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TWISTING CLASSICAL SOLUTIONS

arXiv:hep-th/9109038v1 20 Sep 1991TIFR/TH/91-40September, 1991TWISTING CLASSICAL SOLUTIONSIN HETEROTIC STRING THEORYS. F. Hassan and Ashoke Sen⋆Tata Institute of Fundamental Research, HomiBhabha Road, Bombay 400005, IndiaABSTRACTWe show that, given a classical solution of the heterotic string theory whichis independent of d of the space time directions, and for which the gauge fieldconfiguration lies in a subgroup that commutes with p of the U(1) generators ofthe gauge group, there is an O(d) ⊗O(d + p) transformation, which, acting on thesolution, generates new classical solutions of the theory.

With the help of thesetransformations we construct black 6-brane solutions in ten dimensional heteroticstring theory carrying independent magnetic, electric and antisymmetric tensorgauge field charge, by starting from a black 6-brane solution that carries magneticcharge but no electric or antisymmetric tensor gauge field charge. The electric andthe magnetic charges point in different directions in the gauge group.Prefitem⋆e-mail addresses: FAWAD@TIFRVAX.BITNET, SEN@TIFRVAX.BITNET1

1. INTRODUCTIONIt has been shown previously [1 −5] that in any string theory, if we look forsolutions that are independent of d of the space-time coordinates ˆY m, then thespace of such solutions has an O(d) ⊗O(d) or O(d −1, 1) ⊗O(d −1, 1) symmetry,depending on whether the coordinates ˆY m have Euclidean or Minkowski signature.

(For definiteness, we shall assume from now on that the coordinates ˆY m haveMinkowski signature). Similar symmetries had been seen earlier in the context ofsupergravity theories [6], and invariance of classical equations of motion of the twodimensional σ model under such transformations of the background was shown inrefs.

[7][8] following an earlier work of Gaillard and Zumino [9]. Under the actionof this O(d −1, 1) ⊗O(d −1, 1) transformation a given solution is in generalmapped to an inequivalent solution.

In this paper we shall show that in heteroticstring theory, if we look for solutions that are independent of d of the space-timecoordinates, and for which the background gauge field lies in a subgroup thatcommutes with p of the U(1) generators of the gauge group, then the space of suchsolutions has an O(d −1, 1) ⊗O(d + p −1, 1) symmetry.† We shall also use thisO(d −1, 1) ⊗O(d + p −1, 1) transformation to generate new classical solutions ofheterotic string theory starting from the known ones.The plan of the paper is as follows. In sect.

2 we present general argumentsshowing the existence of O(d −1, 1) ⊗O(d + p −1, 1) symmetry in the heteroticstring theory for restricted class of backgrounds of the type mentioned above. Insect.

3 we study the manifestation of this symmetry in the low energy effectivefield theory. Section 4 contains application of this O(d −1, 1) ⊗O(d + p −1, 1)transformation on the known solutions of the heterotic string theory, namely, theblack p-brane solution in ten dimensions carrying magnetic charge [10].

We showthat by applying the O(d−1, 1)⊗O(d+p−1, 1) transformation on this solution wecan generate a black p-brane solution carrying electric, magnetic and antisymmetric† Invariance of the classical equations of motion in the two dimensional σ-model under suchtransformations of the background was discussed in ref. [7].2

tensor gauge field charge. We summarise our results in sect.

5.2. O(d −1, 1) ⊗O(d + p −1, 1) SYMMETRYIN HETEROTIC STRING THEORYThe origin of the O(d −1, 1) ⊗O(d −1, 1) symmetry discussed in refs.

[1]- [5]can be traced to the fact [3] that if we restrict to backgrounds that are independentof d coordinates ˆY m, then the interaction involving such backgrounds is governedby correlation functions of vertex operators carrying zero ˆY m momenta in the twodimensional field theory of d scalar fields ˆY m. Such correlation functions factoriseinto products of correlation functions in the holomorphic sector and the antiholo-morphic sector, each of which is separately invariant under the Lorentz transfor-mations involving the coordinates ˆY m. In other words, these correlation functionshave an O(d −1, 1) ⊗O(d −1, 1) symmetry.As a result, the action involvingˆY m independent background also has an O(d −1, 1) ⊗O(d −1, 1) symmetry. Themanifestation of this symmetry in the context of low energy effective field theorywas found in refs.

