianchi identities.
For further use, we also define It is easy to check that and • The superfield strength φ ++ also satisfies the on-shell analyticity conditions which follow as the Bianchi identities from (4.5) and can be explicitly checked using the relations given above, the N = (1, 0) analyticity of q +A , D + a q +A = 0, and the N = (1, 0) superfield equations of motion.
• After all gauge-fixings and solving the constraints, the only residual gauge free- As the last steps in constructing the formalism of the differential superspace geometry of N = (1, 1) SYM theory, one defines, in the standard way, the negatively charged covariant spinor derivatives and the covariant vector derivatives Note that the identification of the vector connections obtained in two different ways plays an important role in finding the explicit solution of the constraints (4.5) presented above. Also note that, taking into account the expression (4.12), the spinor derivative ∇ − a defined in (4.18) has the very simple form In what follows, we will also use the relations which can be easily deduced from the previous relations and definitions. Some other useful relations following as Bianchi identities from the basic ones read

Generalities
The building-blocks of all higher-dimension invariants of N = (1, 1) SYM theory in the bi-harmonic approach are the superfield strength φ ++ , its various harmonic projections obtained by acting on it by the gauge-covariant harmonic derivatives ∇ ±± , ∇±±, as well as the superfields obtained through acting of the spinor and vector gauge-covariant derivatives ∇ ± a , D+ a , ∇− a and ∇ ab on the harmonic projections just mentioned. These objects live either on the entire N = (1, 1) bi-HSS or on its analytic subspaces. To ensure gauge invariance, one is led to take traces of the products of such elementary matrix blocks, each block being valued in the adjoint representation of the gauge group algebra (like φ ++ ). As shown in [26], given some covariant N = (1, 1) superfield Φ(Ẑ), the transformations of hidden N = (0, 1) supersymmetry on the N = (1, 0) superfield constituents of such products, before taking traces, are induced by the following generic transformation of Φ(Ẑ) Here, δ stand is the standard N = (0, 1) superfield transformation in the analytic basis induced by the proper shifts of the superspace coordinates, while δ comp (1) and δ comp (2) are compensating gauge transformations with analytic parameters Λ (1) ,Λ (2) needed to preserve various superfield gauges imposed in the process of solving the constraints (4.5) where the higher-order θ+ a terms inΛ (1) are of no interest for us here. Actually, the same formula for variation is applicable not only to gauge-covariant expressions but equally for the various gauge connections. Applying it to the basic entity φ ++ defined in (4.9), we find the following N = (0, 1) transformation of the basic N = (1, 0) superfield objects We start by recalling some invariants obtained in [26]. In what follows, the dimension d of diverse invariants is understood as the canonical dimension (in the mass units) of their component Lagrangian. So the dimension of the "microscopic" action S (1,1) is d = 4. Note that it makes no sense to ask whether this simplest invariant can be represented as an integral over N = (1, 1) superspace since an essential feature of our formalism is that it is on-shell. The relevant N = (1, 0) superfields are assumed to satisfy the equations of motion (3.28), (3.29) following just from S (1,1) , so the latter vanishes on the shell of these equations. It will be useful to list the dimensions of the basic building blocks: It is worth to have in mind that, due to the presence of trace in the possible invariants, it is legal to integrate by parts with respect to the harmonic, spinor and x-derivatives, e.g., Tr∇ ++ Ω = D ++ TrΩ, etc. The same is true for the invariants the Lagrangian densities of which are representable as products of few traces. For what follows, it will be useful to give the general definition of various invariants as integrals over the full bi-harmonic superspace and its various subspaces (5.8)

Examples
As promised, we start with recalling the results of [26]. and taking into account that the factor (D+) 4 in (5.15) acts only on φ −− in view of the hat-analyticity of φ ++ , it is rather straightforward to find, using, in particular, the relations (4.24), that Hence, On the other hand, using the integration by parts with respect to ∇++, one can show that Note an essential difference between the representations (5.20) and (5.22). When passing from (5.15) to (5.20), no use of the N = (1, 0) superfield equations was made, so N = (1, 0) supersymmetry can be still treated as an off-shell supersymmetry, the equations of motion are needed only for demonstrating the hidden N = (0, 1) supersymmetry. The Lagrangian in (5.20) is manifestly hat-analytic since the superfield strengths φ ±+ do not depend on θ− a . On the other hand, when passing to the representation (5.22), it is imperative to use the equations of motion, because, e.g., the property ∇ + a φ ++ = 0 is valid only on shell. So the Lagrangian in (5.22) is N = (1, 0) analytic only on shell, and N = (1, 0) supersymmetry is valid also on shell, like in the case of d = 8 invariant (5.12). In other words, the equivalency between the initial full bi-harmonic superspace single-trace invariant (5.15) and the invariant (5.20) is valid only on shell. Thus (5.20) provides another example of N = (1, 1) invariants in which both N = (1, 0) and N = (0, 1) supersymmetries are on shell. This peculiarity was not noticed in [26].
