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Strings on Curved Spacetimes:

arXiv:hep-th/9202092v1 29 Feb 1992hepth@xxx/9202092LA-UR-92-640NEIP92-001Strings on Curved Spacetimes:Black Holes, Torsion, and DualityPaul Ginsparg1 and Fernando Quevedo2Los Alamos National LaboratoryTheoretical Division, MS-B285Los Alamos NM 87545We present a general discussion of strings propagating on noncompact coset spaces G/Hin terms of gauged WZW models, emphasizing the role played by isometries in the exis-tence of target space duality. Fixed points of the gauged transformations induce metricsingularities and, in the case of abelian subgroups H, become horizons in a dual geometry.We also give a classification of models with a single timelike coordinate together with anexplicit list for dimensions D ≤10.

We study in detail the class of models described by thecosets SL(2, IR) ⊗SO(1, 1)D−2/SO(1, 1). For D ≥2 each coset represents two differentspacetime geometries: (2D black hole)⊗IRD−2 and (3D black string)⊗IRD−3 with nonvan-ishing torsion.

They are shown to be dual in such a way that the singularity of the formergeometry (which is not due to a fixed point) is mapped to a regular surface (i.e. not evena horizon) in the latter .

These cosets also lead to the conformal field theory descriptionof known and new cosmological string models.2/921 email: ginsparg@xxx.lanl.gov2 Present address: Institut de Physique, Universit´e de Neuchˆatel, Rue A.-L. Br´eguet 1 CH-2000, Switzerland. email: phquevedo@cnedcu51.bitnet.

Supported by Swiss National Foundation.

1. IntroductionIssues of principle in quantum gravity are among the most important questions thatmay eventually be addressed in the context of string theory.

The study of curved spacetimesas string backgrounds could be used to investigate the string theoretical approach to physicsat the Planck scale where classical methods are known to fail. The singularities in blackhole and cosmological backgrounds are especially interesting.The description of string backgrounds has been studied extensively by means of con-formal field theory (CFT), but most of the effort thus far has been directed to the casewhere the noncompact part of the spacetime is flat, i.e.

described by a trivial CFT, andonly the internal space requires nontrivial CFT techniques. Coset models provide a richclass of explicit CFT’s for this case and lead to an understanding of the space of statictree-level vacua.Noncompact coset models provide a natural framework for nonstaticbackgrounds and other nontrivial spacetimes which have recently received more attention,especially in the context of 2D gravity.In this article we expand on previous approaches [1,2] to study noncompact cosets inthe Wess-Zumino-Witten (WZW) formalism [3,4].

We present a general discussion of suchspaces, classifying all those with a single timelike coordinate and any number of spacelikecoordinates. These will provide the natural background for consistent string propagation.We also discuss the structure of singularities that occur in gauged WZW models, as well asthe geometrical interpretation of the spaces obtained in this way.

The metric obtained fromthe WZW model is singular and we show that there are singularities at fixed points of thegauge transformation. In cases where there is a dual gauging, these same group elementswill provide horizons of the dual metric, generalizing the horizon/singularity duality.We shall then discuss in detail a simple class of models that provides an example forany spacetime dimension: SL(2, IR) ⊗SO(1, 1)D−2/SO(1, 1).

For D = 2, this is knownto describe a single self-dual black hole geometry.We find that for D > 2, there aretwo geometries described by the two anomaly-free gaugings (vector and axial). Unlike theLorentzian D = 2 case in which both gaugings give dual versions of the same black hole,the two geometries in this case are seemingly different.

They are nonetheless dual to eachother in a manner similar to mirror manifolds in string compactifications, where a singleCFT can describe different target space geometries. For the axial gauging, we find thegeometry (2D black hole)⊗IRD−2, or a D −2 black brane [5].

For the vector gauging,on the other hand, we find (3D black string)⊗IRD−3 with nonvanishing torsion which is1

a different black brane. We explicitly verify that the large k (Kac-Moody level) limit ofthe gauged WZW model gives a solution of the field equations for dilaton, graviton andantisymmetric tensor field to lowest order in α′.

Unlike the 2D case, however, we are able tofind a more general solution of those equations. The explicit form of the curvature scalar isused to clarify the nature of the singularities leading to the above black hole interpretation.Duality acts in a nontrivial way for these geometries, in particular the singularity for theaxial gauging occurs for elements of the coset that are not fixed points, and its dual isat a regular surface in the asymptotically flat region of the vector gauged geometry.

Thisillustrates the possibility that strings do not necessarily preclude spacetime singularitiesbut may nonetheless be better behaved than expected on them.In section 2 we present a general discussion of the geometry of noncompact coset spacesand their associated Kac-Moody algebras, and their description in terms of gauged WZWmodels. We discuss the conditions for anomaly-free gaugings and the conditions for theexistence of isometries.

