target-superspace in 2d dilatonic supergravity

5 August 1999

Ž .Physics Letters B 460 1999 87–93

Target-superspace in 2d dilatonic supergravity

Thomas StroblInstitut fur Theoretische Physik, RWTH Aachen, D-52056 Aachen, Germany¨

Received 30 April 1999Editor: R. Gatto

Abstract

The Ns1 supersymmetric version of generalized 2d dilaton gravity can be cast into the form of a Poisson s-model,where the target space and its Poisson bracket are graded. The target space consists of a 1q1 superspace and the dilaton,which is the generator of Lorentz boosts therein. The Poisson bracket on the target space induces the invariance of the

Ž .worldsheet theory against both diffeomorphisms and local supersymmetry transformations superdiffeomorphisms . Themachinery of Poisson s-models is then used to find the general local solution to the field equations. As a byproduct,classical equivalence between the bosonic theory and its supersymmetric extension is found. q 1999 Published by ElsevierScience B.V. All rights reserved.

The most general 2d gravity action for a metric gand a dilaton field f yielding second order differen-

w xtial equations is of the form 1 :

2 'Ls d x ygHMM

=2U f RqV f =f qW f , 1Ž . Ž . Ž . Ž . Ž .

where R denotes the Ricci scalar and U, V, and WŽ .are some arbitrary reasonable functions of the dila-

w xton. Its supersymmetrization, considered first in 2 ,may be obtained in a most straightforward manner

w xby using the superfield formalism of 3 . In thisframework the action takes the same form as above,where, however, each term is replaced by an appro-priately constrained supersymmetric extension and,simultaneously, the volume form d2 x is replaced by

Ž . 2 2its worldsheet superspace analog d x d u . A termŽ . Ž .such as U f , e.g., is replaced by U F , where F is

the superfield Fsfq iujq iuu f with f being thebosonic dilaton field, j a Majorana spinorial super-

Žpartner, and f an auxiliary bosonic scalar. Forfurther details the reader is referred to the literature

.cited above .For many practical calculations it is necessary to

reexpress the supersymmetric extension of L in termsŽ .of its component fields f, j , f etc. . The resulting

action and all the more its field equations becomelengthy and their analysis involved. In the presentletter we propose a different formulation, whichgreatly simplifies not only the notation but also theanalysis of the supersymmetric theory.

This latter formulation is provided by a Poissonw xs-model 4 , the definition of which we will briefly

recapitulate now, generalizing it to the case of gradedPoisson manifolds, before we then come to its rela-

Ž . Žtion to 1 for an introduction to bosonic Poissonw x.s-models cf. 5 . The action is a functional of n

0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 99 00649-8

( )T. StroblrPhysics Letters B 460 1999 87–9388

scalar fields X i, is1, . . . ,n, on the one hand, whichmay be viewed as coordinates in an n-dimensionalŽ .not necessarily linear manifold N, as well as of noneforms A 'A dx m, on the other hand, whichi m i

may be regarded as oneforms on the worldsheetŽ m . )coordinates x , ms0,1 taking values in T NŽ Ž .more precisely, A is a oneform on the worldi

sheet which is simultaneously the pullback of a) Ž ..section of T N by the map X x . In the present

context we allow N to carry also a Z -grading, i.e.2

some of the fields X i and A may be Grassmannm iŽ i jvalued, s denoting their respective parity so X Xi

Ž .s i s j j i .s y1 X X etc. .To define an action we require N to be equipped

Ž .with a graded Poisson bracket

� i j4 i jX , X 'PP . 2Ž .

Note that the Poisson bracket is anti-symmetric onlyfor the case that at least one of its entries is an evenŽ . i jcommuting quantity; in general one has PP sŽ .s i s jq1 ji Ž i jy1 PP while PP itself has grading s qi. i js . In terms of the two-tensor PP the standard,j

Ž w x.graded Jacobi identity cf., e.g., 6 may be broughtinto the form

§

Es si k i j sky1 PP PP qcycl ijk s0 , 3Ž . Ž . Ž .sž /E X

Žwhere a sum over the index s is understood but not.over i or k in the first of the three cyclic terms .

