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ABJ anomalies in supersymmetric Yang-Mills theories
Volume 156B, n u m b e r 5,6
PHYSICS LETTERS
27 J u n e 1985
ABJ A N O M A L I E S IN S U P E R S Y M M E T R I C Y A N G - M I L L S T H E O R I E S L. BONORA, P. PASTI and M. T O N I N Dipartimento di Fisica "'G. Gafilei'" Via Marzolo 8, I-35100 Padua, Italy and lstituto Nazionale di Fisica Nucleare, Sezione di Padova, 1-35100 Padua, Italy Received 12 December 1984; revised manuscript received 21 February 1985
Cohomological methods are used in order to obtain the structure of the ABJ anomaly in SYM theories in an algebraic form: the ABJ anomaly consists of a chiral gauge anomaly which breaks rigid supersymmetry and a supersymmetric partner required by consistency.
Abelian and non-abelian chiral anomalies in supersymmetric gauge theories have been studied extensively [ 1-3]. Their superfield formulation is simple and well known in the abelian case, while it is not so clear in the non-abelian case. The consistent non-abelian anomaly has been obtained by Piguet and Sibold [2] in terms of a recursive relation in the vector superfield V. Another expression for the non-abelian anomaly is given in ref. [3] where a heat kernel technique is used. However a simple algebraic superfield expression of the consistent anomaly is still lacking. In this paper we cope with the problem of deriving the consistent chiral anomaly in supersymmetric gauge theories using cohomological methods. To this end we study the coupled cohomology associated with (chiral) gauge invariance and rigid supersymmetry and find a solution for this coupled cohomology problem. Our solution is rather peculiar. We f'md a chiral anomaly AG which breaks supersymmetry and requires, for consistency, a supersymmetric partner AS, i.e. a cocycle of the cohomological operator associated with rigid supersymmetry. A s is non-invariant under the BRS transformations associated with gauge invariance. We work in a superspace with coordinates z M = (x m, 0u, ~7~i) (m = 0,1,2, 3;/.t, ti = 1,2). Small latin letters denote space-time indices, small Greek letters spinor indices and capital letters both. Letters from the beginning of the alphabet are devoted to tangent space indices and from the middle to world indices. 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
The rigid vielbeins e A are 1-superforms def'med by
de A = ieC eB oBcA ' where OABCis the.rigid torsion. It vanishes except for a --a al~ " oa~= o~a and o a = (1, o)a •, where a are the Pauh matrices. The NW-SE summation convention is adopted for Greek indices. We shall split the exterior differential d in the superspace as follows
(l) where cl = eaOa, D = eaDa, b = e&Da and I) = D + D. Finally ~ denotes the Lie algebra of the Lie group G. SYM theories are described by the ~-valued 1-superform ¢ = eA~A with the addition of the constraints [4] Fao = Fag[ = Fat~ = 0 ,
(2)
where F = d~b+ ~b = eAeBFBA is the ~-valued curvature 2-superform. The curvature constraints, eq. (2), imply
Faa= OajW~,
Fa&=~aa#W# ,
Fab = ( l/2i)(oab)#'Y A¢W.y+ ( l/2i)(~ab)#~t AtlWq ,
(3,4) (5)
and AaW a + ~&W &= 0 ,
(6)
where Wa(W &) are covarianfly chiral (antichiral) superfields, Aa and A f denote gauge-covariant derivatives, fx stands for the integral over the ordinary space341
Volume 156B, number 5,6
PHYSICS LETTERS
time only, while fz denotes the integral over the whole superspace. The coboundary operator associated with (superlocal) gauge invariance (BRS operator) is, as usual, the nilpotent operator
27 June 1985
(~G + ~s)(AG + AS) = 0 , that is ~GAG = 0 ,
ZGAs + ZSAG = 0 ,
(10)
~sAs = o.