The classification of all coset WZW models with a single timelikecoordinate is a purely group theoretical question which we solve using the known propertiesof general coset spaces. Finally we discuss briefly the appearance of singularities in thelarge k limit of gauged WZW models and their relation to fixed points of the correspondinggauge transformation.In section 3 we construct explicitly the metric for the models SL(2, IR)⊗SO(1, 1)D−2/SO(1, 1)for the two anomaly-free gaugings and obtain the geometries mentioned above.

We explic-itly verify that the background fields obtained from the large k limit of the WZW modelsatisfy the string equations to lowest order in α′. We also find the most general solutionof those equations with the same number of isometries, and argue that it can be obtainedfrom marginal perturbations of the coset CFT.

Finally in section 4 we discuss duality ofthese solutions. We briefly review the duality of σ–models whenever there is an isometry(following reference [6]), and generalize to the case of several commuting isometries.

Thissymmetry in particular relates spaces with torsion to spaces without torsion. We provethe relation between the two different geometries by identifying the vector transformation(g →hgh−1) as the isometry when gauging the axial transformation (g →hgh), and vice-versa.

We end with some final comments and future developments, and compare our workwith other recent papers on the subject.2

2. Noncompact coset CFT’sThe study of noncompact coset CFT’s was undertaken in [7] for SL(2,IR)/U(1) currentalgebra via the conventional GKO construction.

The formalism was later generalized toany coset in [8]. Given a level k Kac-Moody algebra for a noncompact group G,JA(z) JB(w) ∼−k ηAB/2(z −w)2 + i fABC JC(w)(z −w)(2.1)(where g ηAB = fACD fBDC is the Cartan metric and g is the Coxeter number of G), thestress-energy tensor for a CFT based on G is given by the Sugawara fromTG(z) = ηAB : JA(z) JB(z):(−k + g).

(2.2)The corresponding central charge is cG = k dim G/(k −g). For the coset G/H with stress-energy tensor TG/H = TG −TH, the central charge is cG/H = cG −cH.

The only changesfrom the compact case are the sign of k and of course the use of noncompact structureconstants fABC. (The extension to supersymmetric coset models, discussed in [8], will notbe considered here.) The spectrum and the corresponding elimination of negative normstates is not yet entirely understood for these models, and more progress is needed beforewe can properly treat the string vacua obtained from this approach.In [2], it was shown that the SL(2,IR)/SO(1, 1) current algebra could be interpretedas a two dimensional black hole spacetime.

An implicit prescription for assembling theholomorphic and anti-holomorphic representations was given in terms of a gauged WZWmodel. In fig.

1, we reproduce the causal structure of this spacetime. In this section weshall provide some background on this construction, emphasizing the semi-classical limitin which various aspects of the geometry can be visualized, and which provides a geometricinterpretation for the GKO current algebra construction.

For an exact incorporation of allquantum corrections, however, we would need to return to the full conformal field theory/ current algebra approach.2.1. AnomaliesThe WZW action in complex coordinates is writtenL(g) = k4πZd2z tr(g−1∂g g−1∂g) −k12πZBtr(g−1dg ∧g−1dg ∧g−1dg) ,(2.3)3

IIIIIIIVVVIFig. 1: The causal structure of the two dimensional black hole spacetime of [2].Regions I,IV are asymptotic regions, regions II,III are inside the horizon, andregions V,VI are beyond the singularities.where the boundary of B is the 2D worldsheet.To promote the global g →h−1L g hRinvariance to a local g →h−1L (z) g hR(z) invariance, we let ∂g →∂g+Ag, and ∂g →∂g−g ¯A.The gauge fields transform as A →h−1L (A + ∂)hL and ¯A →h−1R ( ¯A + ∂)hR (so that Dg →h−1L Dg hR for D equal to either holomorphic or anti-holomorphic covariant derivative).Vector gauge transformations correspond to hL = hR, and axial gauge transformations tohL = h−1R .

Substituting covariant derivatives in (2.3) gives the gauged actionL(g, A) = L(g) + k2πZd2z trA ∂gg−1 −¯A g−1∂g −g−1Ag ¯A. (2.4)Under the infinitesimal transformations hL ≈1 + α, hR = 1 + β, we have δA =∂α + [A, α] and δ ¯A = ∂β + [ ¯A, β].

The anomalous variation of the effective action is (seee.g. [9] for a review)δW = k2π (α∂A + β∂¯A) .

(2.5)The variation of the (LR→VA) counterterm trA ¯A, on the other hand, isδ(trA ¯A) = tr−β∂A −α∂¯A + (α −β) ¯A, A. (2.6)For the abelian case, we see that (2.6) can compensate the variation (2.5) for either α = ±βsince the commutator term automatically vanishes.Thus both vector and axial-vectorgauging are allowed.