™ ŽUsing left derivatives EsE in contrast to the above§.right derivatives E , this equation may be written

Ž .s i sk i s jk Ž .equivalently as y1 PP E PP qcycl ijk s0.s

The action of the 2d theory is

1i jiSs A ndX y A nA PP , 4Ž .H i i j2

where the order of the terms and indices has beenchosen so as to avoid unnecessary signs in theconsiderations to follow while simultaneously S co-incides with its purely bosonic counterpart in theprevious literature.

Ž .Before further investigating the general model 4 ,we now discuss its relation to Ns1 supersymmetricdilaton gravity. For this purpose we first restrict our

Ž .attention to the case where U'f and V'0 in 1 .This is not a serious restriction since the general

Žaction can be brought into this form always at least.locally by a dilaton-dependent conformal rescaling

of the metric g and a simultaneous change of theŽ . w xdilaton field f™F f 1 ; the information about U

and V is then ‘‘stored’’ in the relation to the originalŽvariables. In this process global information may be

lost, as happens, e.g., if U has critical points. In thefollowing we thus restrict our attention to suchchoices of the potentials U,V,W where the abovetransformation is sufficiently global; otherwise theconsiderations to follow are of local nature and theglobal information has to be restored in a subsequent

w x .step, cf. also 7 for details. Following a recentw xpaper by Izquierdo 8 , the explicit component form

Ž .of the supersymmetric extension of 1 , rewritten inEinstein-Cartan variables, takes the form 1

a a b aLsf dvqX de q´ v e q2 icg cŽ .a b

iuXX

X a b 3y 2uu y xx ´ e e q4 iucg cabž /16X 1aq iu x e g cq ix dcq vg c . 5Ž .Ž .a 32

a ŽHere e is the zweibein of the metric g or the. aconformally rescaled metric, respectively , ´ v isb

the spin connection, and c is a oneform-valuedŽ .Majorana spinor. x is a zeroform-valued Majorana

fermion, which, such as c , is of odd Grassmanna Žparity, and X denotes a pair of scalar functions the

Lorentz index a running over two values in two. aspacetime dimensions . The fields X are introduced

in the bosonic theory as Lagrange multipliers so asto determine v through the vanishing torsion con-straint; as the field equations for v determine X a

uniquely in terms of the remaining variables, thea Ž .fields X may be eliminated from or introduced to

Žthe action without changing the theory at least on

1 After completion of this letter, I became aware that the actionw x Ž w xbelow was found already in 9 which was not noticed in 8 ,

where the latter of the two works was cited in a different context. w xonly . 9 contains also parts of the considerations to follow, but,

such as in the case of bosonic Poisson s-models, the hiddenŽ . Žstructure of a graded Poisson manifold is not noted and, conse-

quently, no statement about, e.g., the classical solutions could be.obtained there .

( )T. StroblrPhysics Letters B 460 1999 87–93 89

.the classical level . u is a function of the dilaton f,Ž . Ž 2 .Xwhich relates to W in 1 through Ws4 u , the

prime denoting differentiation with respect to thew xargument f. For the conventions we conform to 8 :

Ž .the Lorentz metric has signature y,q , Lorentz0 1indices are underlined, ´ sq1, and the gamma

matrices are related to the usual Pauli matrices si0 1 3 0 1according to: g syis , g ss , and g 'g g2 1

ss .3

We now collect the one- and zeroforms into twomultiplets:

i a aX :s X ,x ,f , A :s e ,ic ,v . 6Ž . Ž . Ž . Ž .Ž .i a a

Ž .After a simple partial integration, the action 5 isŽ .seen to take the form 4 and the coefficient matrix

PP i j may be read off by straightforward comparison:

1X XXab ab aPP sy´ 4uu q u x x ,až /8i

aXaa aPP su g x ,Ž .ab a ba b 3 aPP sy8iu g y4 iX g ,Ž .Ž . a

� a 4 a bX ,f s´ X ,b

a1a 3� 4x ,f sy g x , 7Ž .Ž .2

Ž a .a Ž a.a bwhere g x ' g x , spinor indices have beenbab Ž braised and lowered by means of ´ x s´ xa a b

01 .with ´ :s1 , and, in the last line, the identificationŽ .2 was used.

