Superfield realization of hidden $R$-symmetry in extended supersymmetric gauge theories and its applications

We present the explicit superfield realizations of the hidden $SU(4)$ and $O(5)$ $R$-symmetries in $4D, {\cal N}=4$ and $5D, {\cal N}=2$ supersymmetric Yang-Mills theories in the harmonic superspace approach. The $R$-symmetry transformations are constructed and their algebraic structure is studied. It is shown that such transformations are consistent both with manifest and with hidden supersymmetry transformations. These symmetries can serve as an alternative tool for constructing the relevant complete low-energy superfield effective actions determined earlier from the hidden supersymmetry considerations.


Introduction
Classical and quantum supersymmetric gauge (SYM) theories with 16 supercharges in diverse dimensions play an important role in modern field theory(see, e.g., [1]). The main source of interest in them is the property that they describe low-energy limit of some brane-like compactifications of type II string theory and thereby provide a bridge between superstring theory and supersymmetric field theory. Most interesting and elaborated are 4D, N = 4, 5D, N = 2 and 6D, N = (1, 1) SYM theories. Some of these models admit superfield formulations with half of the underlying supersymmetry being manifest and off-shell, namely, with 4D, N = 2, 5D, N = 1 and 6D, N = (1, 0) off-shell supersymmetries. These formulations were constructed within the relevant harmonic superspaces (see [2,3] for review).
In such formulations, the second half of the total supersymmetry is realized on the basic superfields of the theory as a hidden (or implicit) supersymmetry, which forms a closed Lie bracket structure with the manifest supersymmetry only on shell. While inspecting the superfield quantum effective actions, the role of this hidden half of supersymmetry turns out to be very restrictive: requiring the effective action to enjoy such a supersymmetry fixes, in most known cases, its structure up to an overall coefficient which is further explicitly calculated from the relevant superfield quantum perturbation theory. Review of this approach is given in Refs. [4,5,6] The basic example of applying such a strategy for constructing the quantum effective action is provided by 4D, N = 4 SYM effective action, which was constructed as a hypermultiplet completion of non-holomorphic N = 2 gauge superfield potential in [7]. Later on, the same effective action was reproduced in various harmonic superspaces [5,8]. These works revealed a correspondence between the N = 4 SYM low-energy effective action and the leading terms in the effective action of D3 brane on the AdS 5 × S 5 background. One more example is related to 5D, N = 2 SYM theory. Its low-energy effective action was constructed as a hypermultiplet completion of the 5D, N = 1 SYM effective action [9].
In this parer we propose another way to determine the low-energy effective action of 4D, N = 4 and 5D, N = 2 SYM theories in harmonic superspace. It is based upon exploiting a hidden bosonic R-symmetry of these theories instead of hidden supersymmetry. To be more precise, we suggest a new possibility to construct the superspace functionals depending on all fields of the corresponding supermultiplet, beginning with a functional which involves only part of such fields. The point is that the supersymmetry algebra possesses the automorphism group which is called the R-symmetry group. We will show that such R-symmetry can be realized directly on harmonic superfields. While some subgroups of R-symmetry are realized linearly and manifestly, the rest of its transformations proves to possess a highly non-trivial realization mixing all fields of the extended supermultiplet. As a result, we gain a possibility to impose the condition of invariance under the total R-symmetry on a superspace functional in order to specify its dependence on all fields of such a supermultiplet.
In the harmonic superspace formulation, the full multiplets of 4D, N = 4 or 5D, N = 2 SYM theories consist of the gauge vector multiplet and hypermultiplet. Not only half of 4D, N = 4 and 5D, N = 2 supersymmetries are realized in an implicit way, but also that part of total R-symmetries of these theories, viz., of SU (4) and O(5), which mixes the hidden and manifest supersymmetry transformations. It is also realized by some implicit transformations. In this paper we find the precise form of the hidden R-symmetry transformations, which extend the manifest R-symmetry groups, namely U (2) × SU (2) and SU (2) × SU (2), to SU (4) ∼ O (6) or Although the low-energy effective action might be found by direct quantum computations in harmonic superspace or by using the hidden supersymmetry transformations, we determine it here in a different way. Namely, we construct the hypermultiplet completions of 4D, N = 2 or of 5D, N = 1 leading terms by imposing requirement of just full R-symmetry invariance. The effective action corresponds to the Coulomb branch, with the gauge group being broken to some abelian subgroup. For simplicity we concentrate on SU (2) gauge group broken to U (1) and consider, as usual in Coulomb phase, only that part of the effective action which depends on the fields of massless U (1) N = 2 gauge multiplet and its neutral hypermultiplet partner which form together 4D, N = 4 or 5D, N = 2 abelian U (1) gauge multiplets.
In addition, we explicitly show that the 4D, N = 4 SYM effective action respects superconformal invariance.