In the non-abelian case, only the vector gauging α = β is allowed. (An integrated form of this argument may be found in [10]: essentially the relevant π34

obstruction is not captured by the trivial topology of U(1).) If we change sign ¯A →−¯Afor the axial gauged case (to give A and ¯A the same transformation properties), then thegauged action may be writtenL(g, A) = L(g) + k2πZd2z trA ∂gg−1 ∓¯A g−1∂g + A ¯A ∓g−1Ag ¯A,(2.7)where the upper and lower signs represent respectively vector (g →hgh−1) and axial-vector(g →hgh) gauging.It is intuitively reasonable that gauging g →hgh∓1 should result in a combination ofholomorphic and antiholomorphic representations of G/H algebras: since the equations ofmotion for the ungauged model result in g(z, z) = a(z) b(z), gauging left multiplication byh(z) and right multiplication by h∓1(z) properly removes the H degrees of freedom fromboth sides.2.2.

Semi-classical limitWe now consider some naive properties of the geometry described by (2.7) in thelarge k (semi-classical) limit. Writing A = Aaσa in terms of the generators σa of H, andintegrating out the components Aa classically gives the effective actionL = L(g) ± k2πZd2z tr(σbg−1∂g) tr(σa∂gg−1) Λ−1ab ,(2.8)with Λab ≡tr(σaσb∓σagσbg−1).

Notice that singularities of Λ occur at least at fixed pointsof the gauge transformation g →hgh∓1. This is because for infinitesimal h ≈1 + αa σa,we see that a fixed point g satisfies σa g ∓gσa = 0.

Multiplying by g−1σb and taking thetrace, we see that Λ = 0 at a fixed point. (In the euclidean case it is easy to show theconverse, i.e.

that Λ = 0 implies a fixed point, whereas this is not true in the lorentziancase. )From the transformation properties of the gauge fields and (2.6), we note that inthe case of H abelian the ungauged axial or vector symmetry remains a global symmetry,i.e.

an isometry of the spacetime geometry. In the non-abelian case, not even a globalvestige of the ungauged symmetry remains.

In the abelian case, this implies that a fixedpoint of the ungauged symmetry corresponds to a point with vanishing Killing vector.For lorentzian signature, the surface carried into the fixed point by the isometry will bea null surface (the norm of the Killing vector is conserved), in general nonsingular andhence a horizon. We see that fixed points of symmetry transformations generically give5

rise to metric singularities when the symmetry is gauged and to horizons when ungauged.This general property is the origin of the singularity/horizon duality of the 2D blackhole of [2]. For the vector gauging, the metric can be written ds2 = −da db/(1 −ab),and the fixed point of the vector transformation (the gauged symmetry) corresponds toab = 1, which is the singularity.

The fixed point of the axial transformation (the isometry)is a = b = 0 indicating that the invariant surface ab = 0 is null, and provides the eventhorizon illustrated in fig. 1.

For the axial gauging, the metric is identical (i.e. the geometryis self-dual) but the role of the fixed points is exchanged, implying the horizon/singularityduality pointed out in [11].We now try to visualize in more detail the naive properties of the geometry describedby (2.7) for G = SL(2, IR) in the large k (semi-classical) limit.

We take SL(2,IR) groupelements parametrized as1g = w1 + xσ1 + iyσ2 + zσ3 =w + zx + yx −yw −z. (2.9)The condition that det g = 1 requires w2 +y2 −x2 −z2 = 1, so we see that x, z parametrizethe IR2 and w, y the S1 of the IR2 × S1 topology of SL(2,IR).

We shall consider the actionsg →hgh∓1 for h = hE,L, wherehE ≡eiασ2 ,hL ≡eασ3 . (2.10)Modding out by hE gives a Euclidean signature metric, and modding out by hL gives aLorentzian signature metric.For the euclidean case h = hE, the action g →hE g h−1E is easily seen to be rotationin the 1,3 = x, z plane.

Thus we can always “gauge-fix” any point to, say, the positive xaxis (fig. 2).

The origin is a fixed point of the transformation and results in a singularityof the metric. Crossing this half-line with endpoint singularity back together with the w, ycircle, we see that the spacetime takes the form of a “trumpet” (fig.

3).The other action g →hE g hE is simply rotation in the w, y circle. (This is easily seenby reparametrizing g = ˜gσ1, which exchanges (x, z) ↔(w, y), and noting that g →hgh ⇒˜g →h˜gh−1.) By rotation of the w, y circle, we can always “gauge-fix” every point to, say,the point (1,0), thus eliminating the S1 entirely.

Since the action has no fixed points the1 The compact group SU(2) has group elements g = x01 + i⃗x · ⃗σ so we identify σ1 and σ3 asthe noncompact generators, giving signature (– + –).6

...........................Fig. 2: Under rotation about the origin, any point in the plane can be gauge-fixedto the positive real axis.

The origin is left fixed.rFig. 3: The two dual versions of the Euclidean black hole.

The upper “trumpet”version has a singularity at r = 0, reflecting the fixed point at the origin of fig. 2.The lower “cigar” version is free of singularities.metric on the remaining IR2 can be entirely regular.

For metrical reasons, we change tor, θ coordinates in which this IR2 is naturally depicted as a “cigar”. Fig.

