Volume 247, number 1 PHYSICS LETTERS B 6 September 1990

Topological supergravity in two dimensions

David Montano ‘s2 California Institute of Technology, Pasadena, CA 91125, USA

Kenichiro Aoki 3,4 and Jacob Sonnenschein 5,6

Department ofPhysics, University of California. Los Angeles, CA 90024-1547, USA

Received 27 May 1990

The topological theory of OSp(2 1 1; W) flat gauge connections is shown to be equivalent to two-dimensional supergravity. By

choosing a particular gauge fixing of the SL( 2; IR) (OSp( 2 11; R) ) invariance we make contact with the Liouville theory approach

to two-dimensional gravity (supergravity). We discuss the scaling and multicritical behavior of two-dimensional supergravity.

1. Introduction

Following ref. [ 11, we pursue topological gauge theories to describe two-dimensional gravity. Since the first papers on topological gravity [ 2-41, there has been progress in understanding the relation be- tween these theories and the more standard ap- proaches to two-dimensional quantum gravity such as those based on the matrix models [ 51, and the Liouville theory [6,7]. Matrix models were pro- posed as nonperturbative definitions of two-dimen- sional quantum gravity [ 81. This caused some ex- citement and has led to further investigation in two- dimensional gravity. Witten has recently derived re- cursion relations for correlation functions in two-di- mensional topological gravity which are in agree- ment with the one-matrix model results [ 93. Distler then made the connection between the Liouville theory and topological gravity by showing that the

Work supported in part by the US Department of Energy Con-

tract DEAC-03-8 1 ER40050.

’ Email address: montanoQtheory3.caltech.edu

3 Work supported in part by National Science Foundation Grant

PHY-86-13201.

4 Email address: aoki@uclahep.bitnet ’ Work supported by the “Julian Schwinger Post-Doctoral

Fellowship”.

6 Email address: cobi@uclahep.bitnet

Liouville theory coupled to c= - 2 “matter” is equiv- alent to topological gravity [ lo].

These three approaches to two-dimensional grav- ity have their advantages and disadvantages. The strength of the matrix models is, of course, that their correlation functions may be computed for Riemann surfaces of any genus. Unfortunately, the interpreta- tion of these correlation functions is not always transparent. Topological gravity (in SL( 2; E? ) for- mulation [ 1 ] ) has the advantage that one knows pre- cisely what is being calculated: integrals of wedge products of forms on the moduli space of Riemann surfaces (see refs. [ 2,9 ] ). Until recently, however, it has been bogged down by computational difficulties. The Liouville theory is troubled by both difficulties: interpretation and computational problems. For c= -2 and c--t --GO matter Liouville theory may be solved because in one case it is a topological theory and in the other saddle point techniques may be used, since one is then in the weak coupling regime [ 10,111.

At first sight, considering topological supergravity might seem like an uninteresting thing to do: we might naively expect that the topology of super moduli space will be the same as that of moduli space, in analogy with the super manifolds of De Witt topology [ 121. Firstly, it is not clear that this is the case, given the results of the genus one case in ref. [ 13 1. In that case, the super moduli space of the super torus with odd spin structure is almost the moduli space with an odd

64 0370-2693/90/$ 03.50 0 1990 - Elsevier Science Publishers B.V. (North-Holland)

Volume 247, number 1 PHYSICS LETTERS B 6 September 1990

direction, but there is an orbifold identification in the odd direction. Secondly, as referred to above, topo- logical gravity has been shown to describe quan tum gravity in two dimensions. We would expect topolog- ical supergravity in two dimensions to describe quan- tum supergravity in two dimensions, which is known to display a different behavior f rom the bose case [6,7].

In this paper we show that the OSp (211; E) topo- logical gauge theory describes N = 1 topological su- pergravity. So far the matrix model techniques have failed to describe supergravity because of the diffi- culties in triangulating super Riemann surfaces. We also show that the different versions of topological gravity of ref. [3] and ref. [4] are just gauge-fixed versions of SL (2; E ) topological field theory.