[1][2] (see also refs. [6][7][8]) and was applied in refs.

[4][5][11] togenerate new classical solutions in string theory from the known ones.In heterotic string theory, besides the usual space-time coordinates we also have16 internal coordinates which have only right moving (anti-holomorphic) compo-nent but no left moving (holomorphic) component. If we consider backgroundswhich are independent of d of the space-time coordinates, and carry zero momen-tum (and winding number) in p of the 16 internal coordinates, then the previousargument can be generalized easily to conclude that the space of such solutionsshould have an O(d −1, 1) ⊗O(d + p −1, 1) symmetry.

For the sake of clarity, weshall now present this argument in some detail.The argument is best presented in the language of string field theory, so letus first assume that there is an underlying string field theory that governs thedynamics of heterotic string theory, and that the vertices in this string field theory3

are given in terms of correlation functions in the conformal field theory describ-ing the first quantised heterotic string theory. We consider heterotic string theoryformulated in 10 dimensional flat space time, although the argument can easily begeneralised to the case where some of the dimensions are replaced by an arbitrary(1,0) super-conformal field theory of the correct central charge.

Let {Φa} denotethe set of basis states in the conformal field theory (including the ghost part). Thestring field is given by |Ψ⟩= Pa ψa|Φa⟩, with ψa’s as the dynamical variablesof the string field theory.

Then the general N point interaction vertex is givenby ψa1 . .

. ψaNfN(a1, .

. .

aN), where fN(a1, . .

. aN) denotes a quantity constructedout of an N point correlation function of some conformal transform of the fieldsΦa1, .

. .

ΦaN.‡ For describing the string field configurations which are independentof d of the space-time coordinates (say ˆY m) and carry zero momentum in p of theinternal directions associated with the coordinates ˇY R (say), only those compo-nents ψa will be non-zero, for which the corresponding basis states |Φa⟩have zeromomentum in these d space-time directions, and also zero momentum in these pinternal directions. A basis of such states in the conformal field theory can bechosen in the form |χl⟩⊗|¯χ¯l⟩⊗|Φ′a′⟩, where |χl⟩and ¯χ¯l⟩are the basis states in theholomorphic and the antiholomorphic sectors respectively of the conformal fieldtheory described by the coordinates ˆY m and ˇY R, and |Φ′a′⟩denote the basis ofstates in the conformal field theory describing the rest of the system.

The correla-tion functions involving these basis states on the sphere factorise into correlationfunctions involving the states |χl⟩, correlation functions involving the states |¯χ¯l⟩and correlation functions involving the states |Φ′a′⟩. Thus fN also has the factorizedform:fN(l1, ¯l1, a′1, .

. .

lN, ¯lN, a′N) = f(1)N (l1, . .

. lN)f(2)N (¯l1, .

. .

¯lN)f(3)N (a′1, . .

. a′N)(2.1)We now note that f(1) and f(2) are separately invariant under O(d −1, 1) andO(d −1 + p, 1) transformations respectively, which simply correspond to Lorentz‡ Such a representation for the string field theory action is known explicitly for the bosonicstring theory [12 −14] , but unfortunately not for the heterotic or the super-string theory.4

transformations acting on the coordinates ˆY m and ˇY R. (Although the general cor-relation functions in the conformal field theory describing the internal coordinatesˇY R has no rotational symmetry due to the fact that the torus is not invariant un-der an arbitrary rotation, in the zero momemtum sector the correlation functionsare completely ignorant of the compactification of the coordinates ˇY R, and as aresult the correlation functions are not only symmetric under a rotation amongthe coordinates ˇY R, but also the ones which mix ˇY R and ˆY m.) This, in turn,implies that restricted to such backgrounds, the string field theory action will havean O(d −1, 1) ⊗O(d + p −1, 1) symmetry, with the O(d −1, 1) transformationsacting on the index l of ψl,¯l,a′, and the O(d −1 + p, 1) transformations acting onthe index ¯l of ψl,¯l,a′.In the absence of a field theory describing the heterotic string theory, the abovearguments can be used to establish an O(d−1, 1)⊗O(d+p−1, 1) symmetry of theS-matrix elements when the external states are restricted to carry zero momentumin d of the space-time directions and zero charge under p of the U(1) generatorsof the gauge group. Since the effective action of the theory is constructed fromthe S-matrix, the O(d −1, 1) ⊗O(d + p −1, 1) symmetry must manifest itself as asymmetry of the effective action also.

