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Chiral anomalies in N = 1 supersymmetric Yang-Mills theories
Volume 156B, number 3,4
PHYSICS LETTERS
20 June 1985
CHIRAL ANOMALIES IN N = 1 SUPERSYMMETRIC Y A N G - M I L L S T H E O R I E S
G. G I R A R D I , R. G R I M M and R. S T O R A LAPP, Annecy-le-Vteux, France
Received 13 March 1985
We estabhsh a mamfestly supersymmetnc, compact, formula for the chtral anomahes of supersymmetnc gauge theories. Tlus result is obtained by comblmng superspace geometry with the usual algebra of anomahes Except for a Wess-Zumlno type term, we obtain an expression which ~s polynormal m the coefficients of the superconnectlon form.
1. Introduction. It has been shown by Piguet and Sibold [1] that chiral anomalies i n N = 1 supersymmetric gauge theories are parametrized by the d-symbols of the structure group, just the same as in the conventional case [2]. So far no closed expression for these anomalies has been deduced from the existence and uniqueness proof. In the supersymmetric case, the calculation of one-loop graphs is difficult because keeping control over the fulfillment of supersymmetry prevents one from using, say, the Wess-Zumino gauge [3] in which the particle content is transparent. Such a calculation has been performed [4], however, and the result has recently been reduced [5] to the simple integral over a real positive parameter of a rather involved algebraic expression. Several authors [6] have tried to find the anomaly in the form of a polynomial in the coefficients of the superconnections on which the theory is based, but this has resulted into a no-go theorem [7]. Recently, Bonora, Pasti and Tonin [8] have found a pair of anomalies A S and A G for the supersymmetry and gauge Ward identities, respectively, which fulfdl the correct consistency conditions. The expressions for A s and A G ar e polynomial in the lowest components of the (real) superconnection, subject to the usual constraints. Their method follows the more or less conventional Russian [9] algebra used in the construction of gravitational anomalies [10] and succeeds thanks to algebraic properties which follow from the constraints. In order to make contact with the usual supersymmetric anomalies, it is necessary to express A S as the supersymmetry variation of a local functional and it is at this point that the non-polynomial character of the algebra shows up [ 11 ]. The purpose of this letter is to construct explicitly the supersymmetric gauge anomaly and to show that its non-polynomial structure comes from Wess-Zumino type terms whose algebraic properties are now well under control [9]. We will fired it however more expedient to stay from the start within the scheme where only the chiral gauge group is involved. No explicit contact is made here with the result of ref. [8]. Section 2 is devoted to a descnption of the notations and main properties of superconnections and gauge groups in rigid superspace. Section 3 sketches out the main steps leading to the Final formula for the supersymmetric anomaly. Section 4 gathers a few concluding remarks. 2. N o t a t i o n x We start with the Lie G-valued superconnection forms *~ = Ea~oa + Ea~oa,
~ = EaCpa + E d ~ -& ,
(1)
where the E are the rigid vielbein forms, and
.1 All the superspace notations are those of ref. [3]. 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
203
Volume 156B, number 3,4 ~oa= - W - 1 D a W ,
PHYSICS LETTERS
Up&= - W D & W -1 ,
~0o~i= t2iDa~0a,
20 June 1985
•ad, = -~iDa~a,
(2)
with
w = eV,
(3)
V being a real superfield. The superconnection forms are related by (4)
~o= W - I c w - w - l d W .
Gauge transformations are defmed by
w-~ w'= ~ w z ,
(5)
where Y~belongs to the chiral gauge group ~ pertaining to G: ~a~ = Da~ = 0.
(6)
Accordingly ~ (respectively ~) transforms like a connection under the chiral (respectively antichiral) gauge group. The corresponding Slavnov transformations read: s~o= -d¢o-~o¢o-¢o~o ,
s~o = - d ~ - ~ o w - ¢ o ~ o ,
sW = - W w + ~ W ,
s¢o = - 6 o 6 o ,
s~ =-6o6o,
(7)
where w (respectively ~ ) is the Faddeev-Popov chiral (respectively antichiral) ghost superfield with values in Lie G D&w = D a ~ = 0 .