In section 2 we show that the fiat connections of OSp (211; E ) parametr ize N = 1 two-dimensional su- pergravity. We derive the lagrangian for topological supergravity. In section 3 we gauge-fix the SL(2; ~ ) theory of ref. [ 1 ] and the OSp (211; R) theory. We show that a gauge-fixed version of SL (2; ~ ) is equiv- alent to the theories in ref. [ 3 ] and ref. [4 ]. In sec- tion 4 we discuss the scaling and multicritical behav- ior of topological supergravity. Finally, in section 5 we conclude with some comments on the SL(N; R) and O S p ( N I M ; E) generalizations and their inter- pretation as topological WN gravity and supergravity [14].

2. Flat OSp(211; R) TQFT and topological supergravity

In ref. [ 1 ], T Q F T of the space of flat SO ( 1, 2), I S O ( I , 1 ) and SO(2, I ) gauge connections ( T F C ) was shown to describe the topological properties of the moduli space of Riemann surfaces of genus g = 0, g = 1 and g > 1 respectively. We now want to examine the similar correspondence between the moduli spaces of super Riemann surfaces for the above genera and the analogous field theories for the graded groups OSp(1, 111;~) , IOSp(1 , 111; R) and OSp(2I 1 ;~) . We denote the gauge fields associated with these graded Lie algebras by

- - a + A u - e u J a +¢9uJ3 +Zu-J+_ , (2.1)

where Ja ( a = z , ~) and J3 are Grassmann even gen-

erators and J_+ are Grassmann odd generators, which obey the following algebra:

[-/2, J=l = - 222J3,

[J~, J_ ] = - 2 / + ,

[J3, J+_ ] = + ½J_+,

[ J_ , J _ ] + = 2 J e ,

[J3, Jz] =&, [J3, J,] = - J z ,

[J,, J+ ] =2/_,

[J+ , J+ ] + = 2Jz,

[ J+ , J _ ] + =22./3. (2.2)

The parameter )2 takes a positive, vanishing and negative values for the OSp (211; ~ ), IOSp ( 1, 1 [ 1; ~ ) and OSp(1, I l l ; R ) algebras, respectively. The in- variant quadratic form which is consistent with a non- degenerate Casimir operator (apart f rom the IOSp(1, 111; R) case ofref. [1 ]) is

( J 3 , J 3 ) = 2 , ~ 2 , (Jz , J ~ ) = ( J e , J z ) = l ,

( J _ , J+ ) = - ( J + , J_ ) = ½2. (2.3)

For the non-supersymmetr ic case it was shown in ref. [ 1 ] that the relevant gauge configurations are fiat SO(2, 1) connections modulo gauge t ransforma- tions. It is thus natural to consider in the present case the restricted space of flat OSp (211; D ) connections. The condit ion for a vanishing field strength, F, ,e= 0, translates to the following equations for e~, o2 u and Xu":

0[~e~l + ¢olue~] --Zlu+Z~l+ = 0 ,

--Oglue,l - X I u - Z v l - = 0 ,

0[aZvl + 1 + ~ OgluX, l + -2e~tuZ~l- = 0 ,

l - +2e~.z . ] + = 0 O[uZvl- _ 20g[~Zvl 2 z g 0laW, ] - 22 e lue,) - 22Zt~+Z,l - = 0 . (2.4)

I f we interpret e~ as the zweibein, w u as the spin con- nection and Zu +- as the gravitino, the equation for e u can be identified as the s tandard torsion equation in supergravity and the other equations are exactly the conditions for constant super curvature R + _ = - 2i2, up to trivial rescalings ,l. (For N = 1 supergrav- ity in two dimensions and super Riemann surface theory, we refer the reader to refs. [ 15-17 ] and ref- erences therein.) The constant super curvature ge- ometry can be used to describe the moduli space of

Denoting the corresponding fields in ref. [ 17 ] by tilded fields, the identifications are to,,=idou ' eua__eu,-,, Zuo,=22u,~ and A= -2i2 .

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super Riemann surfaces. The generators of the graded Lie groups can be scaled in such a way that 22= - 1, 0 and 1. Since interesting topological properties on the moduli space exist mainly for g > 1, we concen- trate here only on the last case corresponding to OSp(211; ~) . The discussion of the other two cases follows similar lines.