3 thus depicts thetwo dual semiclassical geometries of the Euclidean 2D black hole constructed in [2].For the Lorentzian case h = hL, on the other hand, the action g →hgh−1 is a boost inthe (x, y) coordinates, and the action g →hgh is a boost in the (w, z) coordinates. (Thelatter result is easily seen by the same reparametrization g = ˜gσ1 mentioned above.) Upto trivial reparametrization, these two actions are identical so we see that the Lorentzianversion of the 2D black hole is self-dual.

Instead of the compact action of fig. 2 and thegauge fixing to a single ray, for the Lorentz boost we have the action depicted in fig.

4,7

Fig. 4: Under Lorentz boost, the plane is partitioned into four regions boundedby the light cones.

The points in each region can be gauge-fixed to the indicateddotted lines. The origin is left fixed.which partitions the vicinity of the origin into four disjoint regions and leaves the originfixed.Modding out by the action g →hgh, we can gauge-fix the action of the (w, z) boost tolines with z = 0 and w = 0 (which correspond respectively to a = ±b in the parametrizationg = au−vbused in [2]), as shown in fig.

4. In fig.

5, we transcribe this picture to theSL(2,IR) hyperboloid (with the x coordinate suppressed). The region z = 0 interpolatesbetween the two fixed lines ±(iσ2 cosh α+σ1 sinh α), passing along the x direction throughthe points g = ±iσ2 at y = ±1 in fig.

5. This z = 0 region encompasses two copies ofregions I–IV of fig.

1. (The origins of the lightcones of fig.

1 appear at the points g = ±1,i.e. w = ±1, of fig.

5 with the x direction restored.) The region w = 0 encompasses twocopies apiece of regions V and VI.

As mentioned earlier, the (self-) duality in this casecorresponds to interchanging w ↔y, z ↔x, so we can see from fig. 5 how the dualityexchanges region I (which is the asymptotic region moving upwards along the x directionfrom w = 1) with region V, and similarly IV↔VI (see [11]).

Region II (which lies betweenw = 1 and y = 1) is mapped into itself, as is region III. (Finally, for comparison with the semiclassical limit of compact cosets, we pointout that the case of SU(2)/U(1) gives a disk with a singular boundary.Recall thatg ∈SU(2) can be parametrized as g = cos χ + iˆn · ⃗σ sin χ, where χ ∈[0, π] denotes anazimuthal angle on S3 and ˆn parametrizes latitudinal S2’s with radius sin χ.

The actiong →hgh−1 with h = eiασ3/2 simply rotates ˆn by angle α about the 3-direction, i.e. forˆn = (sin θ cos φ, sin θ sin φ, cos θ) the action is the translation φ →φ + α. Modding out bythis action simply removes the φ coordinate and squashes each latitudinal S2 parametrized8

z , (x)wya=ba=-ba=-bVII-IVVFig. 5: The SL(2,IR) hyperboloid with the x coordinate suppressed.The twoblack dots represent the fixed points of fig.

4, and the gray lines represent thegauge-fixing.The two lightcones are the intersections of the hyperboloid withplanes perpendicular to the y axis at y = ±1.by ˆn to an interval θ ∈[0, π] with size still proportional to sin χ. The result is a disk whoseboundary, given by the circle eiασ3/2, is a line fixed under the group action and conse-quently a singularity of the induced metric on the disk.

By the argument used above inthe noncompact case, modding out by the dual action g →hgh results in an equivalentpicture. (Write g = x01 + i⃗x · ⃗σ as i˜gσ1, which interchanges (x0, x3) ↔(x1, x2), and has˜g →h˜gh−1.) The U(1) gauged SU(2) WZW model thus gives another example of a theorythat is self-dual in this sense.)2.3.

Enumeration of possibilitiesWe have seen that the gauged “G/H” WZW models considered here are not the usualleft or right G/H coset spaces with standard coset metric, as considered in the mathemat-ical literature [12] and in standard treatments of coset space nonlinear σ–models[13]. Thisis because we gauge g →hgh∓1 type symmetries rather than g →gh or g →hg, and as wellwe include a Wess-Zumino term which can add a torsion piece to the metric.

Gauging theH subgroup nonetheless eliminates the H degrees of freedom, and it is easily verified thatthe signature of the resulting metric is the same as that of the standard coset metric. It is9

therefore straightforward to impose the phenomenological restriction to spaces with only asingle timelike coordinate [1]. The only subtlety is that the level k appears in front of theaction.

Positive k, in our sign conventions, results in a metric whose compact generatorscorrespond to timelike directions and noncompact generators to spacelike directions. Fornegative k (when allowed by unitarity), the roles of compact and noncompact generatorsare interchanged in the correspondence.To classify all the coset CFT’s with a single timelike coordinate we consider first thecase k positive and examine the differenceN ≡|G|c −|H|c ,for all possible cosets, where |G, H|c denote the number of compact generators.

To this endwe employ the known classification [12] of symmetric spaces G/H (where H is a maximalsubgroup and G is simple). From this list we eliminate all cases with N > 1, since for agiven G modding out by smaller (non-maximal) subgroups increases the value of N. ForN = 1, this leaves only the case SO(D −1, 2)/SO(D −1, 1) ([1]).