Note that this argument holds to all ordersin the α′ expansion, since nowhere we had to assume that the momenta carried bythe external states in directions other than these d directions are small.Let us now see how this transformation acts on some specific components ofthe string field. Let hµν, bµν and aµR denote the components of the stri ng fieldwhich couple to the graviton, the antisymmetric tensor and the gauge field vertexoperators respectively.

We shall choose normalizations such that hmn+bmn couplesto the vertex operator c¯c∂ˆY m ¯∂ˆY nei˜k. ˜Y and amR couples to the vertex operatorc¯c∂ˆY m ¯∂ˇY Rei˜k.

˜Y , where ˜Y α denote the set of coordinates other than ˆY m. Let Sand R be the O(d −1, 1) and O(d −1 + p, 1) matrices associated with the Lorentztransformations involving the unbarred and the barred indices respectively. Then5

the O(d −1, 1) ⊗O(d + p −1, 1) transformation acts on these fields as:( (h′ + b′)a′ ) = S ( (h + b)a ) RT(2.2)where ( (h + b)a ) is regarded as a d × (d + p) matrix. Similarly, if (hmα + bmα)couples to the vertex operator c¯c∂ˆY m ¯∂˜Y αei˜k.

˜Y , (hαm + bαm) couples to the vertexoperator c¯c∂˜Y α ¯∂ˆY mei˜k. ˜Y , and aαR couples to the vertex operator c¯c∂˜Y α ¯∂ˇY Rei˜k.

˜Y ,then these fields transform as,h′mα + b′mα =Smn(hnα + bnα)h′αm + b′αm =(hαn + bαn)Rnm + aαRRRma′αR =(hαn + bαn)RnR + aαSRSR(2.3)Similar transformation laws can be derived for other fields as well, but we shall notlist them here.Note that the O(d−1, 1)⊗O(d+p−1, 1) ‘symmetry’ described above holds forany background that is independent of d of the space-time coordinates, and is neu-tral under p of the U(1) subgroups of the gauge group. This includes backgroundmassive fields as well.3.

SYMMETRY OF THE LOW ENERGY EFFECTIVE ACTIONAlthough the general argument guarantees the existence of an O(d −1, 1) ⊗O(d + p −1, 1) symmetry, realisation of this symmetry in terms of fields thatappear in the low energy effective action is somewhat non-trivial, since the explicitrelationship between the string field components hµν, bµν and the fields that appearin the low energy effective action is not known. What we would like to do nowis to see how these transformations can be realised in the context of low energyeffective field theory.

To do this we start with the low energy effective action of6

heterotic string theory and rewrite it in such a way that its symmetry becomesmanifest. The action is given by,S = −ZdDx√det Ge−Φ(Λ−R(D)(G)+ 112HµνρHµνρ−Gµν∂µΦ∂νΦ+18XaF aµνF aµν))(3.1)where Gµν, Bµν, Aaµ and Φ denote the graviton, the antisymmetric tensor field,the gauge field, and the dilaton, respectively, F aµν = ∂µAaν −∂νAaµ + fabcAaµAbν,Hµνρ = ∂µBνρ + cyclic permutations −(Ω(3)A )µνρ, R(D) denotes the D dimensionalRicci scalar, and Λ is the cosmological constant equal to (D −10)/2 for heteroticstring.

Ω(3)A is the gauge Chern-Simons term given by (Ω(3)A )µνρ = (1/4)(AaµF aνρ +cyclic permutations −fabcAaµAbνAcρ). The full effective action also involves LorentzChern Simons term, but these are higher derivative terms and can be ignoredto this order.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 further concentrate on backgrounds where the gauge field background liesin a subgroup that commutes with p of the right moving U(1) generators ¯∂XIassociated with the internal coordinates XI of the heterotic string theory.

Let usdenote the corresponding internal coordinates by ˇY R (1 ≤R ≤p). Let ˜a denotethe gauge indices corresponding to the gauge generators that commute with theU(1)p subgroup, and lie outside this subgroup.