The correspondence between the gauge and geo- metrical pictures is further supported by comparing of the gauge t ransformations with the local Lorentz, d i f feomorphism and N = 1 supersymmetry transfor- mations. The gauge transformation, 6Au=D~,e = 3~,~ + [A u, e ], with e - VaJa -}- (.a Jot "l- l J3 translate to

~e~ = V, v ~ - 2Z, + ~+ - l e~ ,

6e~u = Vu v e - 2Zu- ~- + le~ ,

~Zu + =2vZzu - +V,~ + - 2 e ~ e - - ½lzu + ,

6X,- = -2v~)c~ + +V~,e - +2e~e + + ½ l x / ,

6o)~, = Bu l - 22 2 (euvZ ~ -euvZ ~ ) - 22 (Z, + e - +Zu- e + ) .

(2.5) Here V~, denotes the covariant derivative with re- spect to reparametr izat ions defined using the spin connection. The transformations involving e and l can be respectively identified with N = 1 local supersym- metry t ransformations and local Lorentz transfor- mations. The t ransformations involving v a can be identified with diffeomorphisms up to a local N = 1 supersymmetry t ransformation with parameter e± =v~;G ± and a local Lorentz t ransformation with parameter l= v~o~, in the constant super curvature geometry (2.4).

The identification of the flat connections of the OSp (211; ~ ) gauge theory with the geometry of reg- ular super Riemann surfaces can only be made when eg is invertible. The flat connections naturally in- clude zweibeins which are not invertible, which should correspond to degenerate surfaces needed for "compact i fying" the moduli space of super Riemann surfaces in the sense of ref. [ 18 ].

To construct a consistent field theoretical frame- work for the flat OSp(2I 1; N) connections, we use the "s tandard procedure" for formulating a T Q F T [ 19-21 ]. Our starting point is the "topological sym- metry", 6 T A u ( x ) = 0~,(X), which is assumed to be an invariance of the original action, (for instance for L~=5¢(A) = 0 ) , and which is used to "gauge away"

all local properties of the gauge configurations. Using a BRST procedure we project onto configurations of fiat connections by gauge fixing this symmetry with the gauge condition, F , p = 0. The resulting lagrangian is

~ ( x ) = ½i~T [ ~,~sTr [ ~F,~p] ]

= ½i~x'~sTr [BF,~p - 5rq2), 5v~]

= i ~ { B a [ V,e~ - (X,~Xp) a ]

+ B3 ( C~o~ o),a + 1 ~. abea a eab _ 2X,~ +Xa- )

+ B ± [ V ,Za + T (e , xa) -+ ) ]

- i e"fl{ ~a [ Va ~/1% "[- ~ a b ~ t 3 e p b _ 2 (~u.Xfl) a ]

-t- I~ 3 ( Oa ~3 B "1- eabv/aaeBb "1- 2qJgZ~ + 2gtgZ~ )

1 j~t3 ~ ± - - + + ~ ± [V,,~,~ -T- ~v.,zp + (e ,~ua) - + ( ~ . Z p ) ± ] } , (2.6)

where (Z~)z= (Z+;¢ +), (ez) + =eZz - and similarly for Z and - . The ghost ~v, which corresponds to the topological symmetry 6TA~ = i~dTA. = ighv~ (with an ant icommuting global parameter ) , has the follow- ing gauge multiplet structure:

}Pa .= ~a Ja -t'- ~13 J 3 -}- ~+ J± , (2.7)

the other fields are similarly parametrized. The BRST transformation of the antighost is 6v ~P= B. This la- grangian is further invariant under a local "ghost symmetry" [ 21 ]

d v T V , = D , q b , dTB=- - i [q ) , ~P], (2.8)

and the OSp(2I 1; R) local gauge transformations. The complete quantum lagrangian is derived by gauge fixing these remaining local symmetries. By impos- ing the gauge conditions D,~ T'~ = 0 and D,~4 '~ = 0 one obtains

~ = ~(7( 1 ) .q_ ~ ( 2 )

