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The chiral anomaly in supersymmetric gauge theories coupled to supergravity
Volume 167B, number 2
PHYSICS LETTERS
6 February 1986
T H E CHIRAL ANOMALY IN S U P E R S Y M M E T R I C GAUGE T H E O R I E S C O U P L E D T O SUPERGRAVITY L. B O N O R A ,
P. PASTI and M. T O N I N
Dipartimento di Fisica "G. Galilei'" Via Marzolo, 8, 1-35131 Padua, Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Padova, 1-35100 Padua, Italy
Received 7 August 1985 We extend a previouslydeveloped algebraic-cohomologicalmethod to calculate an explicit form of the consistent chiral anomaly in a SYM theory coupled to N = 1 supergravity.
The structure of the consistent chiral anomaly in supersymmetric Yang-Mills theories (SYMT's) has recently been clarified [ 1 - 7 ] . In particular in ref. [3] (I, in the following) a non-trivial solution of the coupled cohomological problem for chiral gauge symmetry and rigid supersymmetry has been found. The result is the local functional A 0 = A0G + A0 where A O, A0 are one-cocycles of the coboundary operators ~;O. ~S0 associated to gauge symmetry and rigid supersymmetry, respectively. From the theorem of ref. [8],it follows that A 0 is a coboundary of Z0 i.e. A 0 = Z0C for some local functional C. Then, by subtracting the counterterm ~C, from the effective action, one gets a supersymmetric invariant anomaly ~O = A0G _ zOC.The counterterm C and the ex~h~cit formula for AO have been found in ref. [5]. A O is non-polynomial in the components of the superconnection according to ref. [6]. Formulae for the supersymmetric invariant chiral anomaly have been obtained also in ref. [2] through explicit calculations of the regularized fermion determinant and in ref. [4] using algebraic-cohomological methods. The anomalies found in refs. [2-5] must be equivalent (i.e. they belong to the same cohomological class) because of the uniqueness theorem, proved in ref. [1], for chiral anomalies in SYMT's. In this letter we shall extend the method of I to get a non-trivial solution of the coupled cohomology associated to chiral gauge symmetry and local supersymmetry (supergauge transformations). In this way we shall obtain an explicit formula for the chiral anomaly in SYMT's coupled to N = 1 supergravity. 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
As usual, superspace coordinates are denoted by
zM =--(xm,ou-) , ~_=0z,/J),
#,/)---1,2,
m=0,1,2,3.
Latin, Greek and Capital letters describe spacetime, spinor-like, and both kinds of indices, respectively. Letters from the middle (the beginning) of the alphabet are reserved to curved (tangent) superspace. In the space-time sector of the tangent superspace, the metric is the Minkowski one and in the spinor sector, is the tensor e-~. The NW-SE summation convention is adopted for Greek indices. Supergravity is described by the vielbeins e A (z) = dzMeMA (z) and the Lorentz connection
wAB(z ) = dzM wMAB(z) = eC(z)6ocAB(z) , together with the torsion constraints T~_~7= ioa_~Z,
T~bc = Tg 2 = T_~ = 0 ,
where the torsion is
TA (z) = dseA (z) + eB(z)6oBA (z) = eB(z)eC(z)TAB(z) d s is the total differential in the superspace and O~BC -------~ '
o t3~ a = ~a,
= 0 otherwise, qs the rigid torsion. SYMT's are described by the Gvalued superconnection ~ z ) = e A (z)¢ A (z) where the curvature 5r = ds¢ + ~ is constrained to ~rat3 = 0. G is the Lie algebra of the gauge group. 191
Volume 167B, number 2
PHYSICS LETTERS
As in I, any n-superform Xn(Z) can be decomposed as
The BRST transformations associated to chiral gauge symmetry are 6G~b=--dsc- [q~,c]+,
Xn(Z) =
