Quantum duality under the θ-exact Seiberg-Witten map

We show that in the perturbative regime defined by the coupling constant, the θ-exact Seiberg-Witten map applied to the noncommutative U(N) Yang-Mills — with or without Supersymmetry — gives an ordinary gauge theory which is, at the quantum level, dual to the former. We do so by using the on-shell DeWitt effective action and dimensional regularization. We explicitly compute the one-loop two-point function contribution to the on-shell DeWitt effective action of the ordinary U(1) theory furnished by the θ-exact Seiberg-Witten map. We find that the non-local UV divergences found in the propagator in the Feynman gauge all but disappear, so that they are not physically relevant. We also show that the quadratic noncommutative IR divergences are gauge-fixing independent and go away in the Supersymmetric version of the U(1) theory.


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It was found in [1] that noncommutative U(N) Yang-Mills theories admit two dual formulations at the classical level: one, in terms of noncommutative gauge fields, the other by using ordinary gauge fields. One moves between these two dual descriptions of the same theory by employing the so-called Seiberg-Witten (SW) map. This map maps ordinary U(N) gauge fields into noncommutative U(N) gauge fields and viceversa [2,3]. Whether this duality persists at the quantum level is still an open issue, even at the one-loop level. Indeed, a key feature of noncommutative field theories defined by using noncommutative fields is the UV/IR mixing phenomenon unveiled in [4][5][6][7]; a phenomenon which cannot be seen by defining the action of the would-be dual ordinary theory as the expansion in the noncommutativity matrix, θ µν , furnished by the Seiberg-Witten map, with the latter constructed as a formal power series in θ µν . It was not known how to reproduce the UV/IR mixing effect in the formulation of noncommutative gauge theory in terms of ordinary fields until the paper in [8] was issued. In accord with the very essence of the perturbative coupling constant description of the quantum field theory our approach to the Seiberg-Witten map issue, is to built the SW map by using the expansion in terms of the coupling constant [8,9] (and no expansion in θ µν is carried out). Thus, the θ µν dependence of the perturbative, in the coupling constant, definition of the theory is treated in an exact way and, then, the UV/IR mixing effect pops up. The occurrence of the UV/IR mixing phenomenon in both these quantum field theories gives strong support to the idea that they are dual descriptions of the same underlying quantum field theory, at least in the perturbative regime defined by the coupling constant. And yet, in U(1) Yang-Mills theory, the UV divergent part of the two-point function of the noncommutative gauge field is local, whereas the UV divergent bit of the two-point function of the ordinary theory obtained by using the θ-exact Seiberg-Witten map contains unusual θ-dependent nonlocal contributions, at least in the Feynman gauge. These nonlocal contributions where unearthed in [10][11][12], and their existence casts doubts on the truth of the quantum duality conjecture at hand. Of course, UV divergent contributions to the two-point function are, in general, gauge dependent; so to decide whether the duality conjecture is right or wrong, there remains to be seen whether or not those nonlocal terms are really gauge dependent; since the gauge dependent contributions are not physically relevant. It is known that in the θ-unexpanded noncommutative nonsupersymmetric gauge theory defined in terms of the noncommutative fields, the noncommutative quadratic IR divergence induced by UV/IR mixing signals an IR instability [13]; and that this IR instability can be cured by making the theory supersymmetric, since supersymmetry removes the corresponding quadratic noncommutative IR divergences [14,15]. Furthermore, it was shown in [16] that if the noncommutative fields carry a linear realization of Supersymmetry their ordinary duals under the Seiberg-Witten map carry a nonlinear realization of Supersymmetry. Hence, it is far from trivial that the Supersymmetry cancelation mechanism between the one-loop noncommutative quadratic IR divergences coming from bosonic and fermionic degrees of freedom works when the classical noncommutative theory is formulated, first, in terms of the ordinary fields and then quantized. And yet, it has been shown in [17] that the