4D, N = 4 SYM theory
In this section we present the superfield realization of the hidden part of the R-symmetry transformations for 4D, N = 4 SYM theory with the gauge group SU (2) and construct lowenergy effective action. As an additional exercise, we show its superconformal invariance. We follow the notation and conventions of Refs. [3,5].

Classical action
When formulated in N = 2 harmonic superspace, N = 4 vector gauge multiplet can be viewed as a "direct sum" of gauge N = 2 multiplet V ++ and the hypermultiplet q + a = (q + , −q + ). Both these multiplets belong to the same adjoint representation of the gauge group. N = 2 gauge multiplet V ++ is described dy the classical action [10] where integration goes over total harmonic superspace and the harmonic distributions , · · · are defined in [3]. This action yields the following equations of motion where (D + ) 2 = D +α D + α , D + α is the harmonic plus projection of the gauge-covariant spinor derivative in the so called "λ" frame, in which D + α andD + α require no gauge connection terms, W ,W are the chiral and antichiral gauge superfield strengths. They can de expressed in terms of the non-analytic harmonic gauge connection where V −− is related to V ++ by the harmonic flatness condition The classical action for the analytic hypermultiplet in the adjoint representation reads [2] where dζ −4 is the measure of integration over the analytical harmonic superspace. This action is invariant under an extra SU (2) PG symmetry transforming q +a as a doublet. Both actions (2.1) and (2.5) are invariant under the standard linear automorphism group SU (2) R which rotates the doublet indices of the harmonic variables. In addition, both actions are invariant under the separate R-symmetry U (1) R which transforms θ ± andθ ± by the conjugated phase factors. Correspondingly, W andW defined in (2.3) are also transformed by the appropriate mutually conjugated phase factors, q +a is the U (1) R singlet. The action of N = 4 SYM theory in N = 2 harmonic superspace is the sum of the actions (2.1) and (2.5), The total action is invariant under the following hidden N = 2 supersymmetry transformations which complement the manifest N = 2 supersymmetry to the full N = 4 supersymmetry with¯ aα and α a as new anticommuting parameters. The algebra of these transformations is closed modulo terms proportional to the classical equations of motion. Therefore, in this formulation only the manifest N = 2 supersymmetry is off-shell closed.