Up to now it may seem that we have not gainedmuch and in particular it is by no means clear thatthe matrix PP obtained above indeed satisfies theŽ . Ž .graded Jacobi identity 3 , which, however, is atthe heart of Poisson s-models.

w x Ž .In 8 it was shown that the field equations of 5Ž . w xform a free differential algebra FDA 10 . In the

Ž .present formulation 4 the field equations take thecompact form

§1i ji lkdX yA PP s0 , dA y A A PP E s0 .j i k l2 ž /i

8Ž .

Applying an exterior derivative to the first set of§ji ji kŽ .these equations, we obtain dA PP yA PP E dXj j k

k Ž .s0. Eliminating dA and dX by means of 8 , wej

end up with an expression bilinear in the AXs. Bydefinition of an FDA, the resulting equations have tobe fulfilled identically, without any restriction to theoneforms A . It is a simple exercise to show that thisi

requirement is fulfilled if and only if PP i j satisfiesŽ . w xEqs. 3 . Thus, using the result of 8 , the validity of

the graded Jacobi identities is proven.Certainly the Jacobi identities may be verified

also by a direct calculation using the specific formŽ .7 of the bracket. This is in fact simpler thanproving the FDA property of the field equations of

Ž . Žthe specific action 5 cf. also first sentence of the.following paragraph . Seeking brackets fulfilling the

graded Jacobi identities with restrictions specifiedbelow will automatically provide 2d supergravitytheories and to us this route seems to be the techni-cally simplest for the construction of such models.This idea will be made clearer in what follows.

Applying an exterior derivative to the second setof equations, the requirement for an FDA does not

Ž .lead to any further relations beside 3 . Thus we mayconclude that the field equations of a general graded

Ž .Poisson s-model 4 form an FDA, iff the tensor PPŽis a Poisson tensor i.e., by definition, iff PP satisfies

Ž ..3 . Alternatively, the Jacobi identity may be veri-fied to be the necessary and sufficient condition for

Ž .the constraints in a Hamiltonian formulation of 4 tobe of first class. This is tantamount to requiring themodel to have maximal local symmetries.

It is a nice and simple exercise to show that dueŽ .to 3 the variations

§i ji k jd X se PP , d A sde yA e PP E 9Ž .j i i j k ž /i

change the action only by a total divergence: dSsŽ i.Hd e dX . Thus there is a local symmetry for anyi

Ž i .pair of fields X , A . Since the action is of firsti

order in these fields, this implies that there are atŽ .most a finite number of physical gauge invariant

Ždegrees of freedom. More precisely, being a one-form, A has two components for each value of i;i

however, the ‘‘time component’’ A of the oneform0 i

A enters a Hamiltonian formulation as Lagrangei

multiplier for the constraints only and therefore it.must not be included in the above naive counting .


Ž . Ž .In the context of 5 resp. 7 there are fiveŽ .independent local symmetries contained in 9 : ef

Žgenerates local Lorentz symmetries cf. last line inŽ ..Eq. 7 . The Grassmann spinor e , on the othera

hand, generates precisely the local supersymmetryw xtransformations of 8 . The remaining two symme-

tries correspond to the obvious diffeomorphism in-variance of the action. Indeed, a simple calculationshows that the Lie derivative of X i and A along ai

Ž .spacetime vector field Õ differs from 9 with theŽ . Ž m .specific choice e :s Õ A only by an additivei m i

Ž .term proportional to the field equations 8 . Forinvertible zweibein A the first two entries establisha

a bijection between any possible choice of Õ and e ,a

moreover. As this will be of some importance lateron, we note, however, that while diffeomorphismscannot change an invertible matrix A into a nonin-ma

Ž . 2vertible one and vice versa, the symmetries 9 can .We return to the structure found in the target

space N of the theory. The target space is spannedby a Lorentz vector X a, a Majorana spinor x a, andthe dilaton f. X a and x a may be combined into aŽ .1q1 -dimensional superspace. f, on the otherhand, generates Lorentz boosts in this superspace by

Ž .means of the Poisson brackets 7 , last line. Indeed,a Ž´ is the Lie algebra element of the one-dimen-b

.sional Lorentz group in the fundamental representa-tion and g 3 is easily identified with the generator ofLorentz boosts in a two-dimensional spinor space:

13 0 1 0 1w x Žg 'g g s g ,g irrespective of the choice of20 1presentation for the generators g and g in the

.Clifford algebra .This structure in the target space will remain also

within generalizations to more general 2d supergrav-ity theories including the supersymmetrization oftheories with nontrivial torsion, such as the

Ž . w xKatanaev–Volovich KV model 12 , for which asupersymmetrization has not been provided in the

Ž .literature yet. Using the same fields 6 as before,such theories may be found by searching for other

Ž .solutions to the Jacobi identity 3 . However, the

2 Since the difference in the symmetries occurs only for nonde-generate metrics, which, however, in a gravitational theory areusually excluded, this difference is mostly irrelevant for what

w xconcerns the relevant factor spaces; cf., however, 11 for coun-terexamples to this somewhat naive picture.

then yet unknown Poisson tensor should be restrictedŽ .to agree with the last line in 7 . This is due to the

relation of the Poisson bracket on the target spaceŽ .and the local symmetries 9 and thus implicitly

required by local Lorentz invariance, present in anygravity theory when formulated in Einstein–Cartanvariables.

It is worth noting that restricting the PoissonŽ .tensor only by the last line of 7 , the Jacobi identi-

Ž .ties 3 with one of the indices corresponding to f

requires the Poisson tensor components PP ab, PP aa,and PP ab to transform covariantly under Lorentztransformations! E.g. for PP aa the Jacobi identities

1aa a ba 3 a ab� 4 Ž .require PP ,f s´ PP y g PP . Thus tob b2

obtain the most general supergravity theory that fitsinto the present framework, we can proceed as fol-lows: It must be possible to build the unknowntensor components of PP by means of the Lorentz

a a ab Ž a.ab Ž 3.abcovariant quantities X , x , ´ , g , and g

Ž ab´ is incorporated automatically by raising and.lowering spinor indices with coefficients that are

Lorentz invariant functions, i.e. functions of X X a,a

x x a, and f. Thus, e.g., the antisymmetric tensoraab ab abŽ a .PP must be of the form PP s´ F qx x F ,1 a 2

where F are functions of the two arguments X X a1,2 a

and f. The remaining Jacobi identities then reduceŽ .to a comparatively simple set of differential equa-

tions for these coefficient functions.Proceeding in this way, e.g., by replacing all

Ž .five coefficients in its first two lines of the bracketŽ .7 by yet undetermined coefficient functions ofX aX and f, one can show that the remaining Jacobia

Ž .identities 3 force the coefficients to agree withŽ . Žthose provided already in 7 except for a simultane-

.ous global prefactor . More general theories can thusbe obtained only by using further covariant entities

aa Ž . abto build PP and possibly also PP . Indeed,Lagrangians quadratic in torsion require an extra

Ž a . aŽ 3.a b aa w xadditive term F X X ,f X g x in PP 13 ,a b

also perfectly compatible with Lorentz covariance. Inw x14 more general supergravity theories will be con-structed by the above method. By construction, theresulting theories will be invariant against superdif-

Ž .feomorphisms incorporated within 9 , thus allowingfor an interpretation as supergravity theory. Thesupersymmetrization of the KV-model will be con-tained as a particular example in the class of modelsconstructed in this way.


We finally turn to the solution of the field equa-tions. Beside its notational compactness the main

Ž . Ž .advantage of the formulation 4 as opposed to 5 isits inherent target space covariance. Thus we maychange coordinates in the target space of the theoryso as to simplify the tensor PP, while the fieldequations in the new variables still will take the formŽ . Ž8 but then with the transformed, simplified Pois-

.son matrix . We will first use this method to showthat locally the space of solutions to the field equa-

Ž .tions of 5 modulo gauge symmetries is just one-di-mensional. Thereafter we will provide a representa-tive of this one-parameter family in terms of the

Ž .original variables used in 5 . As a byproduct we willfind that locally the space of solutions is identical tothe one of the bosonic theory; all the fermionic fieldsmay be put to zero by gauge transformations. Sinceany global solution can be obtained by patchingtogether local solutions, we conclude that the local

Ž .equivalence between the bosonic theory 1 and itssupersymmetrization holds also on a global level. Itwould be interesting, however, to confirm this resultin a more direct way.

