- Email: [email protected]
Superfield description of N=2 topological supergravity
Physics Letters B 268 ( 1991 ) 197-202 North-Holland
P H ¥51C S L ETTE R$ B
Superfield description of N = 2 topological supergravity J. G o r n i s i a n d J. R o c a 2 Departament d'Estructura i Constituents de la Matkria, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Catalonia, Spain
Received 15 July 1991
A free-field formulation of N= 2 topological supergravity is obtained with a suitable gaugefixing of two-dimensional topological Yang-Mills theory, taking as the gauge group a contraction of Osp (2 [2). We also give a superfield description of the action and its symmetry currents.
1. Introduction The study o f non-critical strings in d < 1 is mainly approached by using the formulations o f (i) matrix models [1], ( i i ) L i o u v i l l e theory [2], (iii) twodimensional topological gravity [3] and (iv) KdVhierarchy [4]. Two-dimensional topological gravity is particularly interesting due to its relation with matrix models [5,6 ] and Liouville theory [ 7 ]. It can be formulated as an SL(2, R) topological gauge theory and, with a suitable contraction of the gauge algebra, the gaugefixed theory is finally written in terms of a set of free conformal fields [6 ]. The coupling with matter can be done through a twisting o f N = 2 superconformal theories [8 ]. On the other hand, an alternative formulation o f topological gravity in terms o f a truncated N = 2 superconformal theory was recently proposed [ 9 ]. Extensions to N = I [ 10 ] and N = 2 [ 1 1 ] topological supergravity have been obtained by changing the gauge group to O s p ( 2 1 1 ) and Osp(212), respectively. With an analogous procedure of contraction o f the gauge algebra, the Osp(2l 1 ) is shown to lead to a conformal invariant theory [ 12 ]. In this paper we follow the same lines to obtain a free-field realization o f N = 2 topological supergravity. All expressions are rewritten in terms of super1 Electronic mail address: [email protected]. 2 Electronic mail address: [email protected].
fields, which makes the N = 2 supersymmetry manifest. The study of the N = 2 two-dimensional supergravity may be interesting because it can shed light on the construction of a matrix model approach to supergravity. It can also be useful in order to understand if the N = 2 string [ 13-15 ] is itself a topological theory.
2. Osp(212) gauge theory N = 2 topological supergravity can be obtained from the lagrangian LPo = ½ B a e " n F ~ a - Z a e " B ( D , ~ u n )
(2.1)
a,
with the gauge group Osp (212) [ 11 ]. This lagrangian has been previously used to obtain free-field formulations of pure topological gravity and N = 1 topological supergravity by choosing suitable contractions o f the gauge groups SL(2, R) [ 6 ] and Osp (211 ) [ 12 ], respectively. It is invariant under standard Yang-Mills gauge transformations a
--
a
a
__
.
Due to the topological nature o f this theory, (2.1) is invariant under a topological Q-symmetry g s A L--~ , ,a
6s~/~ = 0 ,
(2.2)
as well as under a local fermionic counterpart of the YM symmetry
0370-2693/91/$03.50© 1991 Elsevier Science Publishers B.V. All rights reserved.
197
Volume 268, number 2
PHYSICS LETTERSB
10 October 1991 -I-
6FA~ =0,
dFg~ = (D~{) ~ .
(2.3)
The Lagrange multipliers B~ and Za impose the equations of motion a
--
F~-0,
+
-+
+
d~e + = O~e +-+_co~e ± -T-eft l - f g u - - f ~ u +- , 8 a f g = O . u ± +- ½co.u +-+ a . u +-T-!zfg l - f g s , ~Gfg = a~a + + ½o).a +-" a . a +--Y- ½f+ l+ f + s •
(2.5 cont'd)
(Dt,~,m)~=0,
i.e., vanishing of the field-strength and of its Qpartner. The Osp(212) algebra is spanned by the SL(2, R) generators L_+, Lo, two pairs of fermionic generators of supersymmetry transformations G+, Ge and a U ( 1 ) generator J. The connection A, is expanded in terms of the gauge generators of Osp (212), A,=c&~Lo+ ~ (e+L+ + f + G + + f + d + ) + o , ~ J .
It remains to show that the above transformations (2.5) are equivalent to those 0 f N = 2 supergravity. It is well known that two sets of gauge transformations, i a ~(~i =Rae. ,
~'~i=R~e'a,
are equivalent if they are related by redefinitions of the gauge parameters Ea plus antisymmetric combinations of the equations of motion, i.e.,
++_
Here ~o, is the spin connection, eg the zweibein, f ~ a n d f ~ the gravitino fields and a , is a U( 1 ) vector field. To obtain a free conformal invariant action for N = 2 topological supergravity we have to rescale and contract the gauge algebra, which implies locally setting the curvature to zero. We redefine L± ~ 2 - ' L ~ ,
G±----~%/~//~-IG+,
G±--, 2x/2~ G+_ ,
and takes the limit 2--+0. The non-zero (anti-)commutators are [Lo, L + ] = _ + L ± , [Lo, G ± ] = + ½ G + ,
[G+,G±]=L+, [Lo, G ± ] = + k G _ + ,
In components, the flatness condition F~z= 0 reads 0=0[.az~,
0 = 0 t ~ e ~ _+o)t.e~l - f ? . f ~ 3 , 1 + 0=0t~fgj+ + ~og[~f~ +a[,~f~~-1 , 1 0 = a [ ~ f h + ~o9[, f~]+ - a t ~ f ~+ ] .