As for the double-trace invariants of the dimension d = 10, they can be defined only in the 3/4 analytic superspaces, First of all, it is important to realize that it is impossible to construct, out of the elementary "bricks" φ ±± , φ ∓± and those obtained from them through action of the covariant differential operators ∇ ± a , D+ a , ∇− a and ∇ ab , the gauge invariant and manifestly analytic objects possessing the charges (+4,+4), (+4, 0) or (0,+4) and canonical dimensions indicated in (6.1) 8 . So we are left with the chargeless general bi-HSS superfield densities L of the dimension 4 as the only candidates for the invariants we are interested in.

Invariants without derivatives
We start with the invariants containing only superfield strengths φ ±± , φ ∓± . One can construct ten chargeless superfield invariants of the dimension 4 from these quantities 2) Some of them are related via the integration by parts with respect to harmonic derivatives. To extract the irreducible set, one needs to take off some derivatives with "hat" and/or without "hats" from some superfield factors in (6.2) and (6.3) and place them on the other factors in the same expression, using the relations (4.14), (4.16) and the on-shell harmonic constraints (4.13), (4.15). This can be done in a few equivalent ways. It will be convenient for us to consecutively take off the derivatives ∇ −− , ∇−− from all negatively charged factors in the densities J 1 , . . . , J 4 . For instance, representing the third factor φ −− in J 1 under trace as φ −− = ∇ −− φ +− = ∇−−φ −+ and integrating by parts with respect to ∇ −− and ∇−−, we obtain, respectively, means "up to a total harmonic derivative". In this way we obtain the following set of linear equations (we replace by = for brevity) J 1 + I 5 + I 6 = 0 , J 1 + I 2 + I 1 = 0 , J 1 + I 3 + I 2 = 0 , J 1 + I 4 + I 5 = 0 , (6.5) J 2 + I 4 + I 1 = 0 , J 2 + I 3 + I 6 = 0 , (6.6) J 3 + I 2 + I 6 = 0 , J 3 + I 3 + I 5 = 0 , J 3 + I 1 + I 5 = 0 , J 3 + I 2 + I 4 = 0 , (6.7) It is direct to check that applying the same procedure to the densities I 1 , . . . , I 6 produces no new relations. Moreover, the system (6.5) -(6.8) is underdetermined and in fact amounts to eight independent equations for ten unknowns. As a result, we are left with the two independent densities. As such it is convenient to choose J 1 and J 2 : where it is assumed as before that the equalities are valid up to total ∇ −− or ∇−− derivatives. So in dimension d = 12 we have two independent single-trace superfield invariants without derivatives, Analogously, one can define two independent double-trace invariants of the same type: (6.11)

Invariants with derivatives
Next we consider possible superfield densities containing spinor and/or vector covariant derivatives. The only possibilities to ensure the total dimension four are the densities with three covariant strengths and two spinor derivatives, or densities with two strengths and either two vector, or four spinor, or two spinor and one vector derivatives properly distributed among these strengths. It is clear that the double-trace invariants of this kind do not exist in view of the tracelessness of the gauge algebra generators. We start with the three-strength case. Taking into account the on-shell analyticities of various harmonic projections of the superfield strength and the possibility to integrate by parts under the trace, we can construct two options Integrating by parts with respect to D+ a in the first expression and using {∇ + a , D+ a } = 4φ ++ , we obtain For constructing densities with two superfield strengths there are much more opportunities, but, once again, they are reduced, modulo the invariants already presented, to the following single invariant (6.14) As an example of such a reduction, consider, e.g., the density An evident chain of transformations involving integrations by parts, yields where we made use of the relation ∇ − b φ −− = 0 . In a similar way one can handle the density Tr can be reduced to (6.15) by using the commutation relations (4.25), where we made use of the relation (6.13).