For N = 0, whichcorresponds to maximal compact subgroup embeddings, the possibilities are listed in table1. From this table we identify the cases for which H has a U(1) factor, H = H′ × U(1)(hermitian symmetric spaces), so that G/H′ has an additional compact generator, henceone timelike coordinate.

These latter cases are listed in table 2, and exhaust all possibilitiesin which G is a simple group. For k negative, we consider instead the difference N =|G|nc −|H|nc of noncompact generators, and find that the only solution with N = 1 isSO(D, 1)/SO(D −1, 1).For G a product of simple groups and U(1) factors, there are several possibilities toconsider:(i)G = G1 ⊗G2 ⊗G3 and H = H1 ⊗H2 ⊗H3 where G1/H1 is in table 2, G2/H2 is intable 1 (or products thereof) and G3/H3 is a (product of) compact coset(s).

(ii)G = G′ ⊗IR where G′/H has N = 0 (products of cases from table 1 and compact).In this case IR provides the timelike coordinate.These are the most general cases. Possibilities such as products of cases in table 2 moddedout by several U(1)’s, for example, are already included in case (i).

In table 3, we list allsuch cases with coset dimension ≤10 (and due to space limitations omit those with Gcompact). In this table, it is implicitly understood that all possible embeddings H ⊂Gare to be considered.The number of possible models so obtained is relatively small,particularly for lower dimensions.10

Other possibilities may be obtained by enlarging consideration from semisimple groupsG to non-semisimple groups of potential relevance, including of course the Poincar´e group.In 3D, for example, ignoring the non-semisimple cases leaves only U(1)3, SL(2,IR) andSU(2), whereas including them gives the nine groups corresponding to the Bianchi modelsconsidered in cosmology.We discuss briefly how to treat other potentially interestingcases involving non-semisimple groups.2 Let G be non-semisimple, then it is a semidirectproduct of S (a semisimple piece) and R (the radical, or maximal invariant subgroup) [12].The algebra takes the form[S, S] = S ,[S, R] ∈R ,[R, R] = R′ ,(2.11)where R′ ⊂N ⊂R. The Cartan matrix has zero eigenvalues, so the group manifold itselfis not interesting, but modding out by subgroups may eliminate these zeroes to give asensible space with a well-defined metric.

The number of zeroes is the dimension of N, themaximal nilpotent subalgebra of R. The possible cosets are(1)G/N. Since G/N = S ⊗R/N and R/N is an abelian invariant subalgebra, the zeroesare not eliminated but instead are equal in number to the dimension of R/N.

(2)G/R = S. This case gives all the semisimple groups. Those with a single timelikecoordinate are SL(2, R) ⊗C (we could also include C ⊗SO(1, 1)n), where C is anycompact semisimple group and one of the SO(1, 1)’s has negative level k to providethe timelike direction.

(3)G/S = R. In general R is an invariant subalgebra and will have abelian subalgebraswith associated zero eigenvalues, so the zeroes are not eliminated. The only exceptionis when R itself is abelian, so the Cartan metric is not defined from the regularrepresentation and will be nonsingular.

This leads to interesting cases such as ISO(d−1, 1)/SO(d−1, 1), but the general classification is not known. Whenever R as a grouphas only a single timelike coordinate, the zeroes can be eliminated but since R isabelian the only choices are products of SO(1, 1)’s and U(1)’s.

(4)(G/S)/N = R/N. As mentioned above this is an abelian subalgebra, so the Cartanmetric is not defined from the regular representation and we are left with the situationof (3).

(5)A more general situation would be to mod out by different subgroups not in the abovedecomposition, but this probably gives nothing new since R is a semidirect product2 We thank V. Kaplunovsky for a question that prompted this discussion.11

of abelian groups whose survival in the coset G/H (for any H) would result in zeroeigenvalues of the metric unless everything remaining is abelian as in (3) and (4)above. If they are all eliminated then we revert to case (2) above.A general means of obtaining non-semisimple cosets is by group contractions, so it maywell be possible to find a more systematic and complete procedure to generate all non-semisimple cosets using the properties of the semisimple cases.3.

SL(2, IR) ⊗SO(1, 1)D−2SO(1, 1) ModelsWe now consider the simplest class of coset models with a single timelike coordinateand any number of spacelike coordinates. In order to find the metric in the large k limit,we employ the standard procedure in nonlinear σ–models [13], as outlined in section 2:i.e.

find a parametrization of the G group elements, impose a unitary type gauge on thefields in the σ–model action and then solve for the (non-propagating) H-gauge fields toderive the G/H worldsheet action. From that action we can read of the correspondingbackground fields.

