In this paper we explicitly prove the invariance of the time-dependent string
gravity Lagrangian with up to four derivatives under the global $O(d,d)$
symmetry.Comment: 10 pages, LATEX file, several misprints corrected, two references
  added, the final version submitted to Phys.Lett.

Bakas

Gasperini

Giveon

Gross

Hassan

Hull

Jack

Krzysztof A. Meissner

Mavromatos

Meissner

Metsaev

Panvel

Shapere

Tseytlin

Veneziano

Zanon

Physics Letters B

English

arXiv

Crossref

Symmetries of higher-order string gravity actions

In this paper we explicitly prove the invariance of the time-dependent string gravity Lagrangian with up to four derivatives under the global $O(d,d)$ symmetry

Meissner, K A

CERN Document Server

CERN-TH/96-291hep-th/9610131Symmetries of higher-order string gravity actionsKrzysztof A. Meissner1CERN, 1211 Geneva 23, Switzerland2ABSTRACTIn this paper we explicitly prove the invariance of the time-dependent string gravityLagrangian with up to four derivatives under the global O(d; d) symmetry.CERN-TH/96-291October 19961Partially supported by a Polish KBN grant.2Permanent address: Institute for Theoretical Physics, Ho_za 69, 00-689 Warszawa, Poland.E-mail: meissner@cern.ch1 IntroductionGlobal, continuous symmetries not connected with the dieomorphism group are veryrare in gravitational systems. The rst example was discovered by Ehlers for the caseof four-dimensional pure gravity with one Killing vector. Later, it was shown byGeroch that in the case of two Killing vectors the symmetry gets enhanced to aninnite Kac-Moody algebra. In string theory, the gravitational multiplet contains notonly the graviton, but also a scalar (dilaton) and the antisymmetric tensor (oftenreferred to as torsion). The symmetries of the Ehlers and Geroch type were alsoshown in this case [4]. Another type of symmetry in such a system was discoveredin [1] (without the torsion eld the discrete symmetry of the action was discoveredin [2] and [3]). It was shown that, for the case of elds depending only on time inan arbitrary number of dimensions (1 time, d space dimensions), the lowest-orderLagrangian exhibits continuous, global O(d; d) symmetry. The symmetry was laterextended to the presence of matter [5] or gauge elds [6] and seems to be presentin a large number of string-inspired theories containing gravity. In [1] argumentwas given that the symmetry should be present to all orders in 0in the -modelexpansion (another argument was given in [7]). In [3] it was argued that for thecase without the torsion eld there should be corrections to elds in the next orderin 0to ensure vanishing of the -functions and in [8] it was demonstrated on onespecic example. Since the inclusion of the next-order terms (like curvature squared)can be very important for the stability of the solutions (as was recently discussed forthe case with no torsion in [9]), it is the purpose of the present paper to show thatthe O(d; d) symmetry is explicit in the order 0Lagrangian of gravity coupled to thedilaton and the antisymmetric tensor elds. There is quite a number of authors thathave calculated the higher-order eective action coming from string amplitudes orfrom loop calculations in the -models (see for example [10, 11]) that sometimes donot agree with one another. We assumed throughout this paper that the result of [10]is correct, and it turned out that with this assumption the O(d; d) symmetry of thequartic action can be proved.2 The O(d; d) symmetry in the lowest orderIn [1] it was shown that the lowest-order string gravity (gravity coupled to dilatonand the antisymmetric tensor) Lagrangian, for elds depending only on cosmic time,possesses explicit O(d; d) invariance, where d is the number of space dimensions. Wewill recall here this construction to set the notation. The lowest-order Lagrangianreads (throughout this paper we use the string frame with e 2out front, since the2symmetry is most simply realized there) (0)=Zdd+1xp g e 2R + 4(@)2 112H2: (2.1)Our metric is ( ;+; : : : ;+) andR= @   : : : ; R= R; H= @B+ cyclic (2.2)When elds depend only on time, it is possible to bring g and B to the formg=  1 00 G(t)!; B= 0 00 B(t)!: (2.3)It was shown in [1] that the action (2.1) can then be rewritten as 0=  Zdte _2+18Tr[_M0_M0]; (2.4)where = 2 12ln detG; (2.5) is the metric for the O(d; d) group in non-diagonal form: = 0 11 0!(2.6)andM0= G 1 G 1BBG 1G BG 1B!: (2.7)This M0has two important properties [1]: it is symmetric and it belongs to the O(d; d)group:MT0= M0; M0M0= : (2.8)The action (2.4) is explicitly symmetric under the action of the O(d; d) group:M0! TM0;  ! ; (2.9)where  belongs to the O(d; d):T = : (2.10)The general O(d; d) element connected to the identity can be written as: = exp A1A2A3 AT1!AT2=  A2; AT3=  A3: (2.11)33 The symmetry in the next order without torsionWe start with the following form of fourth order in derivatives action in the stringframe (formula 3.24 in [10])  =Zdd+1xp g e 2R + 4(@)2 112H2 00RR 12RHH+124HHHH 18HHHH+ O(02)(3.1)(0=  18for the heterotic string,  14for the Bose string and 0 for the superstring).This is the simplest possible form of the string eective action. If one makes localredenitions of elds, it does not change the equations of motion (in the redenedelds); however, the symmetry can be easily seen for one choice but impossible toguess for another. Thus we have to try all possible redenitions to see whether we canbring the action to some suitable form. There are two guidelines for the search. Therst one is that the action (when expressed in terms of time derivatives of elds andwith all integrations by parts used) contains only rst derivatives of elds. It turnsout that this can always be done. The second one is that the whole action can bewritten in terms of  and M0dened before (but with possible corrections of order0), since the symmetry is then explicit.We start to show the techniques involved with the simpler case of vanishing H(then, of course, we do not have the full O(d; d) symmetry but only some discretesubgroup); temporarily, we use the Lagrangian:  =Zdd+1xp g e 2nR + 4(@)2  00RR+ O(02)o(3.2)Requiring only rst time derivatives allows for the four structures of order 0:Zdd+1xp ge 2a1R2GB+ a2R 12gR@@ + a32(@)2+ a4(@)4(3.3)where R2GBis the Gauss-Bonnet termR2GB= RR  4RR+ R2(3.4)In order to transform (3.2) to the form (3.3) we use the redenitionsg= 0[b1R+ b2@@ + g(b3R + b4(@)2+ b52)] = 0[c1R + c2(@)2+ c32]: (3.5)4Under these redenitions the action (3.2) is corrected by  =  Zdd+1xp ge 2R+ 2D@ 12g(R + 42  4(@)2)g++ 2(R + 42  4(@)2)o(3.6)Plugging (3.5) into (3.6) we get the form (3.3) wheng=  400R =  1200R + 200(@)2; (3.7)the action then becomes (1)=Zp ge 200 R2GB+ 16R 12gR@@  162(@)2+ 16(@)4(3.8)In order to write the action for elds depending only on cosmic time, we introducethe matrixW := G 1_G: (3.9)Then we have  =Zdte  _2+14TrW2 0018TrW4 116(TrW2)2+13TrW3_ +12(TrW2)_2 13_4:(3.10)It is now necessary to list all possible O(d; d) invariants with rst time derivatives.There are only four of them:A1Tr(_M0)4+ A2(Tr(_M0)2)2+ A3Tr(_M0)2_2+ A4_4: (3.11)Since (we still suppress the B-dependence!)Tr(_M0)4= 2TrW4; Tr(_M0)2=  2TrW2; (3.12)we see that there is one term in the action (3.10) that does not belong to this class.In order to make the symmetry explicit we change the denition of M by adding toM0a term of order 0:M = M0  00  G 1_GG 1_GG 100_GG 1_G!