(8)
One defines as usual ¢=~o+ o~,
~=¢+~,
A=d+s,
(9)
related through 4= W-I~w_
(10)
w-lAW.
One has the Russian formula ~r ___9r(4) = A¢ + 44 = F(~o) --- d~ + ~o~o,
(11)
and (12)
~ 7 ( ~ ) = W ~ W -1 •
3. Construction o f the supersymmetric anomaly. We now consider a symmetric invariant polynomial J ( , , ), of degree 3 on Lie G and write the corresponding Chern-Weil formula
J ( 5 r, 5r,5 r) = AQs(4, 0) = AQ5(~, 0 ) ,
(13)
where 1
Q5(41, 42) = 3
f dt J(41 -
42, 7 t , 7 t ) ,
(14)
0 with 5rt = 5r(¢t) = ~r(t¢ 1 + (1 - t) ¢2). Recall that Q5(41, 42) is invariant under simultaneous gauge transformations of 41 and ¢2. 204
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Volume 156B, number 3,4
PHYSICS LETTERS
20 June 1985
Q5(¢, 0) expands in powers of w as
Q5(dp, O)=QO + QI + Q2 + Q3 + Q4 + Q5,
(16)
where the upper index denotes the power of ~o ,2 and the lower index denotes the form degree in superspace. Expanding out eq. (13) we get a hierarchy of identities of which we shall need sQ°+ d Q l = 0 ,
sQl+ d Q 2 = 0 ,
(17a, b)
which in superspace read: (1/5!) EA EBECEDEE(sQ0 EDCBA+ 5DE Q1DCBA -- I OTEDFQ1FCBA) = O,
(18a)
2 (1/4!) EAEBECEDts~I k ~Z4DCBA+ 4DDQ2CBA + 6TDCFQ3FBA) = O,
(18b)
featuring the constant torsion of rigid superspace. We shall now split the form degree p into a multidegree:
p-rklm,
k+l+m=p,
(19)
where k is the number of vector indices, I the number of undotted spinor indices and m the number of dotted spinor indices. Of eq. (18b) we keep a .~2a= sQloo+ a~g300 O, (20) where Qloo = (1/4!) edcba'~l~d400dcba = edcbatr[Co~d(~cab ~Oa-- ~Oc~gb~Oa)],
2d
dcba-.2
Q300 = (1/3!) e
(21)
~d300cba •
We furthermore extract Qloo by repeated use of eq. (18a) and of the explicit form of the torsion coefficients: 16iQ~o0 _ D 2 Q 1 0 2 _ ~2(Q120 - ~a.I D ~Q121 1 o.) + . . . .
(22)
where from now on the dots denote gauge variations and total derivatives and 0102 = - - ~ll-. e o-ha,) ~ d..d 2 0 2 ~qha,
Q120 = --~(eoba)6"rQ120,.rba,
Q1218 =g( 1 eoa~ 1 ) gQ121n-r a.
(23)
Depending on % Q102 is chiral,
QI02 = -~s(~o, ~2~, ~2¢o3,
(24)
and the second term in eq. (22) can be shown to be
~)2(Q12011D6Q~216)=~2QT220(~) + zID 1 . - 2 D 2{9111 1 + ...,
(25)
with Q120(~p) = _3__~j(~ ' O2~&, D2~5 ,
0~11 = - - ; (1e e ) a "r. 1 30111.r a.