Thus the allowed non-vanishingcomponents of the gauge fields are A˜aµ and ARµ .To begin with, we shall further restrict to background field configurationsfor which Gmα = Bmα = ARα = A˜am = 0; i.e.to backgrounds of the formG = ˆGmn00˜Gαβ!, B = ˆBmn00˜Bαβ!, Aaµ = { ˜A˜aα, ˆARm}.⋆Afterwards weshall see how to write down the transformation laws in the general case when suchrestrictions are not there. In this case, after an integration by parts, the action⋆For such backgrounds, the equations of motion obtained by varying the action with respectto the field components that we have set to zero are satisfied identically.

Hence any invari-ance of the action for such restricted set of backgrounds will also imply invariance of thecomplete set of equations of motion.7

(3.1) can be shown to take the form:S = −Zdd ˆYZdD−d ˜Ypdet ˜Ge−χhΛ −˜Gαβ ˜∂αχ˜∂βχ −132˜GαβTr(˜∂αML˜∂βML)−˜R(D−d)( ˜G) + 112˜Hαβγ ˜Hαβγ + 18X˜a˜F ˜aαβ ˜F ˜aαβi(3.2)where,L = ηd00−ηd+p! (3.3)χ = Φ −lnpdet ˆG(3.4)and,M =(KT −ηd) ˆG−1(K −ηd)(KT −ηd) ˆG−1(K + ηd)−(KT −ηd) ˆG−1 ˆA(KT + ηd) ˆG−1(K −ηd)(KT + ηd) ˆG−1(K + ηd)−(KT + ηd) ˆG−1 ˆA−ˆAT ˆG−1(K −ηd)−ˆAT ˆG−1(K + ηd)ˆAT ˆG−1 ˆA(3.5)Here ηm denotes the m dimensional Minkowski metric diag(−1, 1, .

. .

1), ˆA is thematrix ˆAmR ≡ˆARm, and,K = −ˆB −ˆG −(1/4) ˆA ˆAT(3.6)The action (3.2) is manifestly invariant under,M →ΩMΩT(3.7)χ →χ,˜Gαβ →˜Gαβ,˜Bαβ →˜Bαβ,˜A˜aα →˜A˜aα(3.8)where,Ω= SR! (3.9)S and R being O(d−1, 1) and O(d+p−1, 1) matrices respectively.

At the linearisedlevel, ˆGmn = ηmn + hmn, ˆBmn = bmn and ˆARm = amR, and the transformations8

given above agree with the transformations of h, b and a given in eq.(2.2). Thetransformation law of Φ can also be shown to agree with the linearised transfor-mations [3].

Also note that the action is in fact invariant under any O(d, d + p)transformation generated by the matrices Ωsatisfying ΩLΩT = L, but the trans-formations outside the O(d −1, 1) ⊗O(d + p −1, 1) subgroup can be shown to bepure gauge deformations [3].One way to derive the transformation laws given in eq. (3.7) under the O(d −1, 1)⊗O(d+p−1, 1) transformation is as follows.⋆Let us imagine for the time beingthat all the dimensions associated with the coordinates ˆY m have been compactified(the effective action does not depend on whether these dimensions are compact ornot).

In that case the low energy effective field theory involving the moduli of thiscompact space (together with the moduli associated with the internal coordinates)is governed by the Zamolodchikov metric for these moduli, which, in turn, is in-variant under the O(d, d + p) group introduced in refs.[16]. Thus the action of theO(d −1, 1) ⊗O(d + p −1, 1) group on the various fields may be obtained from theaction of this O(d, d + p) group on the moduli space.

This action, in turn, may beread out directly from the analysis of ref. [17] and is given by,M →ΩMΩT(3.10)where M is the same matrix as given in eq.

(3.5) and Ωis an O(d, d + p) matrixwhich preserves the matrix diag(ηd, −ηd+p).Let us now consider the case where Gαm, Bαm and AαR are non zero. Forsimplicity, we shall assume here that the background gauge field is Abelian, andbelongs to the U(1)16 subgroup of the gauge group.