Two more examples are the following 'trial' densities Using the relations (4.23), it is easy to reduce them to the expressions which are related to each other via integration by parts with respect to the harmonic derivative ∇ ++ . All other possible expressions, e.g., ∼ ε abcd Tr can be transformed in a similar manner.
For reader's convenience, we finish this section by listing the complete set of independent N = (1, 1) superfield invariants corresponding to the component Lagrangian densities of the canonical dimension d = 12 .

Single-trace invariants
It is worth to point out that, as distinct from the d = 10 single-trace invariant, the d = 12 ones cannot be equivalently rewritten as integrals of the manifestly analytic Lagrangian densities over the appropriate analytic subspaces. It would be interesting to understand possible implications of this property for the structure of quantum corrections in N = (1, 1) SYM theory. Anyway, in all these invariants N = (1, 0) supersymmetry can be treated as an off-shell one and all of them could in principle appear as divergences in the four-loop order of 6D, N = (1, 0) HSS superfield perturbation calculations. Note that in the abelian case all invariants without derivatives become identical to each other. 7 Passing to N = (1, 0) superfields Here we consider the N = (1, 0) superfield form of some invariants listed in the previous sections. For simplicity, we will confine our attention to the contributions of the N = (1, 0) gauge superfield and ignore the hypermultiplet pieces. So our input for the basic N = (1, 1) superfields defined in (4.9), (4.10) will be the following φ ++ → −θ+ a W +a , V++ → iθ+ a θ+ b A ab . (7.1) We also obtain To compute the remaining N = (1, 1) SYM quantities, we need the explicit expression for V−− which is defined by the hat-flatness condition (4.11). It is easy to see from the latter that the θ− a expansion of V−− starts with the second-order terms. So we parametrize V−− as where the coefficients are functions of (θ+ a , u± A ) and of the N = (1, 0) coordinates Z = (x ab , θ ±a , u ±i ). From (4.11) we obtain the following equations for these coefficients These equations fully fix the (θ+ a , u± A ) dependence of the coefficients and we obtain the rather simple expression for V−− in the approximation considered: where In what follows we will be interested in the on-shell form of the invariants. The coefficients in (7.5) are simplified on shell and can be expressed through the spinor superfield strengths W ±a . We can use one of the equivalent forms of the equations of motion for W ±a , and represent, using the relation (3.15), the second coefficient in the expansion (7.5) as Also, after simple algebra, using the on-shell cyclic identity (3.32), we derive Now one can compute the total vector connection and also the rest of the harmonic projections of the covariant superfield strength As the first example of passing to N = (1, 0) superfield notation, we rewrite the on-shell invariant (5.22). Using the representation dζ (−4,0) II = dζ (−4) dû(D−) 4 (D+) 4 , (7.13) and the relations (D±) 4 Ψ∓ 4 = −4! (plus a total x-derivative), we find (7.14) It is straightforward to check the implicit on-shell analyticity of L +4 (10) , This highlights the property that the d = 10 single-trace invariant in the form (5.22) is both N = (1, 0) and N = (0, 1) supersymmetric only on shell. Note that the same Lagrangian, up to a total derivative, can be rewritten as the trace of two (anti)commutators from which it follows, in particular, that this invariant vanishes in the abelian limit. Analogously, one can find the N = (1, 0) superfield core of the on-shell double-trace invariant (5.24): The on-shell analyticity of this Lagrangian can also be easily checked. In the abelian limit this expression is reduced to Recall that in the abelian limit these invariants become identical to each other up to numerical coefficients. They can be reduced to the expression S (abel) which is not a total derivative even on shell, with ∂ gf ∂ gf W ±d = 0. Hence this sort of invariants could be relevant to the Coulomb branch of the theory, with the original gauge symmetry being broken down to some abelian subgroup (e.g., Cartan subgroup). Before going to the invariants with the spinor and vector derivatives, it will be convenient to define the set of N = (1, 0) SYM invariants in terms of which the N = (1, 0) SYM cores of the single-trace on-shell N = (1, 1) invariants considered here admit a succinct compact expression. Using the on-shell conditions and integrating by parts with respect to various derivatives (including the harmonic ones), it is possible to show that there exist only five independent structures of this kind. It is convenient to choose them as (7.28) Using the list of integrals given in Appendix, the explicit expressions for the N = (1, 0) SYM cores of the