6GC=-CC,
where c(z) is the G-valued anticommuting FP ghost. eA (z) and COBA (z) are of course inert under 6G" We shall call ]~G the nilpotent generator which represents these transformations in the space of the functionals of superfields. Supergauge transformations [9,10] are a special composition of superdiffeomorphisms and fielddependent Lorentz transformations in the tangent superspace. Their associated BRST transformations are
~
p+q=n
Xp,q(a),
where X p , q ( g ) represents a superform proportional to p Latin indexed vielbeins, ea, and q Greek indexed vielbeins, e -~. Let us call F and G the superforms Sr2 ,0 and 5r1,1 in which the curvature 5rcan be decomposed, so that
~=F+G if is the internal product with respect to (M(z) =
~'A (z)eAM(z), where eAM(z ) are the inverse vielbeins. The integration of a four-superform X4 over the space-time manifold c/~ 4 contained in the superspace, is defined by
s¢(z ) = ~A (Z)~A ¢(z), 6SeM A (z) = eMB (z)[~ B~A (z) + ~C(z) TAcB ] ,
f X4(z)=4~ f d 4 x e mlm2m3m4 crY4
6 S COMAB(z) = ~C(Z)RcMA B(z ) ' 6 S~A (z) = -- ~B(z)~C(z)TAB,
6 February 1986
(1)
where ~o(z)is any superfield without indices (such as e(z))where
RCMA B = eMDRcDA B(_I )qC(qD +qM) and RCDA B are the intrinsic components of the cur-
× era1A e m2 B e ma Cem 4D.,ADCBA • In the rigid case, this definition coincides with that given in I. As in I, the starting point is the extended transgression formula
vature
str(9 r, 5r, 7 ) = dsa5 ,
RA B = dscoA B + coA CcocB .
where
The c-/)A are the intrinsic components of the covariant differential D s = eACDA with respect to the intemal Lorentz group. The U1(z) are the FP ghosts associated to supergauge so that ~-(z)(~a(z))are commuting (anticommuting) superfields. E S will denote the generator of the transformations (1) in the space of the functionals of superfields. We remark that ES is nilpotent only modulo field-dependent internal Lorentz transformations with parameters [9,10]
VA B(z) = ½~C(z)(-D(Z)RDcAB(z). Therefore it is nilpotent if it acts on Lorentz invariant functionals. Since only such functionals willbe needed, Y'S can be considered nilpotent for our purposes. Moreover ~ G ~ S + ]~SZG = 0.
192
(2)
1 5 Q5 = 3 f dtstr(q~,~rt,~)= ~ Q P _ p , 0 p=O d s = d s +SG,
$=~+c,
~rt = tds~ + t 2 ~ ,
0 ~< t <~ 1 .
(3)
str denotes the completely symmetric trace of the matrix generators of G in a given representation and Q~_p are 5 - p superforms with ghost number n G = p, defined unambiguously through eq. (3). Since Q5 is invariant under internal Lorentz transformations, d s in eq. (2) can be replaced by 15s = D s + 8 G . Moreover it is convenient to decompose D s as Ds=D+~+T where Trepresents the action of D s on the vielbeins, that is TeA = eBeCTAcB and D = eacDa, ~ = e a ~ a act only on the intrinsic components of the superforms.
Volume 167B, number 2
PHYSICS LETTERS
T in turn can be decomposed as
6 February 1986
+ aai b. ° - d & O 1 =0,
(8)
T-- TI + T2 + T 3 , 6GQ 1 + dsQ 2 = 0 ,
where Tlea(z ) = 2ioa~ea(z)e~(z),
Tle ~- -- o,
T2e~-(z) = T~eb(z)e~-(z),
T2 ea = 0 ,
where £~ = i~ds - dsi~ is the covariant Lie derivative along ~M(z). ~Q1 and 2(i~£~ - ~i~)Q~ are just the supergauge transformations of Q41 and i~Q O, respectively. This can be verified immediately by taking into account eq. (1). By integrating eqs. (7)-(9) o v e r Q?~4one gets
T3e-a(z) = T~c(z)eC(z)eb(z ) , T3ea = O . Another useful operator is S; on any couple of spinor-like vielbeins eae~ it acts as follows:
]~sAs=0,
See@ = ~ea oad~ ,
~ G A s + ~ s A G =0,
str(fir, fir, fir)= I)sX4,1 ,
(4)
ac =c//t4 f ol(z)lo=a=o, as= c/~ f 4 i~Q.°(Z)lo =o =o.
