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Gauge-fixing redundant symmetries in the superparticle
Volume 228, number 1
PHYSICS LETTERS B
7 September 1989
GAUGE-FIXING REDUNDANT SYMMETRIES IN THE SUPERPARTICLE U. LINDSTRC)M Institute of Theoretical Physics, University of Stockholm, Vanadisva'gen 9, S- 11346 Stockholm, Sweden M. RO(~EK i, W. S I E G E L J, P. VAN N I E U W E N H U I Z E N 1 Institute for Theoretical Physics, State University of New York, Stony Brook, NY 11794-3840, USA and A.E. VAN DE V E N 2 Department of Physics, New York University, New York, NY 10003, USA Received 27 June 1989
We consider the Batalin-Vilkovisky quantization of the superparticle including the redundant "third" symmetry that is important for the quantization of the type II superstring. We find that the third symmetry can be gauge-fixed consistently.
Recently, we presented a new covariant a p p r o a c h to the q u a n t i z a t i o n o f the superparticle [ 1,2 ] a n d the heterotic superstring [ 3 - 5 ] based on the B a t a l i n Vilkovisky q u a n t i z a t i o n procedure [ 6 ] and a novel covariant gauge choice ~1. Similar results, following our a p p r o a c h as presented in refs. [1,3], have appeared in refs. [ 9,10] (the BV procedure had previously been applied to the heterotic superstring in refs. [ 11,12 ], but without gauge fixing, a n d with some errors). O u r a p p r o a c h has not yet been applied to type I and II superstrings because the gauge algebra o f symmetries has a m o r e c o m p l i c a t e d structure: It involves a " n e w " s y m m e t r y [13] that can be interpreted as a x - s y m m e t r y with a nonlocal, field dependent parameter. ( P a r t i a l results, including the complete classical gauge algebra, have been given in ref. [ 14 ] ; a less complete discussion o f the possible structure o f such results has been given in ref. [ 15 ]. ) Here we perform a q u a n t i z a t i o n o f the superparticle Work supported in part by NSF grant No. PHY 85-07627. 2 Work supported in part by NSF grant No. PHY 87-15995. ~ A subtle problem in our approach has been pointed out in refs. [ 7,8 ]. It is unclear at the present how serious this is. 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )
where we include the " n e w " symmetry; it need not be included for the superparticle, as it does not arise in the algebra o f x-symmetry and world-line diffeomorphisms, but it is a s y m m e t r y and m a y be included. We find that our final results agree with ref. [ 1 ], but along the way we uncover a lot o f the structure that will be needed for the superstring. As in ref. [ 1 ], we use the first-order superparticle action: S o = f d z [ p ( J c - O T O ) - ½ g p 2] ,
(1)
which gives rise to the following field equations:
p=0, ~b=0, p2=0, g p - Jc+ OyO=O .
(2)
The action ( 1 ) is invariant under the following diff e o m o r p h i s m s and local x-symmetry: 60=0,
60=/kx,
,~x=@, ,~x=O~,,~o, ~p=O, ~p=O, 5g=~,
~g=40K.
(3) 53
Volume 228, number 1
PHYSICS LETTERS B
As in ref. [ 1 ], we use modified diffeomorphisms. In addition, the action is invariant under the following "third" symmetry [ 13,1 ]:
60=~0, 6x=--(dt~)70,
6p=dg=O.
(4)
This symmetry is not independent, since, modulo field equations, it can expressed as a particular Ktransformation with a nonlocal field dependent parameter:
0= ~
1
[~(0O)+~(~O)]
7 September 1989
ASI (higher levels) = - ~ c7 dir/. i=2
(9)
Note that (7) contains both the i = 1 and i = 0 terms of this series. Following the BV procedure, we now find $2, i.e., the terms quadratic in the antifields. We begin with (S~, S t ) + 2 ( S o , $2)=0, which determines all the terms in $2 that contain at least one classical antifield. This gives
(5) S 2 = 2g* [ x* (el yc~+ flee2) + 0*c2
for arbitrary n. The commutator algebra of the diffeomorphisms and local K-symmetry closes (on-shell) without the third symmetry, and is given in our notation in ref. [ 1 ]; the commutator algebra of the third symmetry with itself and the other symmetries is
[ G ( x ) , 6 : ( 0 ] = G ( ~ ) +6¢( - 4~b-x), [6:(~), 6~(~) ] = 0 .