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Supersymetry cancelation mechanism just mentioned works for all the two-point functions when we have N = 1, 2 and 4 Supersymmetry. This result gives further robustness to the quantum duality conjecture between the formulation in terms of ordinary fields and the description in terms of noncommutative fields. However, the nonlocal UV divergent structure still persists after introducing Supersymmetry in the game. But, by using two different gauge-fixing terms, it was shown in [17] that the nonlocal UV divergent contributions are gauge dependent and, therefore, it could be possible to remove them. This is unlike the noncommutative quadratic IR divergences which do not change with the gauge-fixing term as proved in [15], in the noncommutative field description, and in [17], in the ordinary field formulation, respectively. Now, since it is known that the on-shell DeWitt effective action is independent of the gauge-fixing term used to define the path integral, thus this action can be used to compute the S matrix elements. Hence, by using on-shell DeWitt effective action [18][19][20][21], one could settle the question of the physical relevance of the nonlocal UV divergent terms found in the two-point functions of the noncommutative theory formulated in terms of the ordinary fields, and as a bonus obtain a complete proof of the gauge-fixing independence of the UV/IR mixing phenomenon and also the cancelation of the noncommutative quadratic IR divergences achieved by introducing Supersymmetry.
Let S NCYM Â µ denote the classical action of noncommutative U(N) Yang-Mills theory, whereÂ µ is a noncommutative gauge field configuration. LetΓ DeW B µ stand for the onshell DeWitt effective action of noncommutative field theory whose action is S NCYM Â µ . Let Γ DeW B µ be the symbol for the on-shell DeWitt effective action of the ordinaryi.e., defined in terms of ordinary fields -U(N) gauge theory whose classical action is S NCYM Â µ A µ ,Â µ A µ being the Seiberg-Witten map that relates the ordinary U(N) gauge field A µ with the noncommutative U(N) gauge fieldÂ µ . Then, the purpose of this paper is to show that, at any loop order and in dimensional regularization, whereB µ [B µ ] stands for the Seiberg-Witten map between the ordinary U(N) gauge field B µ with the noncommutative U(N) gauge fieldB µ . Thus proving that the conjecture that Seiberg-Witten map provides two dual formulations of the same underlying quantum theory is true for the noncommutative U(N) Yang-Mills theories, with or without Supersymmetry. Let us warn the reader that to avoid any clashes with unitarity [22], we shall always consider θ µν -the noncommutativity matrix -such that θ 0i = 0, i = 1, 2, 3.
The layout of this paper is as follows. In section 2, we introduce the on-shell DeWitt action for noncommutative U(N) Yang-Mill theory and the corresponding ordinary U(N) gauge theory defined by means of the θ-exact Seiberg-Witten map. In the same section, we also discuss the effect and some properties of the Seiberg-Witten map applied to the ordinary background-field splitting. We establish the quantum equivalence of the field theories defined in the previous section by performing the appropriate changes of variables in the path integral in section 3, while in section 4, we check the conclusion reached in section 3 by direct computation -i.e., by using the Feynman rules (FR's) derived from JHEP09(2016)052 classical action -of the one-loop two-point function of the on-shell DeWitt action of dual ordinary U(1) gauge theory: we show by explicit computation that, in particular, all the ugly non-local UV divergences -see [17] -that occur in the propagator in the Feynman gauge all but go away. The noncommutative quadratic IR divergences remain unless the noncommutative theory is supersymmetric with N = 1, 2 and 4 Supersymmetry. Of course, there is no UV divergence nor any noncommutative IR divergence (quadratic or logarithmic) for N = 4 U(1) super Yang-Mills. We have included appendices to help understanding the central body of the paper.

Seiberg-Witten map and DeWitt effective action
In this and next section we provide an expanded proof/review of the equivalence between DeWitt effective actions in terms of noncommutative and ordinary fields via θ-exact Seiberg-Witten map indicated in [23].