invariants S (1) I,II found earlier can be expressed through the independent integrals as Note that the only combination of Y 1 and Y 2 which vanishes on shell in the abelian limit is 30) It appears just in the difference S (1) II , which can be transformed on shell to the form ∼ Tr dẐ [φ ++ , φ −− ] 2 that vanishes in the abelian limit (the superfield densities of the structures E 1,2,3 are reduced to total D + a derivatives in this limit). Now we can proceed to the remaining invariants S (1) (12)(3) and S (1) (12)(2) defined in (6.22). It will be convenient to rewrite, using various on-shell properties of φ ±± and integrating by parts, the Lagrangian density L (1) and also to take into account that Rather boring (albeit straightforward) calculations yield We observe that here just the combination (7.30) appears, so the N = (1, 0) core of the 3-superfield invariant is on-shell vanishing in the abelian limit. One way to calculate the N = (1, 0) limit of the Lagrangian density of the remaining invariant S (1) (12)(2) is to use the explicit form of the vector connection (7.10). Another, simpler way is to note that this Lagrangian can be conveniently rearranged as Once again, rather cumbersome time-consuming calculations finally yield the expression from which it follows, in particular, that this invariant also does not contribute to the abelian limit. Finally, to give a feeling how these invariants look in terms of components, we will present the component form of the abelian N = (1, 0) superfield Lagrangian (7.23), with all fields, except for the gauge field A ab (x), being omitted. The corresponding form of V ++ in WZ gauge and in the analytic basis is just V ++ = iθ +a θ +b A ab . (7.35) The relevant non-analytic harmonic connection V −− is easily restored from the abelian version of the flatness condition (3.3): That part of W −a = D −− W +a which contributes to the Berezin integral in (7.23) can also be easily found Then it is straightforward to perform the Berezin integration and to find S (abel) The same expression in the vector notation looks rather bulky. Passing to the vector notation according to eqs. (2.6), (2.7), we obtain 9 It is interesting that there exist no d = 12 invariants which could yield, in the abelian gauge theory limit, the expression like ∼ F 6 , as distinct from the d = 8 invariants which contain F 4 term in such a limit [26]. It is likely that the abelian F 6 terms do not admit even N = (1, 0) completion.

Summary and outlook
In this paper we continued the applications of the on-shell 6D, N = (1, 1) HSS for constructing higher-derivative invariants of the maximally supersymmetric 6D SYM theory. We have explicitly constructed new, dimension d = 12, single-trace invariants (6.22) and the double-trace invariants (6.23), studied their N = (1, 0) SYM cores and the component gauge-field structure of their abelian limit. All these invariants are given by integrals over the total bi-harmonic 6D superspace and we did not find any "no-go"-type conjectures which would forbid the appearance of the four-loop logarithmic divergences of such a form within the off-shell 6D, N = (1, 0) HSS supergraph calculations in the quantum N = (1, 1) SYM theory. The leading in momenta contributions should come from (6.23) and the invariants in the first line of (6.22), while the rest of the single-trace invariants, including additional derivatives, should be responsible for some sub-leading terms (divergent and/or finite) in the relevant order of the total effective action.
It would be interesting to look for the invariants of the dimension d = 14 and d = 16 corresponding to the five-and six-loop contributions. The leading terms should be a direct generalization of those in (6.22) and (6.23). E.g., one of the possible candidates for such invariants of the dimension d = 14 reads ∼ Tr dẐ (φ ++ ) 3 (φ −− ) 3 , and it seems that finding the full set of independent on-shell invariants of this sort could be a comparatively easy straightforward task. On the other hand, according to the arguments of ref. [28], already at the dimension d = 16 there should be taken into account deformations of the conventional mass shell associated with the superfield equations of motion (3.29) and (3.28). So the formalism of the on-shell 6D, N = (1, 1) HSS should undergo a proper modification. It would be extremely interesting to explicitly learn how this would happen, how the initial constraints of 6D, N = (1, 1) SYM theory are to be corrected and which geometric principle could stand behind such a modification.
(1, 1) invariants in Sect. 7. While transforming the relevant integrands, we integrate by parts and use the on-shell conditions D + a W +d = D − a W −d = 0 , D + a W −d = −D − a W +d , D + a W −a = 0 , ∇ ±± W ±a = 0 , ∇ ±± W ∓a = W ±a . In this way we derive the following list of the on-shell N = (1, 0) SYM integrals