+2Z*0*c2q+x*[ ~qb*ct +Z* (4dr yct + 2[~/C2)
+ O*yc, + i=, ~" c*yci+ l lrl "
(10)
(6)
As in ref. [ 1 ], we quantize using the BV formalism [6]. The leading terms merely express the classical algebra s, = x* [ p z + & ( ~ c , - 0~) ] + o* [tkc, - O~]
+g* [40c, +21 +Z* [2(~ (/~c, -2Ot/) l
+q* [r///] --c1'[d~ . ] ,
(7)
where the ghosts Z, c~, q correspond to the parameters ~, x, ~ of diffeomorphisms, x-symmetry and the third symmetry, respectively. Let us for the moment ignore the last term in St; then the remaining terms in Sj are invariant under transformations of the x-ghost 6Cl =~x2 modulo the classical field equations. This would imply that the system has ghosts-for-ghosts; indeed, there is an infinite tower of such ghosts, and at this level they can all be incorporated by adding S, (higher levels) = ~ c*~ci+ l . i=1
(8)
Note that the O*I~c~term in (7) effectively extends this sum to i=0. If we now consider the last term in S~, we can cancel its variation by adding a second infinite sum: 54
+4)(*G c2 + i=, ~ c*ci+21
We determine the rest of $2 by requiring that (S], $2) = 0. After some calculation, we find S~xtra=2)~ * ~ i=1
C*Ci+2t].
(1 1)
A straightforward calculation shows that ($2, $2) = 0. This completes the determination of the minimal solution to the master equation in the BV formalism. For convenience, we collect all the terms below (i.e., (1), ( 7 ) - ( 1 1 ) ) :
S=p( dc- 070) - ½gp2+ x*px + g* (2- 4e~ O) "Jl-~*~0--N)t"lcetj~Cl "Jr ~, C~(~Ci+ 1 --Ci~]) i=0
+2(g"l:--~?I)fi~=oC~Ci+a+X~f1ft~Ct + X * q [ ,=o~C*yC'+'--2Z*gtYC'I'
(12,
where c~ = 0* + x * & + 4 ~ e , . To be able to gauge-fix, we need to add nonminimal terms to the action (12). Those which do not involve the third symmetry take the form [ 1 ]
Volume 228, number 1
S . . . . in.=2*rr¢+ £
PHYSICS LETTERS B
£ cS,*zd,:,
(13)
i=1 j=l
where c o = ci and Co o = 0, and the remaining cS,. and all the rri, as well as 2 and he, are nonminimal. Since the third symmetry is not really an independent symmetry, it is not clear if we should gauge-fix it; here we consider both options, and see that it makes no basic difference. A nonminimal term for the third symmetry is (14)
S~onmin" = ~Tgq .
Finally, we eliminate the antifields by introducing a gauge fermion ~t, and imposing q~*= 0¢t/00. The gauge fermion o f ref. [ 1 ] is ~t=2(g-1)-t- £
£ e}+~+'b~.
(15)
i=Oj=O
Again, we consider a term for the third symmetry: ~u"=t/i t/.
(16)
If we do not fix the third symmetry (i.e., without (14) and ( 1 6 ) ) , we find the following results for the antifields: 2"= (g-1),
g*=2,
ci* = - di+~+~ - (~-~
(17)
(all others zero), and the following gauge-fixed action: S~, = p ( 2 - 070) - ½gp2,l,+2(2_ 4g, O)
- E ~i++I(~c,+,-6.) i=0
--
22 £
C~+~ i+ i Ci+2
+(g-1)~
i=o
i=lj=l
This appears to depend on the third symmetry ghost q, but such dependence can be eliminated by the shift / ~J ni--rci( - ) / [ ( 1 - a ) ? ° 2 j + ( - j~iOl~'2j--i]~ t.2j_i.lq,
This arbitrariness is a reflection o f one described in ref. [8].
a
nated from ( 18 ), we see that there really is a residual gauge invariance; however, as it is algebraic, it is harmless. Alternatively, if we include the terms for the third symmetry (14) and (16), we find additional terms
&,+~,~O+~n..