(26,27)
0~11 -fa will be defined in a short while and will be split as O l l l = I~111- X~11,
(28)
,2 When~o,to are used we put a bar on the upper index. 205
Volume 156B, number 3,4
PHYSICS LETTERS
20 June 1985
with ,3 XII1 = ~iJ(w-l~w,~oe~,D2hoo)-~iJ(W¢olll-l,e&,O2~-a)+¼J((oo, X~), ~oCX&,X.o)-~J(w,X'
. o,a, (Xe~,,X'o))
- ~+((~o,x ~ , x., x ~ - ¼+((~,~ , +=a, ~oJ + ~+(~, 2r°a, ( ~ , x.)) + ~s((~, 2°a), x~, xoJ,
(29)
and 1
rhl ---~ f
[J(W]-1 dt Wt, ( a t , Xta), (.It'?, X t ~)) + J ( W t 1d t W,,
(at, X~), (X~',Xt oa))
o
+ J ( W t 1dt Wt, (~2t, Xt ~), (Xf ~,Xt&))] •
(30)
These expressions are obtained from the identity QI(~) _ Qi(~p) = d®1 + sO 0 •
(31)
O111 is related to O~ as usual, cf. eq. (19). Eq. (31) is the term linear in the ghosts of the splitting [9] Q5(¢), 0) - 05(~, 0) = --AX4(~, - w - l A w , 0) + QS(-W-1A I¢, 0)
(32)
with X4( , , ) the triangular function 1
l-t1
X4(¢,-W-1AW, O)=6 f dtl f 0
dt2d($,-W-1Aw,~r(tlq)-t2w-lAw))
(33)
0
and I
Q5(- I¢-IA W, 0) = AI~ ,
I ~= ~ f J(wf-ldtwt, ( w t l A w t ) 2, ( w t l A w t ) 2 ) ,
(34)
0 where F is the Wess-Zumino-Witten action and
Wt = e tv •
(35)
Use has been made of eq. (10) and of the invariance of Qs( , ) under simultaneous gauge transformations of both arguments. The form of (91 [cf. eq. (28)] follows: 1
1
1
03 = --X3 + P3.
(36)
To sum up we may rewrite 64Q~00 = D2D2 s~ + ...,
(37)
¢3 We make use of the def'mitions: X.~---W-ldlt] ,
=~_=-Wdl¥ - 1 '
1-~=-I¥-1sI¥,
Xt__~-lyFldlyt,,
~f-=--ll/F1slYt,
Wt=_e t V .
In addition, for a A (respectively cB) a p (respectively q) form with supers'pace indices A (respectively B) the graded commutation relation is defined as: ( a A cB) ~ a A r B _ ( _ ) p . q ( _ ) a b c B a A "
206
Volume 156B, number 3,4
PHYSICS LETTERS
20 June 1985
: sLij(6o, ~pa, ~2¢a) _ LsiJ(W6ow - l , ~,~, D2~-d) _ ~J(co, X a~, (Xt~, X6)) - ~-J((6o,Xa6), Xa, X&) + ¼J((6o, X~), ~&, X'a) - ~iJ(~, t~, D2~--~ + ~ i J ( w - l ~ w , ~a, ~2tpo~) + ~ j ( ~ , ~,,*&, (-'~&,-~tz)) + t8J((~, X~a), Xa, X~) - ¼J((~, X'o), ~ ,
Xo) - F I l l ,
(38)
with I~11 given by eq. (30). Finally, the supersymmetric anomaly is given by
=fd4xd40 ~ (t.o, ~, e V -
1),
(39)
where s~ is given in terms of superfields.
4. Concluding remarks. We have verified directly that the final formula fulfills the Wess-Zumino consistency condition s 93 = 0 .
(40)
The proof that 93 is non-trivial is a consequence of the supersymmetric version of the usual argument: Q1 cannot be written as Q1 = sQ° + dQ 1 .
(41)
Indeed, by the supersymmetric version of the algebraic Poincar6 lemma, it would follow that Q5 is of the form
Q5=sQ4.
(42)
But this is tantamount to say that J(co, (co, co), (¢o, 6o)) is a trivial cohomology class of Lie ~. We recall that from the anomaly we can construct the corresponding Wess-Zumino lagrangian 1
r(g, ~, V)=
-fd4x d40 f at m ( s t 1at st, s t
ldt St, ~ t l e v~, t - 1),
(43)
0 with S t = e t~ .