(11)
In conclusion the local functional AT = AG + AS
where X4,1 = ~s[str(G, G, G)]
(5)
is a gauge invafiant superform. Indeed it holds str(F, G, G)= c-/)X4,1,
str(G,G,G)= T1X4,1 .
(D s+ 6G)Q 5 = 0 , 5
-p
P,q p+q=5
Of course X4 1, affects only 00 1, the remaining terms ~5 p ' q a being equal to' Q~ p q,q. Eq. (6) contains many identities, one for each sector with given ghost number. Looking at the identifies in the sectors with ghost number 0, 1,2 one gets:
i~i~dsQ°=O,
i~dsQ 1 +6Gi~QO=o, = o,
or, equivalently
( i ~ -- ~i~)Q 0 + dsi~i~Q0 = O,
=
01 Q4-q ,q
,
(13)
4
where
?0e
is a non-trivial cocycle for the coupled cohomology associated with chiral gauge and supergauge transformations. Qa1 is proportional to the chiral gauge ghost e(z) and i 0 0 to the supergauge ghosts ~A (z) Moreover they can be decomposed as follows:
Ql (6)
= =
(12)
4
Moreover 6GX4,1 = 0 since 7,4,1 is gauge invariant and DX4,1, T2X4,1 and T3X4,1 vanish, being fiveforms or six-forms in 9g 4. Then eq. (3) becomes
Qs-Qs-)(4,1
]~GAG= 0 , (10)
where
it gives 0 otherwise and anticommutes with eA . The remarkable fact about eq. (2) is that its LHS can also be written as
+ dsO
(9)
(7)
i~55 = q~=O(i~O0) 4-q'q "
(14)
All the terms on the RHS of eqs. (13), (14) contribute to AQ and AS, respectively. The leading terms QI,o and (i~Q°)4,0 give contributions which differ from the rigid ones only for the presence, in the integral, of the determinant of the space-time vielbeins ema(X). (Of course, in addition, here the ghosts ~A (x) are fields whereas in the rigid case they are global parameters.) We remark that ~a(x) does not appear in the leading term (i~00)4,0.1 The non-leading terms Q4-q,q and (i~Q°)4 _q,q (q ¢ 0) are new terms which are q-linear in the gravitino fields ema--(x). The anomaly AT in eq. (12) is a polynomial functional in the superfields CA(z) and their derivatives. Through a Bardeen-Zumino-like transformation [11], as used in ref. [6], one can get an equivalent anomaly expressed in terms of the Gvalued real prepotential V(z). 193
Volume 167B, number 2
PHYSICS LETTERS
As already remarked in the rigid case the supersymmetric partner, A 0, of the chiral anomaly is a trivial cocycle of the coboundary operator ~0. In the local case, a theorem analogous to the one of ref. [8] is lacking and the question arises if A S is a coboundary of Z S or not. An interesting possibility is that the answer to this question be negative. In this case the fact that no consistent, supersymmetry preserving regularization has yet been discovered could find its raison d'gtre. The situation would be similar to the one met in ordinary chiral gauge theories. In such a case, in fact, a consistent, chiral preserving regularization does not exist in general; moreover, disregarding global and topological effects, rigid chirality is preserved but local chirality is anomalous in some models. We must remark that A S vanishes if and only if A G vanishes so that A s, even if non-trivial, should be considered as a part of the chiral ABJ anomaly, rather than as a new supergauge anomaly. In any, case, the question if A s is a trivial or a non-trivial cocycle of NS, is still open and deserves further investigations. A final comment is in order concerning superdiffeomorphisms. Eq. (8) implies that also the invariance under superdiffeomorphisms (sdffs) is broken. Therefore it is natural to look for a partner A D of the sdffs such that an overall consistency condition be satisfied. As is easy to see A D coincides exactly with A s provided that we interpret ~M(z) in ~A (7,) = ~M(z) X eMA(z ) as the ghosts of the sdffs [10]. This is the result we would have found had we ignored supergauge transformations and carried out the analysis by considering only sdffs. Of course, since the constraints on T and fir are expressed in terms of flat intrinsic components it is much easier to work with supergauge transformations.