(20)
In this result, the third symmetry ghost and antighost are essentially an auxiliary system; after we eliminate them by their field equation, we again reproduce the result ofref. [ 1 ]. There is a slight subtlety: the action (20) is still i nvariant under shifts of the antighost q~ (equivalently, t/~ can be eliminated by the shift ~z,= ~, -qT 0)- This is of course a less problematic algebraic symmetry, as, unlike (19), it does not involve an infinite number of terms. However, we have been unable to eliminate this further invariance, as adding further generations of ghosts always leads to a new, trivial, algebraic invariance. Indeed, the situation is analogous to x-symmetry, with the significant simplification that we are dealing with algebraic symmetries. We have found that the third symmetry should be gauge-fixed, though if it is not, it gives rise only to a harmless algebraic residual gauge freedom. However, in the superparticle, the third symmetry is not essential to close the algebra of diffeomorphisms and xsymmetry. In the superstring, it may well be quite important to treat the third symmetry correctly. In ref. [ 14 ], it is claimed that the third symmetry is auxiliary after gauge fixing as we found above; in ref. [ 15 ], it is claimed that the third symmetry should not be gauge-fixed. Since neither reference finds the complete BV action, it remains unclear what the correct procedure is. Our results should illuminate the basic structure of the complete action, and suggest that it is always correct to gauge-fix any symmetry in the action, even a dependent one. M.R. is happy to thank the ITP at the University o f Stockholm for its hospitality.
(19)
which leads to the same result as in ref. [ I ]. Here a is an arbitrary parameter ,2. Since r/ can be elimi~2
7 September 1989
symmetry analogous to
References
[ 1] u. Lindstr6m, M. Ro6ek, W. Siegel, P. van Nieuwenhuizen and A.E. van de Ven, Phys. Lett. B 224 (1989) 285. 55
Volume 228, number 1
PHYSICS LETTERS B
[2] M. Ro~ek, W. Siegel, P. van Nieuwenhuizen and A.E. van de Ven, Phys. Lett. B 227 (1989) 87. [3] W. Siegel, Lorentz-covariant gauges for Green-Schwarz superstrings, talk presented at Strings '89 (College Station, TX, March 1989), Stony Brook preprint ITP-SB-89-19 (March 1989). [4] S.J. Gates, M.T. Grisaru, U. Lindstr6m, M. Ro~ek, W. Siegel, P. van Nieuwenhuizen and A.E. van de Ven, Phys. Lett. B 225 (1989) 44. [ 5 ] U. Lindstrtim, M. Ro~ek, W. Siegel, P. van Nieuwenhuizen and A.E. van de Ven, Construction of the covariantly quantized heterotic superstring, Stony Brook preprint ITPSB-89-38 (May 1989). [6] I,A. Batalin and G.A. Vilkovisky, Phys. Lett. B 102 (1983) 27; B 120 (1983) 166; Phys. Rev. D 28 (1983) 2567; D 30 (1984) 508; Nucl. Phys. B 234 (1984) 106; J. Math. Phys. 26 (1985) 172.
56
7 September 1989
[7] F. Bastianelli, G.W. Delius and E. Laenen, in preparation. [8] J.M.L. Fisch and M. Henneaux, A note on the covariant BRST quantization of the superparticle, Universit6 Libre de Bruxelles preprint ULB TH2/89-04. [9] M.B. Green and C.M. Hull, Phys. Lett. B 225 (1989) 57. [ 10] R.E. Kallosh, Phys. Lett. B 224 (1989) 273. [ 11 ] R.E. Kallosh, Phys. Lett. B 195 (1987) 369. [ 12] A.H. Diaz and F. Toppan, Phys. Lett. B 211 (1988) 285. [ 13] W. Siegel, Phys. Lett. B 128 (1983) 397; M.B. Green and J.H. Schwarz, Phys. Lett. B 136 (1984) 367. [ 14] D.A. Depireux, S.J. Gates, P. Majumdar, B. Radak and S. Vashakidze, Towards covariantly quantized D = 10 type-II Green-Schwarz a-models, Maryland preprint UMDEPP 89169 (May 1989). [15] R.E. Kallosh, Phys. Lett. B 225 (1989) 49.