Since this subgroup commuteswith all the 16 U(1) generators, we can take p = 16. In this case, we shall define⋆This argument was pointed out by C. Vafa [15].9

a (2D + 16) × (2D + 16) matrix M as:M =(KT −ηD)G−1(K −ηD)(KT −ηD)G−1(K + ηD)−(KT −ηD)G−1A(KT + ηD)G−1(K −ηD)(KT + ηD)G−1(K + ηD)−(KT + ηD)G−1A−AT G−1(K −ηD)−AT G−1(K + ηD)AT G−1A(3.11)where,K = −B −G −(1/4)AAT(3.12)The gauge index of A now runs over all the 16 coordinates. In this case, the fullaction can be expressed as,−Zdd ˆYZdD−d ˜Y e−ψhΛ −Gαβ ˜∂αψ ˜∂βψ −132GαβTr(˜∂αML˜∂βML)+ ˜∂αψ ˜∂βGαβ −12P βα(αβ)i(3.13)where,L = ηD00−ηD+16!

(3.14)and,ψ = Φ −ln√det G(3.15)P βα(αβ) is defined as follows. We first define the matrices:V =ηD/√2−ηD/√201/√21/√20001(3.16)ˇM = 12V MV T =G−1−G−1KG−1A/√2−KT G−1KT G−1K−KT G−1A/√2AT G−1/√2−AT G−1K/√2ATG−1A/2(3.17)10

ˇL = V LV T =01D01D0000−116(3.18)We now define the matrices P(αβ), . .

. , Z(αβ) through the relations:(∂α ˇM ˇL( ˇM −ˇL) ˇL∂β ˇM) =P(αβ)Q(αβ)R(αβ)S(αβ)T(αβ)W(αβ)X(αβ)Y(αβ)Z(αβ)(3.19)In the above, P(αβ) is a D×D matrix.

We now define P µν(αβ) to be the µν componentof this matrix.The action given in eq. (3.13) can be shown to be invariant under a transfor-mation of the form:M →ΩMΩT ,ψ →ψ(3.20)with,Ω=1D−dS1D−dR(3.21)where S and R are the O(d−1, 1) and O(d+p−1, 1) matrices discussed previously(with p = 16).

Note that when Gmα, Bmα and AαR are zero, these transformationlaws are identical to those given in eq.(3.10). Also, these transformations reduceto the ones given in eqs.

(2.2) and (2.3) when Gµν −ηµν, Bµν and AµR are smalland hence can be identified with hµν, bµν and aµR respectively. The invariance ofthe action given in eq.

(3.13) under the symmetry transformation given in eq. (3.21)follows from the fact that Gαβ and P βα(αβ) remain invariant under these transforma-tions.Before we conclude this section, let us remark that although the general ar-guments of sect.

2 guarantee the existence of an O(d −1, 1) ⊗O(d + p −1, 1)11

‘symmetry’ of the string theory for appropriate backgrounds to all orders in α′, itdoes not guarantee that the transformation laws, when expressed in terms of thefields Gµν, Bµν and Φ will remain unchanged when we include corrections that arehigher order in α′. This is due to the fact that the functional relationship betweenthe string fields and the fields that appear in the effective field theory may undergomodification when we include the effect of higher derivative terms.

Evidence ofsuch modification in the transformation laws has already been seen [18] [19] [3].4. APPLICATION OF THEO(d −1, 1) ⊗O(d + p −1, 1) TRANSFORMATIONWe shall now apply the above transformations to known solutions of heteroticstring theory to generate new solutions.

In particular we shall take our startingsolution to be the black six brane solution of ref. [10] carrying a magnetic charge.

(For related work see refs. [20 −35] .) The solution is given by the following formof the metric and other fields:ds2 = −(1 −r+/r)(1 −r−/r)dt2 +dr2(1 −r+/r)(1 −r−/r) + r2dΩ22 +6Xi=1dXidXi(4.1)Φ = −ln(1 −r−/r) + Φ0(4.2)F 1 = 2√2QMǫ2(4.3)where, dΩ2 is the line element on a two sphere, and ǫ2 is the volume form on thesame two sphere.

Φ0, r+ and r−are the three independent parameters labellingthe solution (r+ > r−), and QM is the (quantised) magnetic charge carried by theblack hole, given by,QM =pr+r−/2(4.4)For definiteness, we have taken the magnetic field to lie in the U(1) subgroup gen-erated by the first internal coordinate. We shall now perform the O(d −1, 1) ⊗12

O(d + p −1, 1) transformation on this solution to generate new solutions. To thisend, note that the solution is independent of the coordinate t and also the sixcoordinates Xi, thus here d = 7.