(44)
Unfortunately at the moment direct comparisons of the present result with those of refs. [8,5] look prohibitive ly complex but we are guaranteed by the work of Piguet and Sibold [1] that we are describing the chiral anomaly. (Details about the present derivation as well as a further analysis of the result will be found in an article in preparation.) This work was motivated by the recent article by L. Bonora, P. Pasti and M. Tonin. We warmly thank these authors for communicating their work prior to publication. We have also benefitted from abundant discussions with S. Ferrara, L. Girardello, O. Piguet and J. Wess.
References [1] O. Piguet and K. Sibold, NucLPhys. B247 (1984) 484. [2] C. Becchi,A. Rouet and R. Stora in: Field theory quantizalaon and statistical physics, ed. E. Tirapegui(Reidel, Dordrecht, 1981). [3 ] J. Wessand J. Bagger,Supersymmetry and supergravity,Princeton Series in Physics (Princeton U.P., Princeton, 1983). [4] N.K. Nielsen, NucLPhys. B244 (1984) 499. [5] E. Guadagnini, K. Konishi and M. Mintchev,Pisa preprint IFUP TH-10/85 (February 1985). 207
Volume 156B, number 3,4
PHYSICS LETTERS
20 June 1985
[6] J. Wess, private communication. [7] S. Ferrara, L. Girardello, O. Piguet and R. Stora, CERN-TH. 4134/85, LAPP-TH-131, to be published. [8] L. Bonota, P. Past1 and M. Tonin, Phys Lett. 156B, No 5,6 (1985). [9] J. Manes, R. Stora and B. Zumino, in preparation; lg Zumino in: Relativity, groups and topology II, eds. B.S. De Witt and R. Stora (North-Holland, Amsterdam, 1984); 1L Stora, in: Recent progress in gauge theories, eds. H. Lehmann et ah, NATO ASI Series B, Physics, Vol. 115 (Plenum, New York, 1984). [10] L. Baulieu and J. Thierry-Mieg, Phys. Let*. 145B (1984) 53; F. Langouche, T. Schiicker and R. Stora, Phys. Let*. 145B (1984) 342. [11] O. Piguet, M. Schweda and K. Sibold, NucL Phys. B174 (1980) 183.
208
PHYSICS LETTERS
20 June 1985
CHIRAL ANOMALIES IN N = 1 SUPERSYMMETRIC Y A N G - M I L L S T H E O R I E S
G. G I R A R D I , R. G R I M M and R. S T O R A LAPP, Annecy-le-Vteux, France
Received 13 March 1985
We estabhsh a mamfestly supersymmetnc, compact, formula for the chtral anomahes of supersymmetnc gauge theories. Tlus result is obtained by comblmng superspace geometry with the usual algebra of anomahes Except for a Wess-Zumlno type term, we obtain an expression which ~s polynormal m the coefficients of the superconnectlon form.
1. Introduction. It has been shown by Piguet and Sibold [1] that chiral anomalies i n N = 1 supersymmetric gauge theories are parametrized by the d-symbols of the structure group, just the same as in the conventional case [2]. So far no closed expression for these anomalies has been deduced from the existence and uniqueness proof. In the supersymmetric case, the calculation of one-loop graphs is difficult because keeping control over the fulfillment of supersymmetry prevents one from using, say, the Wess-Zumino gauge [3] in which the particle content is transparent. Such a calculation has been performed [4], however, and the result has recently been reduced [5] to the simple integral over a real positive parameter of a rather involved algebraic expression. Several authors [6] have tried to find the anomaly in the form of a polynomial in the coefficients of the superconnections on which the theory is based, but this has resulted into a no-go theorem [7]. Recently, Bonora, Pasti and Tonin [8] have found a pair of anomalies A S and A G for the supersymmetry and gauge Ward identities, respectively, which fulfdl the correct consistency conditions. The expressions for A s and A G ar e polynomial in the lowest components of the (real) superconnection, subject to the usual constraints. Their method follows the more or less conventional Russian [9] algebra used in the construction of gravitational anomalies [10] and