JHEP03(2016)136 Springer March 11, 2016 March 21, 2016 : : December 8, 2015 : Accepted Published 10.1007/JHEP03(2016)136 Received Doi: Published for SISSA By

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JHEP03(2016)136 Springer March 11, 2016 March 21, 2016 : : December 8, 2015 : Accepted Published 10.1007/JHEP03(2016)136 Received doi: Published for SISSA by . 3 1509.02916 The Authors. = 1 supersymmetric gauge theory in conﬁning phase, using the localization c Supersymmetric gauge theory, Nonperturbative Eﬀects, Solitons Monopoles

N In this note we exactly compute the gaugino condensation of an arbitrary four , [email protected] Yukawa Institute for Theoretical Physics,Kyoto Kyoto 606-8502, University, Japan E-mail: Open Access Article funded by SCOAP and Instantons, Confinement ArXiv ePrint: Abstract: dimensional technique. This result gives a nonperturbative proof of theKeywords: Dijkgraaf-Vafa conjecture. Seiji Terashima A nonperturbative proof of Dijkgraaf-Vafa conjecture JHEP03(2016)136 1 ]. = 1 23 N 2 ], the superpotential , the symmetry breaking 1 ]), after the important R 32 S 21 = 1 SUSY gauge theories = 1 SUSY gauge theories ]–[ 3 limit. (The gaugino condensation ]. N N 30 ]–[ → ∞ 24 R , the deformed action will not be weak coupling ˜ Λ which is the effective dynamical scale determined R 7 – 1 – only (and the coupling constants in the original S and then taking the ˜ Λ which remain strong coupling. 1 R = 1 SUSY Yang-Mills theories in confining phase [ S 1 × N 3 R This integration of the chiral multiplets can be done perturbatively ]. It should be stressed that using the localization technique we can 2 .) With the non-trivial v.e.v. of the Wilson line around ]. ], is a general way to compute them exactly. Recently, this technique is . Withtout this infra-red cut-off 22 R 2 23 /R , ]. 1 32 ] and [ After the integration of the chiral multiplets, we have In this paper, we will compute the gaugino condensation of four dimensional The gaugino condensation wereHere, computed we various consider ways, the see theory [ on 31 2 1 does not depend on will occur at the very highby scale the compared with deformed the action. scale much below Thus, the the 1 computationsbecause reduced of the to low 3d energy Abelian modes theory below in the the low energy region is a function ofsuperpotential). the For this gaugino theory, bi-linear wecan will compute compute the the gaugino gaugino condensation.with condensation Therefore, general in we chiral multiplets four and dimensional a superpotential. weak coupling [ and we only need thein effective superpotential. [ Thus, this can be done by the methodswith used only vector multiplets and a superpotential. As shown in [ condensation in four dimensional SUSY gauge theories with general chiralIn multiplets and order a to superpotential in domultiplets. confining this, phase. we This first isgaugino integrate condensation consistent out (i.e. using because the the localization chiral we technique) multiplets, such can that while the deform theory keeping is the the arbitrary theory vector without changing the supersymmetric (SUSY) field theories.originated The in localization [ technique for SUSYapplied field to various theories, kinds ofwork SUSY field by theories Pestun (for [ examples,compute [ non-topological quantities. In particular, using it, we can compute the gaugino 1 Introduction and summary Analytic computations in quantum fieldImportant theories quantum are field important, but theories very in hard, which in general. we can compute some quantities exactly are 3 A proof of Dijkgraaf-Vafa conjecture Contents 1 Introduction and summary 2 Gaugino condensation in theory with a generic superpotential JHEP03(2016)136 ] 32 = 1 (2.1) where ) is a N i i S and the S, g = 2 SUSY ( G ], however, F N 32 ) is the glueball ] for