(3.13)The redened M satises (to order 0) the properties (2.8).5With this new denition the total action can be rewritten as (but still without theantisymmetric tensor in M):  =Zdte  _2 18Tr(_M)2 00116Tr(_M)4 164(Tr(_M)2)2 14(Tr(_M)2)_2 13_4: (3.14)4 The full O(d; d) symmetryWe now set to prove that (3.14) is actually the proper form of the action after inclusionof the antisymmetric tensor. We have to try all possible redenitions of the action(3.1) that give only rst time derivatives. We may now use, in addition to (3.7), thefollowing redenitions:g= 00(b6H2+ b7GH2) = 00c4H2B= 00(d1DH+ d2H@); (4.1)whereH2= HH; and H2= HH(4.2)and the Lagrangian changes as follows:  =  Zdd+1xp ge 2fR+ 2D@ 14(H2) 124g(12R + 482  48(@)2 H2)g++16(12R + 482  48(@)2 H2)+12(2@H DH)B: (4.3)The requirement of only rst time derivatives allows for, in addition to (3.3), thefollowing structures: (2)=Zdd+1xp ge 2a1R2GB+ a2R 12gR@@+ a32(@)2+a4(@)4+ a5RHH  2RH2+13RH2+ a6H2(@)2+a7D@H2 132H2+ a8HHHH+ a9H2H2+a10(H2)2+ a11H2@@ + a12DHH@ +162H2: (4.4)6Starting from the action (3.1) and trying dierent redenitions, we nally arrive atthe following form of the action:  =Zp ge 2R + 4(@)2 112H2(4.5)+00 R2GB+ 16R 12gR@@  162(@)2+ 16(@)4+12RHH  2RH2+13RH2  2D@H2 132H2 23H2(@)2 124HHHH+18H2H2 1144(H2)2:Making the eld redenitions of order 0is equivalent to all possible substitutionsof the lowest-order equations of motion in the higher-order Lagrangian. The lowest-order equations of motion are:R+ 2D@ 14H2= 0; 2  2(@)2+112H2= 0R + 42  4(@)2 112H2= 0; DH  2H@ = 0 (4.6)Using them and the Bianchi identities for curvature and torsion it is relatively straight-forward to show the equivalence \on-shell" of (4.5) with (3.1). The action (4.5) cor-responds to the choice in (4.1):b6= 0; b7= 0; c4=  124; d1= 0; d2= 4 (4.7)To write the result (4.5) for the case of elds depending only on time, we introduce(in addition to W dened before) the matrix Y :Y := G 1_B: (4.8)We have  =Zdte  _2+14TrW2 14TrY2+00 18TrW4+116(TrW2)2 13TrW3_ 12(TrW2)_2+13_4+12Tr(W2Y2) +14Tr(WYWY ) 18TrW2TrY2+_Tr(WY2) +12_2TrY2+38TrY4+116(TrY2)2: (4.9)In order to compare it to the O(d; d) symmetric form, we need the expressionsTr(_M0)4= 2TrW4+ 2TrY4  8Tr(W2Y2) + 4Tr(WYWY )Tr(_M0)2_2= ( 2TrW2+ 2TrY2)_2(Tr(_M0)2)2= ( 2TrW2+ 2TrY2)2: (4.10)7We see that our result (4.9) contains a number of terms that are not of this form, sowe redene M :M = M0  00  T!; (4.11)where =  G 1_GG 1_GG 1+ G 1_BG 1_BG 1 = G 1(_GG 1_G _BG 1_B)G 1B  G 1(_GG 1_B +_BG 1_G) (4.12) =_GG 1_G _BG 1_B   (_GG 1_B +_BG 1_G)G 1B B( G 1_GG 1_GG 1+ G 1_BG 1_BG 1)B   BG 1(_GG 1_B +_BG 1_G):To order 0the redened M satises (2.8), so that the redenition is itself (time- andeld-dependent) an O(d; d) rotation. To make the properties (2.8) manifest, we writethe redenition (4.11) asM = !TM0!; (4.13)where ! is in the form (2.11), with:A1=  00 12_GG 1_GG 1+12_BG 1_BG 1A2=  00 _GG 1_B  _BG 1_G +12(_GG 1_G _BG 1_B)G 1B+12BG 1(_GG 1_G _BG 1_B)A3= 0: (4.14)With this new M , the action (4.9) is exactly in the form anticipated before in eq.