DeWitt action and the path integral in terms of noncommutative fields
LetÂ µ =B µ + 1 2Q µ be the standard splitting of the noncommutative U(N) gauge field A µ in a noncommutative backgroundB µ and a noncommutative quantum fieldQ µ . We shall assume thatB µ satisfies the classical noncommutative equations of motion (EOM), which readD µF µν = 0. Then the on-shell DeWitt effective action [21],Γ DeW B µ , is given by the following path integral The gauge-fixing term S BFG B µ ,Q µ ,F ,Ĉ,Ĉ is 1 whereδ BRS stands for the noncommutative BRS operator, which acts on the noncommutative fields as follows: Although -as shown in [24] by using BRS techniques for ordinary theories, a proof which remains valid in the case at hand -Γ DeW B µ does not depend on the choice of gaugefixing term, we have chosen the background field gauge (BFG) for convenience.

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2.2 DeWitt effective action of the dual classical ordinary theory The next task will be the background field quantization of the ordinary theory with action is the θ-exact Seiberg-Witten map which expressed the noncommutative field A µ in terms of its ordinary counterpart A µ . But before carrying out the background field quantization, we need to discuss -since we are dealing with a nonlinear map -how the Seiberg-Witten map acts on the background-field-quantization splitting, A µ = B µ + 1 2 Q µ , of the ordinary gauge field; this we do next.
Let us begin with A (n) µ (a 1 , µ 1 , p 1 ), . . . , (a n , µ n , p n ); θ . In [25], it was shown that the θexact Seiberg-Witten map can be constructed by setting h = 1 in the following formal series where A (n) µ [a ρ ; hθ] is of order n in the number of classical fields and it is given by the . Now, it is easily seen by inspection of the formulae given in [25] µ [a ρ ; hθ] there is no polynomial dependence in h, but, we shall allow for the possibility that for higher n there is a cancelation among phase factors that gives rise upon integration over t to positive powers of h. Before we go on, let us recall that, for all integers s ≥ 0, we have Next, let us assume that, for all m < n we have that, a) A µ [a ρ ; hθ], so that mathematical induction leads to the conclusion that a) and b) also hold for any n; which in turn implies that A (n) µ (a 1 , µ 1 , p 1 ), . . . , (a n , µ n , p n ); θ is a linear combination of functions of the type (2.24), for whatever value of n.
It is plain that the same kind of reasoning can be carried out to show that C (n) (a 1 , µ 1 , p 1 ), . . . , (a n , µ n , p n ); (a, p); θ , is also a linear combination of functions of the type (2.24).

DeWitt effective action for ordinary fields
We are now ready to quantize the classical ordinary U(N) gauge theory which is dual, under the θ-exact Seiberg-Witten map, to the noncommutative U(N) Yang-Mills theory.
To quantize the ordinary theory in question, we shall use the background-field splitting; so the classical action that defines the ordinary theory reads Let us first introduce two extra fields,F =F a T a ,Ĉ =Ĉ a T a , on which the ordinary δ BRS , and noncommutativeδ BRS , operators act by the following definition: (2.32) HereĈ is a Grassmann field andF is a boson field. Recall that T a is U(N) generator in the fundamental representation.
We shall assume from now on that B µ is a solution to the classical equation of motion of the theory with action S NCYM [B µ ], as defined previously. Then, as shown in the appendix A, B µ satisfies:D The on-shell DeWitt action, Γ DeW [B µ ], of ordinary theory now reads where S gf B µ , Q µ ,F ,Ĉ, C is the gauge-fixing term, which is BRS-exact -thus benefits the BRS quantization method: Here δ BRS is the ordinary BRS operator which acts on the fields B µ , Q µ as defined in (2.21) and onĈ andF as defined in (2.32). The X gf B µ , Q µ ,F ,Ĉ, C is an arbitrary functionalwith ghost number -1 -of the fields, which can be expressed as formal series of the fields.

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Taking into account that δ 2 BRS = 0, when acting on B µ , Q µ , C,Ĉ andF , respectively, one concludes that δ BRS S gf = 0. Hence, the results presented in [24] also apply here, so that Γ DeW B µ does not depend on the X gf B µ , Q µ ,F ,C, C that one chooses. For instance, one may choose the standard background field gauge of the ordinary fields, i.e., but this gauge-fixing term will not suit our purpose. We shall choose the following term instead. Note thatD µ Â µ B µ + 1 2 Q µ is the noncommutative covariant derivative. Now, taking into account (2.21) and (2.32), one concludes that our choice of gauge-fixing term reads which is the gauge-fixing term corresponding to the noncommutative background field gauge.