5S
aOJ
14o-
lgJi
(2.6)
The transformations generated by ! and s are exactly local Lorentz and U ( 1 ) gauge transformations. To obtain the standard supersymmetry transformations we only have to add combinations of the equations of motion ( 2.4 ). Reparametrizations with gauge parameter v" are obtained by a combined action of all gauge transformations in (2.5), i.e.,
+ (equations of m o t i o n ) ,
[J,G--]=G+_, [J,G+]=-G.. 0=Ot~o)zj,
R~=R~A~+M~
where we choose e ± =e+~ v% 1= o)~v", u +-= f ~ v", a ± = f ~ v" and s = a~v ~. Since we have added combinations of the equations of motion 8S/5B a to obtain supersymmetry and reparametrizations from (2.5) for the supergravity multiplet, we should keep in mind that the new gauge transformations for B a will also have to contain combinations of the equations of motion 8S/~A~ as dictated by the antisymmetry condition (2.6).
(2.4)
We can also expand the YM gauge parameter,
3. Gauge fixing
e = l L o + ~ (e*-L~ + u Z G ± + a - - G - ) + s J , T
and write the associated transformations of the component fields,
Let us change to new sets of variables and gauge parameters which are more suitable for the gauge fixing we will impose,
~ o9,~= 0,~l, ~oa,~ = 0~s,
e +=exp(~b_+),
198
(2.5)
h_+_+=exp(-½•_+)e + ,
(3.1)
Volume 268, number 2
PHYSICS LETTERSB
g-+-+ = e x p ( - ¼¢+ T- ½ p ) f + , g-+ -+ = e x p ( - 1¢-+ _+½p)f+ , • + = e x p ( - ~ ¢ + - T - ~1 P ) f i+, _ + ~5+ = e x p ( -¼0-+ 1+ ~p)f~-,
a+ = ½(O-+'c_+ O_+p), E-+=exp(½¢_+) v-+ ,
u-+=exp(¼¢-+ +½p) ((-++@+v -+) , ~-+ =exp(¼¢-+ T ½p) ( ( - + + ~ + v - + ) .
(3.1 cont'd)
All transformations above are algebraic except for the expression of the vector field a~ which is given in terms of derivatives of two scalars z and p. The effect of this transformation is two-fold. First, to take into account the associated jacobian J = d e t 0+ det 0_ in the path-integral formulation, we will have to introduce two pairs o f " b - c - t y p e " fermionic ghost systems of conformal weight ( 1, 0). They will be written explicitly in the ghost lagrangian after we complete the gauge fixing. Second, the change from cr, to two scalar fields introduces two new chiral transformations generated by parameters s -+ satisfying 0~s -+=0: &r-+ = 0 - + s ~ O T = 2 s + ~ T-s-+, -+
-+
The gauge symmetries (2.5) are conveniently fixed by choosing the superconformal gauge ¢+=¢_=¢,
10 October 1991
supersymmetry transformations, which close to a transformation of reparametrizations only after using the equations of motion. In general this causes that a naive BRST quantization fails to give a nilpotent BRST charge. The appropriate framework for quantization of these systems is the BatalinVilkovisky formalism [ 16 ] which is an extension of the usual BRST quantization and gives the right answer when the gauge algebra is not closed. I n our specific case the gauge algebra for the fields defined in (3.1), and using (3.3) wherever the spin connection co+ appears, is indeed not closed. However, due to the peculiar structure of the non-closure coefficients and the use of a gauge fixing not involving the Lagrange multiplier fields, the naive BRST quantization and the more general BV procedure may be shown to coincide and lead to the same nilpotent BRST transformation. Thus, we proceed with the usual quantization method. We introduce the standard Faddeev-Popov ghosts and auxiliary fields associated with these gauge fixing conditions: bo-co for local Lorentz, b-c for reparametrizations, B-v--C +-for supersymmetry and b~ca for the U ( 1 ) symmetry. Integration over the auxiliary fields imposes the gauge conditions (3.2). Since the Lorentz transformation is algebraic, the integration over bo allows to write Coas ~1
1(Oc+cO¢-~Z+
Co= ~
c~¢-+-(c-c.)
)
-
h-+-+=0,
g_+-+ =g-+-+ =0,
r=0.
(3.2)
From (2.4) we see that the spin connection co,~can be algebraically solved in terms of the zweibein and gravitino fields. On the gauge slice co= has the form co-+ = +½ 0-+¢.
(3.3)
So the only contribution to the curvature in the gauge (3.2) comes from the Liouville field ~. The algebra of the gauge transformations (2.5) is closed (it is just a rewriting of Yang-Mills transformations which are indeed closed). However, after redefinition of fields and gauge parameters (3.1) and addition of the equations of motion (3.3), the resulting gauge algebra may close only on-shell, i.e., using the equations of motion. This is a common feature of
The U ( 1 ) transformation of z is also algebraic and the b l - c x system can be completely eliminated: bl=cl=0. Besides the ordinary Yang-Mills transformations we also have to fix the transformations (2.3). In order that the gauge-fixed action retains the topological Q-symmetry (2.2), we impose as gauge fixing of the fermionic symmetry the Q-transformations of the superconformal gauge (3.2). All the expressions are then simply Q-variations of the previous results. For example, the ghost 70 associated with fermionic Lorentz transformations is ~ We have changed notation from + to holomorphic and antiholomorphic components. The upper + indices refer to the N = 2 supersymmetry.