succeeds thanks to algebraic properties which follow from the constraints. In order to make contact with the usual supersymmetric anomalies, it is necessary to express A S as the supersymmetry variation of a local functional and it is at this point that the non-polynomial character of the algebra shows up [ 11 ]. The purpose of this letter is to construct explicitly the supersymmetric gauge anomaly and to show that its non-polynomial structure comes from Wess-Zumino type terms whose algebraic properties are now well under control [9]. We will fired it however more expedient to stay from the start within the scheme where only the chiral gauge group is involved. No explicit contact is made here with the result of ref. [8]. Section 2 is devoted to a descnption of the notations and main properties of superconnections and gauge groups in rigid superspace. Section 3 sketches out the main steps leading to the Final formula for the supersymmetric anomaly. Section 4 gathers a few concluding remarks. 2. N o t a t i o n x We start with the Lie G-valued superconnection forms *~ = Ea~oa + Ea~oa,
~ = EaCpa + E d ~ -& ,
(1)
where the E are the rigid vielbein forms, and
.1 All the superspace notations are those of ref. [3]. 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
203
Volume 156B, number 3,4 ~oa= - W - 1 D a W ,
PHYSICS LETTERS
Up&= - W D & W -1 ,
~0o~i= t2iDa~0a,
20 June 1985
•ad, = -~iDa~a,
(2)
with
w = eV,
(3)
V being a real superfield. The superconnection forms are related by (4)
~o= W - I c w - w - l d W .
Gauge transformations are defmed by
w-~ w'= ~ w z ,
(5)
where Y~belongs to the chiral gauge group ~ pertaining to G: ~a~ = Da~ = 0.
(6)
Accordingly ~ (respectively ~) transforms like a connection under the chiral (respectively antichiral) gauge group. The corresponding Slavnov transformations read: s~o= -d¢o-~o¢o-¢o~o ,
s~o = - d ~ - ~ o w - ¢ o ~ o ,
sW = - W w + ~ W ,
s¢o = - 6 o 6 o ,
s~ =-6o6o,
(7)
where w (respectively ~ ) is the Faddeev-Popov chiral (respectively antichiral) ghost superfield with values in Lie G D&w = D a ~ = 0 .
(8)
One defines as usual ¢=~o+ o~,
~=¢+~,
A=d+s,
(9)
related through 4= W-I~w_
(10)
w-lAW.
One has the Russian formula ~r ___9r(4) = A¢ + 44 = F(~o) --- d~ + ~o~o,
(11)
and (12)
~ 7 ( ~ ) = W ~ W -1 •
3. Construction o f the supersymmetric anomaly. We now consider a symmetric invariant polynomial J ( , , ), of degree 3 on Lie G and write the corresponding Chern-Weil formula
J ( 5 r, 5r,5 r) = AQs(4, 0) = AQ5(~, 0 ) ,
(13)
where 1
Q5(41, 42) = 3
f dt J(41 -
42, 7 t , 7 t ) ,
(14)
0 with 5rt = 5r(¢t) = ~r(t¢ 1 + (1 - t) ¢2). Recall that Q5(41, 42) is invariant under simultaneous gauge transformations of 41 and ¢2. 204
(15)
Volume 156B, number 3,4
PHYSICS LETTERS
20 June 1985
Q5(¢, 0) expands in powers of w as
Q5(dp, O)=QO + QI + Q2 + Q3 + Q4 + Q5,
(16)
where the upper index denotes the power of ~o ,2 and the lower index denotes the form degree in superspace. Expanding out eq. (13) we get a hierarchy of identities of which we shall need sQ°+ d Q l = 0 ,
sQl+ d Q 2 = 0 ,
(17a, b)
which in superspace read: (1/5!) EA EBECEDEE(sQ0 EDCBA+ 5DE Q1DCBA -- I OTEDFQ1FCBA) = O,
(18a)
2 (1/4!) EAEBECEDts~I k ~Z4DCBA+ 4DDQ2CBA + 6TDCFQ3FBA) = O,
(18b)
featuring the constant torsion of rigid superspace. We shall now split the form degree p into a multidegree:
p-rklm,
k+l+m=p,
(19)
where k is the number of vector indices, I the number of undotted spinor indices and m the number of dotted spinor indices. Of eq. (18b) we keep a .~2a= sQloo+ a~g300 O, (20) where Qloo = (1/4!) edcba'~l~d400dcba = edcbatr[Co~d(~cab ~Oa-- ~Oc~gb~Oa)],
2d
dcba-.2
Q300 = (1/3!) e
(21)
~d300cba •
We furthermore extract Qloo by repeated use of eq. (18a) and of the explicit form of the torsion coefficients: 16iQ~o0 _ D 2 Q 1 0 2 _ ~2(Q120 - ~a.I D ~Q121 1 o.) + . . . .