(3.14):  =ZdtNe  _2 18Tr(_M)2 00N2116Tr(_M)4 164(Tr(_M)2)2 14(Tr(_M)2)_2 13_4): (4.15)We have introduced the lapse function N (in the rst order in derivatives action, it isa trivial replacement dt ! Ndt), since it gives one more equation of motion (calledthe \g00" equation in [1]) and only afterwards we put N to 1.This action is explicitly O(d; d)-invariant under (2.9). It looks, however, like a littlemiracle that the coecients in (3.1) coming from the string amplitudes are exactlysuch that they give the explicit O(d; d) symmetry of (4.15). In comparison with thelowest-order case, now the O(d; d) symmetry acts in a more complicated way, as arotation of not only elds but elds with their derivatives.8The form of the action (3.8) needed to exhibit the symmetry is remarkably thesame as the unique (\o-shell") form of the action found in [13] (eq. (20) there).The comparison of the full action (4.5) with the result of [14] is more dicult sincethere are apparent contradictions between this reference and [10, 13]. However, ourredenition (4.7) is exactly the same as the redenition used in [14] and we suspectthat the \o-shell" conformal invariance also leads to the unique action (4.5) which isa remarkable feature pointing out to a deeper structure behind the O(d; d) symmetry.Since the O(d; d) symmetry is continuous and global, it has an associated conservedcurrent, which means, for a theory depending only on time, that the current shouldbe constant (it is an \integrated once" equation of motion for M). In analogy to [1]we call this constant A:A = const = e M_M + 20012M(_M)3 18M_MTr(_M)2 M_M_2(4.16)where AT=  A and AM =  MA ([1]).The N equation reads:0 =  _2 18Tr(_M)2 300116Tr(_M)4 164(Tr(_M)2)2 14(Tr(_M)2)_2 13_4(4.17)Equations (4.16) and (4.17) are non-linear in elds but (as a result of the existenceof symmetry) rst order in derivatives. The analysis of these equations and theirsolutions will appear in a subsequent publication [15].AcknowledgementsI am very grateful to Gabriele Veneziano for earlier collaboration, for discussionson O(d; d) symmetry, and for a critical reading of the manuscript. I wish to thankIoannis Bakas and Maurizio Gasperini for discussions. I also thank the organizers ofthe ESI Workshop in Vienna, in August 1996, where part of this work has been done.9References[1] K.A.Meissner and G.Veneziano, Phys. Lett. B267 (1991) 33,Mod. Phys. Lett. A6 (1991) 3397.[2] G.Veneziano, Phys. Lett. B265 (1991) 287.[3] A.A. Tseytlin Mod. Phys. Lett A6 (1991) 1721.[4] I.Bakas, Nucl. Phys. B428 (1994) 374.[5] M.Gasperini and G.Veneziano Phys. Lett. B277 (1992) 256.[6] S.F.Hassan and A.Sen Nucl. Phys. B375 (1992) 103.[7] A.Sen, Phys. Lett. B271 (1991) 295.[8] J.Panvel, Phys. Lett. B284 (1992) 50.[9] M.Gasperini, M.Maggiore and G.Veneziano, \A non-singular Pre-Big-Bang Cos-mology?", CERN-TH/96-267.[10] R.R.Metsaev, A.A.Tseytlin, Nucl. Phys. B293 (1987) 385[11] D.J.Gross and J.H.Sloan, Nucl. Phys. B291 (1987) 41;C.M. Hull and P.K. Townsend, Phys. Lett. B191 (1987) 115;D.Zanon, Phys. Lett. B191 (1987) 354;I.Jack and D.R.T. Jones, Phys. Lett. bf B193 (1987) 449;K.A. Meissner, J. Pawe lczyk and S. Pokorski, Phys. Rev. D38 (1988) 1144.[12] A.Giveon, D.Friedan, E.J.Martinec and M.J. Perry, Nucl. Phys. B322 (1989)167;A.Shapere and F.Wilczek, Nucl. Phys. B320 (1988) 669.[13] N.E.Mavromatos and J.L.Miramontes, Phys. Lett. B201 (1988) 473[14] I.Jack and D.R.T.Jones, Nucl. Phys. 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Symmetries of higher-order string gravity actions

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