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Volume 268, number 2
PHYSICSLETTERSB
1(
~,o=asCo=-~ o~,+~,,O0-cO~,-½Z c • ~-+ +
-~ Zr~o-+-(c.c.))._+
currents in a compact way within a superfield formalism. The N = 2 superspace is described in terms of bosonic (z, z) and fermionic (0 -+, 0-+ ) coordinates ~2. We can define covariant derivatives D -~-
4. Superfield description of N = 2 topological supergravity The final gauge-fixed action can be split into Liouville
10 October 1991
0 0 + 0 ± 0, I) + - - + 0 + 0 . 06 = 00 •
The action for the Liouville sector can be written in terms of four superfields S = and T +, satisfying the constraints D - S = =I)-S-+ =0,
D+T-* =I)+T-+ = 0 .
If we consider the expansions
&= f (7r000+ ~ [//~ ~O-++(c.c.)]+K~Op +zOO~u+2_+[X-v0'/-'±+ (c.c.)l+aOOrt), and ghost sectors
S ±=9=p+O-O ±+O-¢b +-+O-O-E = , T == - ~(Tr+to)- ½0+H+-- ½0+17+-+ O+O+V*-, the action for the Liouville sector SL can be rewritten /
S= [ dZzd0 + dO+dO- dO- ~ T•S *,1
Sgh= f (bOc+ ~ B~ OC+-+dOa+ flO~ + ~A*_+ OF±+fiOa+(c.c.)). Here (0; • -+; p) are the components of the Liouville supermultiplet and ( ~ ~u-+; q) its respective Qpartners. The auxiliary fields (Tr;H-~; Jc) and (Z; X~; #) are also related by Q-symmetry and come from the components of B~ and Za which survive after gauge fixing: The ghost lagrangian is simply a collection of "b-c-type" systems corresponding to the gauge symmetries (2.5) and its Q-transformed. The d-a and fia systems, however, are not associated to any gauge symmetry but come from the jacobian for the transformation from a,~ to the scalar fields z and p. As we mentioned above, this transformation contains time derivatives; which in general leads to a change of the physical spectrum. With the introduction of these ghosts and an associated "BRST charge" [ 17] - together with the ordinary ghosts for the gauge symmetries - we ensure that the physical cohomology remains the same. Thus, the final gauge-fixed action S=&+Sgh is a free, conformal invariant theory just as in N = 0 and N = 1 topological gravity. It will be useful to rewrite the action and symmetry 200
z
We can define for the ghost sector the expansions
C=c+ ~ O~-C++O-O+a, B=d+ ~ ++_O=B++O-O+b, F = ~ - ~ O~F+-+O-O+a, A = ~ + ~ +_O~-A++O-O+fl.
(4.1)
Now, the ghost action Sgh is
Sgh= S dZzd0+ dO- (BOC+A OF)+ (c.c.). The non-zero fundamental operator product expansions are ¢,( Z,~)H( Z,)) ~ ~ ( Z a ) X ( Zb ) ~ C( Z,,)B( Zb ) ~F(Z~)A(Zb) ~
o~o~ , Z~b
where Zab--za--Zb--(O+OF +OgO~ ) and Oyb=+ Og+ --0~. The expansion of the holomorphic ghost superfields C(Z), B(Z), F(Z) and A(Z) is given in (4.1) and the expansion of the Liouville-sector superfields is ~2 We followthe notation and conventionsof ref. [ 15].
Volume 268, .number2
PHYSICS LETTERSB
~ ( z ) = O ( z ) + 2 o~q'~(z)+O-O + Op(z), //(z) =x(z) + y, -7-0~//-+ (z) - 0 - 0 + a~(z), ~(z) =~,(z)- Z 0~ ~-+ (z) +o-o+ a~(z), x ( z ) = ~ ( z ) + ~ ~o-vx+(z)_O.O + az(z). The symmetries of S factorize into holomorphic and anti-holomorphic currents. Here we only write the holomorphic parts. They are generated by the super stress-energy tensor T = t + 52___0 ~ t + + 0 : 0 + T and its Q-partner G = g + 52_+0 = g + + 0 - 0 + G, Which split into Liouville and ghost sectors: T L ( Z ) = ½ Z D : ~ / ] D + ~ + O / / - ½ Z D'TXD-+ ~ , ,.
_+
+
Tgh(Z)=-3(BC)-½ y., DT-BD+C +
-O(AF)+½ ~ D~-AD+-F, GL(Z)=½ ~ D V X D - + ~ - 0 X , +__
G , ( Z ) = - O(AC) + ½Z D-r-A D+C.
10 October 1991
where f D Z = ~ (dz/2•i) f dO + f dO-. We have [Qs, G ( Z ) ] = T ( Z ) for either the Liouville and ghost sectors. There is also a ghost-number current J,
J= - X~'- B C - 2AF , which assigns ghost numbers (0, 0; 1, - 1; 1, - 1; 2, - 2) to ( 4, H; ~, X; C, B; F, A). These assignations arise in a natural way when the original lagrangian ~eo (2.1) is interpreted as a partial gauge fixing of L~ (A,) =0. The fermionic field g& plays the role of a first- generation ghost - with ghost-number one - and (2.3) are gauge transformations of the ghost lagrangian which are conveniently fixed with the introduction of new ghosts - of ghost-number two - associated with the gauge parameter ~ , following a standard ghost-for-ghost mechanism. The total nilpotent BRST charge is taken to be the addition of the two BRST charges obtained in the gauge-fixing procedure from 3¢=0, i.e., we write QBP.ST= QS + QV, where Qv is given by
Qv = (fl DZ[C(TL + ½Tgh)-F(GL + ½Ggh)].