Establishing quantum equivalence by changing variables in the path integral
LetQ a µ = tr(Q µ T a ) andĈ a = tr(Ĉ C a ), whereQ µ andĈ are given by the Seiberg-Witten map in (2.17) and (2.18), respectively. Let J 1 [B a , Q a ] and J 2 [B a , Q a ] be the following Jacobian determinants By changing variables in the path integral in (2.34): C a →Ĉ a and Q a µ →Q a µ , we obtain whereB µ andQ µ are expressed in terms of B µ and Q µ and

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To continue we start with the following proposition: If J 1 [B, Q] = 1 and J 2 [B, Q] = 1, then the right hand side of (3.2) equals the right hand side of (2.1), leading to (1.1), that is: This result is valid on-shell sinceB µ [B µ ] satisfies the noncommutative Yang-Mills equations (2.33), and the reason is the on-shell uniqueness of DeWitt effective action [19,24].
In summary, if we are able to show that proposition holds, that would prove that both theories defined in terms of noncommutative fields and in terms of ordinary fields, through the Seiberg-Witten map, have the same on-shell DeWitt effective action and, therefore, they are dual -i.e., they are different descriptions of the same underlying theory -to each other at the quantum level. We shall show below that, indeed, in dimensional regularization, and in the perturbative regime defined by the coupling constant, our proposition holds.

No one-loop two-point contribution coming from
Before we plunge into the general proof that proposition holds, we shall show that by employing dimensional regularization the one-loop two-point contribution to vanishes. By working out this simple instance, we shall acquaint ourselves with the techniques that we shall employ in the general case, as well as the type of dimensionally regularized integrals one has to face. From (2.17) and δQ a µ (p) . . , (a n , µ n , p n ); θ]

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Now, substituting the previous results in (3.3), one gets where and and Tr (3.10) Let us show now that in dimensional regularization Tr M 1 = 0. The term A (2) µ [(a 1 , µ 1 , p 1 ), (a, µ, q); θ] can be obtained from A (2) µ in section III of ref. [25]: (3.11) Hence, in dimensional degularization the loop integral -the integral over q - One may actually use δ(p 1 ) to further simplify the argument, as only the second vanishing identity above would be needed. We have included the discussion of the vanishing of the previous type of integrals due to the employment of dimensional regularization in the appendix B.
Next, by integrating out the Dirac delta function, δ(p 1 + p 2 ) in (3.9), one comes to the conclusion that to work out TrM 2 , one has to compute the following dimensionally regularized integral µ , one concludes that the integral in (3.14) is a linear combination of the following types of dimensionally regularized integrals: where Q(q) denotes symbolically a monomial in q (i.e. Q ≡ q µ 1 . . . q µr ), and

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Hence Putting it all together one reaches the conclusion that all the integrals listed above are of the type which vanish -see appendix B for details -in dimensional regularization. We have thus shown that Tr M 2 = 0, in dimensional regularization. Let us finally show that Tr M 1 M 1 = 0 in dimensional regularization. The loop integral over q, contributing to Tr M 1 M 1 -as seen from (3.10) -is: , (a, µ 2 , p 1 + q); θ ; (3.18) but this integral vanishes since it is, again, a linear combination of integrals of the type However, these integrals -appendix B -are equal to zero in dimensional regularization.
In summary, we have just shown that in dimensional regularization Tr M 1 = 0, Tr M 2 = 0 and Tr M 1 M 1 = 0, and, hence -see (3.7) and (3.3) -one obtains Finally, since the same types of integral contribute to J 2 [B, Q] it is plain that also holds, and therefore, the one-loop two-point contribution to Γ DeW [B µ ] does not receive Later in this paper a head-on -i.e., by using the Feynman rules for the ordinary fields and not changing variables in the path integral -computation of the same twopoint function will be performed. We are now ready to show that there are no nontrivial contribution either to

Triviality of the full Jacobian determinants
It is shown in the appendix B that Note that the definition of splitting in the momentum space readsÃ . . , m + 1, as the following sums then, by taking into account (3.20) and carrying out a lengthy straightforward computation -see appendix B for details -one gets The general structure of the master integral (3.22) above can be visualized by a 1-loop diagram, as given in figure 1.