ŽLocally more precisely, in the neighborhood of a. Ž .generic point any bosonic Poisson manifold allows

Ž . Ž A I .for Casimir–Darboux CD coordinates C ,Q , PJw x A15 ; constant values of the Casimir functions Clabel symplectic leaves in N, on which the remain-ing coordinates are standard Darboux coordinates:� I 4 I Ž .Q , P sd all other brackets vanishing . Accord-J J

w xing to 6 , Darboux coordinates exist also for super-Žsymplectic manifolds manifolds with a graded, non-

.degenerate Poisson bracket . We thus assume thatCD coordinates exist in the case of general, gradedPoisson manifolds. However, at least in the case of

Ž .the bracket 7 , they definitely do: Although we didnot succeed to find such coordinates explicitly in thepresent paper, we will provide a Casimir function

Žbelow. On its level surfaces defined by the appropri-.ate quotient algebra of superfunctions the Poisson

Ž .bracket is almost everywhere nondegenerate andthe result on supersymplectic manifolds may be ap-plied.

We now are in the position to show that locallythere is only a oneparameter family of gauge in-

Ž .equivalent solutions to the field equations of 5 . Forthis purpose we only need to know about the local

˜ iexistence of CD coordinates X . In these coordi-

3 Ž .nates the second set of field equations 8 reduce to˜ ˜dA s0. Thus locally A sdf for some functionsi i iŽ . Ž .f x . However, the local symmetries 9 also sim-i

˜plify dramatically in these new field variables: d Ai

sd´ . This infinitesimal formula may be integrated˜i

easily showing that all the functions f may be put toi

zero identically. But then we learn from the first set˜ iŽ .of the field equations 8 that all the functions X

are constant. All of the constant values of Q I and PI

may be put to an arbitrary value by means of theŽ . Ž .residual gauge freedom in 9 constant e . Whati

remains as gauge invariant information is only theconstant values of the Casimir functions C A. Since

Ž . Žthe Poisson tensor 7 has rank four almost every-. 4 Ž .where , the model defined by Eq. 5 has just one

Casimir function and its space of local solutions isthus indeed one-dimensional only.

The local solution obtained above in terms ofCD-coordinates may be transformed back easily toany choice of target space coordinates. We find thatalso in the original variables: A '0 and X i sconst i,i

where the latter constants may be chosen at will aslong as they are compatible with the constant values

Ž . A Ž .of the Casimir s C , which characterize the localsolution. As it stands, this solution corresponds to asolution with vanishing zweibein and metric. In a

Ž .gravitational theory, the metric and zweibein isrequired to be a nondegenerate matrix, however. Thevanishing zweibein is a result of using the symme-

Ž .tries 9 , which, in contrast to diffeomorphisms, con-nect the degenerate with the nondegenerate sector ofthe theory. A similar phenomenon occurs, e.g., also

Ž .within the Chern–Simons formulation of 2q1 -gravity. The problem may be cured by applying a

Ž .gauge transformation 9 to the local solution A '0i

so as to obtain a solution with nondegeneratezweibein. However, in contrast to Chern–Simons

3 To be sure: These are coordinates on the target space, not onthe worldsheet spacetime. From the point of view of the field

i ˜ itheory a change of coordinates X ™ X , which induces the§ j i˜ ˜Ž . Ž .change A ™ A ' A E X rE X , corresponds to a local changei i j

of field variables.4 To determine the rank of the matrix PP i j, we may concentrate

on the rank of the the two by two matrix PP ab and the three bythree matrix in the purely bosonic sector; PP aa, being linear inGrassmann variables, cannot contribute to the rank of the matrix,

w xcf. 6 .


Žtheory, where the behavior of A under finite non-.abelian gauge transformations is known, the in-

Ž .finitesimal gauge symmetries 9 cannot be inte-Žgrated in general except for the case where the

Ž . iPoisson tensor is at most linear in the fields X andŽ . .the theory reduces to a non abelian gauge theory .