= e'~asTr [ ½iBF, p - i gqD~ ~ ]

+ xfg sTr{ qSD,~ D'~q~- iqD,~ 7 v'~ + c-D,~ D'~c

- i b D , A " - i c ~ D ~ ~ - ½i(~[ ~,~, ~ " ] + ~P[@, ~P]

+ ½iq[qb, r / ]_ q3[qb, [~ , ~ ] ] ) } . (2.9)

The various fields in eq. (2.9) , (A, ~, ~P, B, qb, qS, r/, c, e, b), have ghost numbers (0, 1, - 1, 0, 2, - 2 , - 1, 1, - 1, 0). The complete BRST algebra and a further discussion of this action for an arbitrary gauge group

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can be found in ref. [ 221. One can introduce a cou- pling constant by resealing the various fields. In this case all the interaction terms are proportional to higher power of the coupling [ 2 1,231. Using the weak coupling limit which is in fact an exact limit, (for cal- culating coupling independent correlators) [ 23 ] one can then rewrite the above lagrangian as

Y= @sTr [ iBd,A, -i !&I, ‘y,]

+~gsTr[~a,a~~-~a,a”c-iba,A”

-i(c+q)a, Y”] . (2.10)

It is thus obvious that the BRST invariance guaran- tees that the terms with even and odd Grassmann variables are identical and therefore the total confor- ma1 anomaly is zero.

To further examine the relations between the mod- uli space of flat OSp (2 11; I?) connections and the moduli space of super Riemann surfaces we compute now the even dim,& and odd dim,&’ dimensions of the former. This is determined easily by the index of the ( y” p, q) system

dim,&=#(t&, &)(O)-#(V~, qO, I,?~, q3)co)

=3(2g-2) )

(2.11)

where ( ) (O) stands for a zero mode. We thus realize

that the even and odd dimensions are identical to those of the moduli space of super Riemann surfaces.

Flat OSp(2 1 I; I?) connections on super Riemann surfaces

In this subsection, we show that flat connections on a super Riemann surface correspond to constant super curvature geometries in a formulation which is manifestly covariant under super reparametriza- tions. Define the gauge field #2

AM=EMAJA+L&J3, (A=z,.q +, -). (2.12)

UsingtheOSp(21 l;[R) algebra (2.2),

w Indices M N . A B, denote Einstein, Lorentz indices, , 1 > , respectively.

F,w,v = &A, - ( - 1 )MN&4, + [AM, AN] f

= [ (a,+Q,)E,=-2( -l)“‘E,+E,+]J,

+ [ (a,-&)E,‘-2( -l)“‘E,yE,y]J,

+ [ (d,++&)E,+-,IE,=E,-]J+

+ [ (a,-$2M)EN-+1EM’EN+]J_

+ [dMQN-2~2EM=ENz-( - l)N2tiM+EN-]J3

- ( - 1 )MN(Mc*N) . (2.13)

Let us consider the generic case when EMA is inverti- ble. The covariant derivative acting on tensors of weight n #3 is defined as VA &AM@,+, - ns2,). The torsion, TABc and the curvature, RAB, are defined by the following action of the covariant derivatives on an n-tensor:

[VA, VB ] 2 = TABqc + idA, . (2.14)

The condition F MN=O, (2.13), can now be cast into a more transparent form. Defining FAB = ( - 1) (B+ N’“EAMEBNFMN,

FAB = ( - TAB’+ 462 8,’ ) J, + ( - TABi+ 4626, ) J,-

+(-TAB’ -M;&, +Ad,&)J+

+ (-TAB- +M;6,+ -M,‘d;)J_

+[-iRAB- 2n2(s:s;-s;&#)

+U(S,‘S, +S,Bs+)]J3. (2.15)

TAB”, RAB determined by FAB= 0 are precisely equal to the torsion and curvature tensors in the constant super curvature geometry obtained using the stan- dard torsion constraints and the Bianchi identities.

As in the component formulation, the gauge trans- formations are identical to super reparametrizations and local Lorentz rotations in the constant super cur- vature geometry.