±
Primary N = 2 conformal fields, ~hq(Z), are characterized by their conformal weight, h, and their charge, q. They have the following OPE with the stress-energy tensor, T ( Z ) :
One easily finds that the full stress-energy tensor is a BRST commutator
Oa-~Oo5~(zb)T(Zo) ~,~(Zb)~h ~-~--
where the anti-ghost B differs from G = GL+ Gsh by a BRST-exact operator,
~q ~(z~)
T--- TL -t- Tgh = [ QBRST,B ] ,
B - G = [ QBRST,A ] .
+~
1
-
+
~+ +--O~bD~~P~(Zb)+ ~O~bO.bOz~~hq(Zb) " _
Here, all superfields have zero charge, q~, H, ~Uand X have also h = 0, whereas the ghosts C, F and B, A have h = - 1 and h = 1, respectively. The supercurrents Tand G are themselves primary fields with conformal spin h = 1. On the other hand, the expansion of the product of two G's is non-singular. Thus, no central extension arises in their operator product expansions and the superfields T and G form a closed operator algebra. The Q-symmetry is realized by the charge Qs,
Qs = (~ DZ ( - H ~ + BF) , ,J
In summary, a formulation of N = 2 topological supergravity is obtained by taking a two-dimensional topological gauge theory with a contraction of the gauge group Osp(212). The resulting gauge-fixed theory is free and can be fully described in terms of N = 2 superfields. The coupling of this theory with matter systems is a subject presently under study.
Acknowledgement
The authors would like to thank J. Distler, D. Montano and P.K. Townsend for discussions. This work has been partially supported by a NATO Collaborative Research Grant (0763/87) and CICYT 201
Volume 268, number 2
PHYSICS LETTERS B
p r o j e c t no. A E N 8 9 . 0 3 4 7 . J.R. a c k n o w l e d g e s a fellowship f r o m M i n i s t e r i o de E d u c a c i 6 n y C i e n c i a o f the Spanish g o v e r n m e n t .
References [1]D.J. Gross and A.A. Migdal, Phys. Rev. Lett. 64 (1990) 717; M. Douglas and S. Shenker, Nucl. Phys. B 335 (1990) 635; E. Br6zin and V.A. Kazakov, Phys. Lett. B 236 (1990) 144. [2] V.G. Knizhnik, A.M. Polyakov and A.B. Zamolodchikov, Mod. Phys. Lett. A 3 (1988) 819; F. David, Mod. Phys. Lett. A 3 (1988) 1651; J.Distler and H. Kawai, Nucl. Phys. B 321 ( 1989 ) 509. [3 ] J:M.F. Labastida, M. Pernici and E. Witten, Nucl. Phys. B 310 (1988) 611; D. Montana and J. Sonnenschein, Nucl. Phys. B 3 13 ( 1989 ) 258. [ 4 ] D.J. Gross and A.A. Migdal, Nucl. Phys. B 340 (1990) 333; T. Banks, M. Douglas, N. Seiberg and S. Shenker, Phys. Lett. B 238 (1990) 279; M. Douglas, Phys. Lett. B 238 (1990) 176; P. Di Francesco and D. Kutasov, Nucl. Phys. B 342 (1990) 589;
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M. Fukuma, H. Kawai and R. Nakayama, Intern. J. Mod. Phys. A 6 (1991) 1385. [5 ] E. Witten, Nucl. Phys. B 340 (1990) 281; R. Dijkgraaf and E. Witten, Nucl. Phys. B 342 (1990) 486. [ 6] E. Veflinde and H. Verlinde, Nucl. Phys. B 348 (1991 ) 457; R. Dijkgraaf, E. Verlinde and H. Verlinde, Notes on topological string theory and 2D quantum gravity, Princeton preprint IASSNS-HEP-90/80 (1990). [ 7 ] J. Distler, Nucl. Phys. B 342 (1990) 523. [8] T. Eguchi and S. Yang, Mod. Phys. Lett. A 5 (1990) 1693. [ 9] J. Distler and P. Nelson, Phys. Rev. Lett. 66 (1991 ) 1955. [ 10 ] D. Montana, K. Aoki and J. Sonnenschein, Phys. Lett. B 247 (1990) 64. [ 11 ] K. Li, Nucl. Phys. B 346 (t990) 329. [ 12] J. Hughes and K. Li, Phys. Lett. B 261 ( 1991 ) 269. [ 13 ] M. Ademollo et al., Nucl. Phys. B 111 (1976) 77. [ 14] See among others: A. Bilal, Phys. Lett~ B 180 (1986) 255; A.R. Bogojevic and Z. Hlousek, Phys. Lett. B179 (1986) 69; S.D. Mathur and S. Mukhi, Phys. Rev. D 36 (1987) 465. [ 15 ] J. Gomis; Phys. Rev. D 40 ( 1989 ) 408. [ 16 ] I.A. Batalin and G.A. Vilkovisky, Phys. Lett. B 102 ( 1981 ) 27; Phys. Rev. D 28 ( 1983 ) 2567; J. Math. Phys. 26 ( 1985 ) 172. [ 17] A.A. Slavnov, Phys. Lett. B 258 (1991) 391.