general y 32 ( ) and 2 ] it was also noted that ] and [ λλ 32 θθS 31 ] and [ Tr( 31 ) + ∼ y ( 0 1 and terms with derivatives 1 S , ) In this paper, we show that the θS i 3 n > ) + S, g = 1 SUSY gauge theory with vector ( y ( with a simple gauge group F 0 N S 1 R + ) with S n S ) 0 ) = × λλ 3 πiτ y, θ – 2 – R ( ]. Thus, this superpotential represents a general . Here we do not assume the Kählerpotential is S i 32 g ) = 2 = 1 SUSY gauge theory with only vector multiplets i , g N 0 ], an off shell extension of the vacuum expectation value was used to τ ( 38 V It would be interesting to study applications of our method. ], however, it would be difficult to have the superpotential in an explicit 5 W 40 , ]. In [ 4 39 37 ] perturbative integrations of chiral multiplets were computed and ] using the generalized Konishi anomaly equation to the 1PI effective action written 32 32 ]. There are “proofs” of this conjecture, i.e. [ are complex constants, 35 i – and coupling constants g 33 S ] and [ and 31 = 1 SUSY gauge theories (with a Lagrangian and in confining phase) according to . We thank Y. Nakayama for the useful discussions on this point. In [ ], the gaugino condensation was computed in the way, which is different from ours and is related = 2 Seiberg-Witten theory. The close connection of the gaugino condensation to Seiberg-Witten i 0 S τ 36 N i N ) gauge theory deformed by a superpotential is computed by a corresponding ma- we can justify the addition of the Veneziano-Yankielowicz superpotential. However, as stressed in [ The organization of this paper is as follows: in section 2 we compute the gaugino It should be noted that we can compute the gaugino condensation for any four dimen- The Dijkgraaf-Vafa conjecture is that the glueball superpotential of the P The perturbative superpotential should be computed using the results in [ More precisely, in [ In [ S 5 3 4 N = to the theory was also discussed incompute [ the gaugino condensation and itconjecture. was claimed to give a non-perturbative proof of the Dijkgraaf-Vafa gauge theories, for examples, [ closed form. by this only works for the caseS without symmetry breaking becausethe there are generalized no Konishi coupling anomaly constants would to have the higher loop corrections. superfield whose lowest componentfunction is of the gaugino bilinear canonical. Note thatare terms regarded containing as Tr(( thesuperpotential Kählerpotential for [ a theory without chiral multiplets. multiplets only (no chiralfollowing multiplets) superpotential: on where of the Dijkgraaf-Vafa conjecture. 2 Gaugino condensation inIn theory this with section, a we generic will superpotential consider four dimensional the discussions in this paper. We hope to return this problems in future. condensation for four dimensional and a generic action. In section 3 we show that our results imply the nonperturbative proof was obtained by the integrationperturbative of superpotential the with chiral the multiplets. the Veneziano-Yankielowicz correct superpotential gaugino indeed condensation. gives Dijkgraaf-Vafa This conjecture. can be regarded as a nonperturbative proofsional of the U( trix model [ both in [ the nonperturbative dynamicsadding of the the Veneziano-Yankielowicz superpotential gauge to fields the non-trivial were superpotential (implicitly) which assumed to be just JHEP03(2016)136 (2.8) (2.4) (2.5) (2.6) (2.7) (2.2) (2.3) . i S h . and ) 0 x , τ ( 0 0 ˜ S , i i ) ) 0 i i ¯ i S : ,g ,g includes the fermions. and a Kählerpotential = ) ) ˜ of the theory with the ) S S y 0 y (

t ( τ F y,θ 0 ) 0 , ( ˜ S Λ ˜ S ¯ S ( S S ( i ( 0 F ¯ ) S dS 2 0 θF δ = ¯ S 2 1 dF S , i

yd yδ ) ): ,g + 4 4 S ) x ( d d y ¯ ( dS S ( R R dF 0 S = e e , ˜ S S ) ) ˙ ( + α