For the sake of generality, we write down the integrated action for ageneric, not necessarily simple, group G and a subgroup H, not necessarily abelian.We first write the gauged WZW action (2.3),(2.7) asL(g, A) = L(g) + 12πXikiZd2z trA ∂gg−1 ∓¯A g−1∂g + A ¯A ∓g−1Ag ¯Ai ,(3.1)where ∓represents respectively vector (g →hgh−1) and axial-vector (g →hgh) gauging.The ungauged action isL(g) = 14πXikiZd2z tr(g−1∂g g−1∂g)i −112πXiZBtr(g−1dg ∧g−1dg ∧g−1dg)i , (3.2)where i runs over the simple group factors in G = ⊗iGi. Writing A = Aaσa in terms of thegenerators σa of H, and integrating out the components Aa classically gives the effectiveactionL = L(g) ± 12πXi,jkikjZd2z tr(σbg−1∂g)i tr(σa∂gg−1)j Λ−1ab ,(3.3)withΛab ≡Xlkl tr(σaσb ∓σagσbg−1) .

(3.4)12

Notice that at the singular points of Λ the classical integration of the gauge fields fails,and hence (according to the discussion following (2.8)) where singularities of the targetspace metric are expected.For the SL(2, IR)⊗SO(1, 1)D−2SO(1, 1) models, we parametrize the group elementsasg =g00. .

.00g1. .

.0............00. . .gD−2,(3.5)whereg0 =au−vb(with ab + uv = 1)(3.6)andgi =cosh risinh risinh ricosh riwithi = 1, ..., D −2 .

(3.7)We choose the embedding such that the generator of H = SO(1, 1) isσ =s00. .

.00s1. .

.0............00. . .sD−2,(3.8)wheres0 = q0100−1andsi = qi0110,(3.9)with coefficients normalized to PD−2i=0 q2i = 1.3.1.

Vector gaugingUnder the infinitesimal vector gauge transformations δg = ε(σg −gσ), the parameterstransform as δa = δb = δri = 0, and δu = 2εq0u, δv = −2εq0v. The choices u = ±v thusfix the gauge completely.

From ±u2 = 1 −ab, we are left with the parameters a, b, ri asthe D spacetime coordinates. Substituting (3.5) and (3.8) into (3.3), we find the action(for both gauge choices u = ±v):L = k02πZd2z−∂a∂b + ∂b∂a2(1 −ab)+Xiκiδij + κjηiηj1 −ab∂ri∂rj+κiηi2(1 −ab)(b∂a −a∂b)∂ri + ∂ri(b∂a −a∂b)(3.10)13

where κi ≡ki/k0 and ηi ≡qi/q0. (From (3.9) we see that the ηi’s parametrize theembedding of SO(1, 1) into the factored SO(1, 1)’s in G.)This action can be identified with a σ–model action of the formS =Zd2zGMN + BMN∂XM∂XN(3.11)to read offthe background metric and antisymmetric tensor field (torsion).

We see that(3.10) gives for D = 2 the (dual) black hole metric of [2]ds2 = −da db/(1 −ab). Forκi →0, it reduces as expected to the 2D black hole and for ηi →0 gives the 2D blackhole times D −2 flat coordinates, again as expected since in this limit H = SO(1, 1) iscompletely embedded in SL(2, IR).

Note that for any D there is no torsion, in particularthe WZ term can be seen to be a total derivative for our choice of gauge. Furthermorewe can observe that there are at least D −2 isometries since the metric does not dependexplicitly on the coordinates ri.

Finally, note that the metric blows up only at the fixedpoint ab = 1 which is the point where (3.4) vanishes and the classical integration is notjustified. The fixed point of the isometry g →hgh is at ab = 0, which we expect to leadto a horizon.To further analyze this metric, we change to coordinates in which it is diagonal (suchcoordinates are to be expected due to the large number of isometries).

We consider (as inthe 2D case) the regions bounded by the horizon and singularity (fig. 1):(i) 0 < ab < 1 ,(ii) ab < 0 ,(iii) ab > 1 .

(3.12)(i) corresponds to the interior regions II, III; (ii) to the asymptotic regions I, IV ; and (iii)to the additional regions V,VI.In the interior regions (i), we can change to coordinates t, X0, Xi by defininga = sin t e(X0+mXD−2)b = sin t e−(X0+mXD−2)ri = Nij Xj ,(3.13)withNij =−ρjρi√κii = j + 1√κj+1 ηi ηj+1ρj+1ρjj ≥iηiρj(ρ2j + 1)1/2i ≤j = D −20otherwise ,(3.14)14

m = −q0ρ2 ,(3.15)andρ2l ≡lXi=1κi η2i ,andρ ≡ρD−2 . (3.16)The matrix elements Nij satisfy the relationsXlκlNliNlj = δiji, j ̸= D −2 ,Xlκl N 2lD−2 = 1/(ρ2 + 1) ,andXlκl ηlNlj = 0 for j ̸= D −2 .

(3.17)In these coordinates the metric takes the diagonal formds2 = k02π−dt2 + tan2 t dX20 +D−2Xi=1dX2i. (3.18)The remaining regions are described similarly.

For the asymptotic regions (ii), we usea = sinh R eX0+mXD−2b = −sinh R e−(X0+mXD−2)ri = Nij Xj ,(3.19)with the same m and Nij as above. In these coordinates the metric takes the formds2 = k02πdR2 −tanh2 R dX20 +D−2Xi=1dX2i.