(22)
where from now on the dots denote gauge variations and total derivatives and 0102 = - - ~ll-. e o-ha,) ~ d..d 2 0 2 ~qha,
Q120 = --~(eoba)6"rQ120,.rba,
Q1218 =g( 1 eoa~ 1 ) gQ121n-r a.
(23)
Depending on % Q102 is chiral,
QI02 = -~s(~o, ~2~, ~2¢o3,
(24)
and the second term in eq. (22) can be shown to be
~)2(Q12011D6Q~216)=~2QT220(~) + zID 1 . - 2 D 2{9111 1 + ...,
(25)
with Q120(~p) = _3__~j(~ ' O2~&, D2~5 ,
0~11 = - - ; (1e e ) a "r. 1 30111.r a.
(26,27)
0~11 -fa will be defined in a short while and will be split as O l l l = I~111- X~11,
(28)
,2 When~o,to are used we put a bar on the upper index. 205
Volume 156B, number 3,4
PHYSICS LETTERS
20 June 1985
with ,3 XII1 = ~iJ(w-l~w,~oe~,D2hoo)-~iJ(W¢olll-l,e&,O2~-a)+¼J((oo, X~), ~oCX&,X.o)-~J(w,X'
. o,a, (Xe~,,X'o))
- ~+((~o,x ~ , x., x ~ - ¼+((~,~ , +=a, ~oJ + ~+(~, 2r°a, ( ~ , x.)) + ~s((~, 2°a), x~, xoJ,
(29)
and 1
rhl ---~ f
[J(W]-1 dt Wt, ( a t , Xta), (.It'?, X t ~)) + J ( W t 1d t W,,
(at, X~), (X~',Xt oa))
o
+ J ( W t 1dt Wt, (~2t, Xt ~), (Xf ~,Xt&))] •
(30)
These expressions are obtained from the identity QI(~) _ Qi(~p) = d®1 + sO 0 •
(31)
O111 is related to O~ as usual, cf. eq. (19). Eq. (31) is the term linear in the ghosts of the splitting [9] Q5(¢), 0) - 05(~, 0) = --AX4(~, - w - l A w , 0) + QS(-W-1A I¢, 0)
(32)
with X4( , , ) the triangular function 1
l-t1
X4(¢,-W-1AW, O)=6 f dtl f 0
dt2d($,-W-1Aw,~r(tlq)-t2w-lAw))
(33)
0
and I
Q5(- I¢-IA W, 0) = AI~ ,
I ~= ~ f J(wf-ldtwt, ( w t l A w t ) 2, ( w t l A w t ) 2 ) ,
(34)
0 where F is the Wess-Zumino-Witten action and
Wt = e tv •
(35)
Use has been made of eq. (10) and of the invariance of Qs( , ) under simultaneous gauge transformations of both arguments. The form of (91 [cf. eq. (28)] follows: 1
1
1
03 = --X3 + P3.
(36)
To sum up we may rewrite 64Q~00 = D2D2 s~ + ...,
(37)
¢3 We make use of the def'mitions: X.~---W-ldlt] ,
=~_=-Wdl¥ - 1 '
1-~=-I¥-1sI¥,
Xt__~-lyFldlyt,,
~f-=--ll/F1slYt,
Wt=_e t V .