¯ x x ¯ of S S, ) G which are SUSY transformations, corre- ( ( ¯ 2 = large enough depending on W S ¯ δ λ λ S S 2 ˙ − ( 1 | α t ) ) . ) ) i b b ¯ , δ F yδ S W x 1 4 ( S With this deformed action, what we need to ( − − d ¯ δ θ F as 2 S 6 R x x – 3 – − ≡ h e 2 ( ( δ ) ) xd ≡ X λ λ i ¯ 1 S 4 x . The dynamical scale h h ) ( d yδ , g in the superpotential are composite operators and x 4 λ ) ( = = d ) x Z ˜ b S S i R ( t ) ∞ e ˜ S − x ) t ( x + λ ( ) ¯ ··· λ S . We also used b ( h ( ) as a constant shift F − F x = ( ≡ h ]. plays as a infra-red regulator, which makes the deformed theory x ˜ ≡ S ( ¯ S X ), we can express 1 23 R ], we can compute them in the weak Yang-Mills coupling constant λ ) i by subtracting the zeroth and linear term in ) x ¯ S S ( 23 ) can be arbitrary low. , 0 i ≡ h F ˜ of ··· S ( . In order to evaluate this, it seems to have to consider ASD connections 2.3 , g h X 2 ) R x , θ ( from 1 ], in order to determine the vacua and evaluate the gaugino condensation, we ˜ S means the expectation value with the gauge coupling θ ( G 0 23 G h· · ·i First, we introduce As in [ We will compute the effective superpotential for this theory and determine the vacua Note that the polynomials of The dynamical scale can be arbitrary low by taking 6 where Then, in terms of To do this explicitly is interesting, however, in this paper we will use a different method. and define where without the superpotential sponding to with many fermion zeromodes because the superpotential term would like to compute the expectation value of the gaugino bi-linear: to the anti-superpotential with additional term ( do is only semi-classicalthat computations the around the radius anti-selfindeed dual weak (ASD) coupling connections. [ Note Using the result ofby [ the localization technique. In a superspace, this is realized by adding should be defined withconstraints a are regularization, not for imposed example, on a these point composites. splitting. Thus,and the then classical compute the gaugino condensation JHEP03(2016)136 )+ = 0, = 0 x = 0, with (2.9) ( i ¯ (2.10) (2.12) (2.13) (2.11) S ) 0 n i x I S ) ( i Then we 2 ¯ α i δ . S a h 7 ) = ) | → ∞ + x = ( α j α i 0 a ··· ) | x ( S x ( 2 ( µ , satisfies 0 δ s in the expansion 1 ∂ is not close to any ¯ S ˜ 1 S µ n S i | n i δ h I ia I δ 2 α i h e ¯ δ a !! = 0. Furthermore, the ) + , i.e. α i i ¯ ) + S ) α a i = 0, , x is the SUSY transformation x ) will be written by a linear ( | ··· , for later convenience. This + i 0 x ¯ S (   ¯ ( δ ¯ S if S i ) 2 α h X ˜ ) S j δ , x i h α i ) 1 a ( ) = Then, we will use the clustering indeed gives the gaugino conden- δ a n 0 α i 1 i 0. ˜ + ¯ S a ) ¯ )) S δ 8 in a product of + . x ) which means x α x → ( + ( ( , i.e. n . =1 α x m i t Y S 0 0 I ( ˜ ˜ x S h S α ˜ Λ . It will be a linear combination of 1 and 0 , S n ( S . Then, ) ) + x | 0 ¯ ( 0 S S ( ˜ C n α j x ˜ i S µ n S n =1 ˜ I ( a ) S | ∂ h Y j

0 | → ∞ h C x µ b 2 n ( S ( , which is strictly smaller than the number of δ ia 2 µ ) will be a self-consistent equation. . We can easliy see that C S 1 e δ − ∂ – 4 – h M δ ) 1 µ G   δ α α j 2.9 ia 2 = µ a xδ x e | δ 4 4 ¯ 1 for each 1 S ¯ d d δσ − as a background constant chiral superﬁeld. Note that ( j )) such that n i xδ -closed correlators, which we are considering, we can ) in terms of a Z Z 4 ¯ i S )) ¯ 2.3 d δ to satisfy

x = ] is a space derivative. Here 2.7 ) = ( )) and ¯ 2 S Z 0 n x =1 δ M ( I Y S 1 α in ( 0 ( k, β → ) ) , δ S n x i 0 n µ ( ¯ ¯ δ S C limit (in ( = 0 by the definition of ∂ I , term, which vanishes for constant ( λ i 6= ( µ as a chiral superfield, we have t ) 2 ¯ ¯ ,g S . The condition ( S ia b δ ) contains only linear term in ) 0 = 0. We will see later that this ) is determined by i i e i = 1 τ y ) ) ( ¯ ¯ δ ν − S S S α i x 0 | . Furthermore, we can show i , g i ∂ ˜ ( j, α i ) a S ) ) x 0 ∂a ( ¯ θ ( S ¯ x S x (( S G are not Lorentz invariant. h 4 ( + ( λ ( 2 ··· h d ˜ 2 δ ) = 0 and [ n