PHYSICS LETTERS B
7 September 1989
GAUGE-FIXING REDUNDANT SYMMETRIES IN THE SUPERPARTICLE U. LINDSTRC)M Institute of Theoretical Physics, University of Stockholm, Vanadisva'gen 9, S- 11346 Stockholm, Sweden M. RO(~EK i, W. S I E G E L J, P. VAN N I E U W E N H U I Z E N 1 Institute for Theoretical Physics, State University of New York, Stony Brook, NY 11794-3840, USA and A.E. VAN DE V E N 2 Department of Physics, New York University, New York, NY 10003, USA Received 27 June 1989
We consider the Batalin-Vilkovisky quantization of the superparticle including the redundant "third" symmetry that is important for the quantization of the type II superstring. We find that the third symmetry can be gauge-fixed consistently.
Recently, we presented a new covariant a p p r o a c h to the q u a n t i z a t i o n o f the superparticle [ 1,2 ] a n d the heterotic superstring [ 3 - 5 ] based on the B a t a l i n Vilkovisky q u a n t i z a t i o n procedure [ 6 ] and a novel covariant gauge choice ~1. Similar results, following our a p p r o a c h as presented in refs. [1,3], have appeared in refs. [ 9,10] (the BV procedure had previously been applied to the heterotic superstring in refs. [ 11,12 ], but without gauge fixing, a n d with some errors). O u r a p p r o a c h has not yet been applied to type I and II superstrings because the gauge algebra o f symmetries has a m o r e c o m p l i c a t e d structure: It involves a " n e w " s y m m e t r y [13] that can be interpreted as a x - s y m m e t r y with a nonlocal, field dependent parameter. ( P a r t i a l results, including the complete classical gauge algebra, have been given in ref. [ 14 ] ; a less complete discussion o f the possible structure o f such results has been given in ref. [ 15 ]. ) Here we perform a q u a n t i z a t i o n o f the superparticle Work supported in part by NSF grant No. PHY 85-07627. 2 Work supported in part by NSF grant No. PHY 87-15995. ~ A subtle problem in our approach has been pointed out in refs. [ 7,8 ]. It is unclear at the present how serious this is. 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )
where we include the " n e w " symmetry; it need not be included for the superparticle, as it does not arise in the algebra o f x-symmetry and world-line diffeomorphisms, but it is a s y m m e t r y and m a y be included. We find that our final results agree with ref. [ 1 ], but along the way we uncover a lot o f the structure that will be needed for the superstring. As in ref. [ 1 ], we use the first-order superparticle action: S o = f d z [ p ( J c - O T O ) - ½ g p 2] ,
(1)
which gives rise to the following field equations:
p=0, ~b=0, p2=0, g p - Jc+ OyO=O .
(2)
The action ( 1 ) is invariant under the following diff e o m o r p h i s m s and local x-symmetry: 60=0,
60=/kx,
,~x=@, ,~x=O~,,~o, ~p=O, ~p=O, 5g=~,
~g=40K.
(3) 53
Volume 228, number 1
PHYSICS LETTERS B
As in ref. [ 1 ], we use modified diffeomorphisms. In addition, the action is invariant under the following "third" symmetry [ 13,1 ]:
60=~0, 6x=--(dt~)70,
6p=dg=O.
(4)
This symmetry is not independent, since, modulo field equations, it can expressed as a particular Ktransformation with a nonlocal field dependent parameter:
0= ~
1
[~(0O)+~(~O)]
7 September 1989
ASI (higher levels) = - ~ c7 dir/. i=2
(9)
Note that (7) contains both the i = 1 and i = 0 terms of this series. Following the BV procedure, we now find $2, i.e., the terms quadratic in the antifields. We begin with (S~, S t ) + 2 ( S o , $2)=0, which determines all the terms in $2 that contain at least one classical antifield. This gives
(5) S 2 = 2g* [ x* (el yc~+ flee2) + 0*c2
for arbitrary n. The commutator algebra of the diffeomorphisms and local K-symmetry closes (on-shell) without the third symmetry, and is given in our notation in ref. [ 1 ]; the commutator algebra of the third symmetry with itself and the other symmetries is
[ G ( x ) , 6 : ( 0 ] = G ( ~ ) +6¢( - 4~b-x), [6:(~), 6~(~) ] = 0 .