R-symmetry transformations
We define the additional R-symmetry transformations of the vector multiplet and hypermultiplet harmonic superfields as follows are the commuting dimensionless complex parameters. These transformations extend the R-symmetry group from U (2) R × SU (2) PG to SU (4). The direct check shows that the action (2.6) is invariant under the transformations (2.8). The form of (2.8) is almost uniquely specified by the dimensionality and analyticity reasonings, together with requiring both sides to have the same harmonic U (1) charges. To avoid a possible confusion, we point out that the superfields in (2.8) are not subject to any onshell conditions, which should be taken into account only when inspecting the closure properties of these transformations (see below).
Further in this section we consider the case of abelian gauge group, since the effective action we will deal with depends only on the superfields of the abelian U (1) gauge multiplet. The equations of motion implied by the action (2.6) read In addition, hypermultiplet q + a obeys the off-shell analyticity constraints Superfield strengths W ,W are chiral and antichiral 11) and they satisfy the off-shell constraints which follow from the harmonic flatness condition (2.4) and the analyticity of V ++ . When superfields W ,W and q + a obey the on-shell constraints (2.9), the transformations of hidden N = 2 supersymmetry (2.7) are simplified to In this case R-symmetry transformations (2.8) are also simplified: (2.14) One may verify that the commutators of R-symmetry transformations (2.14) with manifest and hidden supersymmetry transformations give the consistent results. The variation of general superfield under the manifest supersymmetry readŝ Let us first consider the commutators of the hidden supersymmetry transformations (2.13) with the R-symmetry transformations (2.14). One can show, by a direct computation, that Hence, the on-shell commutator of the hidden supersymmetry transformations (2.13) with the R-symmetry transformations (2.14) gives the manifest one (2.15) with the bracket parameter λ + c αc , in agreement with N = 4 supersymmetry algebra.
Let us now evaluate the commutators of the R-symmetry transformations (2.14) with the manifest supersymmetry transformations (2.15). We obtain In other words, the on-shell commutator of the R-symmetry transformations (2.14) with the manifest supersymmetry transformations (2.15) yields the hidden supersymmetry transformations (2.13) with the bracket parameter λ ia αi .
Finally, we consider the commutator of the R-symmetry transformations with itself. We have Here, the parameters λ ij correspond to SU (2) R transformations, the parameter λ (λ = −λ) corresponds to the additional U (1) R symmetry and λ ab (PG) are associated with the SU (2) PG symmetry commuting with N = 2 supersymmetry and U (2) R symmetry.
Thus the on-shell closure of the implicit R-symmetry transformations yields the linear U (2) R and SU (2) PG transformations, once again in agreement with the action of the coset part of the full automorphism symmetry SU (4) R on the N = 4 supersymmetry algebra.
Note that the calculation of the brackets (2.18) is not as straightforward as that of the previous Lie brackets. Some details of it are collected in Appendix.

Effective action
The leading low-energy term in the effective action of N = 2 SYM theory in N = 2 superspace has the form (see, e.g., the reviews [4,5]): where Λ is an arbitrary scale 1 . The complete N = 4 SYM effective action is an extension of the effective action (2.20) by hypermultiplet-dependent terms. It was first found in [8] and reads where Li 2 (Z) is the Euler dilogarithm function. The part containing q +a is fixed by the requirement that the effective action Γ be invariant under both manifest N = 2 supersymmetry and hidden on-shell N = 2 supersymmetry. As a result, the effective action (2.21) is invariant of N = 4 supersymmetry and depends on all fields of N = 4 vector multiplet. Now we will show that the effective action (2.21) can be alternatively derived from (2.20) by imposing the requirement of R-symmetry instead of invariance under the hidden N = 2 supersymmetry. To this end, let us consider the variation of (2.20) under transformations (2.14), (2.23) Based on reality reasonings, one can concentrate only on that part of the transformations which involves the parameter λ ia . Due to the property that d 12 zdu q +a λ − ā W = 0, the variation (2.23) 1 In fact, the action does not depend on Λ in virtue of the (anti)chirality of (W )W .
can be canceled by adding the new term to H: Evaluating the variation of the sum H + L 1 , we obtain (2.25) Consider the last term in some detail (2.26) Here we have used various properties of the involved superfields (chirality, analyticity), as well as the integration by parts with respect to spinor derivative in the second line and cyclic identities for the SU (2) doublet indices in the third line. Observe that the last term in the third line is equal, modulo a minus sign, to the variation we started with. Hence, Plugging (2.27) into (2.25), we obtain (2.28) This variation is canceled by adding the new proper term to (2.20), (2.29) Continuing the iterative process, one can find that L n = c n 2 (n + 1) Summing up all L n , one recovers the effective action (2.21).
One can directly verify that (2.21) is invariant under R-symmetry transformations (2.14). Once again, we limit our attention to the terms with parameter λ ia : .
This expression can be simplified, using the identity which is deduced by integrating by parts with respect to spinor derivative and applying to the definition of the function L(Z) (2.21). Thus, we obtain (2.33) To summarize, the requirement of invariance under R-symmetry transformations allowed us to completely restore the hypermultiplet dependence of N = 4 supersymmetric effective action.