We thus need to introduce one further step.Ž .For the Poisson brackets 7 a possible choice for

� 4a Casimir function C, C,P '0, is

iX1 a 2 aCs X X q2u y ux x . 10Ž .a a2 8

This expression basically coincides with the Casimirfound in the bosonic theory: as there the last two

0 1terms are the integral of PP with respect to f,x x a Poisson commuting with all bosonic vari-a

Žables. We now choose new coordinates on patches. Ž a .of N where X /0 according to

i q a< <X :s C ,ln X ,x ,f , 11Ž .Ž ." 1 0 'Ž . Žwith the null coordinates X :s X .X r 2 . An

argumentation similar to the following one may be< q< q Ž ..applied also if ln X is replaced by X in 11 .

Ž < q< .C,ln X ,f provides a CD coordinate system inŽ w x.the purely bosonic sector cf. also 7 ; however, in

the five-dimensional target space N, these coordi-nates are far from being CD, several brackets still

Ž .containing the potential u f . It thus seems ratherdifficult to solve the field equations for field vari-

Ž .ables 11 . However, from our considerations above,we know that up to Poisson-s gauge transformationsthe local solution always takes the from A '0,i

< q< aCsconst, ln X sx sfs0. In these field vari-Ž .ables, induced by the coordinates 11 , it is now

possible to gauge transform this explicitly to a solu-tion with nondegenerate zweibein and, simultane-ously, the resulting solution may be transformed

Ž .back to the original variables used in 5 – this is˜ ipossible since in contrast to the CD coordinates X

iused above, the coordinates X are known explicitlyin terms of the original variables.

It is straightforward to verify that on the aboÕeŽ .solutions the infinitesimal gauge transformations 9

Ž . Ž .with e :s e ,e ,0,0,0 are simply:i C q

d A sde , d A sde , dfse , 12Ž .C C q q q

with all other variations vanishing. Note that, e.g., inthe second relation we dropped terms proportional toA , since A can be kept zero consistently by thea a

above transformations due to d A s0. It is thusa

Ž .possible to integrate the gauge symmetries 12 :ŽA ™A qdf , f™fq f , A ™A qdf allC C 1 2 q q 2

.other fields remaining unaltered where f is an1,2

arbitrary pair of functions on the 2d spacetime. Thedegenerate solution is then transformed into A sC

q a< <df , A sdf , A sA s0, Csconst, ln X sx1 q 2 a f

s0, and fs f . Using f and f as coordinates x12 1 2

and x 0 on the worldsheet, respectively, and trans-forming these solutions back to the original variables

§ j iŽ . Ž Ž ..6 using A sA E X rE X , we obtain:i j

eq,ey,vŽ .

1 X1 0 0 1 0 1s dx ,dx q h x dx ,yh x dx ,Ž . Ž .Ž .2

1q y 0 0X , X ,f s 1, h x , x , 13Ž . Ž . Ž .Ž .2

Ž 0. 2Ž 0.where h x 'Cy2u x , C being the constantŽ .value of the Casimir 10 . All the fermionic vari-

ables vanish identically.Thus, up to gauge transformations, the local solu-

tion agrees completely with the one found in theŽ .purely bosonic dilaton theory 1 . This applies at

least to those patches where the above coordinatesystems are applicable. Since, e.g., all the fixed

Ž .points of the supersymmetric bracket 7 lie entirelywithin the bosonic sector of the target space, weexpect that there are also no exceptional solutions,

Ž .containing necessarily nonvanishing fermionicfields. Moreover, the subsequent global analysis ofw x Ž .7 may be applied to the solutions 13 withoutchange.

So the characterization of the dynamics of theŽ .general supersymmetric extension of 1 turns out to

w xbe less difficult than it appeared at the time when 2Ž Ž .was written cf. the remarks following Eq. 50 of

.that paper . Rather, it appears that the supersymmet-Ž . ( )ric extension of 1 is triÕial on-shell , at least at the

classical level.It would be interesting to check this result by

some other method and to possibly establish it in aless indirect way. It is to be expected, moreover, thata similar result holds also on the quantum level.


Let us finally remark that the supersymmetricextension may still be of some value even on the

w xpurely classical level: In 2 it was used, e.g., toŽ .establish the positivity of some notion of the

Ž‘‘mass’’ presumably closely related to the Casimirw x. ŽC above, cf. 16 in a large class of nonsupersym-

. Ž .metric models 1 coupled to matter fields. Furtherinvestigations of 2d dilatonic supergravity theories,including generalizations to theories with nontrivialtorsion and a comparison to the existing literaturew x w x17 is in preparation 14 .