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Now, in a view of previous equations (3.21) and (3.22), to complete computation of ln J 1 [B, Q], one has to work out the following dimensionally regularized type of integrals over the internal momenta q µ : 1 , ν 1,1 , p 1,1 ), . . . , (b 1,n 1 −1 , ν 1,n 1 −1 , p 1,n 1 −1 ), (a 1 , µ 1 , q); θ] However, according to the discussion at the end of subsection 2.2.1, the previous integral is a linear combination of integrals of the type where Q(q) = q ρ 1 q ρ 2 · · · q ρn , qθk i = q µ θ µν k iν , i = 1, . . . , s and k i θk j = k iµ θ µν k jν , i, j = 1, . . . , s. Here n and s run over all relevant momenta other than q, in general. It is important to stress that Q(q) is a monomial on q ρ and that the function I in the integrand of the previous integral is a function of the variables qθk i and k i θk j only, and, hence, as shown in the appendix C, one concludes that I = 0 and V = 0.

Incorporating adjoint matter
The θ-exact Seiberg-Witten map for scalar or fermion fields in the adjoint reads -see [26]: µ 1 , p 1 ), . . . , (a n , µ n , p n ); (a, p); θ] ·Ã a 1 µ 1 (p 1 ) . . .Ã an µn (p n )Φ a (p), (3.27) JHEP09 (2016)052 where Φ = Φ a T a denotes an ordinary scalar or fermion field transforming under the adjoint of U(N). Now, taking into account the recursive equations -see section III of ref. [26] which yieldΦ [A µ , Φ, θ] (x) have similar θ and momentum structure to the one forÂ µ [A µ , θ], it is easy to see that F (n) [(a 1 , µ 1 , p 1 ), . . . , (a n , µ n , p n ); (a, p); θ] is also a linear combination of functions of the type in (2.24). Hence, one also concludes that the Jacobian determinant for the change of variables Φ a →Φ a (x) = tr(Φ(x)T a ) = tr(T aΦ [A µ , Φ, θ] (x)) in the path integral over the fields Φ a is one, i.e.: det δΦ a (x) δΦ a (y) = 1. (3.28) Hence the inclusion of matter fields in the adjoint -and for that matter any type of matter fields -does not change the conclusion that we have reached above for gauge fields, i.e., that the θ-exact Seiberg-Witten map associates every quantum field theory, with gauge group U(N) and formulated in terms of noncommutative fields, to an ordinary gauge theory, with gauge group U(N), which is dual to the former at the quantum level, because, indeed, they have the same on-shell DeWitt effective action. Note that the massless tadpole integrals also vanish in the dimensional-reduction scheme, which preserves supersymmetry manifestly in the one loop. Therefore our conclusion here should be also valid for the noncommutative supersymmetric Yang-Mills theories (NCSYMs).

Testing the quantum equivalence by direct computation: the one-loop two-point function
We choose to test the formal equivalence established in the last section by computing explicitly one-loop quantum correction to the quadratic part of the effective action in the noncommutative U(1) gauge theory prior to and after the Seiberg-Witten map. At this specific order, the general equivalence relation (1.1) reduces to a much simpler relation whenB ν (p) is placed on-shell, because only the zeroth order of the SW map counts here. We start by reviewing the standard procedure for computng the DeWitt effective action of U(1) gauge theory perturbatively in the background field formalism/method (BFM) [19,20], which evaluates all 1-PI diagrams with all background field external legs and all integrand fields (Q µ ,Ĉ,Ĉ,F ) internal line using the following actionŜ loop

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Once the SW map is employed, one may choose to map the action above, making it or to map the classical gauge-fixed action then subtract the equations of motion with respect to the commutative/ordinary fields, i.e.
These two actions are equivalent on-shell as long as the Seiberg-Witten map is invertible, as proven in the appendix A, yet they are but not identical to each other because of the additional field redefinition factor. We choose to proceed with S loop in the computations presented below. As we will see soon, this choice leads to result directly identical to the computation using noncommutative fields, i.e. 2 Γ µν (p) = Γ µν (p). (4.5) We are going to use the extended version of dimensional regularization scheme as in [17], which we know to be compatible with the prescriptions used in subsection 3.1. To simplify the computation we choose α = 1 and have the auxiliary field F integrated out.