Even though the formulation in terms of super- fields retains manifest supersymmetry and is more compact, there are some complications: First, not all the torsion and curvature tensors are independent. Put another way, FAe= 0 is 20 I 20 conditions whereas AM only has 10 I10 components. This will make the gauge fixing procedure slightly more involved. Sec-

” The conventions are that V’, V + have weights - 1, - f,

respectively.

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ondly, A M contains ½ forms on the super Riemann surface, whose properties differ for even and odd spin structure [ 17 ]. Henceforth, we shall work with the component formulation.

3. TFC and the other formulations of gravity and supergravity

Pure gravity and supergravity in the cont inuum have been formulated in several different ways [ 2 - 4,7,10], which we refer to as "metric formulations", in addition to the TFC description [ 1 ]. Let us now examine to what extent the latter is equivalent to the former. The action of (2.10) and the actions of the "metr ic" formulations show little if no resemblance. The question is, therefore, whether one can modify the construction of the TFC action and redefine the degrees of freedom so that they would match those of the "metr ic" action. The first stage of gauge fixing cannot be altered, since not only is it the only gauge invariant condition, but different choices of gauge are known to lead to different moduli spaces, in general. It is thus only the fixing of the residual local symme- tries that one may change.

Let us first consider the SL(2; ~) case. All the steps from (2.1) to (2.6) apply for this case when one sets the odd generators J+ to zero. To fix the SL(2; R) and the associated "ghost symmetry" we take a gauge condition which is the analog of the covariant gauge in fixing the reparametrization symmetry, namely

5o(2)= _ i x / ~ " a a ~o a(0) 6 T [ b a ( e , ~ - - e eo~ )

+ fl'~ ( ~Ug -- ¢e~eg ~°) ) ]

= ~v/g{ - - id'~ ( e a - e~'eg ( ° ) )

+ ba~( - i~u~ + V , c a - ~abeotbC3 )

- i6~ (~,~ - ~ue~eg ~°) )

"+ f la[vaoa-}-eabeab03 Al-f. ab( ~l/otC b --~llab c3 ) ]} ,

(3.1)

where eg <°) and ~ug <°) satisfy F , a = 0 and D t , q / ~ ° ) = 0, respectively and d T b a - - d a , ~ '~= 6a ~. Setting e~ to e~eg <°) implies fixing the traceless part of eg; i.e. three out of the four degrees of free- dom of SL (2; N) gauge transformation. Accordingly,

b~ is traceless. Obviously, the same argument holds for ~,g and fl~. The antisymmetric parts of b~ and fi~ are set to zero by integrating over the auxiliary fields c 3 and 0 3. We also integrate over d~ and 6g, and that fixes the traceless parts of eg and ~,,~. Sub- stituting these configurations into (2.6), we can ex- press (0,~ and ~u 3 in terms of ~0 and q/. All together the final action has the form

q_B 3 [ V~Va ~.q_ R (o) exp( - 2~) + 1 ]

-t- q)3(V°~Tag/+ 2 ~ ) ] . (3.2)

It is now easy to see that the action becomes that of an anticommuting and commuting spin (2, - 1 ) sys- tems; i.e. a b - c and fl-0 system (often denoted asf l=7 system). In addition, we have Grassmann even and odd scalars, ~0 and gt, which obey Liouville type equa- tions found in ref. [ 3 ]. Indeed, whenever one is gauge- fixing to a constant curvature constraint, these extra scalars will appear. They are consequences o f having fixed the Weyl symmetry by the constant curvature constraint. But, as far as the observables in the theory are concerned, these scalars are of no consequence. They correspond to a zero-dimensional moduli space; there is a unique solution to the Liouville equation and there are no zero mode. In other words, there ex- ists a unique metric of constant curvature (a hyper- bolic metric). The non-zero modes cancel each other in a topological field theory. This simply verifies that our original gauge-slice, flat SL(2; N) connections, is a complete one.

Bosonizing the fl-7 system into a scalar field theory with background charge (identified with the Liouville field) and a c = - 2 matter system reproduces the Liouville theory of refs. [ 10,7]. We have now shown that the theories of SL(2; ~) flat connections [ 1 ], topological gravity [ 3,4], and Liouville with c = - 2 [ 10,71, are equivalent.