Superconformal invariance of the effective action
The effective action (2.21) is evidently scale-invariant. In this section we prove that it is actually invariant under 4D, N = 4 superconformal group SU (2, 2|4).
Due to the structure of 4D, N = 4 superconformal algebra and the R-symmetry invariance of the effective action it suffices to show only its 4D, N = 2 superconformal invariance. Moreover, it is enough to check just invariance under conformal boosts.
To prove the superconformal invariance we should use the transformation rules of the harmonic superspace coordinate, as well as of the harmonic and spinor derivatives. The harmonic superspace coordinate transformations under conformal boosts in the analytic basis read [3] δx αα = x βα k ββ x αβ , where k αα is the corresponding 4 vector parameter. The transformation law of the harmonics is [3] δu +k = Λ ++ u − k , δu − k = 0, Λ ++ = 4iθ +α k ααθ +α , Next, let us write the superconformal transformations of the harmonic and spinor derivatives Using these transformation rules it is easy to establish the transformation of the superspace integration measure dZ = d 12 zdu = d 4 xd 4 θ + d 4 θ − du, Using the transformation rules (2.36) and (2.38), it is easy to obtain the transformation law of the superfield strengths W ,W The transformation of the hypermultiplet q ±a under conformal boost reads [3] δq +a = −k αα x αα q +a , Now we are prepared to show superconformal invariance of the effective action (2.21). Let us first show invariance of the logarithmic term where we made use of some properties of W andW (2.11)-(2.12) and the equations of motion (2.9). Then we check invariance of the generic term in the power expansion of the function L(z) Here, once again, we used the properties of W andW (2.11)-(2.12) and equations of motion (2.9).
So we have proved that the effective action (2.21) is superconformally invariant.

5D, N = 2 SYM theory
In this section, we introduce the R-symmetry transformations for N = 2 SYM theory with the gauge group SU (2) and construct the complete low-energy effective action by requiring invariance under these R-symmetry transformations. We use the notations and conventions of Refs. [3,11].

Classical action
N = 2 gauge multiplet in 5D, N = 1 harmonic superspace is described by N = 1 gauge multiplet and hypermultiplet q +a . The N = 1 gauge multiplet classical action reads [10] where g is a coupling constant of mass-dimension −1/2. The superfield strength is defined in the analytic λ-frame as where (D + ) 2 = D +α D + α = ΩαβD + β D + α , Ωαβ is the Spin(1, 4) invariant symplectic "metric" and V −− is a non-analytic gauge potential related to V ++ by the harmonic flatness condition The classical action of the hypermultiplet q +a = (q + , −q + ) in the adjoint representation of the gauge group is written as [2] S q = 1 2g 2 tr dζ −4 q + a ∇ ++ q +a = 1 where dζ −4 is the analytic superspace integration measure. In addition, this action is invariant under SU (2) PG symmetry, which transforms q +a as a doublet. The action of N = 2 gauge multiplet in 5D, N = 1 harmonic superspace is just the sum of (3.1) and (3.4), The action is invariant under the implicit N = 1 supersymmetry completing manifest N = 1 supersymmetry to the whole N = 2 supersymmetry where a aα is the relevant anticommuting parameter, and