Model definition
As our first test we choose S gf to be the background field gauge with respect to the noncommutative fields The U(1) theory version of (4.3) then reads (4.7) To perform the one-loop computation we must expand this action up to the BBQQ order, which is worked out in details in the appendix D. In the end we get 3

S
(1) 2 Our prior computation in [17] would actually correspond to the same evaluation but with S loop , which is, because of the proof in the appendix A, equivalent to the results here on-shell. 3 We assume g = = 1 from now on for simplicity, actually coupling is g for BQQ and g 2 for BBQQ.
As a convention interactions with subindex 2 are derived from S gf , while those with subindex 1 are from the rest of S loop . We assumeĈ ≡C from now on, too.
OperatorsQ µ andĈ are defined in (D.4) and (D.6), respectively. Here and later "irrelevant" denotes those four-field-interaction terms which do not generate nontrivial nonlocal factor and/or denominator in the tadpole diagrams. Their contributions to tadpole is then zero under dimensional regularization because of the reasons given in the section 3 and the appendix C. They would still be needed for loop corrections to the three and higher point functions. The interaction Feynman rules are read out from the interactions listed above, and given in the appendix E.1.

One-loop quantum corrections in the background field gauge
The one-loop photon 1-PI two point function computation in the background field gauge consists four diagrams (figures 2-5): the photon self-interacting bubble B µν BFG photon and tadpole T µν BFG photon , as well as the ghost bubble B µν BFG ghost and tadpole T µν BFG ghost :  with One can prove that 4  Explicit computation then yields

One-loop corrections in the noncommutative Feynman gauge
We perform a second test on the gauge-fixing (in-)dependence by shifting from the background field gauge fixingD µ [B µ ]Q µ to the NC Feynman gauge fixing (NCFG) ∂ µQ µ . The standard background field method procedure then leads us to a modification to the following action The resulted Feynman rules are listed in the appendix E.2. In analogy to the background field gauge, we have the following one-loop contributions Explicit computation then yields

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This result matches the computations in the Feynman gauge without Seiberg-Witten map in the literature [4,6,7]. Since the result without Seiberg-Witten map is equivalent to the background field gauge result on shell [19,24], we conclude that the Seiberg-Witten mapped result here fulfills this equivalence too.

One-loop corrections in the noncommutative U(1) Super Yang-Mils
We also investigate whether our method can be used to remove non-polynomial UV divergences in the 1-PI two point functions of the superpartners, i.e. the photinos and adjoint scalars. Our starting actions are as follows In this case after the background-field splittingλ =λ B +λ Q andφ =φ B +φ Q we must subtract both the equations of motion of superpartner fields, and their contributions as source of the photon equations of motion, the resulted actions for loop computation are listed below The relevant SW map can be derived using the background-field splitting method in the subsection 2.2 and results [17]. Once we start reading out Feynman rules our first observation is that the superpartner's contribution to the photon effective action is identical to the results in [17]. Therefore we have the same quadratic IR divergence cancellation. The total UV divergence in the background field gauge is now  Out of figures 6-9 we read out the following loop integrals for N = 1 photino 52) and the following for the minimally coupled adjoint scalar

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Explicit computation then yields Since all other diagrams in the superpartner two point function computation in the SW mapped U(1) NCSYM are identical to the diagrams in the unexpanded theory [17] (see also the short summary in the appendix F.2), we conclude that the full 1-PI two point functions/quadratic part of the background field effective actions are identical up to one-loop in U(1) NCSYM with and without SW map.