194
6 February 1986
This comment seems to suggest that we could use the results of ref. [10] to conclude that A D must be a coboundary for, there, it was proven that sdffs are anomaly.free. However this conclusion is not permitted because the results of ref. [10] were obtained under the essential proviso that the other classical symmetries of the theory (in particular, gauge invariance) be anomaly-free ,1 ,1 An anomaly of local supersymmetry has been found in two dimensions in ref. [12].
References [1 ] O. Piguet and K. Sibold, Nucl. Phys. B247 (1984) 484. [2] N.K. Nielsen, Nuel. Phys. B244 (1984) 499; E. Guadagnini, K. Konishi and M. Mintchev, Phys. Lett. 157B (1985) 37; M. Pernici and M. Riva, prepdnt ITP-5B-85-10. [3] L. Bonora, P. Pasti and M. Ton.in, Phys. Lett. 156B (1985) 341. [4] G. Girardi, R. Grimm and R. Stora, Phys. Lett. 156B (1985) 203. [5] L. Bonora, P. Pasti and M. Tordn, Nuel. Phys. B261 (1985) 249. [6] S. Ferrara, L. GirardeUo, O. Piguet and R. Stora, Phys. Lett. 157B (1985) 179. [7] H. Itoyama, V.P. Nair and H. Ren, Princeton prepdnt (March 1985). [8] O. Piguet, K. Sibold and M. Schweda, Nucl. Phys. B174 (1980) 183. [9] J. Wess and J. Bagger, Supersymmetry and supergravity (Princeton U.P., Princeton, NJ, 1983). [10] L. Bonora, P. Pasti and M. Tonin, Nucl. Phys. B252 (1985) 458. [11 ] W2k. Bardeen and B. Zumino, Nucl. Phys. B244 (1984) 421. [12] Y. Tanii, Local SUSY anomaly in two dimensions, preprint Tokyo Institute of Technology/HEP]85 (January 1985).
PHYSICS LETTERS
6 February 1986
T H E CHIRAL ANOMALY IN S U P E R S Y M M E T R I C GAUGE T H E O R I E S C O U P L E D T O SUPERGRAVITY L. B O N O R A ,
P. PASTI and M. T O N I N
Dipartimento di Fisica "G. Galilei'" Via Marzolo, 8, 1-35131 Padua, Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Padova, 1-35100 Padua, Italy
Received 7 August 1985 We extend a previouslydeveloped algebraic-cohomologicalmethod to calculate an explicit form of the consistent chiral anomaly in a SYM theory coupled to N = 1 supergravity.
The structure of the consistent chiral anomaly in supersymmetric Yang-Mills theories (SYMT's) has recently been clarified [ 1 - 7 ] . In particular in ref. [3] (I, in the following) a non-trivial solution of the coupled cohomological problem for chiral gauge symmetry and rigid supersymmetry has been found. The result is the local functional A 0 = A0G + A0 where A O, A0 are one-cocycles of the coboundary operators ~;O. ~S0 associated to gauge symmetry and rigid supersymmetry, respectively. From the theorem of ref. [8],it follows that A 0 is a coboundary of Z0 i.e. A 0 = Z0C for some local functional C. Then, by subtracting the counterterm ~C, from the effective action, one gets a supersymmetric invariant anomaly ~O = A0G _ zOC.The counterterm C and the ex~h~cit formula for AO have been found in ref. [5]. A O is non-polynomial in the components of the superconnection according to ref. [6]. Formulae for the supersymmetric invariant chiral anomaly have been obtained also in ref. [2] through explicit calculations of the regularized fermion determinant and in ref. [4] using algebraic-cohomological methods. The anomalies found in refs. [2-5] must be equivalent (i.e. they belong to the same cohomological class) because of the uniqueness theorem, proved in ref. [1], for chiral anomalies in SYMT's. In this letter we shall extend the method of I to get a non-trivial solution of the coupled cohomology associated to chiral gauge symmetry and local supersymmetry (supergauge transformations). In this way we shall obtain an explicit formula for the chiral anomaly in SYMT's coupled to N = 1 supergravity. 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
As usual, superspace coordinates are denoted by
zM =--(xm,ou-) , ~_=0z,/J),
#,/)---1,2,
m=0,1,2,3.