S ( F α 1 = 2. Now we will take a following “large separations” limit: h S h x C x ( Z yδ ( ¯ 0 S 4 0 ≥ ). Therefore, for the = d S ˜ = and S is an arbitrary constant. i R α | → ∞ 2 ¯ n δ 1 n j e δ ··· β k S I m C a 1 a )( δ h δ Note that we can do this replacement for each Now we take the constant − If we do not regard Here, we take the large µ 7 8 α j ¯ δσ a and number of the points of the integration is because where | properties to factorize theother correlator for insertion each points. Here we can see that of the exponentialcombination with of different the following form: where This follows from ( replace where because corresponding to which means sation, i.e. expand term will be relevant ifthe we action regard of where we left JHEP03(2016)136 (2.16) (2.20) (2.22) (2.19) (2.21) (2.14) (2.15) (2.17) (2.18) to 0 τ , ¯ S i ) is the (would-be) moduli. ] by the semi-classical x ( X λ 43 ) – . b ¯ ). We have seen that the S i ¯ − S 41 ) , , , x ) x ), is the path-integral with the . ( ( ¯ 23 S ¯ 0 S (= = 1 λ = 0, where , , i S h , i = ) 2.8 . 2 h ) S c S ¯ ¯ S S | h = µ ,( ( ) . = = ˜ ( 2 S, g ¯ /∂X 0 S ) f S ) ˜ ( τ X c i i i ,

0 , ω S ˜ ) ) τS, ) π ) i 2 + G ( x 2 / ,g S ∂S ( 3 = 0 ∗ j πi ) ( + X, τ ··· 0 k ˜ ( Λ ∂F ( ∂S S − h y,θ ) h eff exp ( S ∂F ˜ τS /∂X 2) S G ) ) ), we see that ) + 2 ( ( − / 0 0 – 5 – πi µ ∂W ¯ and πi 1 F S ∗ j are found as in [ ) ( 1 2 ( 2 is just replacing the coupling constant 2 α ω e S 9 f G ∗ j g X, τ X, τ ¯ ( S 2 ( ( + ˜ ydθ Λ is the dynamical scale in the 1-loop Pauli-Villars k 3 c ¯ ¯ ) + 2 4 F ( = S S 0 µ d ¯ ∂ S τ = r = ( R =0 ˜ does not depend on h· · ·i = Y j W e 0 f = ) ¯ S eff 3 ) = ( x ˜ τ = ¯ ˜ Λ ( S /∂τ W ) = ( λ )) V ) f G ¯ ), the S can be written as b ( ( W e f V − 2.8 − x W ) ( 0 )) = 1, and the λ c h N X, τ = ( 2. Therefore, there is at least an isolated insertion of an operator which eff ]. Note that this also implies ≥ X W (SU( α ( e 42 ∂ , ) vanishes and we find m ) comes from the path-integral measure which is defined with the coupling . ]. More precisely, the definition of 0 µ is the dual coxeter number of τ ( 41 , 2 2.13 45 2 , g c 23 44 Now we will consider the gaugino condensation As we can see from ( In order to find the vacua, we need to evaluate because 9 α i With this and is a solution. Thus, we can think that where constant original superpotential for example, regularization which is defined by where where Thus, the effective superpotential is that the 1-loop factortial in [ the localization techniquesuperpotential only contributes to the Kählerpoten- Thus, the superpotential andcomputations the around vacua the fundamental monopoles which have two fermion zeromodes. Note makes ( from which we canas evaluate in the [ superpotential, the vacua and the gaugino condensation a JHEP03(2016)136 ˜ Λ(Λ). (2.28) (2.30) (2.23) (2.25) (2.26) (2.27) (2.31) (2.29) (2.24) ˜ Λ = is evaluated , i S = 0. Thus, the ) ¯ W S i ( ) f ˜ S ˜ τ ∂ ∂ , + ) . S ( , . )) πiS i i F ). 2 h = 0 , g + a πiS . ) = 2.25 S 2 ) S . ( h ˜ 1 i Λ by using the relation , ( µ 3 0 (polynomials of ( F , ∂S G ˜ 2 h τ Λ − = 0 ) , c ˜ τ ∂F ¯ + w ) 3 ∂ S ¯ πτ and ) ∂ ) S ( ¯ a S ( 2 ¯ + ∂ f S . The superpotential )Λ G S S ( S ) ( ∂F 3 a 0 G S e f + ) + 2 ( , which is the correct one. Therefore, the ( 1 c ¯ ∂S exp ) S e )Λ ¯ S = − e ( ∂F S S πiτ ) ( 3 G 2 – 6 – i f 2 1 ln µ ∂S ( c c )) (2 ) ( 1 ˜ τ e ∂F τ e x ∂ 2 ∂ (˜ 3 = ( g = Λ =1 ˜ S Λ) was given by ( S ¯ ln S a X S Λ 3 ( 3 2 + h eff ¯ 2 µ − S c w ˜ ) which is just a sum of the Veneziano-Yankielowicz Λ τ, c ) (˜ ) = W − ) = = − = G i S πiS ¯ eff ( S 3 ( ). We will also define a dynamical scale Λ in the 1-loop ¯ 2.29 2 S e = , g h Λ W F a 0 S ˜ τ Λ) = Λ is given by = 0 which follows from 2.25 = + ( ˜ ). τ Λ) as a function of ∂ ∂S V S, i ∂ S )) = S ∂W ( ) and ( S, S 2 τ can be computed using G W ( S (˜ c F ˜ τ S ∂ ¯ ¯ ∂ S S W 2.15 h W → = 1 SUSY gauge theory of only vector multiplets with a gauge group τ, (˜ S N eff W W ˜ τ ∂ ∂ is semi-simple and a superpotential ) to a is determined by ( G ¯ S a 2.30 ⊗ We can easily generalize the results to the theory with a semi-simple gauge group. Let Now we see that the following glueball superpotential reproduces the gaugino conden- The result is = us consider a 4d G which is equivalentwith to ( glueball superpotential issuperpotential the and ( the where we can think Indeed, we find where we have used ( sation and the effective superpotential: The relation between Λ and The superpotential is evaluated to where Pauli-Villars regularization of the coupling constant where we have used gaugino condensation then, the effective potential should give JHEP03(2016)136 . ) a a S ( = 0. (3.1) ∂S (2.32) a ∂F S a 2 1 c e 3 a Λ w Then, we can ) a G 12 ( e , ) a = S a ( S F and chiral multiplets couple + and rescaling of the fields. In with G 1 ) a ¯ δV a − t S ( , 3 a ∂S ) ∂F a )Λ a a Φ a , S S i G a g ( ( e P ln tree is the chiral superfields, the theory is expected − – 7 – a ) W a a S S = ( a F and Φ X tree + a 2 10 W c a S − a 2 ] the perturbative computation of the chiral multiplets with c 31 and gaugino condensation is non-trivial, which we will compute. ] it was shown that the effective superpotential obtained by → Λ) = 11 , 32 S a = 1 gauge theory with gauge group S W ( limit with the regularization term S ]. Then, the theory is effectively in weak coupling and the effective N W 23 )[ → ∞ t 2.3 is the coupling constants ) for the vector multiplet is the anti-holomorphic superpotential. Both of them do not affect the i . Here we expand the bosonic fields in the chiral multiplets around the classical g 2.3 . With a generic tree level superpotential ) = 0. Thus, we can add the regularization term of the localization for the vector On the other hand, in [ We can also add a large kinetic terms for the chiral multiplets. We will compute the correlation functions of the operators insertions which satisfy With the chiral multiplets, the Wilson loop will not behave the areThe law. kinetic Thus, terms precisely speaking, for the the chiral superfields are written by the Kählerpotential. The regularization If the low energy theory is a non-trivial conformal fixed point, we will add an arbitrary small pertur- G O 11 12 10 → ∞ ( i ¯ bation to the coupling constants or a smallphase deformation will of the not vacuumsimplicity, be we we choose. a will call confining it phase, confining but phase. aterm phase ( with aeffective mass superpotential gap and the with correlation possible function free of the U(1) operators factors. in the For chiral rings. the vector multiplet background was done byway. deforming the anti-superpotential Furthermore, appropriate inintegrating out [ the chiral superfields can be determined by the generalized Konishi anomaly. regarded as background fields.trivial Thus and the integrations saddle over pointseffective the of superpotential. the saddle It large points will kineticsuperpotential be with terms which interesting the are should exactly superpotential be non- follow a this giveessentially matrix line a the model and zero non-trivial computation find modes because the of the effective the saddle chiral points are multiplets. as a backgroundt because the effectivevacua. gauge Note coupling that the constant 1-loopwe is computaiton take is very the exact small in theour by usual case, taking localization the technique where kinetic terms of the chiral multiplets contain the vector multiplets which is δ multiplets ( dynamical scale can be set to arbitrary low. integrate out the chiral multiplets perturbatively, where the vector multiplets are regarded where to be in a confining phase 3 A proof ofLet Dijkgraaf-Vafa us conjecture