Furthermore, the presence of the magnetic fieldrequires A1 to have a non-vanishing component tangent to the 2-sphere, thus if wewant to satisfy the condition AαR = 0, we must exclude the direction 1 from the setof directions R. Although we have shown that this is not necessary, we shall firstconsider this case. Thus here p = 15.

Although we can involve all the 7 space-timecoordinates, and all the 15 internal coordinates in the transformation, a generaltransformation of this kind will generate solutions which will be related by rotationin the external and/or internal space.⋆Thus the set of inequivalent field config-urations are generated by taking the appropriate ‘Lorentz transformations’amongthe coordinate t, one of the space coordinates (say X1) and one of the internalcoordinates ( say ˇY 2). The symmetry group in this case is O(1, 1) ⊗O(2, 1).

Thediagonal O(1, 1) subgroup corresponds to Lorentz transformation of the solutionin the t −X1 space, we may fix a Lorentz frame by choosing the matrix S to bethe identity matrix. Thus we are left with the O(2, 1) matrix R parametrized bythe three Euler angles.

A further reduction of the parameters may be made bynoting that the original solution is left invariant if we choose R to be a rotationin the X1 −ˇY 2 plane. Thus the general solution is obtained by taking R to be aboost in the t −ˇY 2 direction followed by a boost in the t −X1 direction:R =cosh α2sinh α20sinh α2cosh α20001cosh α10sinh α1010sinh α10cosh α1(4.5)We can now calculate the transformed solution in a straightforward way using⋆For E8 ×E8 heterotic string theory rotation among the 15 internal coordinates can generateinequivalent field configurations since O(16) is not a subgroup of the gauge group.

But thisonly changes the direction of the gauge field in the final solution without modifying theessential properties of the solution.13

eqs. (3.7), (3.8).

The transformed solution is given by,ds2 = −14(r −r0)2(4(r −r+)(r −r−) −(r+ −r−)2β2)dt2 + βr+ −r−(r −r0)dX1dt+6Xi=1dXidXi +dr2(1 −r+/r)(1 −r−/r) + r2dΩ22Bt1 = β r+ −r−2(r −r0)A2t = γ r+ −r−(r −r0)A21 = 0F 1 = 2√2QMǫ2Φ = −ln(1 −r0/r) + Φ0(4.6)where,γ = sinh α1β = cosh α1 sinh α2r0 = 12((r+ + r−) −(r+ −r−)p1 + β2 + γ2)(4.7)This solution is characterized by an electric field as well as an antisymmetric tensorfield strength, given by,F 2rt = ∂rA2t = −γ (r+ −r−)(r −r0)2Hrt1 = ∂rBt1 = −β (r+ −r−)2(r −r0)2(4.8)Hence, besides carrying the magnetic charge, the new solution carries both, electricand antisymmetric tensor gauge field charge, proportional to γ and β respectively.Let us now discuss singularities of the solution (4.6). It can be easily seen thatthe matrix GttGt1Gt1G11!has zero eigenvalues at r = r+ and r = r−.

The component Grr has poles at pre-14

cisely these values of r, as can be seen from eq.(4.1). These singularities representcoordinate singularities, and can be removed by appropriate coordinate choice.

Tosee this, let us first define new coordinates t′, X′ and ρ through the relations:t =t′ cosh θ −X′ sinh θX1 =X′ cosh θ −t′ sinh θr =ρ + r+(4.9)where,tanh θ = β2r+ −r−r+ −r0(4.10)In this coordinate system, the metric near r = r+ takes the form:ds2 = −2 cosh θ sinh θβ(r+ −r0) ρdt2(1 + O(ρ))+ 2(r+ −r0)β(r+ −r−)h1 −β2(r+ −r−)24(r+ −r0)2i2cosh θ sinh θ(dX′)2(1 + O(ρ))−2ρr+ −r0h1 −14β2(r+ −r−)2(r+ −r0)2−r+ −r−r+ −r0icosh θ sinh θdX′dt′(1 + O(ρ))+(r+)2(r+ −r−)ρ(dρ)2(1 + O(ρ)) + (r+ + ρ)2(dΩ2)2 +6Xi=2dXidXi(4.11)From this we see that the metric has a singularity at ρ = 0. This singularity isremoved by defining new coordinates u, v as,u = √ρeat′,v = √ρe−at′(4.12)where,a =scosh θ sinh θ(r+ −r−)2(r+)2(r+ −r0)β(4.13)15