+2Z*0*c2q+x*[ ~qb*ct +Z* (4dr yct + 2[~/C2)
+ O*yc, + i=, ~" c*yci+ l lrl "
(10)
(6)
As in ref. [ 1 ], we quantize using the BV formalism [6]. The leading terms merely express the classical algebra s, = x* [ p z + & ( ~ c , - 0~) ] + o* [tkc, - O~]
+g* [40c, +21 +Z* [2(~ (/~c, -2Ot/) l
+q* [r///] --c1'[d~ . ] ,
(7)
where the ghosts Z, c~, q correspond to the parameters ~, x, ~ of diffeomorphisms, x-symmetry and the third symmetry, respectively. Let us for the moment ignore the last term in St; then the remaining terms in Sj are invariant under transformations of the x-ghost 6Cl =~x2 modulo the classical field equations. This would imply that the system has ghosts-for-ghosts; indeed, there is an infinite tower of such ghosts, and at this level they can all be incorporated by adding S, (higher levels) = ~ c*~ci+ l . i=1
(8)
Note that the O*I~c~term in (7) effectively extends this sum to i=0. If we now consider the last term in S~, we can cancel its variation by adding a second infinite sum: 54
+4)(*G c2 + i=, ~ c*ci+21
We determine the rest of $2 by requiring that (S], $2) = 0. After some calculation, we find S~xtra=2)~ * ~ i=1
C*Ci+2t].
(1 1)
A straightforward calculation shows that ($2, $2) = 0. This completes the determination of the minimal solution to the master equation in the BV formalism. For convenience, we collect all the terms below (i.e., (1), ( 7 ) - ( 1 1 ) ) :
S=p( dc- 070) - ½gp2+ x*px + g* (2- 4e~ O) "Jl-~*~0--N)t"lcetj~Cl "Jr ~, C~(~Ci+ 1 --Ci~]) i=0
+2(g"l:--~?I)fi~=oC~Ci+a+X~f1ft~Ct + X * q [ ,=o~C*yC'+'--2Z*gtYC'I'
(12,
where c~ = 0* + x * & + 4 ~ e , . To be able to gauge-fix, we need to add nonminimal terms to the action (12). Those which do not involve the third symmetry take the form [ 1 ]
Volume 228, number 1
S . . . . in.=2*rr¢+ £
PHYSICS LETTERS B
£ cS,*zd,:,
(13)
i=1 j=l
where c o = ci and Co o = 0, and the remaining cS,. and all the rri, as well as 2 and he, are nonminimal. Since the third symmetry is not really an independent symmetry, it is not clear if we should gauge-fix it; here we consider both options, and see that it makes no basic difference. A nonminimal term for the third symmetry is (14)
S~onmin" = ~Tgq .
Finally, we eliminate the antifields by introducing a gauge fermion ~t, and imposing q~*= 0¢t/00. The gauge fermion o f ref. [ 1 ] is ~t=2(g-1)-t- £
£ e}+~+'b~.
(15)
i=Oj=O
Again, we consider a term for the third symmetry: ~u"=t/i t/.
(16)
If we do not fix the third symmetry (i.e., without (14) and ( 1 6 ) ) , we find the following results for the antifields: 2"= (g-1),
g*=2,
ci* = - di+~+~ - (~-~
(17)
(all others zero), and the following gauge-fixed action: S~, = p ( 2 - 070) - ½gp2,l,+2(2_ 4g, O)
- E ~i++I(~c,+,-6.) i=0
--
22 £
C~+~ i+ i Ci+2
+(g-1)~
i=o
i=lj=l
This appears to depend on the third symmetry ghost q, but such dependence can be eliminated by the shift / ~J ni--rci( - ) / [ ( 1 - a ) ? ° 2 j + ( - j~iOl~'2j--i]~ t.2j_i.lq,
This arbitrariness is a reflection o f one described in ref. [8].
a
nated from ( 18 ), we see that there really is a residual gauge invariance; however, as it is algebraic, it is harmless. Alternatively, if we include the terms for the third symmetry (14) and (16), we find additional terms
&,+~,~O+~n..