consider 4d to which is evaluated to Here, for U(1) gauge group, there is no dynamically generated superpotential and Following the previous discussions, we can easily see that JHEP03(2016)136 at a ), we . The will be 0 , it was G G 2.31 G , we have the a S ] because corresponding is the value of Φ 32 a ) is the one conjectured ¯ Φ 2.31 ) where Φ is the chiral superfield, In terms of α ). after integration of the chiral W where a α 13 a ] and taking the decoupling limit to get 2.31 W ¯ n Φ 32 − a for any 0 τ = Φ 0 a , Dijkgraaf-Vafa conjecture was proved by using the = 0 a G 0 – 8 – τ = G for the original theory as stressed in [ a ), with to the effective action ( S 0 ), which permits any use, distribution and reproduction in G 2.31 ,( V W are possible to be defined for this setting because the gauge sym- is given by just adding the Veneziano-Yankielowicz superpotentials a CC-BY 4.0 ] that the perturbative effective superpotential for the chiral multiplet i S a 32 This article is distributed under the terms of the Creative Commons S h 14 a P ] and [ = 31 could be interpreted as a composite operator, like Tr(Φ i ] . Therefore, this can be regarded as a nonperturbative proof of the Dijkgraaf- a S S h 35 – In particular, if we consider a chiral multiplet of the adjoint representation of Then, applying the discussion in the previous section to the effective action ( Depending on the choice of the classical vacuum, the original gauge group Here, the chiral multiplets with classical superpotential can have a non-trivial moduli The For the unbroken gauge group case, i.e. 33 14 13 although we will notThere study is this no in effective this superpotentialcoupling paper. constants fot Note are that absent. such relations are just forgeneralized Konishi the anomaly expectation equation values. ofthe the Veneziano-Yankielowicz terms. full effective action [ Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. Acknowledgments S.T. would like to thank Masatoand Taki useful for his dicussions collaboration and atwould like the Yu to early Nakayama stage thank for of K. this important Lee, project K. comments Ohta, and S. Rey, discussions. N. Sakai and S.T. P. Yi also for useful discussions. integral of the vector multipletsthe discussion gives in just the the previous section. Veneziano-Yankielowiczin terms The [ according final effective to action ( Vafa conjecture. sation for all simple gauge groups in shown in [ is equivalent to the one of the matrix model of the Dijkgraaf and Vafa. Then, the path- the gauge theory whicheffective is superpotential lowered by thesuperfields. regulator term. conclude that the effective superpotential from which we can compute the gaugino conden- Then, we redefine thethe chiral classical superfields vacuum as we have Φ chosen. The perturbative calculationbroken is to done around a this. semi-simpleglueball gauge superfields group withmetry U(1) is factors, broken which at we very will high denote energy scale compared to the effective dynamical scale of those methods. space of vacua.the We moduli have space discrete needby set not giving of a to vacua small be with deformation discrete. a of superpotential, generic Here for superpotential, we example assume although, a the mass term, moduli if space it is is discrete needed. Thus, in this paper, we assume that the integrating out the chiral multiplets is done by JHEP03(2016)136 ] D , 06 5 , ]. ] JHEP JHEP and , , (2013) 2 JHEP B 716 S , SPIRE 05 IN ]. (2012) 159 (2012) 033 ][ (2014) 125001 05 09 (1988) 353 JHEP partition function arXiv:1206.6008 SPIRE (1982) 253 ]. , [ 3 IN supersymmetric gauge D 89 Phys. Lett. S 117 ][ , arXiv:1211.0364 JHEP JHEP [ = 2 , SPIRE , B 202 YM IN D gauge theories on ][ D ]. 2 (2012) 157 2) ]. , arXiv:0909.4559 Phys. Rev. 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