ZG = J ( s ~ 6 / ~
Our problem is finding a non-trivial cohomology class of the coupled cohomology defined by eqs. (8)-(10). The starting point of our analysis is the identity
+ sc~/Sc),
Z
where s¢ = - ( d c + [~, c ] ) ,
sc =
-cc
Str(F, F, P0 = (d + s) ~2,
are the BRS transformations, and the g-valued superfield c(z) represents the Faddeev-Popov ghost fields. Wherever we mention the ghost number #g, of a monomial we refer to the number of superfields c(z) which are contained in it. More generally we shall speak of the #g of a polynomial if it is the sum of monomials with the same #g. Clearly the coboundary operator ~G has #g = 1. In order to define the coboundary operator ZS corresponding to rigid supersymmetry, we introduce the "global".Faddeev-Popov ghosts e A = (0, e a, e--d): e a and ~a commute with one another (and, otherwise, keep their usual grading properties). The action of the functional operator ZS acting on superfields if(z) integrated over space-time is defined by ZS f ~k(z) = f ( e a D a + ~4~b&) ~k(z). X
X
~S is nilpotent when acting on space-time integrals of superfields. Moreover ~G~S + ~ S Z G = 0 , therefore (~S + ~G) 2 = 0 with the above proviso for the domain of Y'S. The coupled Ward identity for BRS and rigid supersymmetry transformations can be written (at one-loop order) as (Y'G + ZS) 1-' = ~(A G + AS) + OQi2),
(7)
where F is the vertex generating functional and AG, AS are space-time integrals of local products of component fields and their derivatives, linear in the component fields of c(z) and in e A, respectively. Since ZG + Y'S is nilpotent, one gets the consistency conditions: 342
(8,9)
(11)
where
1
n = 3 f d, Sir(0, F,, F,),
(12)
0 and ~ = ~ + c, fit = t(d + s) ~ + t 2 ~ with 0 ~< t ~< 1. Str denotes the completely symmetric trace of the matrix generators of ~ in a given representation. The LHS ofeq. (11) is a 6-superform with ghost number 0. I2 is a sum of (5 -p)-superforms with #g = p (p = 0 ..... 5) 5 ~2 = ~ coP_p. (13) p=0 Eq. (11) is the starting point to display the algebraic and geometric properties of the ABJ anomaly [5,6] in ordinary space-time. The same equation is true also in superspace. However in superspace we have additional structure: since e a and e-a[_a= (a, t0] have different mass-dimensions (dim [ea] = _ 1, dim [e-a] = - 1 / 2 ) any n-superform splits into different sectors characterized by their canonical dimensions. The sector with canonical dimension n - q / 2 contains n - q factors e a and q factors e-a. In other words, i f x is an n-superform wehave the decomposition X = ~'~=OXn-q,q, where X n - q , q has definite canonical dimension n - q/2, and X n - q , q = e a-1 ... e a-qeal ... e a n - q Xan_q...al~q...Ctl. Similarly for wPn_p n-p
,4-p =
,,,P q=0
-P-q'q
'
where t.OP_p_q,q has canonical dimension n - q - p/2. Consequently eq. (11) contains different sets of identities, one for each sector. Let us consider the follow-
Volume 156B, number 5,6 ing
PHYSICS LETTERS If we defme
,1 :
5 S t r ( F , F , F ) 6 , 0 = ( d + s) ~ 60P p=0 5 - p , 0 , Str (F,
27 June 1985
^0 =600, 1 _~b0, 1 , 604,1 (14)
5- q = (d + S) p=O ~-J ¢OP-P-q'q
eq. (18) becomes [I ~ 0 , 1 1 4 , 2 + ~600, 2 = 0 .
F, F)6_q, q 6- q
+ p~-O- [Dt"gP-P]6-p-q'q'
q = 1 ..... 6 .
(15)
The above equations contain pieces with different ghost number. Therefore they split into independent equations characterized by a fixed #g. We pick out the following: from eq, (14) 0 = s601,0 + d602,0,
(16)
(21)
(22)
The integral over space-time of a generic space-time four-form X4 is def'med as follows: let X4 = efeCebea × Xabcf, then fX4 =
fea%bcf.
X
X
(23)
Now let us call ie the interior product with respect to the vector field eM, where eIn = eA eA M and eM are the inverse vielbeins. We remark that i2 does not vanish due to the commuting properties of e a and ~a. Applying twice ie to eq. (22) and observing that ie [Die &O, 1] 4, 1 = {ie ie [ b ~ O , 1] 4, 2, we obtain
from eq. (15) f o r q = 1 0 = S600,1 + a60~, 1 + [b~14,014,1,
(17)
(18)
The important point about eq. (18) is that, due to the curvature constraints, eq. (2), and their implications eq. (3)-(6), the LHS o f e q . (18) can be written as Str(F, F, t04, 2 = ( I ~ 0 , 1 ) 4 , 2,
(19)
where k0,1 = (i/24)
(24)
x
from eq. (15) for q = 2 Str(F, F, F)4, 2 = d600, 2 + [D600,114,2.