In this coordinate system the metric takes the form:ds2 = 4(r+)2(r+ −r−)dudv + 2(r+ −r0)β(r+ −r−)h1 −β2(r+ −r−)24(r+ −r0)2i2cosh θ sinh θ(dX′)2+ (r+)2(dΩ2)2 +6Xi=2dXidXi + O(u, v)(4.14)From this we see that the metric is non-singular in this coordinate system at u = 0or v = 0. It can also be seen easily that both the electric and the antisymmetrictensor fields are non-singular at r = r+ in the new coordinate system.

Similarchange of coordinates can also be carried out near r = r−to show that thisalso represents a coordinate singularity.⋆On the other hand, the point r = r0as well as r = 0 represents genuine singularities of the solution. (Φ →±∞near these points.

)Since for real β and γ, r0 ≤r±, we see that the solutionrepresents a genuine singularity surrounded by two horizons. Solvable conformalfield theories corresponding to black string solutions with two horizons have beenfound previously by Horne and Horowitz [35].Note that if we take β = 0, then the solution represents the direct productof a four dimensional black hole carrying magnetic and electric charge, and asix dimensional flat space described by the coordinates Xi (1 ≤i ≤6).

If wecompactify the coordinates Xi (say on a Calabi-Yau manifold, or a six dimensionaltorus), the result would be a four dimensional black hole carrying electric andmagnetic charge. (The full solution, on the other hand, may be regarded as ablack string in 5 dimensions by compactifying the coordinates X2, .

. .

X6.) Thetwo charges, however, lie in different U(1) subgroups of the gauge group.

These⋆In this case we again look for a coordinate transformation of the form t = t′′ cosh φ −X′′ sinh φ, X1 = X′′ cosh φ −t′′ sinh φ, r = r−+ ρ′ as in eq. (4.9), so as to bring the metricin the standard form near the singular surface.

It can be easily seen that if |β| < γ2/2, thenit is possible to find a φ (tanh φ = β(r+ −r−)/2(r−−r0)) for which Gt′′t′′ and GX′′t′′ areof order ρ′, and GX′′X′′ is of order 1 as r →r−. On the other hand, if |β| > γ2/2, then itis possible to find a φ (coth φ = β(r+ −r−)/2(r−−r0)) for which GX′′X′′ and GX′′t′′ areof order ρ′ and Gt′′t′′ is of order unity as r →r−.

Thus the global structure resembles thatof a Reissner-Nordstrom black hole in the first case, and that of the black string solution ofref. [35][4] in the second case.16

solutions are different from the ones discussed in ref. [33] in that in their solution theelectric and the magnetic charge lie in the same U(1) subgroups of the gauge group.On the other hand, these solutions can be identified to the black hole solutions ofGibbons and Maeda [28] carrying electric and magnetic charge, if we interprete theelectric and magnetic charge in their solution to belong to different U(1) subgroupsof the gauge group.

(Note that this is the only way to interprete the solutions ofGibbons and Maeda in the context of string theory, since if the electric and themagnetic fields belong to the same U(1) subgroup, we need to take into account theeffect of the gauge Chern Simons term coupling to the antisymmetric tensor gaugefield strength, which was not included in the analysis of ref.[28].) If we further setthe magnetic charge QM to zero, the solution reduces to the charged black holesolution of ref.[22].

(Note that the metric ˆds2 considered in refs. [28][22] is relatedto the metric ds2 given in eq.

(4.6) through the relation ˆds2 = e−Φds2 [10]. )Since we have derived the transformation laws of various fields under O(d −1, 1) ⊗O(d + p −1, 1) transformation even when Gmα, Bmα and AαR are nonzero, we could, in principle, perform an O(d −1, 1) ⊗O(d + p −1, 1) rotation thatincludes the 1 direction of the gauge field.

Note, however, that in this case, theinitial gauge potential needs to be defined in separate coordinate patches; and arerelated by a gauge transformation on the overlap. This, in general, implies thatthe transformed fields also need to be defined in separate coordinate patches, andare related by gauge and general coordinate transformation on the overlap.