(20)
In this result, the third symmetry ghost and antighost are essentially an auxiliary system; after we eliminate them by their field equation, we again reproduce the result ofref. [ 1 ]. There is a slight subtlety: the action (20) is still i nvariant under shifts of the antighost q~ (equivalently, t/~ can be eliminated by the shift ~z,= ~, -qT 0)- This is of course a less problematic algebraic symmetry, as, unlike (19), it does not involve an infinite number of terms. However, we have been unable to eliminate this further invariance, as adding further generations of ghosts always leads to a new, trivial, algebraic invariance. Indeed, the situation is analogous to x-symmetry, with the significant simplification that we are dealing with algebraic symmetries. We have found that the third symmetry should be gauge-fixed, though if it is not, it gives rise only to a harmless algebraic residual gauge freedom. However, in the superparticle, the third symmetry is not essential to close the algebra of diffeomorphisms and xsymmetry. In the superstring, it may well be quite important to treat the third symmetry correctly. In ref. [ 14 ], it is claimed that the third symmetry is auxiliary after gauge fixing as we found above; in ref. [ 15 ], it is claimed that the third symmetry should not be gauge-fixed. Since neither reference finds the complete BV action, it remains unclear what the correct procedure is. Our results should illuminate the basic structure of the complete action, and suggest that it is always correct to gauge-fix any symmetry in the action, even a dependent one. M.R. is happy to thank the ITP at the University o f Stockholm for its hospitality.
(19)
which leads to the same result as in ref. [ I ]. Here a is an arbitrary parameter ,2. Since r/ can be elimi~2
7 September 1989
symmetry analogous to
References
[ 1] u. Lindstr6m, M. Ro6ek, W. Siegel, P. van Nieuwenhuizen and A.E. van de Ven, Phys. Lett. B 224 (1989) 285. 55
Volume 228, number 1
PHYSICS LETTERS B
[2] M. Ro~ek, W. Siegel, P. van Nieuwenhuizen and A.E. van de Ven, Phys. Lett. B 227 (1989) 87. [3] W. Siegel, Lorentz-covariant gauges for Green-Schwarz superstrings, talk presented at Strings '89 (College Station, TX, March 1989), Stony Brook preprint ITP-SB-89-19 (March 1989). [4] S.J. Gates, M.T. Grisaru, U. Lindstr6m, M. Ro~ek, W. Siegel, P. van Nieuwenhuizen and A.E. van de Ven, Phys. Lett. B 225 (1989) 44. [ 5 ] U. Lindstrtim, M. Ro~ek, W. Siegel, P. van Nieuwenhuizen and A.E. van de Ven, Construction of the covariantly quantized heterotic superstring, Stony Brook preprint ITPSB-89-38 (May 1989). [6] I,A. Batalin and G.A. Vilkovisky, Phys. Lett. B 102 (1983) 27; B 120 (1983) 166; Phys. Rev. D 28 (1983) 2567; D 30 (1984) 508; Nucl. Phys. B 234 (1984) 106; J. Math. Phys. 26 (1985) 172.
56
7 September 1989
[7] F. Bastianelli, G.W. Delius and E. Laenen, in preparation. [8] J.M.L. Fisch and M. Henneaux, A note on the covariant BRST quantization of the superparticle, Universit6 Libre de Bruxelles preprint ULB TH2/89-04. [9] M.B. Green and C.M. Hull, Phys. Lett. B 225 (1989) 57. [ 10] R.E. Kallosh, Phys. Lett. B 224 (1989) 273. [ 11 ] R.E. Kallosh, Phys. Lett. B 195 (1987) 369. [ 12] A.H. Diaz and F. Toppan, Phys. Lett. B 211 (1988) 285. [ 13] W. Siegel, Phys. Lett. B 128 (1983) 397; M.B. Green and J.H. Schwarz, Phys. Lett. B 136 (1984) 367. [ 14] D.A. Depireux, S.J. Gates, P. Majumdar, B. Radak and S. Vashakidze, Towards covariantly quantized D = 10 type-II Green-Schwarz a-models, Maryland preprint UMDEPP 89169 (May 1989). [15] R.E. Kallosh, Phys. Lett. B 225 (1989) 49.