Zs feb0 = 0,
AO • ^0 where 604 = te 604,1. Moreover since the components of ~0, 1 are gauge-invariant superfields,
~G f
~0
4 = Y'G fie600,1
x
(25)
X
Applying ie to eq. (17), integrating over space-time and using eq. (25) we obtain
efeCebe a [e ~ Str(Foa, F~b, OcT"rF~/f) ZG f ~ O _ ZS f601,0=0 "
-e&Str(F~a,F~tb, °c~F~/f)] .
(20)
x
(26)
X
Finally after integrating over space-time eq. (16) becomes .1 We recall that applying f) to a superform, as in eq. (15), is not the same as applying D to its intrinsic components since D, when acting on the rigid vielbe~s e a produces terms proportional to the rigid torsion: Dea = 2ieaeao~sc However such terms can be present only in sectors with at least one dotted (e a) and one undotted (ea) vielbein. Therefore they do notappeax in eq. (17) below. They are also absent in sectors [Dx ]4,2 (which is the case of eqs. (18) and (19) since the corresponding contribution would contain intrinsic components with f'we Latin indices, which vanish in four dimensions. Therefore contributions from rigid torsion are absent in all the equations we shall use.
X G f o o l , 0= 0 .
(27)
X
From eqs. (24), (26) and (27) it follows that the couple formed by A G = fz 00-0-060~,0 and A s = --fz 0000¢o~ satisfies the consistency conditions (8)-(10). The same argument used in ordinary space-time [5,6] shows that A G belongs to a non-trivial cohomology class of AG. Let us discuss briefly the meaning of our result. One must take into account that we have 343
Volume 156B, number 5,6
PHYSICS LETTERS
worked in a framework where the fundamental fields are the components ¢A of a constrained superconnection, and the gauge parameters are real superfields. The standard formulation involves instead an unconstrained -valued real vector superfield V and the gauge parameters are chiral and antichiral superfields. It is possible to express our anomaly in the standard framework by performing a suitable Bardeen-Zumino-like transformation [7] (which amounts to subtracting a suitable local counterterm to the action). This transformation will be worked out elsewhere. Here it is enough to specify that after this transformationhas been perfor2ned, our anomaly giG + giS becomes AG + AS, where A G is linear in the chiral and antichiral ghost superfields and both 7,G and mS are local functionals o f V. On the basis of the theorem quoted in ref. [8], ~S should be a coboundary of l~S, i.e. ~S = ZS C ,
(28)
where C is a local functional of V. Then, by subtracting from the action the counterterm C, one ends up with a purely gauge, sul~ersymmetry preserving anomaly 7~G - ~GC, where ~G is the cohomology operator associated with gauge symmetry in the standard framework. So far we have not been able to fred the counterterm C. We will return on this subject in a forthcoming paper. One of us (M.T.) would like to acknowledge a valuable discussion he had with O. Piguet and A. Sibold. We would like to thank S. Ferrara, L. Girardello, R. Grimm and, especially, R. Stora for their remarks on a preliminary version o f this paper.
344
27 June 1985
After completion of this work we have received a paper by Nair and Namazie [9] where it is shown that i n N = 2 non-abelian SYM theories rigid supersymmetry is anomaly-free.
Note added in proof. We have recently determined the counterterm C of eq. (28). We have also received a paper from Girardi, Grimm and Stora [10] who derive a supersymmetric invariant expression for the anomaly A G. References [1] O. Piguet and K. Sibold, Nucl. Phys. B197 (1982) 257, 272. [2] O. Piguet and K. Sibold, The anomaly in the Slavnov identity forN = 1 supersymmetric Yang-Mills theory, MPIPAE/PTh 21/84. [3] N.K. Nielsen, Nucl. Phys. B244 (1984) 499. [4] J. Wess, Supersymmetry-supergravity, in: Topics in quantum field theory and gauge theories (Salamanca, 1977) (Springer, Berlin, 1978) p. 81. [5] L. Bonora and P. Cotta-Ramusino, Phys. Lett. 107B (1981) 87. [6] B. Zumino, Y.S. Wu and A. Zee, Nucl. Phys. B239 (1984) 477; Carg6se lectures 1983; R. Stora, Carg~se lectures 1983. [7] W.A. Bardeen and B. Zumino, Nucl. Phys. B244 (1984) 421. [8] O. Piguet, K. Sibold and M. Sehweda, Nucl. Phys. B174 (1980) 183. [9] V.P. Nair and M.A. Namazie, On the absence of supersymmetry anomalies in gauge theories, Princeton preprinL [10] G. Gkardi, R. Grimm and R. Stora, Phys. Lett. 156B (1985) 203.