P H ¥51C S L ETTE R$ B
Superfield description of N = 2 topological supergravity J. G o r n i s i a n d J. R o c a 2 Departament d'Estructura i Constituents de la Matkria, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Catalonia, Spain
Received 15 July 1991
A free-field formulation of N= 2 topological supergravity is obtained with a suitable gaugefixing of two-dimensional topological Yang-Mills theory, taking as the gauge group a contraction of Osp (2 [2). We also give a superfield description of the action and its symmetry currents.
1. Introduction The study o f non-critical strings in d < 1 is mainly approached by using the formulations o f (i) matrix models [1], ( i i ) L i o u v i l l e theory [2], (iii) twodimensional topological gravity [3] and (iv) KdVhierarchy [4]. Two-dimensional topological gravity is particularly interesting due to its relation with matrix models [5,6 ] and Liouville theory [ 7 ]. It can be formulated as an SL(2, R) topological gauge theory and, with a suitable contraction of the gauge algebra, the gaugefixed theory is finally written in terms of a set of free conformal fields [6 ]. The coupling with matter can be done through a twisting o f N = 2 superconformal theories [8 ]. On the other hand, an alternative formulation o f topological gravity in terms o f a truncated N = 2 superconformal theory was recently proposed [ 9 ]. Extensions to N = I [ 10 ] and N = 2 [ 1 1 ] topological supergravity have been obtained by changing the gauge group to O s p ( 2 1 1 ) and Osp(212), respectively. With an analogous procedure of contraction o f the gauge algebra, the Osp(2l 1 ) is shown to lead to a conformal invariant theory [ 12 ]. In this paper we follow the same lines to obtain a free-field realization o f N = 2 topological supergravity. All expressions are rewritten in terms of super1 Electronic mail address: [email protected]. 2 Electronic mail address: [email protected].
fields, which makes the N = 2 supersymmetry manifest. The study of the N = 2 two-dimensional supergravity may be interesting because it can shed light on the construction of a matrix model approach to supergravity. It can also be useful in order to understand if the N = 2 string [ 13-15 ] is itself a topological theory.
2. Osp(212) gauge theory N = 2 topological supergravity can be obtained from the lagrangian LPo = ½ B a e " n F ~ a - Z a e " B ( D , ~ u n )
(2.1)
a,
with the gauge group Osp (212) [ 11 ]. This lagrangian has been previously used to obtain free-field formulations of pure topological gravity and N = 1 topological supergravity by choosing suitable contractions o f the gauge groups SL(2, R) [ 6 ] and Osp (211 ) [ 12 ], respectively. It is invariant under standard Yang-Mills gauge transformations a
--
a
a
__
.
Due to the topological nature o f this theory, (2.1) is invariant under a topological Q-symmetry g s A L--~ , ,a
6s~/~ = 0 ,
(2.2)
as well as under a local fermionic counterpart of the YM symmetry
0370-2693/91/$03.50© 1991 Elsevier Science Publishers B.V. All rights reserved.
197
Volume 268, number 2
PHYSICS LETTERSB
10 October 1991 -I-
6FA~ =0,
dFg~ = (D~{) ~ .
(2.3)
The Lagrange multipliers B~ and Za impose the equations of motion a
--
F~-0,
+
-+
+
d~e + = O~e +-+_co~e ± -T-eft l - f g u - - f ~ u +- , 8 a f g = O . u ± +- ½co.u +-+ a . u +-T-!zfg l - f g s , ~Gfg = a~a + + ½o).a +-" a . a +--Y- ½f+ l+ f + s •
(2.5 cont'd)
(Dt,~,m)~=0,
i.e., vanishing of the field-strength and of its Qpartner. The Osp(212) algebra is spanned by the SL(2, R) generators L_+, Lo, two pairs of fermionic generators of supersymmetry transformations G+, Ge and a U ( 1 ) generator J. The connection A, is expanded in terms of the gauge generators of Osp (212), A,=c&~Lo+ ~ (e+L+ + f + G + + f + d + ) + o , ~ J .
It remains to show that the above transformations (2.5) are equivalent to those 0 f N = 2 supergravity. It is well known that two sets of gauge transformations, i a ~(~i =Rae. ,
~'~i=R~e'a,
are equivalent if they are related by redefinitions of the gauge parameters Ea plus antisymmetric combinations of the equations of motion, i.e.,
++_
Here ~o, is the spin connection, eg the zweibein, f ~ a n d f ~ the gravitino fields and a , is a U( 1 ) vector field. To obtain a free conformal invariant action for N = 2 topological supergravity we have to rescale and contract the gauge algebra, which implies locally setting the curvature to zero. We redefine L± ~ 2 - ' L ~ ,
G±----~%/~//~-IG+,
G±--, 2x/2~ G+_ ,
and takes the limit 2--+0. The non-zero (anti-)commutators are [Lo, L + ] = _ + L ± , [Lo, G ± ] = + ½ G + ,
[G+,G±]=L+, [Lo, G ± ] = + k G _ + ,
In components, the flatness condition F~z= 0 reads 0=0[.az~,
0 = 0 t ~ e ~ _+o)t.e~l - f ? . f ~ 3 , 1 + 0=0t~fgj+ + ~og[~f~ +a[,~f~~-1 , 1 0 = a [ ~ f h + ~o9[, f~]+ - a t ~ f ~+ ] .