(3.20)Finally, in the regions (iii) beyond the singularity the new variables are defined bya = cosh Re(X0+mXD−2)b = cosh Re−(X0+mXD−2)ri = NijXj(3.21)with metricds2 = k02πdR2 −coth2 R dX20 +D−2Xi=1dX2i. (3.22)15

Using the symmetry a →−a, b →−b, we identify the geometry (2D blackhole)⊗IRD−2. In particular the isometry generated by g →hgh is now explicit (it is alinear combination of translation in X0 and the Xi’s).

We can see how the associatedKilling vector changes signature on each boundary: it is timelike in (3.20) and (3.22) andspacelike in the region (3.18) in between. In (3.10), this was not explicit in the a, b, ricoordinates.

Although we have chosen a general embedding of H = SO(1, 1) in all of G,the resulting geometry nonetheless coincides with the case ηi = 0, where SO(1, 1) wasembedded only in SL(2, IR). This is as expected since the SO(1, 1) factors in G are areabelian and therefore transform trivially under g →hgh−1.

The spacetime diagram forthe relevant 2D geometry was shown in fig. 1.3.2.

Axial gaugingWe now consider the axial gauging for which things are less trivial. Under the infinites-imal gauge transformation δg = ε(σg + gσ), we see that δu = δv = 0 and δa = 2εq0a,δb = −2εq0b, δri = 2εqi.

A simple choice that fixes the gauge completely is a = ±b. Using±a2 = 1 −uv leaves u, v, and ri as the spacetime coordinates.

The gauged WZW actionfor the axial gauging is (3.1) with the lower (+) signs, and integration over the gauge fieldsgives again (3.3) but with Λ defined by the lower (+) sign in (3.4). Substituting (3.5) and(3.8) into (3.3), and using the above gauge fixing gives the effective actionL = k02πZd2zκi δij −κi κj ηi ηj1 −uv + ρ∂ri∂rj + (u∂v −v∂u)(u∂v −v∂u)4(1 −uv + ρ)−12(∂u∂v + ∂v∂u) −(u∂v + v∂u)(u∂v + v∂u)4(1 −uv)−κi ηi2(1 −uv + ρ)h(u∂v −v∂u)∂ri −∂ri(u∂v −v∂u)i.

(3.23)From this expression we can make the following observations. First, unlike the vectorgauging, there is nonvanishing torsion given by the term in square brackets in (3.23), eventhough the WZ term vanishes for the gauge choice made.

We also can see that the metrichas singularities at uv = 1, which in 2D is the fixed point of the axial transformation, andalso at uv = 1 + ρ, which is not a fixed point. Again the lines uv = 0 represent horizons,and the metric and torsion do not depend on the ri variables so there are also the D −2isometries ri →ri + constant.

As in the vector case, the D = 2 (κi = 0) limit reproducesthe 2D black hole of [2].Furthermore the ηi = 0 limit gives the geometry (2D black16

IIIIIIIVVVIVIIVIIIFig. 6: A two dimensional slice of the three dimensional black string metric (3.23).In addition to the regions of fig.

1, the regions VII,VIII lie between the singularitiesand inner horizons.hole)⊗IRD−2 (with vanishing torsion) as in the vector case, recovering the self-duality ofthose solutions.The general case is more conveniently studied via variables that diagonalize the metricin different regions. We will consider the analog of the three regions (i), (ii), (iii) of thevector case (3.12), but with a, b exchanged for u, v. In principle we could add an additionalregion due to the extra metric singularity which forms the inner horizon (fig.

6), but weshall find it is already included as part of region (iii).It is straightforward to see that the same changes of variables (3.13),(3.19),(3.21) madefor the vector case also diagonalize this metric, but now for m = 0 instead of (3.15). For(i) 0 < uv < 1, we findds2 = k02π−dt2 +1(ρ2 + 1) tan2 t + ρ2dX20 + tan2 t dX2D−2+D−3Xl=1dX2l,(3.24)and the antisymmetric tensor isBX0XD−2 =(ρ2 + 1) tan2 t + ρ2−1 .

(3.25)In the region (ii) uv < 0, we haveds2 = k02πdR2 +1(ρ2 + 1) coth2 R −ρ2 (−dX20 + coth2 R dX2D−2) +D−3Xl=1dX2l,(3.26)17

with torsionBX0XD−2 =(ρ2 + 1) coth2 R −ρ2−1 . (3.27)Finally in the region (iii) uv > 1 the metric isds2 = k02πdR2 +1(ρ2 + 1) tanh2 R −ρ2 (−dX20 + tanh2 R dX2D−2) +D−3Xl=1dX2l,(3.28)with torsionBX0XD−2 =(ρ2 + 1) tanh2 R −ρ2−1 .

(3.29)From these metrics we can compute the corresponding curvature scalar in each of theregions and findR = BX0XD−2 + constant . (3.30)We see that R blows up only in region uv > 1 at the hyperbola uv = 1 + ρ2 which is thereal singularity, whereas the surface uv = 1 is only a metric singularity where the signatureof the metric changes.