Latin, Greek and Capital letters describe spacetime, spinor-like, and both kinds of indices, respectively. Letters from the middle (the beginning) of the alphabet are reserved to curved (tangent) superspace. In the space-time sector of the tangent superspace, the metric is the Minkowski one and in the spinor sector, is the tensor e-~. The NW-SE summation convention is adopted for Greek indices. Supergravity is described by the vielbeins e A (z) = dzMeMA (z) and the Lorentz connection
wAB(z ) = dzM wMAB(z) = eC(z)6ocAB(z) , together with the torsion constraints T~_~7= ioa_~Z,
T~bc = Tg 2 = T_~ = 0 ,
where the torsion is
TA (z) = dseA (z) + eB(z)6oBA (z) = eB(z)eC(z)TAB(z) d s is the total differential in the superspace and O~BC -------~ '
o t3~ a = ~a,
= 0 otherwise, qs the rigid torsion. SYMT's are described by the Gvalued superconnection ~ z ) = e A (z)¢ A (z) where the curvature 5r = ds¢ + ~ is constrained to ~rat3 = 0. G is the Lie algebra of the gauge group. 191
Volume 167B, number 2
PHYSICS LETTERS
As in I, any n-superform Xn(Z) can be decomposed as
The BRST transformations associated to chiral gauge symmetry are 6G~b=--dsc- [q~,c]+,
Xn(Z) =
6GC=-CC,
where c(z) is the G-valued anticommuting FP ghost. eA (z) and COBA (z) are of course inert under 6G" We shall call ]~G the nilpotent generator which represents these transformations in the space of the functionals of superfields. Supergauge transformations [9,10] are a special composition of superdiffeomorphisms and fielddependent Lorentz transformations in the tangent superspace. Their associated BRST transformations are
~
p+q=n
Xp,q(a),
where X p , q ( g ) represents a superform proportional to p Latin indexed vielbeins, ea, and q Greek indexed vielbeins, e -~. Let us call F and G the superforms Sr2 ,0 and 5r1,1 in which the curvature 5rcan be decomposed, so that
~=F+G if is the internal product with respect to (M(z) =
~'A (z)eAM(z), where eAM(z ) are the inverse vielbeins. The integration of a four-superform X4 over the space-time manifold c/~ 4 contained in the superspace, is defined by
s¢(z ) = ~A (Z)~A ¢(z), 6SeM A (z) = eMB (z)[~ B~A (z) + ~C(z) TAcB ] ,
f X4(z)=4~ f d 4 x e mlm2m3m4 crY4
6 S COMAB(z) = ~C(Z)RcMA B(z ) ' 6 S~A (z) = -- ~B(z)~C(z)TAB,
6 February 1986
(1)
where ~o(z)is any superfield without indices (such as e(z))where
RCMA B = eMDRcDA B(_I )qC(qD +qM) and RCDA B are the intrinsic components of the cur-
× era1A e m2 B e ma Cem 4D.,ADCBA • In the rigid case, this definition coincides with that given in I. As in I, the starting point is the extended transgression formula
vature
str(9 r, 5r, 7 ) = dsa5 ,
RA B = dscoA B + coA CcocB .
where
The c-/)A are the intrinsic components of the covariant differential D s = eACDA with respect to the intemal Lorentz group. The U1(z) are the FP ghosts associated to supergauge so that ~-(z)(~a(z))are commuting (anticommuting) superfields. E S will denote the generator of the transformations (1) in the space of the functionals of superfields. We remark that ES is nilpotent only modulo field-dependent internal Lorentz transformations with parameters [9,10]
VA B(z) = ½~C(z)(-D(Z)RDcAB(z). Therefore it is nilpotent if it acts on Lorentz invariant functionals. Since only such functionals willbe needed, Y'S can be considered nilpotent for our purposes. Moreover ~ G ~ S + ]~SZG = 0.