5S
aOJ
14o-
lgJi
(2.6)
The transformations generated by ! and s are exactly local Lorentz and U ( 1 ) gauge transformations. To obtain the standard supersymmetry transformations we only have to add combinations of the equations of motion ( 2.4 ). Reparametrizations with gauge parameter v" are obtained by a combined action of all gauge transformations in (2.5), i.e.,
+ (equations of m o t i o n ) ,
[J,G--]=G+_, [J,G+]=-G.. 0=Ot~o)zj,
R~=R~A~+M~
where we choose e ± =e+~ v% 1= o)~v", u +-= f ~ v", a ± = f ~ v" and s = a~v ~. Since we have added combinations of the equations of motion 8S/5B a to obtain supersymmetry and reparametrizations from (2.5) for the supergravity multiplet, we should keep in mind that the new gauge transformations for B a will also have to contain combinations of the equations of motion 8S/~A~ as dictated by the antisymmetry condition (2.6).
(2.4)
We can also expand the YM gauge parameter,
3. Gauge fixing
e = l L o + ~ (e*-L~ + u Z G ± + a - - G - ) + s J , T
and write the associated transformations of the component fields,
Let us change to new sets of variables and gauge parameters which are more suitable for the gauge fixing we will impose,
~ o9,~= 0,~l, ~oa,~ = 0~s,
e +=exp(~b_+),
198
(2.5)
h_+_+=exp(-½•_+)e + ,
(3.1)
Volume 268, number 2
PHYSICS LETTERSB
g-+-+ = e x p ( - ¼¢+ T- ½ p ) f + , g-+ -+ = e x p ( - 1¢-+ _+½p)f+ , • + = e x p ( - ~ ¢ + - T - ~1 P ) f i+, _ + ~5+ = e x p ( -¼0-+ 1+ ~p)f~-,
a+ = ½(O-+'c_+ O_+p), E-+=exp(½¢_+) v-+ ,
u-+=exp(¼¢-+ +½p) ((-++@+v -+) , ~-+ =exp(¼¢-+ T ½p) ( ( - + + ~ + v - + ) .
(3.1 cont'd)
All transformations above are algebraic except for the expression of the vector field a~ which is given in terms of derivatives of two scalars z and p. The effect of this transformation is two-fold. First, to take into account the associated jacobian J = d e t 0+ det 0_ in the path-integral formulation, we will have to introduce two pairs o f " b - c - t y p e " fermionic ghost systems of conformal weight ( 1, 0). They will be written explicitly in the ghost lagrangian after we complete the gauge fixing. Second, the change from cr, to two scalar fields introduces two new chiral transformations generated by parameters s -+ satisfying 0~s -+=0: &r-+ = 0 - + s ~ O T = 2 s + ~ T-s-+, -+
-+
The gauge symmetries (2.5) are conveniently fixed by choosing the superconformal gauge ¢+=¢_=¢,
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supersymmetry transformations, which close to a transformation of reparametrizations only after using the equations of motion. In general this causes that a naive BRST quantization fails to give a nilpotent BRST charge. The appropriate framework for quantization of these systems is the BatalinVilkovisky formalism [ 16 ] which is an extension of the usual BRST quantization and gives the right answer when the gauge algebra is not closed. I n our specific case the gauge algebra for the fields defined in (3.1), and using (3.3) wherever the spin connection co+ appears, is indeed not closed. However, due to the peculiar structure of the non-closure coefficients and the use of a gauge fixing not involving the Lagrange multiplier fields, the naive BRST quantization and the more general BV procedure may be shown to coincide and lead to the same nilpotent BRST transformation. Thus, we proceed with the usual quantization method. We introduce the standard Faddeev-Popov ghosts and auxiliary fields associated with these gauge fixing conditions: bo-co for local Lorentz, b-c for reparametrizations, B-v--C +-for supersymmetry and b~ca for the U ( 1 ) symmetry. Integration over the auxiliary fields imposes the gauge conditions (3.2). Since the Lorentz transformation is algebraic, the integration over bo allows to write Coas ~1
1(Oc+cO¢-~Z+
Co= ~
c~¢-+-(c-c.)
)
-
h-+-+=0,
g_+-+ =g-+-+ =0,
r=0.
(3.2)
From (2.4) we see that the spin connection co,~can be algebraically solved in terms of the zweibein and gravitino fields. On the gauge slice co= has the form co-+ = +½ 0-+¢.
(3.3)
So the only contribution to the curvature in the gauge (3.2) comes from the Liouville field ~. The algebra of the gauge transformations (2.5) is closed (it is just a rewriting of Yang-Mills transformations which are indeed closed). However, after redefinition of fields and gauge parameters (3.1) and addition of the equations of motion (3.3), the resulting gauge algebra may close only on-shell, i.e., using the equations of motion. This is a common feature of
The U ( 1 ) transformation of z is also algebraic and the b l - c x system can be completely eliminated: bl=cl=0. Besides the ordinary Yang-Mills transformations we also have to fix the transformations (2.3). In order that the gauge-fixed action retains the topological Q-symmetry (2.2), we impose as gauge fixing of the fermionic symmetry the Q-transformations of the superconformal gauge (3.2). All the expressions are then simply Q-variations of the previous results. For example, the ghost 70 associated with fermionic Lorentz transformations is ~ We have changed notation from + to holomorphic and antiholomorphic components. The upper + indices refer to the N = 2 supersymmetry.