PHYSICS LETTERS
27 J u n e 1985
ABJ A N O M A L I E S IN S U P E R S Y M M E T R I C Y A N G - M I L L S T H E O R I E S L. BONORA, P. PASTI and M. T O N I N Dipartimento di Fisica "'G. Gafilei'" Via Marzolo 8, I-35100 Padua, Italy and lstituto Nazionale di Fisica Nucleare, Sezione di Padova, 1-35100 Padua, Italy Received 12 December 1984; revised manuscript received 21 February 1985
Cohomological methods are used in order to obtain the structure of the ABJ anomaly in SYM theories in an algebraic form: the ABJ anomaly consists of a chiral gauge anomaly which breaks rigid supersymmetry and a supersymmetric partner required by consistency.
Abelian and non-abelian chiral anomalies in supersymmetric gauge theories have been studied extensively [ 1-3]. Their superfield formulation is simple and well known in the abelian case, while it is not so clear in the non-abelian case. The consistent non-abelian anomaly has been obtained by Piguet and Sibold [2] in terms of a recursive relation in the vector superfield V. Another expression for the non-abelian anomaly is given in ref. [3] where a heat kernel technique is used. However a simple algebraic superfield expression of the consistent anomaly is still lacking. In this paper we cope with the problem of deriving the consistent chiral anomaly in supersymmetric gauge theories using cohomological methods. To this end we study the coupled cohomology associated with (chiral) gauge invariance and rigid supersymmetry and find a solution for this coupled cohomology problem. Our solution is rather peculiar. We f'md a chiral anomaly AG which breaks supersymmetry and requires, for consistency, a supersymmetric partner AS, i.e. a cocycle of the cohomological operator associated with rigid supersymmetry. A s is non-invariant under the BRS transformations associated with gauge invariance. We work in a superspace with coordinates z M = (x m, 0u, ~7~i) (m = 0,1,2, 3;/.t, ti = 1,2). Small latin letters denote space-time indices, small Greek letters spinor indices and capital letters both. Letters from the beginning of the alphabet are devoted to tangent space indices and from the middle to world indices. 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
The rigid vielbeins e A are 1-superforms def'med by
de A = ieC eB oBcA ' where OABCis the.rigid torsion. It vanishes except for a --a al~ " oa~= o~a and o a = (1, o)a •, where a are the Pauh matrices. The NW-SE summation convention is adopted for Greek indices. We shall split the exterior differential d in the superspace as follows
(l) where cl = eaOa, D = eaDa, b = e&Da and I) = D + D. Finally ~ denotes the Lie algebra of the Lie group G. SYM theories are described by the ~-valued 1-superform ¢ = eA~A with the addition of the constraints [4] Fao = Fag[ = Fat~ = 0 ,
(2)
where F = d~b+ ~b = eAeBFBA is the ~-valued curvature 2-superform. The curvature constraints, eq. (2), imply
Faa= OajW~,
Fa&=~aa#W# ,
Fab = ( l/2i)(oab)#'Y A¢W.y+ ( l/2i)(~ab)#~t AtlWq ,
(3,4) (5)
and AaW a + ~&W &= 0 ,
(6)
where Wa(W &) are covarianfly chiral (antichiral) superfields, Aa and A f denote gauge-covariant derivatives, fx stands for the integral over the ordinary space341
Volume 156B, number 5,6
PHYSICS LETTERS
time only, while fz denotes the integral over the whole superspace. The coboundary operator associated with (superlocal) gauge invariance (BRS operator) is, as usual, the nilpotent operator
27 June 1985
(~G + ~s)(AG + AS) = 0 , that is ~GAG = 0 ,
ZGAs + ZSAG = 0 ,
(10)
~sAs = o.