In addition, for a A (respectively cB) a p (respectively q) form with supers'pace indices A (respectively B) the graded commutation relation is defined as: ( a A cB) ~ a A r B _ ( _ ) p . q ( _ ) a b c B a A "
206
Volume 156B, number 3,4
PHYSICS LETTERS
20 June 1985
: sLij(6o, ~pa, ~2¢a) _ LsiJ(W6ow - l , ~,~, D2~-d) _ ~J(co, X a~, (Xt~, X6)) - ~-J((6o,Xa6), Xa, X&) + ¼J((6o, X~), ~&, X'a) - ~iJ(~, t~, D2~--~ + ~ i J ( w - l ~ w , ~a, ~2tpo~) + ~ j ( ~ , ~,,*&, (-'~&,-~tz)) + t8J((~, X~a), Xa, X~) - ¼J((~, X'o), ~ ,
Xo) - F I l l ,
(38)
with I~11 given by eq. (30). Finally, the supersymmetric anomaly is given by
=fd4xd40 ~ (t.o, ~, e V -
1),
(39)
where s~ is given in terms of superfields.
4. Concluding remarks. We have verified directly that the final formula fulfills the Wess-Zumino consistency condition s 93 = 0 .
(40)
The proof that 93 is non-trivial is a consequence of the supersymmetric version of the usual argument: Q1 cannot be written as Q1 = sQ° + dQ 1 .
(41)
Indeed, by the supersymmetric version of the algebraic Poincar6 lemma, it would follow that Q5 is of the form
Q5=sQ4.
(42)
But this is tantamount to say that J(co, (co, co), (¢o, 6o)) is a trivial cohomology class of Lie ~. We recall that from the anomaly we can construct the corresponding Wess-Zumino lagrangian 1
r(g, ~, V)=
-fd4x d40 f at m ( s t 1at st, s t
ldt St, ~ t l e v~, t - 1),
(43)
0 with S t = e t~ .
(44)
Unfortunately at the moment direct comparisons of the present result with those of refs. [8,5] look prohibitive ly complex but we are guaranteed by the work of Piguet and Sibold [1] that we are describing the chiral anomaly. (Details about the present derivation as well as a further analysis of the result will be found in an article in preparation.) This work was motivated by the recent article by L. Bonora, P. Pasti and M. Tonin. We warmly thank these authors for communicating their work prior to publication. We have also benefitted from abundant discussions with S. Ferrara, L. Girardello, O. Piguet and J. Wess.
References [1] O. Piguet and K. Sibold, NucLPhys. B247 (1984) 484. [2] C. Becchi,A. Rouet and R. Stora in: Field theory quantizalaon and statistical physics, ed. E. Tirapegui(Reidel, Dordrecht, 1981). [3 ] J. Wessand J. Bagger,Supersymmetry and supergravity,Princeton Series in Physics (Princeton U.P., Princeton, 1983). [4] N.K. Nielsen, NucLPhys. B244 (1984) 499. [5] E. Guadagnini, K. Konishi and M. Mintchev,Pisa preprint IFUP TH-10/85 (February 1985). 207
Volume 156B, number 3,4
PHYSICS LETTERS
20 June 1985
[6] J. Wess, private communication. [7] S. Ferrara, L. Girardello, O. Piguet and R. Stora, CERN-TH. 4134/85, LAPP-TH-131, to be published. [8] L. Bonota, P. Past1 and M. Tonin, Phys Lett. 156B, No 5,6 (1985). [9] J. Manes, R. Stora and B. Zumino, in preparation; lg Zumino in: Relativity, groups and topology II, eds. B.S. De Witt and R. Stora (North-Holland, Amsterdam, 1984); 1L Stora, in: Recent progress in gauge theories, eds. H. Lehmann et ah, NATO ASI Series B, Physics, Vol. 115 (Plenum, New York, 1984). [10] L. Baulieu and J. Thierry-Mieg, Phys. Let*. 145B (1984) 53; F. Langouche, T. Schiicker and R. Stora, Phys. Let*. 145B (1984) 342. [11] O. Piguet, M. Schweda and K. Sibold, NucL Phys. B174 (1980) 183.
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