Acknowledgements

The author gratefully acknowledges discussionswith M. Ertl, C. Gutsfeld, W. Kummer, D. Royten-berg and P. Schaller.

References

w x Ž .1 T. Banks, M. O’Loughlin, Nucl. Phys. B 362 1991 649;Ž .S.D. Odintsov, I.L. Shapiro, Phys. Lett. B 263 1991 183;

D. Louis-Martinez, J. Gegenberger, G. Kunstatter, Phys.Ž .Lett. B 321 1994 193.

w x Ž .2 Y. Park, A. Strominger, Phys. Rev. D 47 1993 1569.w x Ž .3 P.S. Howe, J. Phys. A 12 1997 393.w x Ž .4 P. Schaller, T. Strobl, Mod. Phys. Lett. A 9 1994 3129; T.

Strobl, Poisson Structure Induced Field Theories and Modelsof 1q1 Dimensional Gravity, PhD thesis, Vienna 1994; P.Schaller, T. Strobl, hep-thr9411163; A. Alekseev, P.

Ž .Schaller, T. Strobl Phys. Rev. D 52 1995 7146. For older

Ž .related work cf. N. Ikeda, Ann. Phys. 235 1994 435, P.Ž .Schaller, T. Strobl, Lecture Notes in Physics 436 1994 98,

Eds. A. Alekseev et al., Springer. For some recent progresscf. A.S. Cattaneo, G. Felder, A Path Integral Approach to theKontsevich Quantization Formula mathr9902090.

w x5 P. Schaller, T. Strobl, Lecture Notes in Physics 469, p.321-333,’Low-Dimensional Models in Statistical Physics andQuantum Field Theory’, Eds. H. Grosse, L. Pittner, Springer1996, hep-thr9507020.

w x6 M. Henneaux, C. Teitelboim, Quantization of Gauge Sys-tems, Princeton University Press 1992.

w x Ž .7 T. Klosch, T. Strobl, Class. Quantum Grav. 13 1996 965,¨Ž . Ž . Ž .Corr. 14 1997 825; ibid. 13 1996 2395; ibid. 14 1997

1689.w x Ž .8 J.M. Izquierdo, Phys. Rev. D 59 1999 084017.w x Ž .9 N. Ikeda, Int. J. Mod. Phys. A 9 1994 1137; Ann. Phys.

Ž .235 1994 435.´w x Ž .10 D. Sullivan, Inst. des Haut Etud. Sci. Pub. Math. 47 1977

269.w x Ž .11 P. Schaller, T. Strobl, Phys. Letts. B 337 1994 266; H.J.

Matschull, On the relation between 2q1 Einstein gravity andChern Simons theory, gr-qcr9903040.

w x Ž .12 M.O. Katanaev, I.V. Volovich, Phys. Lett. B 175 1986 413;Ž .M.O. Katanaev, J. Math. Phys. 31 1990 882; W. Kummer,

Ž .D.J. Schwarz, Phys. Rev. D 45 1992 3628; P. Schaller, T.Ž .Strobl, Class. Quant. Grav. 11 1994 331.

w x13 W. Kummer, private communication.w x14 M. Ertl, W. Kummer, T. Strobl, in preparation.w x Ž .15 A. Weinstein, J. Diff. Geom. 18 1983 523.w x Ž .16 W. Kummer, S. Lau, Annals Phys. 258 1997 37; J. Gegen-

berg, G. Kunstatter, D. Louis-Martinez, Phys. Rev. D 57Ž .1998 3537.

w x Ž .17 S. Nojiri, I. Oda, Mod. Phys. Lett. A 8 1993 53. A. Bilal,Ž .Phys. Rev. D 48 1993 1665. A.H. Chamseddine, Phys.

Ž .Lett. B 258 1991 97. D. Cangemi, M. Leblanc, Nucl. Phys.Ž . Ž .B 420 1994 363. V.O. Rivelles, Phys. Lett. B 321 1994

189. M. Ertl, M.O. Katanaev, W. Kummer, Nucl. Phys. BŽ .530 1998 457.

target-superspace in 2d dilatonic supergravity

Documents