Discussion and conclusions
We have shown that at the quantum level the θ-exact Seiberg-Witten map providesat least in perturbative theory with respect to the coupling constant -a dual description, in terms of ordinary fields, of the noncommutative U(N) Yang-Mills theory with JHEP09(2016)052 or without Supersymmetry. We have shown that by performing appropriate changes of variables in the path integral defining the on-shell DeWitt effective action in dimensional regularization. We have explicitly computed, by using the Feynmann rules derived from the classsical action, the one-loop two-point contribution to the on-shell DeWitt action for U(1) SuperYang-Mills with N =0, 1, 2 and 4 Supersymmetry and found complete agreement with general result obtained by carrying out changes of variables in the path integral. We have also shown that all the nasty non-local noncommutative UV divergences which occur in the one-loop 1PI functional in the Feynman gauge, computed in [10][11][12]17], are merely offshell gauge artifacts since they do not occur in the one-loop two-point contribution to the on-shell DeWitt action -which is a gauge-fixing independent object -and therefore they do not contribute to any physical quantity. We have also shown that the same quadratic noncommutative IR divergences that occur in nonsupersymmetric noncommutative U(N) gauge theories formulated in terms of noncommutative fields occur in the ordinary theory obtained from the former by using the θ-exact Seiberg-Witten map and that this UV/IR mixing effect -signaling a vacuum instability -is a gauge-fixing independent characteristic of the ordinary gauge theory, in keeping with the duality statement. We have also seen that those quadratic noncommutative IR diverges can be removed by considering supersymmetric versions of the theory, a nontrivial effect since supersymmetry is not linearly realized in terms of the ordinary fields [16]. Finally, there remain to be seen how the results presented here carry over to the nonpertubative regime in the coupling constant. In this regard the analysis of the nonperturbative features of N = 2 and 4 supersymmetric gauge theories looks particularly interesting.

A Classical equations of motion for the noncommutative and ordinary fields
In this subsection we prove that the equations of motion are equivalent for the noncommutative and ordinary fields in the NC U(N) gauge theories. We start with the noncommutative fields. The action reads If, in terms of the component fieldsB µ =B a µ T a , than T a is in the fundamental representation of U(N). The equations of motion forB a µ read µ 1 , p 1 ), . . . , (a n , µ n , p n ); θ] µ 1 , p 1 ), . . . , (a n , µ n , p n ); θ] · n ·B a 1 µ 1 (p 1 ) . . . δB an µn (p n ), has no zero modes (nonzero solutions), i.e.B a µ =B a µ B b ν can be inverted into B a µ = B a µ B b ν . We have that the equation of motion for B a µ with action

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Notice however: 1. For SU(N), SO(N) etc. groups (A.9) is not the equation of motion of B a µ since the dependence of S NCYM on B a µ is not exhausted by the dependence ofB a µ on B a µ .
2. While the equations of motion of noncommutative and ordinary fields are equivalent, they are not exactly identical. This would affect the subtraction of EOM proportional terms when evaluating the background field effective action and lead to nonidentical off-shell results. As described in the main text, one can obtain exactly identical results in direct computations using noncommutative or ordinary fields only by subtracting the identical EOM proportional terms.