ZG = J ( s ~ 6 / ~
Our problem is finding a non-trivial cohomology class of the coupled cohomology defined by eqs. (8)-(10). The starting point of our analysis is the identity
+ sc~/Sc),
Z
where s¢ = - ( d c + [~, c ] ) ,
sc =
-cc
Str(F, F, P0 = (d + s) ~2,
are the BRS transformations, and the g-valued superfield c(z) represents the Faddeev-Popov ghost fields. Wherever we mention the ghost number #g, of a monomial we refer to the number of superfields c(z) which are contained in it. More generally we shall speak of the #g of a polynomial if it is the sum of monomials with the same #g. Clearly the coboundary operator ~G has #g = 1. In order to define the coboundary operator ZS corresponding to rigid supersymmetry, we introduce the "global".Faddeev-Popov ghosts e A = (0, e a, e--d): e a and ~a commute with one another (and, otherwise, keep their usual grading properties). The action of the functional operator ZS acting on superfields if(z) integrated over space-time is defined by ZS f ~k(z) = f ( e a D a + ~4~b&) ~k(z). X
X
~S is nilpotent when acting on space-time integrals of superfields. Moreover ~G~S + ~ S Z G = 0 , therefore (~S + ~G) 2 = 0 with the above proviso for the domain of Y'S. The coupled Ward identity for BRS and rigid supersymmetry transformations can be written (at one-loop order) as (Y'G + ZS) 1-' = ~(A G + AS) + OQi2),
(7)
where F is the vertex generating functional and AG, AS are space-time integrals of local products of component fields and their derivatives, linear in the component fields of c(z) and in e A, respectively. Since ZG + Y'S is nilpotent, one gets the consistency conditions: 342
(8,9)
(11)
where
1
n = 3 f d, Sir(0, F,, F,),
(12)
0 and ~ = ~ + c, fit = t(d + s) ~ + t 2 ~ with 0 ~< t ~< 1. Str denotes the completely symmetric trace of the matrix generators of ~ in a given representation. The LHS ofeq. (11) is a 6-superform with ghost number 0. I2 is a sum of (5 -p)-superforms with #g = p (p = 0 ..... 5) 5 ~2 = ~ coP_p. (13) p=0 Eq. (11) is the starting point to display the algebraic and geometric properties of the ABJ anomaly [5,6] in ordinary space-time. The same equation is true also in superspace. However in superspace we have additional structure: since e a and e-a[_a= (a, t0] have different mass-dimensions (dim [ea] = _ 1, dim [e-a] = - 1 / 2 ) any n-superform splits into different sectors characterized by their canonical dimensions. The sector with canonical dimension n - q / 2 contains n - q factors e a and q factors e-a. In other words, i f x is an n-superform wehave the decomposition X = ~'~=OXn-q,q, where X n - q , q has definite canonical dimension n - q/2, and X n - q , q = e a-1 ... e a-qeal ... e a n - q Xan_q...al~q...Ctl. Similarly for wPn_p n-p
,4-p =
,,,P q=0
-P-q'q
'
where t.OP_p_q,q has canonical dimension n - q - p/2. Consequently eq. (11) contains different sets of identities, one for each sector. Let us consider the follow-
Volume 156B, number 5,6 ing
PHYSICS LETTERS If we defme
,1 :
5 S t r ( F , F , F ) 6 , 0 = ( d + s) ~ 60P p=0 5 - p , 0 , Str (F,
27 June 1985
^0 =600, 1 _~b0, 1 , 604,1 (14)
5- q = (d + S) p=O ~-J ¢OP-P-q'q
eq. (18) becomes [I ~ 0 , 1 1 4 , 2 + ~600, 2 = 0 .
F, F)6_q, q 6- q
+ p~-O- [Dt"gP-P]6-p-q'q'
q = 1 ..... 6 .
(15)
The above equations contain pieces with different ghost number. Therefore they split into independent equations characterized by a fixed #g. We pick out the following: from eq, (14) 0 = s601,0 + d602,0,
(16)
(21)
(22)
The integral over space-time of a generic space-time four-form X4 is def'med as follows: let X4 = efeCebea × Xabcf, then fX4 =
fea%bcf.
X
X
(23)
Now let us call ie the interior product with respect to the vector field eM, where eIn = eA eA M and eM are the inverse vielbeins. We remark that i2 does not vanish due to the commuting properties of e a and ~a. Applying twice ie to eq. (22) and observing that ie [Die &O, 1] 4, 1 = {ie ie [ b ~ O , 1] 4, 2, we obtain
from eq. (15) f o r q = 1 0 = S600,1 + a60~, 1 + [b~14,014,1,
(17)
(18)
The important point about eq. (18) is that, due to the curvature constraints, eq. (2), and their implications eq. (3)-(6), the LHS o f e q . (18) can be written as Str(F, F, t04, 2 = ( I ~ 0 , 1 ) 4 , 2,
(19)
where k0,1 = (i/24)
(24)
x
from eq. (15) for q = 2 Str(F, F, F)4, 2 = d600, 2 + [D600,114,2.