192
(2)
1 5 Q5 = 3 f dtstr(q~,~rt,~)= ~ Q P _ p , 0 p=O d s = d s +SG,
$=~+c,
~rt = tds~ + t 2 ~ ,
0 ~< t <~ 1 .
(3)
str denotes the completely symmetric trace of the matrix generators of G in a given representation and Q~_p are 5 - p superforms with ghost number n G = p, defined unambiguously through eq. (3). Since Q5 is invariant under internal Lorentz transformations, d s in eq. (2) can be replaced by 15s = D s + 8 G . Moreover it is convenient to decompose D s as Ds=D+~+T where Trepresents the action of D s on the vielbeins, that is TeA = eBeCTAcB and D = eacDa, ~ = e a ~ a act only on the intrinsic components of the superforms.
Volume 167B, number 2
PHYSICS LETTERS
T in turn can be decomposed as
6 February 1986
+ aai b. ° - d & O 1 =0,
(8)
T-- TI + T2 + T 3 , 6GQ 1 + dsQ 2 = 0 ,
where Tlea(z ) = 2ioa~ea(z)e~(z),
Tle ~- -- o,
T2e~-(z) = T~eb(z)e~-(z),
T2 ea = 0 ,
where £~ = i~ds - dsi~ is the covariant Lie derivative along ~M(z). ~Q1 and 2(i~£~ - ~i~)Q~ are just the supergauge transformations of Q41 and i~Q O, respectively. This can be verified immediately by taking into account eq. (1). By integrating eqs. (7)-(9) o v e r Q?~4one gets
T3e-a(z) = T~c(z)eC(z)eb(z ) , T3ea = O . Another useful operator is S; on any couple of spinor-like vielbeins eae~ it acts as follows:
]~sAs=0,
See@ = ~ea oad~ ,
~ G A s + ~ s A G =0,
str(fir, fir, fir)= I)sX4,1 ,
(4)
ac =c//t4 f ol(z)lo=a=o, as= c/~ f 4 i~Q.°(Z)lo =o =o.
(11)
In conclusion the local functional AT = AG + AS
where X4,1 = ~s[str(G, G, G)]
(5)
is a gauge invafiant superform. Indeed it holds str(F, G, G)= c-/)X4,1,
str(G,G,G)= T1X4,1 .
(D s+ 6G)Q 5 = 0 , 5
-p
P,q p+q=5
Of course X4 1, affects only 00 1, the remaining terms ~5 p ' q a being equal to' Q~ p q,q. Eq. (6) contains many identities, one for each sector with given ghost number. Looking at the identifies in the sectors with ghost number 0, 1,2 one gets:
i~i~dsQ°=O,
i~dsQ 1 +6Gi~QO=o, = o,
or, equivalently
( i ~ -- ~i~)Q 0 + dsi~i~Q0 = O,
=
01 Q4-q ,q
,
(13)
4
where
?0e
is a non-trivial cocycle for the coupled cohomology associated with chiral gauge and supergauge transformations. Qa1 is proportional to the chiral gauge ghost e(z) and i 0 0 to the supergauge ghosts ~A (z) Moreover they can be decomposed as follows:
Ql (6)
= =
(12)
4
Moreover 6GX4,1 = 0 since 7,4,1 is gauge invariant and DX4,1, T2X4,1 and T3X4,1 vanish, being fiveforms or six-forms in 9g 4. Then eq. (3) becomes
Qs-Qs-)(4,1
]~GAG= 0 , (10)
where
it gives 0 otherwise and anticommutes with eA . The remarkable fact about eq. (2) is that its LHS can also be written as
+ dsO
(9)
(7)
i~55 = q~=O(i~O0) 4-q'q "
(14)
All the terms on the RHS of eqs. (13), (14) contribute to AQ and AS, respectively. The leading terms QI,o and (i~Q°)4,0 give contributions which differ from the rigid ones only for the presence, in the integral, of the determinant of the space-time vielbeins ema(X). (Of course, in addition, here the ghosts ~A (x) are fields whereas in the rigid case they are global parameters.) We remark that ~a(x) does not appear in the leading term (i~00)4,0.1 The non-leading terms Q4-q,q and (i~Q°)4 _q,q (q ¢ 0) are new terms which are q-linear in the gravitino fields ema--(x). The anomaly AT in eq. (12) is a polynomial functional in the superfields CA(z) and their derivatives. Through a Bardeen-Zumino-like transformation [11], as used in ref. [6], one can get an equivalent anomaly expressed in terms of the Gvalued real prepotential V(z). 193