199
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PHYSICSLETTERSB
1(
~,o=asCo=-~ o~,+~,,O0-cO~,-½Z c • ~-+ +
-~ Zr~o-+-(c.c.))._+
currents in a compact way within a superfield formalism. The N = 2 superspace is described in terms of bosonic (z, z) and fermionic (0 -+, 0-+ ) coordinates ~2. We can define covariant derivatives D -~-
4. Superfield description of N = 2 topological supergravity The final gauge-fixed action can be split into Liouville
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0 0 + 0 ± 0, I) + - - + 0 + 0 . 06 = 00 •
The action for the Liouville sector can be written in terms of four superfields S = and T +, satisfying the constraints D - S = =I)-S-+ =0,
D+T-* =I)+T-+ = 0 .
If we consider the expansions
&= f (7r000+ ~ [//~ ~O-++(c.c.)]+K~Op +zOO~u+2_+[X-v0'/-'±+ (c.c.)l+aOOrt), and ghost sectors
S ±=9=p+O-O ±+O-¢b +-+O-O-E = , T == - ~(Tr+to)- ½0+H+-- ½0+17+-+ O+O+V*-, the action for the Liouville sector SL can be rewritten /
S= [ dZzd0 + dO+dO- dO- ~ T•S *,1
Sgh= f (bOc+ ~ B~ OC+-+dOa+ flO~ + ~A*_+ OF±+fiOa+(c.c.)). Here (0; • -+; p) are the components of the Liouville supermultiplet and ( ~ ~u-+; q) its respective Qpartners. The auxiliary fields (Tr;H-~; Jc) and (Z; X~; #) are also related by Q-symmetry and come from the components of B~ and Za which survive after gauge fixing: The ghost lagrangian is simply a collection of "b-c-type" systems corresponding to the gauge symmetries (2.5) and its Q-transformed. The d-a and fia systems, however, are not associated to any gauge symmetry but come from the jacobian for the transformation from a,~ to the scalar fields z and p. As we mentioned above, this transformation contains time derivatives; which in general leads to a change of the physical spectrum. With the introduction of these ghosts and an associated "BRST charge" [ 17] - together with the ordinary ghosts for the gauge symmetries - we ensure that the physical cohomology remains the same. Thus, the final gauge-fixed action S=&+Sgh is a free, conformal invariant theory just as in N = 0 and N = 1 topological gravity. It will be useful to rewrite the action and symmetry 200
z
We can define for the ghost sector the expansions
C=c+ ~ O~-C++O-O+a, B=d+ ~ ++_O=B++O-O+b, F = ~ - ~ O~F+-+O-O+a, A = ~ + ~ +_O~-A++O-O+fl.
(4.1)
Now, the ghost action Sgh is
Sgh= S dZzd0+ dO- (BOC+A OF)+ (c.c.). The non-zero fundamental operator product expansions are ¢,( Z,~)H( Z,)) ~ ~ ( Z a ) X ( Zb ) ~ C( Z,,)B( Zb ) ~F(Z~)A(Zb) ~
o~o~ , Z~b
where Zab--za--Zb--(O+OF +OgO~ ) and Oyb=+ Og+ --0~. The expansion of the holomorphic ghost superfields C(Z), B(Z), F(Z) and A(Z) is given in (4.1) and the expansion of the Liouville-sector superfields is ~2 We followthe notation and conventionsof ref. [ 15].
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PHYSICS LETTERSB
~ ( z ) = O ( z ) + 2 o~q'~(z)+O-O + Op(z), //(z) =x(z) + y, -7-0~//-+ (z) - 0 - 0 + a~(z), ~(z) =~,(z)- Z 0~ ~-+ (z) +o-o+ a~(z), x ( z ) = ~ ( z ) + ~ ~o-vx+(z)_O.O + az(z). The symmetries of S factorize into holomorphic and anti-holomorphic currents. Here we only write the holomorphic parts. They are generated by the super stress-energy tensor T = t + 52___0 ~ t + + 0 : 0 + T and its Q-partner G = g + 52_+0 = g + + 0 - 0 + G, Which split into Liouville and ghost sectors: T L ( Z ) = ½ Z D : ~ / ] D + ~ + O / / - ½ Z D'TXD-+ ~ , ,.
_+
+
Tgh(Z)=-3(BC)-½ y., DT-BD+C +
-O(AF)+½ ~ D~-AD+-F, GL(Z)=½ ~ D V X D - + ~ - 0 X , +__
G , ( Z ) = - O(AC) + ½Z D-r-A D+C.
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where f D Z = ~ (dz/2•i) f dO + f dO-. We have [Qs, G ( Z ) ] = T ( Z ) for either the Liouville and ghost sectors. There is also a ghost-number current J,
J= - X~'- B C - 2AF , which assigns ghost numbers (0, 0; 1, - 1; 1, - 1; 2, - 2) to ( 4, H; ~, X; C, B; F, A). These assignations arise in a natural way when the original lagrangian ~eo (2.1) is interpreted as a partial gauge fixing of L~ (A,) =0. The fermionic field g& plays the role of a first- generation ghost - with ghost-number one - and (2.3) are gauge transformations of the ghost lagrangian which are conveniently fixed with the introduction of new ghosts - of ghost-number two - associated with the gauge parameter ~ , following a standard ghost-for-ghost mechanism. The total nilpotent BRST charge is taken to be the addition of the two BRST charges obtained in the gauge-fixing procedure from 3¢=0, i.e., we write QBP.ST= QS + QV, where Qv is given by
Qv = (fl DZ[C(TL + ½Tgh)-F(GL + ½Ggh)].