Zs feb0 = 0,
AO • ^0 where 604 = te 604,1. Moreover since the components of ~0, 1 are gauge-invariant superfields,
~G f
~0
4 = Y'G fie600,1
x
(25)
X
Applying ie to eq. (17), integrating over space-time and using eq. (25) we obtain
efeCebe a [e ~ Str(Foa, F~b, OcT"rF~/f) ZG f ~ O _ ZS f601,0=0 "
-e&Str(F~a,F~tb, °c~F~/f)] .
(20)
x
(26)
X
Finally after integrating over space-time eq. (16) becomes .1 We recall that applying f) to a superform, as in eq. (15), is not the same as applying D to its intrinsic components since D, when acting on the rigid vielbe~s e a produces terms proportional to the rigid torsion: Dea = 2ieaeao~sc However such terms can be present only in sectors with at least one dotted (e a) and one undotted (ea) vielbein. Therefore they do notappeax in eq. (17) below. They are also absent in sectors [Dx ]4,2 (which is the case of eqs. (18) and (19) since the corresponding contribution would contain intrinsic components with f'we Latin indices, which vanish in four dimensions. Therefore contributions from rigid torsion are absent in all the equations we shall use.
X G f o o l , 0= 0 .
(27)
X
From eqs. (24), (26) and (27) it follows that the couple formed by A G = fz 00-0-060~,0 and A s = --fz 0000¢o~ satisfies the consistency conditions (8)-(10). The same argument used in ordinary space-time [5,6] shows that A G belongs to a non-trivial cohomology class of AG. Let us discuss briefly the meaning of our result. One must take into account that we have 343
Volume 156B, number 5,6
PHYSICS LETTERS
worked in a framework where the fundamental fields are the components ¢A of a constrained superconnection, and the gauge parameters are real superfields. The standard formulation involves instead an unconstrained -valued real vector superfield V and the gauge parameters are chiral and antichiral superfields. It is possible to express our anomaly in the standard framework by performing a suitable Bardeen-Zumino-like transformation [7] (which amounts to subtracting a suitable local counterterm to the action). This transformation will be worked out elsewhere. Here it is enough to specify that after this transformationhas been perfor2ned, our anomaly giG + giS becomes AG + AS, where A G is linear in the chiral and antichiral ghost superfields and both 7,G and mS are local functionals o f V. On the basis of the theorem quoted in ref. [8], ~S should be a coboundary of l~S, i.e. ~S = ZS C ,
(28)
where C is a local functional of V. Then, by subtracting from the action the counterterm C, one ends up with a purely gauge, sul~ersymmetry preserving anomaly 7~G - ~GC, where ~G is the cohomology operator associated with gauge symmetry in the standard framework. So far we have not been able to fred the counterterm C. We will return on this subject in a forthcoming paper. One of us (M.T.) would like to acknowledge a valuable discussion he had with O. Piguet and A. Sibold. We would like to thank S. Ferrara, L. Girardello, R. Grimm and, especially, R. Stora for their remarks on a preliminary version o f this paper.
344
27 June 1985
After completion of this work we have received a paper by Nair and Namazie [9] where it is shown that i n N = 2 non-abelian SYM theories rigid supersymmetry is anomaly-free.
Note added in proof. We have recently determined the counterterm C of eq. (28). We have also received a paper from Girardi, Grimm and Stora [10] who derive a supersymmetric invariant expression for the anomaly A G. References [1] O. Piguet and K. Sibold, Nucl. Phys. B197 (1982) 257, 272. [2] O. Piguet and K. Sibold, The anomaly in the Slavnov identity forN = 1 supersymmetric Yang-Mills theory, MPIPAE/PTh 21/84. [3] N.K. Nielsen, Nucl. Phys. B244 (1984) 499. [4] J. Wess, Supersymmetry-supergravity, in: Topics in quantum field theory and gauge theories (Salamanca, 1977) (Springer, Berlin, 1978) p. 81. [5] L. Bonora and P. Cotta-Ramusino, Phys. Lett. 107B (1981) 87. [6] B. Zumino, Y.S. Wu and A. Zee, Nucl. Phys. B239 (1984) 477; Carg6se lectures 1983; R. Stora, Carg~se lectures 1983. [7] W.A. Bardeen and B. Zumino, Nucl. Phys. B244 (1984) 421. [8] O. Piguet, K. Sibold and M. Sehweda, Nucl. Phys. B174 (1980) 183. [9] V.P. Nair and M.A. Namazie, On the absence of supersymmetry anomalies in gauge theories, Princeton preprinL [10] G. Gkardi, R. Grimm and R. Stora, Phys. Lett. 156B (1985) 203.