R-symmetry transformations
We define the R-symmetry transformations in 5D, N = 1 harmonic superspace as where λ ia is the relevant commuting parameter (λ ia = λ ia , λ +a = λ ia u + i ). These transformations extend R-symmetry group from SU (2) R × SU (2) PG to O(5). The direct check shows that the action (3.5) is invariant under the transformations (3.8).
Further in this section we consider the case of abelian gauge group. The action (3.5) yields the equations of motion In addition, the superfield strength W satisfies the off-shell constraints where q −a = D −− q +a . The R-symmetry transformations (3.8) take the form (3.12) Now we can consider the commutator of supersymmetry transformations with those of Rsymmetry. The variation of general superfield under the manifest supersymmetry readŝ where ± α = â α u ± a is the relevant anticommuting parameter.
Let us first consider the commutators of the hidden supersymmetry transformations (3.11) with the R-symmetry transformations (3.12). One can show that (3.14) Hence, the on-shell commutator of the hidden supersymmetry transformations with Rsymmetry transformations gives the manifest supersymmetry with parameter λ i c αc , as expected.
Let us now consider the commutators of the R-symmetry transformations (3.12) with the manifest supersymmetry transformations (3.17). It leads to In other words, the on-shell commutator of R-symmetry with the manifest supersymmetry yields the hidden supersymmetry with the bracket parameter λ ia α i . At last, one can consider the commutator of R-symmetry transformations (3.12) with itself. It gives (3.16) The details of the derivation are given in Appendix. We observe that the on shell commutator of two hidden R-symmetry transformations yields manifest linear SU (2) R and SU (2) PG transformations, as should be.

Effective action
The part of the superfield N = 1 SYM effective action containing the component four-derivative term of gauge field reads [11] where W is the abelian gauge superfield strength, Λ is a scale parameter and c 0 is a dimensionless constant. The variation of action (3.17) under the transformation (3.12) is as follows (3.18) It can be partially canceled by variation of the extra term Variation of (3.19) reads (3.20) Due to the property d 5|8 zdu λ +a q − a = 0, the first term in (3.20) exactly cancels (3.18), provided that c 1 = −c 0 /4. Hence, (3.21) Consider the last term here in more detail: (3.22) We used the integration by parts with respect to the spinor derivative in the second line and cyclic identities for SU (2) indices in the third line. We observe that the last term in the third line equals the expression we started from, but with a minus sign. Hence, (3.23) Substituting (3.23) into (3.21), we obtain Once again, the variation of (3.24) can be partially canceled by the variation of the additional term Finally, we consider the general expression and H(Z) = 1 + 2 ln 1 + √ 1 + 2Z 2 + 2 3 This expression coincides with the one obtained in [9] by resorting to hidden supersymmetry instead of R-symmetry 2 .
One can also directly verify that (3.28) in invariant under the R-symmetry (3.31) This expression can be simplified with the help of the identity which is derived employing the integration by parts with respect to the spinor derivative and recalling the definition of the function H(Z) (3.30). Thus, we obtain (3.33) We conclude that the condition of invariance under R-symmetry can be employed instead of the demand of hidden supersymmetry in order to construct the complete 5D, N = 2 invariant superspace functional, starting from the functional which is invariant under the manifest N = 1 supersymmetry only.