±
Primary N = 2 conformal fields, ~hq(Z), are characterized by their conformal weight, h, and their charge, q. They have the following OPE with the stress-energy tensor, T ( Z ) :
One easily finds that the full stress-energy tensor is a BRST commutator
Oa-~Oo5~(zb)T(Zo) ~,~(Zb)~h ~-~--
where the anti-ghost B differs from G = GL+ Gsh by a BRST-exact operator,
~q ~(z~)
T--- TL -t- Tgh = [ QBRST,B ] ,
B - G = [ QBRST,A ] .
+~
1
-
+
~+ +--O~bD~~P~(Zb)+ ~O~bO.bOz~~hq(Zb) " _
Here, all superfields have zero charge, q~, H, ~Uand X have also h = 0, whereas the ghosts C, F and B, A have h = - 1 and h = 1, respectively. The supercurrents Tand G are themselves primary fields with conformal spin h = 1. On the other hand, the expansion of the product of two G's is non-singular. Thus, no central extension arises in their operator product expansions and the superfields T and G form a closed operator algebra. The Q-symmetry is realized by the charge Qs,
Qs = (~ DZ ( - H ~ + BF) , ,J
In summary, a formulation of N = 2 topological supergravity is obtained by taking a two-dimensional topological gauge theory with a contraction of the gauge group Osp(212). The resulting gauge-fixed theory is free and can be fully described in terms of N = 2 superfields. The coupling of this theory with matter systems is a subject presently under study.
Acknowledgement
The authors would like to thank J. Distler, D. Montano and P.K. Townsend for discussions. This work has been partially supported by a NATO Collaborative Research Grant (0763/87) and CICYT 201
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PHYSICS LETTERS B
p r o j e c t no. A E N 8 9 . 0 3 4 7 . J.R. a c k n o w l e d g e s a fellowship f r o m M i n i s t e r i o de E d u c a c i 6 n y C i e n c i a o f the Spanish g o v e r n m e n t .
References [1]D.J. Gross and A.A. Migdal, Phys. Rev. Lett. 64 (1990) 717; M. Douglas and S. Shenker, Nucl. Phys. B 335 (1990) 635; E. Br6zin and V.A. Kazakov, Phys. Lett. B 236 (1990) 144. [2] V.G. Knizhnik, A.M. Polyakov and A.B. Zamolodchikov, Mod. Phys. Lett. A 3 (1988) 819; F. David, Mod. Phys. Lett. A 3 (1988) 1651; J.Distler and H. Kawai, Nucl. Phys. B 321 ( 1989 ) 509. [3 ] J:M.F. Labastida, M. Pernici and E. Witten, Nucl. Phys. B 310 (1988) 611; D. Montana and J. Sonnenschein, Nucl. Phys. B 3 13 ( 1989 ) 258. [ 4 ] D.J. Gross and A.A. Migdal, Nucl. Phys. B 340 (1990) 333; T. Banks, M. Douglas, N. Seiberg and S. Shenker, Phys. Lett. B 238 (1990) 279; M. Douglas, Phys. Lett. B 238 (1990) 176; P. Di Francesco and D. Kutasov, Nucl. Phys. B 342 (1990) 589;
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M. Fukuma, H. Kawai and R. Nakayama, Intern. J. Mod. Phys. A 6 (1991) 1385. [5 ] E. Witten, Nucl. Phys. B 340 (1990) 281; R. Dijkgraaf and E. Witten, Nucl. Phys. B 342 (1990) 486. [ 6] E. Veflinde and H. Verlinde, Nucl. Phys. B 348 (1991 ) 457; R. Dijkgraaf, E. Verlinde and H. Verlinde, Notes on topological string theory and 2D quantum gravity, Princeton preprint IASSNS-HEP-90/80 (1990). [ 7 ] J. Distler, Nucl. Phys. B 342 (1990) 523. [8] T. Eguchi and S. Yang, Mod. Phys. Lett. A 5 (1990) 1693. [ 9] J. Distler and P. Nelson, Phys. Rev. Lett. 66 (1991 ) 1955. [ 10 ] D. Montana, K. Aoki and J. Sonnenschein, Phys. Lett. B 247 (1990) 64. [ 11 ] K. Li, Nucl. Phys. B 346 (t990) 329. [ 12] J. Hughes and K. Li, Phys. Lett. B 261 ( 1991 ) 269. [ 13 ] M. Ademollo et al., Nucl. Phys. B 111 (1976) 77. [ 14] See among others: A. Bilal, Phys. Lett~ B 180 (1986) 255; A.R. Bogojevic and Z. Hlousek, Phys. Lett. B179 (1986) 69; S.D. Mathur and S. Mukhi, Phys. Rev. D 36 (1987) 465. [ 15 ] J. Gomis; Phys. Rev. D 40 ( 1989 ) 408. [ 16 ] I.A. Batalin and G.A. Vilkovisky, Phys. Lett. B 102 ( 1981 ) 27; Phys. Rev. D 28 ( 1983 ) 2567; J. Math. Phys. 26 ( 1985 ) 172. [ 17] A.A. Slavnov, Phys. Lett. B 258 (1991) 391.