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SOLUTIONS IN STRING THEORY

arXiv:hep-th/9108011v1 21 Aug 1991TIFR/TH/91-37August, 1991TWISTED BLACK p-BRANESOLUTIONS IN STRING THEORYAshoke Sen⋆Tata Institute of Fundamental Research, HomiBhabha Road, Bombay 400005, IndiaABSTRACTIt has been shown that given a classical background in string theory whichis independent of d of the space-time coordinates, we can generate other classicalbackgrounds by O(d) ⊗O(d) transformation on the solution. We study the effectof this transformation on the known black p-brane solutions in string theory, andshow how these transformations produce new classical solutions labelled by extracontinuous parameters and containing background antisymmetric tensor field.Prefitem⋆e-mail address: SEN@TIFRVAX.BITNET1

In a previous paper [1] we showed that in string theory, if we have an exactclassical solution which is independent of d of the space-time coordinates, thenwe can perform an O(d) ⊗O(d) transformation on the solution, which produces anew configuration of string field, satisfying the classical equations of motion to allorders in the string tension α′. This generalised the result found by Meissner andVeneziano [2] [3] to leading order in α′.

In this paper we shall study the effect of thisO(d) ⊗O(d) transformation on the known black p-brane solutions in string theory,and obtain new classical solutions in string theory labelled by extra continuousparameters.We begin by recalling the general argument of ref. [1] and also by giving ageneralised version of the analysis of ref.[3].

In the language of string field the-ory, looking for solutions which are independent of d of the coordinates (say Y i,1 ≤i ≤d) corresponds to looking for a string state |Ψ⟩carrying zero momentum inthese d directions. Restricting string states of this type gives us the reduced stringfield theory action which governs the classical dynamics in this subspace.

Thisreduced action, in turn, is expressed in terms of correlation functions of vertexoperators carrying zero Y i momentum in the appropriate conformal field theory.In the part of the conformal field theory described by the free scalar fields Y i, thecorrelation functions factorise into the left and the right part, and each part is sep-arately invariant under the rotation group O(d) which acts on these d coordinates.Thus the reduced action has an O(d) ⊗O(d) symmetry, which implies that givena classical solution of the string field theory equations of motion in the subspacecarrying zero Y i momemta, we can generate other solutions by acting with thisO(d) ⊗O(d) transformation. Of this the diagonal O(d) subgroup simply corre-sponds to rotating the solution in the d dimensional space, the other generators ofO(d)⊗O(d) acting on the solution produces inequivalent solutions in general, sinceO(d)⊗O(d) is not a symmetry of the full action.

Although the above argument wasgiven in the context of string field theory, note that the argument is independentof the detailed form of string field theory, and hence the final result is expected tohold for fermionic string theories as well. Note that if one of the coordinates Y m is2

time-like, the O(d)⊗O(d) transformation gets replaced by O(d−1, 1)⊗O(d−1, 1).The low energy manifestation of this symmetry had been discovered in ref. [3].We shall briefly reproduce this analysis in a somewhat more general form than theone in which it was discussed in ref.[3].

Let us consider the low energy effectiveaction of string theory in D space-time dimension. This can be obtained eitherfrom the study of the S-matrix elements in string theory (see ref.

[4] and referencestherein) or from the calculation of the β-function of the σ-model [5 −8] , and isgiven by,S = −ZdDx√det Ge−Φ(Λ −R(D)(G) + 112HµνρHµνρ −Gµν∂µΦ∂νΦ)(1)where Gµν, Bµν and Φ denote the graviton, the dilaton, and the antisymmetrictensor fields respectively, Hµνρ = ∂µBνρ + cyclic permutations, R(D) denotes theD dimensional Ricci scalar, and Λ is the cosmological constant equal to (D−26)/3for bosonic string and (D −10)/2 for fermionic string. (For simplicity we have setthe other background massless fields, which appear in fermionic string theories, tozero.) Let us now split the coordinates Xµ into two sets ˆY m and ˜Y α (1 ≤m ≤d,1 ≤α ≤D −d) and consider backgrounds independent of ˆY m. Let us furtherconcentrate on backgrounds where Gmα = Bmα = 0, i.e.

to backgrounds of theform G = ˆGmn00˜Gαβ!, B = ˆBmn00˜Bαβ!. In this case, after an integrationby parts, the action (1) can be shown to take the form:−Zdd ˆYZdD−d ˜Ypdet ˜Ge−χhΛ −˜Gαβ ˜∂αχ˜∂βχ −18˜GαβTr(˜∂αML˜∂βML)−˜R(D−d)( ˜G) + 112˜Hαβγ ˜Hαβγi(2)where,L = 0110!