JHEP09(2016)052 C Vanishing integrals in dimensional regularization
In this appendix we shall discuss why the integrals over the internal momentum q that arise in the computation of the Jacobian determinants in sections 3.1 and 3.2 vanish in dimensional regularization. These integrals are of the following type where Q(q) = q ρ 1 q ρ 2 · · · q ρn , qθk i = q µ θ µν k iν , i = 1, . . . , s, and k i θk j = k iµ θ µν k jν , i, j = 1, . . . , s. Here n and s runs over all relevant momenta other than q, in general. The function I in the integrand of the previous integral is a function of the variables qθk i and k i θk j only.
We shall define the integral (C.1) by Wick rotating the corresponding integral defined for Euclidean signature, a signature which we shall assume for the time being.
The first problem one has to face when defining, in dimensional regularization, the object in (C.1) is the definition of θ µν in the infinite dimensional space, E ∞ -see section 4.1 of ref. [35] -of which the momenta q µ , k µ i are elements in dimensional regularization. Let us recall that, to avoid problems with unitarity, our θ µν in four dimensions is such that θ 0i = 0, i = 1, 2, 3. Hence, by a rotation, this θ µν in four dimensions can be transformed into an object whose only non-vanishing components are θ 23 and θ 32 . Then, without loss of generality, we shall assume this latter θ µν to be our object in four dimensions. Now, since θ µν is an antisymmetric object, its properties depend on the dimension of spacetime. So, as happens with the Levi-Civita tensor and the γ 5 matrix [35], the only consistent way to define it in dimensional regularization is to keep it essentially fourdimensional, since our physical theory is in four dimensions. This amounts to defining θ µν in the infinite dimensional space -see section 4.1 of ref. [35] -of dimensional regularization: With this definition of our θ µν -object in dimensional regularization, one comes to the conclusion that all the vectors 1 θ θ µν k iν , i = 1, . . . , n, belong to the same two-dimensional subspace, E 2 , of the infinite dimensional space E ∞ . Let us follow ref. [35] and split the vector q µ ∈ E ∞ into two components: where q µ ∈ E 2 and q µ ⊥ ∈ E ⊥ , E ⊥ being the subspace orthogonal to E 2 . Then, using [35], we define the following object in (C.1) where l 1 and l 2 are the coordinates of q µ in an orthonormal basis of E 2 and we have taken into account that qθk i = q θk i .

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Now, I(q θk i , k i θk j ) does not depend on q µ ⊥ , so that But in dimensional regularization tadpole-type integrals -see [35] -vanish: recall that Q(q) is a monomial. We thus conclude that so that the right hand side of equation (C.3) vanishes, which in turn implies that We shall assume thatD µ B µ B µ F µν B µ B µ = 0, then the action is The second line after the third equality can be neglected because the background field satisfies the equations of motion (A.9) (Kallosh formalism). Therefore Now, one extracts the O( 0 ) order terms ofQ µ from (2.17) and deduces that µ 1 , p 1 ), . . . , (a n , µ n , p n ); θ] · n ·B a 1 µ 1 (p 1 ) . . .B a n−1 µ n−1 (p n−1 )Q an µn (p n ).
Similarly, we can expand the gauge fixing action (2.36) up to the 0 order

JHEP09(2016)052 F Evaluation of DeWitt effective action in terms of noncommutative fields
We accumulate the reference results for section 4, i.e. the one-loop quantum corrections to the quadratic part of the DeWitt effective action of the NC U(1) Super Yang Mills. We first give the model setting, then the results of relevant one-loop diagrams.

F.1 The noncommutative Yang-Mils theory
Let's first handle the U(N) NCYM only, then extend the results to its supersymmetrization. The NCYM action is the usual one Background field quantization follows the BRST procedure below: To evaluate it we first expand the corresponding classical actions to the appropriate order NC , (F.14) with S (1) (F.15) Restrict (F.15) to U(1) and α = 1, the 1-loop 1PI photon two point function is then evaluated as the sum over 1-loop 1PI diagrams with allB µ external lines. There are four diagrams in total, which can be separated into two parts: the bubble part which sums over the photon and ghost bubble diagrams and tadpole part which sums over the photon and ghost tadpole diagrams. Consequently the final result is as followŝ Γ µν BFG =B µν BFG +T µν BFG , (F.16) (F.18)

F.2 The U(1) noncommutative Super Yang-Mils theory
Now we shift to the supersymmetrization of the U(1) theory. As discussed in [17], this sector contains the photino(s) for N = 1, 2, 4 and adjoint scalars for N = 2, 4. The interaction between photinos and adjoint scalars remain the same before and after SW map [17], therefore we are not going to repeat them here. Using (4.48) without SW map we obtain one self-energy/bubble diagram figure 19 for photino, as well as a bubble diagram figure 20 and a tadpole diagram figure 21 for adjoint scalar. Explicit computation based on these diagrams then gives the following two point functions (Σα α NCSYM and Σ (φ) NCSYM ) for noncommutative photinoλ and adjoint scalarφ: Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.