Tosee this let us consider the transformation of the fields in the asymptotic regionr →∞, so that Gµν −ηµν, Bµν and AµR are small. If we choose S = 1, and R tobe a O(1, 1) transformation that mixes the t coordinate with the 1 direction in theinternal space, the transformed fields take the form:G′αt = 12 sinh θA1αB′αt = 12 sinh θA1αA1′α = A1α cosh θ(4.15)where α denotes any of the three directions x, y or z on which the original solution17

depends. Let A1α and ¯A1α be the components of the original gauge field in the twodifferent coordinate patches, then A1α −¯A1α = ∂αΛ, where Λ is a function which isnot single valued under a 2π rotation about the z axis (although eieΛ is).

Fromeq. (4.15) we see that G′αt −¯G′αt is now given by (1/2) sinh θ∂αΛ, where G′ and ¯G′denote the transformed metric in the two coordinate patches.

This shows that G′and ¯G′ are related by a coordinate transformation of the form t →t + Λ sinh θ.However, since Λ is not a single valued function of the coordinates, this coordinatetransformation is not globally well defined.We could also have started with the metric which represents black holes carry-ing quantised antisymmetric tensor gauge field charge, instead of magnetic charge.This solution is given by [10]:ds2 = −1 −r2+/r21 −r2−/r2dt2 +dr2(1 −r2+/r2)(1 −r2−/r2) + r2dΩ23 +5Xi=1dXidXiΦ = −ln(1 −r2−/r2) + Φ0˜Hαβγ =Q(ǫ3)αβγ(4.16)where dΩ3 is the line element on a three sphere, ǫ3 is the volume form on thesame three sphere, and Q = r+r−. In this solution, the expressions for Gtt and Φare similar to those given in eqs.

(4.1) and (4.2), except that the ratios r/r± arereplaced by (r/r±)2. As a result, the final transformed solution will have the sameform as given in eqs.

(4.6) and (4.7) with r, r0 and r± replaced by r2, (r0)2 and(r±)2 everywhere in the expression for ˆG, ˆB and ˆA. Thus the final solution willtake the form:18

ds2 = −14(r2 −r20)2(4(r2 −r2+)(r2 −r2−) −(r2+ −r2−)β2)dt2 + βr2+ −r2−r2 −r20dX1dt+5Xi=1dXidXi +dr2(1 −r2+/r2)(1 −r2−/r2) + r2dΩ23ˆBt1 =β r2+ −r2−2(r2 −r20)At =γ r2+ −r2−r2 −r20A1 =0Φ = −ln(1 −r20/r2)˜Hαβγ =Q(ǫ3)αβγ(4.17)where,r20 = 12((r2+ + r2−) −(r2+ −r2−)p1 + β2 + γ2)(4.18)Note that in this case the antisymmetric tensor gauge field has a ‘magnetic’ typecomponent denoted by ˜Hαβγ and also an electric type component denoted byˆHrt1 ≡∂r ˆBt1.Again, by taking the directions X2, . .

. X5 to be compact, thissolution may be regarded as a black string solution in six dimensions.In some cases, one can get solvable conformal field theories describing blackhole solutions [36 −41] [19].

One expects that by twisting these solutions one willget solutions that again correspond to solvable conformal field theories. In factthe black p-brane solution obtained by twisting the solution [3] given in ref [36] arealso described by solvable conformal field theories [34] [35].19

5. CONCLUSIONIn this paper we have shown that given a classical solution of the heteroticstring theory which is independent of d of the space-time coordinates, and forwhich the background gauge field lies in a subgroup that commutes with p of theU(1) generators of the gauge group, we can generate other classical solutions byapplying an O(d−1, 1)⊗O(d+p−1, 1) transformation on the original solution.

Byusing these transformations on the known black 6-brane solution of the heteroticstring theory carrying magnetic charge, we have generated new solutions carryingmagnetic, electric and antisymmtric tensor gauge field charge. These solutions arelabelled by four continuous and one discrete parameters, characterizing the mass,the electric charge, the antisymmetric tensor gauge field charge, the asymptoticvalue of the dilaton field, and the magnetic charge of the 6-brane respectively.

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