(3)χ = Φ −lnpdet ˆG(4)3

and,M = ˆG−1−ˆG−1 ˆBˆB ˆG−1ˆG −ˆB ˆG−1 ˆB! (5)If the coordinates ˆY m are all of Euclidean signature, this action is invariant underan O(d) ⊗O(d) transformation on ˆG, ˆB and Φ, given by,M →14 S + RR −SR −SS + R!M ST + RTST −RTST −RTST + RT!

(6)χ →χ,˜Gαβ →˜Gαβ,˜Bαβ →˜Bαβ(7)where S and R are O(d) rotation matrices, and ST, RT denote the transpose ofthe matrices S, R. In fact, the action is invariant under a general O(d, d) trans-formation which leaves the matrix L invariant [3], but the members of the O(d, d)algebra outside the O(d) ⊗O(d) algebra can be shown to generate pure gauge de-formations [1] if the coordinates ˆY m are non-compact. The transformations givenin eqs.

(6), (7) were shown to agree with the O(d) ⊗O(d) transformation on thestring fields to linearised order. Corrections to the action given in eq.

(1) includinghigher derivative terms are expected to change the transformation laws given ineqs. (6), (7), but the existence of a modified transformation is guaranteed by thestring field theory argument given before.Since in the above analysis we have explicitly set Gmα and Bmα to zero, wemust make sure that the equations of motion obtained by varying the action withrespect to these fields is also O(d) ⊗O(d) invariant.

In this case it is easy to seethat for the backgrounds considered here these equations of motion are satisfiedidentically, hence their O(d) ⊗O(d) invariance is obvious.The above symmetry gets modified to O(d −1, 1) ⊗O(d −1, 1) when one ofthe coordinates ˆY m (say ˆY 1) is time-like. In this case the transformation laws (5)4

get modified to,M →14 η(S + R)ηη(R −S)(R −S)ηS + R!M η(ST + RT )ηη(RT −ST )(RT −ST)η(ST + RT )! (8)where η =diag(−1, 1, .

. .

1), and S and R are O(d −1, 1) matrices satisfying,SηST = η,RηRT = η(9)In all the examples we shall consider, one of the coordinates ˆY m will be time-like, and hence the relevant group will be O(d −1, 1) ⊗O(d −1, 1). We shall nowexamine the transformation laws of various fields under the O(d−1, 1)⊗O(d−1, 1)group in some detail.

In component form the transformed fields ˆG′ij, ˆB′ij and Φ′are given by,( ˆG′−1)ij =14η(S + R)η ˆG−1η(ST + RT)η + η(R −S)( ˆG −ˆB ˆG−1 ˆB)(RT −ST)η−η(S + R)η ˆG−1 ˆB(RT −ST)η + η(R −S) ˆB ˆG−1η(ST + RT )ηijˆB′ij =14(R −S)η ˆG−1η(ST + RT )η + (S + R)( ˆG −ˆB ˆG−1 ˆB)(RT −ST)η+ (S + R) ˆB ˆG−1η(ST + RT )η −(R −S)η ˆG−1 ˆB(RT −ST )η ˆG′ijΦ′ =Φ −12 ln det ˆG + 12 ln det ˆG′(10)The transformation laws of ˆG and ˆB may be expressed in a compact form bydefining the matrix:C ≡ˆG −η ˆG−1η −ˆB ˆG−1 ˆB + ˆB ˆG−1η + η ˆG−1 ˆB(11)The transformation law of C then takes a simple form:C′ = SCRT(12)Let us now turn to specific examples. We start from a simple solution of the5

equation of motion (1) of the form [9] [10]:⋆ds2 = −tanh2 Qr2 dt2 + dr2 +d−1Xi=1dXidXiBµν =0Φ = −ln cosh2 Qr2 + Φ0(13)where Φ0 is an arbitrary parameter, and,Q =√−Λ =r25 −d3for bosonic string=r9 −d2for fermionic string(14)Since the solution is independent of d −1 space-like coordinates, it can be called ad−1 brane solution in the language of ref.[12]. Note that a priori we cannot ignorethe contribution from the higher derivative terms in the action since the scale ofspatial variation of the solution is set by Q which is of order 1.

This problemmay be avoided if instead of just taking d −1 scalar (super)-fields Xi, we taked −1 scalar (super)-fields together with a (super)-conformal field theory of centralcharge c (32c). In this case Q will be given by,Q =r25 −d −c3for bosonic string=r9 −d −c2for fermionic string(15)and we may consider a situation where d+c ≃25 (d+c ≃9) so that Q is small.

Inthis case the effect of higher derivative terms in the effective action will be small,at least away from any singularity.⋆The O(d −1, 1) ⊗O(d −1, 1) transformation may also be applied to the more general classof solutions discussed in ref. [11], but we shall not discuss it here.6

The solution given in eq. (13) has the property that it is independent of thecoordinates t and Xi.

Thus we can make an O(d−1, 1)⊗O(d−1, 1) transformationon the fields so that the new configuration will also be a solution of the equationsof motion. For the background defined in eq.