Volume 167B, number 2
PHYSICS LETTERS
As already remarked in the rigid case the supersymmetric partner, A 0, of the chiral anomaly is a trivial cocycle of the coboundary operator ~0. In the local case, a theorem analogous to the one of ref. [8] is lacking and the question arises if A S is a coboundary of Z S or not. An interesting possibility is that the answer to this question be negative. In this case the fact that no consistent, supersymmetry preserving regularization has yet been discovered could find its raison d'gtre. The situation would be similar to the one met in ordinary chiral gauge theories. In such a case, in fact, a consistent, chiral preserving regularization does not exist in general; moreover, disregarding global and topological effects, rigid chirality is preserved but local chirality is anomalous in some models. We must remark that A S vanishes if and only if A G vanishes so that A s, even if non-trivial, should be considered as a part of the chiral ABJ anomaly, rather than as a new supergauge anomaly. In any, case, the question if A s is a trivial or a non-trivial cocycle of NS, is still open and deserves further investigations. A final comment is in order concerning superdiffeomorphisms. Eq. (8) implies that also the invariance under superdiffeomorphisms (sdffs) is broken. Therefore it is natural to look for a partner A D of the sdffs such that an overall consistency condition be satisfied. As is easy to see A D coincides exactly with A s provided that we interpret ~M(z) in ~A (7,) = ~M(z) X eMA(z ) as the ghosts of the sdffs [10]. This is the result we would have found had we ignored supergauge transformations and carried out the analysis by considering only sdffs. Of course, since the constraints on T and fir are expressed in terms of flat intrinsic components it is much easier to work with supergauge transformations.
194
6 February 1986
This comment seems to suggest that we could use the results of ref. [10] to conclude that A D must be a coboundary for, there, it was proven that sdffs are anomaly.free. However this conclusion is not permitted because the results of ref. [10] were obtained under the essential proviso that the other classical symmetries of the theory (in particular, gauge invariance) be anomaly-free ,1 ,1 An anomaly of local supersymmetry has been found in two dimensions in ref. [12].
References [1 ] O. Piguet and K. Sibold, Nucl. Phys. B247 (1984) 484. [2] N.K. Nielsen, Nuel. Phys. B244 (1984) 499; E. Guadagnini, K. Konishi and M. Mintchev, Phys. Lett. 157B (1985) 37; M. Pernici and M. Riva, prepdnt ITP-5B-85-10. [3] L. Bonora, P. Pasti and M. Ton.in, Phys. Lett. 156B (1985) 341. [4] G. Girardi, R. Grimm and R. Stora, Phys. Lett. 156B (1985) 203. [5] L. Bonora, P. Pasti and M. Tordn, Nuel. Phys. B261 (1985) 249. [6] S. Ferrara, L. GirardeUo, O. Piguet and R. Stora, Phys. Lett. 157B (1985) 179. [7] H. Itoyama, V.P. Nair and H. Ren, Princeton prepdnt (March 1985). [8] O. Piguet, K. Sibold and M. Schweda, Nucl. Phys. B174 (1980) 183. [9] J. Wess and J. Bagger, Supersymmetry and supergravity (Princeton U.P., Princeton, NJ, 1983). [10] L. Bonora, P. Pasti and M. Tonin, Nucl. Phys. B252 (1985) 458. [11 ] W2k. Bardeen and B. Zumino, Nucl. Phys. B244 (1984) 421. [12] Y. Tanii, Local SUSY anomaly in two dimensions, preprint Tokyo Institute of Technology/HEP]85 (January 1985).