One may similarly gauge fix OSp(211; R). In ad- dition to the above commuting and anticommuting spin (2, - 1 ) systems, one obtains from gauge-fixing the gravitino commuting and anticommuting spin (3, _ ½) systems. One may bosonize the commuting spin-2 and anticommuting spin -3 systems by two super scalar fields [24].

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4. Scaling and multicritical behavior of supergravity

Let us now discuss the scaling behavior of the su- pergravity theory following the procedure of ref. [ 141. Consider the general topological supergravity lagrangian

dip=$pT+ 1 t,fY, (4.1) n

where 6”; is the lagrangian of eq. (2.9 ). o” are BRST invariant dimension two operators which can have non-zero ghost number and t, are “coupling con- stants”. In terms of the fields in _I?=, we can construct BRST globally non-trivial invariant operators [ 1,22 ] such as S,sTr( @)+‘sTr( YA Y) which carry ghost number 4n - 2. In the Bose case, Witten identified one of the BRST invariant operators as the puncture op- erator and its coupling constant as the cosmological constant [ 91. This identification may seem a little bi- zarre, for the cosmological constant is the coefficient of a topologically non-invariant quantity, namely the area. In TQFT one circumvents this problem by summing over theories with an arbitrary number of punctures on the Riemann surface. The cosmological constant is then the weighting factor for the number of punctures.

We will now show that it is easy to generalize the scaling and the multicritical behavior of gravity to supergravity if we assume the existence of a general- ized puncture operator, go, which is a linear combi- nation of terms that increase the dimension of the moduli space by 210 or 012. For t,=O (n>O), it is clear that the partition function on the sphere scales as:

Z= SDXexp(&+t,i 0O)

DX$ o” .U >

(4.2)

since only ( (a’) 5 ) survives. We can derive the mul- ticritical behavior by changing the “initial condi- tions” for the recursion relations in refs. [ 9,141. It seems reasonable to assume that this is the only mod- ification, since we are dealing with a theory contain- ing a single “primary field” - just as for SL( 2; [R ), OSp (2 1 1; R) only has one Casimir. The “initial con-

dition” is now

U=(a”Oo)=t;, (4.3)

whose effect is to change to H 1: in the derivation of the multicritical points using the Landau-Ginzburg arguments of ref. [ 141. If the above assumptions are correct, we are lead to the “string susceptibility” for supergravity of -3/k wheres for gravity it is - 1 lk.

5. Discussion

In this paper we have followed the philosophy of ref. [ 1 ] in studying two-dimensional gravity using topological gauge theories. We showed that topologi- cal SL( 2; [R ) and OSp (2 1 1; R ) describe topological gravity and supergravity, respectively. The generali-

zations to SL(N; [R) and OSp(N]M; [R) are conjec- tured to describe topological WN gravity and super- gravity [ 141. This is a subject under current investigation. The theory of flat SL( 3; IR) connec- tions has recently been considered in relation to top- ological W3 gravity [ 2 5 1. It has also been speculated that SL(N, [R) topological gauge theories describe the N- 1 matrix models [ 14 1. It is reasonable to believe that the Casimirs of SL(N; [R) correspond to the pri- mary fields. Hence, topological SL(N, E?) is a theory of N- 1 primary fields coupled to topological gravity.

We have also shown that accepting certain assump- tions about the validity of the recursion relations of ref. [ 91 in the supergravity case, one may derive the multicritical behavior of topological supergravity.

It appears that, at least as far as topologically in- variant observables are concerned, the different for- mulations of two-dimensional gravity are equivalent. Of course, when computing geometrical quantities (i.e. quantities which are not BRST invariant in the topological formulation) the different formulations of two-dimensional quantum gravity will give differ-

ent results.

Acknowledgement

D.M. would like to thank A.S. Schwarz for encour- aging the study of topological OSp (2 ( 1; IR ) gauge theory in relation to the supermoduli space. We also thank R. Brooks, E. D’Hoker, J. Hughes and K. Li for useful discussions.

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