(13) the tensor Cij defined in eq. (11)takes the form:Cij = −δi1δj1(tanh2 Qr2 −coth2 Qr2 )(16)Thus from eq.

(12) we get,C′ij = −Si1Rj1(tanh2 Qr2 −coth2 Qr2 )(17)In order to study the set of inequivalent classical solutions, we note that under aLorentz transformation, C′ →MC′MT where M is an O(d −1, 1) matrix. Thiscorresponds to changing S to MS and R to MR. Thus,Si1 →MikSk1,Ri1 →MikRk1(18)Both the vectors Si1 and Ri1 are normalised to −1 with respect to the metric ηsince they form the first columns of O(d −1, 1) matrices.

Thus by choosing anappropriate M we may bring the vectors Si1 and Rj1 into the form:S11 = cosh θ,S21 = −sinh θ,Si1 = 0 for i ≥3R11 = cosh θ,R21 = sinh θ,Ri1 = 0 for i ≥3(19)For this choice of S and R eq. (10) gives the following form of the transformed fields7

ˆG′, ˆB′ and Φ′:ˆG′ =−sinh2(Qr/2)cosh2(Qr/2)+sinh2 θ00· · ·00cosh2(Qr/2)cosh2(Qr/2)+sinh2 θ0· · ·0001· · ·0···· · ······ · ··000· · ·1ˆB′ =cosh θ sinh θcosh2(Qr/2) + sinh2 θ010· · ·0−100· · ·0000· · ·0···· · ······ · ··000· · ·0Φ′ = −ln(cosh2(Qr/2) + sinh2 θ) + Φ0(20)Thus the full metric now takes the form:ds2 = −sinh2(Qr/2)cosh2(Qr/2) + sinh2 θdt2 + dr2 +cosh2(Qr/2)cosh2(Qr/2) + sinh2 θ(dX1)2+d−1Xi=2dXidXi(21)and has a coordinate singularity at r = 0. This singularity may be removed bychoosing a new coordinate system:u = sinh(Qr/2)eQt/(2 cosh θ),v = sinh(Qr/2)e−Qt/(2 cosh θ)(22)8

In this coordinate system the metric takes the form:ds2 = 2Q2dudvcosh2 θuv + cosh2 θ+1uv + 1−1Q2sinh2 θ(uv + 1)(uv + cosh2 θ)(v2du2 + u2dv2)+uv + 1uv + cosh2 θ(dX1)2 +d−1Xi=2dXidXi(23)Thus we see that at r = 0, i.e. at u = 0 or v = 0, the metric does not haveany singularity in this new coordinate system.

In fact, in this coordinate systemthe metric has finite non-zero eigenvalues in the region uv > −1, u, v finite. Inorder to make sure that the solution is non-singular at r = 0, we must also checkthat the dilaton, as well as the field strength associated with the anti-symmetrictensor field are non-singular at r = 0.

The dilaton given in eq. (20) is clearly non-singular at r = 0.

The antisymmetric tensor field strength Hµνρ = (∂µBνρ + cyclicpermutations) takes the form:Hrt1 = ∂rBt1 = −Q cosh θ sinh θ cosh(Qr/2) sinh(Qr/2)(cosh2(Qr/2) + sinh2 θ)2(24)When transformed to the u −v coordinate system, this becomes,Huv1 = 2Qcosh2 θ sinh θ(uv + cosh2 θ)2(25)which is clearly non-singular at u = 0 or v = 0.So far we have discussed the part of O(d−1, 1)⊗O(d−1, 1) transformation thatis connected to identity. As was discussed in ref.

[1], the effect of the disconnectedpart of the O(d) ⊗O(d) transformation on the solution is to expand the solutionin powers of eQr and reverse the sign of all the odd powers of eQr. This transforms9

the solution given in eq. (20) to,ˆG′ =−cosh2(Qr/2)sinh2(Qr/2)−sinh2 θ00· · ·00sinh2(Qr/2)sinh2(Qr/2)−sinh2 θ0· · ·0001· · ·0···· · ······ · ··000· · ·1ˆB′ = −cosh θ sinh θsinh2(Qr/2) −sinh2 θ010· · ·0−100· · ·0000· · ·0···· · ······ · ··000· · ·0Φ′ = −ln(sinh2(Qr/2) −sinh2 θ) + Φ0(26)Thus in this case the solution is singular at r = 2θ/Q.As was shown by Witten [10] (see also refs.

[13 −17] ), for the solution given ineq. (13), after modification due to higher order terms have been taken into account,the coordinates r and t together describe a solvable conformal field theory basedon the gauged SL(2,R) WZW theory, where a non-compact subgroup of SL(2,R)is gauged.