Summary
In this paper we have found the realization of R-symmetry for 4D, N = 4 and 5D, N = 2 supersymmetric gauge theories as the superfield transformations in the relevant harmonic superspaces. The R-symmetry transformations were defined in the explicit form, and they mix the vector multiplet and hypermultiplet harmonic superfields with each other. It was proved that the microscopic actions of 4D, N = 4 and 5D, N = 2 SYM theories are invariant under these transformations without any on-shell conditions on the superfields involved. Thus, the above transformations constitute an additional invariance of 4D, N = 4 and 5D, N = 2 supersymmetric gauge theories.
The algebraic structure of the harmonic superfield R-symmetry transformations was studied. First, these transformations form the closed algebra only on shell. Second, the R-symmetry transformations are consistent with both manifest and hidden supersymmetry transformations, which are a necessary element of the harmonic superspace formulations of the maximally extended SYM theories. To be more precise, the R-symmetry transformations form a closed algebra with manifest and hidden supersymmetry transformations. This means, in particular, that the R-symmetry transformations and hidden supersymmetry transformations are in a sense interchangeable. If the manifestly invariant superfield functional is invariant under R-symmetry transformations, then it will be automatically invariant under the hidden supersymmetry transformations and vise versa.
The R-symmetry transformations were applied to the problem of hypermultiplet completion of the low-energy effective action of 4D, N = 4 and 5D, N = 2 SYM theories, proceeding from the low-energy effective actions in the gauge multiplet sector. Using these transformations, we constructed the leading low-energy complete effective actions for above theories, beginning with the main terms containing only gauge multiplet contributions. We have shown that the hypermultiplet dependence of the effective actions under consideration is completely specified by the requirement of invariance under the R-symmetry transformations. We focused on the case of the SU (2) gauge group spontaneously broken to U (1). A generalization to other gauge groups is straightforward. An interesting property is that the effective action is not only invariant under R-symmetry transformations but can be fixed by them up to an overall constant.
It would be tempting to reveal other possible implications of hidden R-symmetry in extended superfield gauge theories in diverse dimensions. The maximally supersymmetric gauge theory in six-dimension is N = (1, 1) SYM theory. It has only manifest linear SU (2) R × SU (2) PG symmetry and for this reason the methods of the present paper seem not to be appropriate for analysis of the structure of the quantum effective action of this theory in the N = (1, 0) harmonic superspace formulation. Only considerations based on the hidden N = (0, 1) supersymmetry prove to be adequate [13,14]. On the other hand, the hidden R-symmetry method could be useful in the harmonic superspace formulations of 6D, N = (2, 0) tensor multiplet (see, e.g., [15]). Indeed, only SU (2) R symmetry is manifest there, while the rest of the full U Sp(4) R-symmetry of 6D, N = (2, 0) supersymmetry should be realized as a hidden symmetry. We plan to consider this and some other additional examples of exploiting superfield hidden R-symmetries elsewhere.

A Appendix
A.1 Commutators of R-symmetry transformations in 4D, N = 4 SYM theory In this section we directly calculate the commutator of R-symmetry with itself for 4D, N = 4 SYM theory (2.14).
Let us start with the transformation ofW The right-hand side of this relation can be further worked out as To properly transform this expression, we note first that the full coefficient before D + α D − β W in the next-to-last line in (A.2) is proportional to αβ , while D +α D − α W = 0, as follows from the equation of motion (2.9) for W and the constraints (2.12). Analogously, using the relations (2.11), we can replace D + αD − αW in the last line with −2i∂ ααW . We also introduce the notations where λ ij refer to SU (2) R transformations and λ (λ = −λ) to U (1) R transformations. As a result we obtain Due to the relations (2.12), this expression can be cast in the form The transformation law for W can be obtained through complex conjugation.
Next, we pass to the transformation of q + Thanks to the analyticity of q + a and the equation of motion (2.9) one can replaceD + α D + α q − b in the last line of (A.6) with −2i∂ αα q + b , and similarly for D + is just associated with the SU (2) PG symmetry transformations. Therefore, (A.8) Using the equation of motions (2.9), this expression can be brought in the form where λ is the Lie bracket parameter for U (1) R transformations.
A.2 Commutators of R-symmetry transformations in 5D, N = 2 SYM theory In this section we calculate the commutator of R-symmetry with itself for 5D, N = 2 SYM theory (3.12). We start with the transformation of W (δ λ 1 δ λ 2 − δ λ 2 δ λ 1 )W = 1 4 δ λ 1 2(λ +a 2 q − a − λ −a 2 q + a ) + λ −a 2 θ +α − λ +a 2 θ −α D + α q − a − (λ 1 ↔ λ 2 ). (A.10) Its right-hand side is evaluated to be To bring this expression to a simpler form, we make use of the relation which follows from the equation of motion (3.9) for W , the constraint (3.10) and the definition (3.2) of W 3 . Using this relation, we can replace D + α D − β with i∂αβ in the last line of (A.11) due to the antisymmetry of the full factor in front of it in the indicesα andβ. We also introduce the notation Rewriting eq. (A.11) and substituting the explicit expressions for D − α and D + α , we obtain (A.14) When passing to the last line, we exploited the relations (3.10).
Substituting the explicit expressions for D − α , we finally obtain (A.16) When passing to the last line, we employed the equation of motion (3.9).