The level k of the WZW theory is related to Q by the relation:2 + 3Q2 =3kk −2 −1(27)so that the Q →0 limit corresponds to k →∞. It is thus natural to ask whetherthere are solvable conformal field theories which correspond to the solution givenin eq.(20).

In fact, conformal field theories associated with Euclidean continuationof these solutions have been found in ref. [18] by gauging a linear combination ofa compact U(1) generator of SL(2,R) and the U(1) currents generated by ∂Xi,¯∂Xi.

By taking independent linear combinations of the U(1) currents in the left10

and the right sector one can get the Euclidean continuation of the backgroundgiven in eq. (20) (with θ replaced by iθ).

This is not surprising, since the originalO(d) ⊗O(d) symmetry was due to the freedom of independent rotations in the leftand the right sector.Note that if we take some of the directions Xi to be compact, then even thediagonal O(d −1, 1) subgroup, acting on the solution, generates new solutions,since this is no longer a symmetry of the full theory. This has been exploited inref.

[18] to get new exact solutions of string theory. In the analysis of this paper weshall restrict ourselves to the non-compact case.We shall now consider a second example, where the solution, in general, doesnot correspond to a solvable conformal field theory.

This is the five-brane solutionof the low energy effective field theory in ten dimensional superstring or heteroticstring theory (or, equivalently, the 21-brane solution in 26 dimensional bosonicstring theory) described in ref.[12]. (See also refs.

[19 −29] for related work.) In aparticular coordinate system the solution takes the form [20]:ds2 = −tanh2 rdt2 + [M + δ2(cosh2 r −12)](dr2 + dΩ23) +5Xi=1dXidXiΦ = lnM + δ2(cosh2 r −12)δ2 cosh2 r+ Φ0H =2Q0ǫ3(28)where M and Φ0 are independent continuous parameters,Q0 ≡rM2 −14δ4(29)is quantised, dΩ3 is the line element on the 3-sphere, and ǫ3 is the volume form onthe same 3-sphere.⋆(For bosonic string theory the sum over i runs from 1 to 21⋆The dilaton Φ and the field strength H in our convention are related to that in the conventionof ref.

[12] by a factor of 2.11

instead of from 1 to 5.) Although the solution does not appear to be flat in theasymptotic region r →∞, after a coordinate transformation:y = δ cosh r(30)one gets a metric which reduces to 10 dimensional Minkowski metric in the regionr →∞.Let us note that the solution is independent of the coordinates t and Xi.Thus, as before, we can construct new solutions through O(d −1, 1) ⊗O(d −1, 1)transformation (here d = 6).

Counting of independent parameters labelling thetransformed solution proceeds exactly in the same way as before, and we are leftwith one independent parameter θ. The new solution obtained in this way is givenby:ds2 = −sinh2 rcosh2 r + sinh2 θdt2 +cosh2 rcosh2 r + sinh2 θ(dX1)2+ (M + δ2(cosh2 r −12))(dr2 + dΩ23) +5Xi=2dXidXiΦ = ln M + δ2(cosh2 r −12)δ2(cosh2 r + sinh2 θ)+ Φ0H(int) =2Q0ǫ3ˆB =cosh θ sinh θcosh2 r + sinh2 θ010· · ·0−100· · ·0000· · ·0···· · ······ · ··000· · ·0(31)where H(int) denotes the field strength associated with the antisymmetric tensorfield in the internal 3-sphere, and ˆB denotes the components of the antisymmetrictensor field in the six dimensional space spanned by the coordinates t and Xi.

Asbefore, it is easy to see that the point r = 0 represents a coordinate singularity, and12

the field strength associated with the antisymmetric tensor field is regular at r = 0.Thus we see that using the O(d−1, 1)⊗O(d−1, 1) transformation we can generatea solution of the low energy effective field theory equations of motion characterisedby three continuous parameters Φ0, M and θ, and one discrete parameter Q0.In conclusion, we have demonstrated in this paper that the O(d)⊗O(d) (O(d−1, 1) ⊗O(d −1, 1)) symmetry may be used effectively to generate new classicalsolutions in string theory from the known ones.Although we have consideredonly a few examples, it is clear that this transformation can be applied on otherknown classical solutions as well. It will be interesting to study the the physicalconsequences of continuous parameter family of solutions in string theory impliedby the O(d) ⊗O(d) symmetry, particularly on the cosmology of the early universein string theory [30].Finally, note that for suitable backgrounds, the O(d−1, 1)⊗O(d−1, 1) symme-try of the reduced action may be extended to O(d −1, 1) ⊗O(d + 15, 1) symmetryin the case of heterotic string theory, due to the possibility of including the 16 in-ternal coordinates in the rotation.

This rotation, in general, will produce a chargedblack hole [21] from an uncharged one. Thus besides producing new solutions, ourmethod also opens up the possibility of relating different known solutions in stringtheory by the twisting procedure.13

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