The latter is another horizon, in addition to uv = 0. The geometryis thus (3D black string)⊗IRD−3 with nonvanishing torsion and an inner horizon.

The 2Drepresentation (uv diagram) with the eight different regions separated by the horizons andsingularity, is presented in fig. 6.

It is not surprising that there is a trivial IRD−3 crossed onsince the hgh action of SO(1, 1) only acts on one nontrivial linear combination of SO(1, 1)generators of G.We have thus far given the expressions for the metric and antisymmetric tensor fieldfor both gaugings, but not the expression for the dilaton. This can be found in principleby considering the correct measure in the path integral, but it is technically simpler tofind it by solving the background field equations to lowest order in α′.

This procedure willalso be useful to verify that the expressions we have given are solutions of those equations.This is to be expected since they are valid for large values of k0 (equivalent to 1/α′ in thesigma model expansion).Let us consider the string background equations [14]RMN + DMDNΦ −14HLPM HNLP = 0(3.31)DLHLMN −(DLΦ)HLMN = 0(3.32)R −2Λ −(DΦ)2 + 2DMDMΦ −112HMNP HMNP = 0 ,(3.33)18

where Λ ≡(D −26)/3 is the cosmological constant in the effective string action and asusual HMNP ≡∂[MBNP ]. To check whether the expresions obtained above for the metricand antisymmetric fields satisfy these equations, we can restrict to one of the regions.

Wechoose the “cosmological” region 0 < uv, ab < 1 and assume an ansatzds2 = −dt2 +D−1Xi=1r2i (t) dX2i ,(3.34)Φ = Φ(t), and HMNP = HMNP (t). The two cases above are particular cases of this ansatz.For the vector gauging, ri = constant for i ≥2 and HMNP = 0.

For the axial gaugingri = constant for i ≥3 and HMNP is nonvanishing only for M, N, P = 0, 1, 2. The casewithout torsion was solved in general in [15] and has solutionsri(t) = αi tanpi γt ,Xip2i = 1(3.35)eΦ = β tan2p γt sec2 γt(3.36)with 2p = 1 + P pi, αi and β arbitrary constants and γ2 ≡(26 −D)/6.

It is easy to seethat our solution for the vector gauging is a particular case of this class of solutions withpi = 0, i > 1, as long as we make the shift k0 →k0 −2. This is suggested by the relationc =3k0k0 −2 −1 + (D −2) = 26(3.37)which implies (k0 −2)−1 = 26 −D/6 = γ2).

This representation makes it straightforwardto discuss the limit D →26 (γ →0) of our solutions. Expanding tan γt and rescaling thevariables we see that the metric depends on powers of t [15] in the cosmological regionand similarly for the other regions.

The curvature scalar behaves similarly so there is nosingularity and the black hole picture disappears. For D > 26 (k0 > 2), if allowed byunitarity constraints (which have not yet been entirely clarified for noncompact cosets)mwe can analytically continue the coordinates and get back the same black hole picture asfor D < 26 with the interchanges II↔I(V) and III↔IV(VI).For the axial case the ansatz (3.34) substituted into (3.31)–(3.33) gives the equations−Xi¨riri+ ¨Φ −Xi

˙Φ =¨Bij˙Bij−Xi˙riri(3.40)2Xi

The most general solution of theequations for the case of interest is thusr21 =Xiα2i tan2pi γt−1r22 = r21A tan2p1+2p2 γtB12 = r21XiBi tan2pi γteΦ = eΦ0 ˙B12r1r2= βr21 tanp1+p2−1 γt sec2 γt ,(3.42)where Bi, Φ0, β, and αi are arbitrary, the pi’s are constrained as above, A is given byAα1α2 = α21B2 −α22B1, and the constant γ remains as above.To see that this is the most general solution consistent with the ansatz is to countthe number of independent parameters. Equations (3.38)–(3.40) provide four second orderequations for the variables r1, r2, Φ, B12.

This allows eight free parameters given by theinitial conditions on the variables and their first derivatives. Equation (3.41) gives a non-trivial relation among them, which reduces the number to seven.

These seven parameterscan be extracted from the expressions above, taking into account a seventh parameter t0which results from the freedom t →t + t0 to choose the origin of time (not an isometry).Notice that the symmetry r1 ↔r2 of the field equations (3.38)–(3.41) is realized by thesymmetries pi →−pi and p1 ↔p2 of the circle described by the parameters pi. Againwe can see that the expressions (3.24),(3.25) obtained for the axial gauging are particularsolutions of the above equations for p1 = 1, p2 = 0, verifying that the WZW approachprovides solutions of the field equations to lowest order in α′ if we shift k0 as before.

Notealso that we now have as well a solution for the dilaton field.This solution was only valid for one of the regions of the black hole geometry (0

For uv < 0 we can make the rotations it →R,r2 →ir1 and r1 →r2. A similar rotation, including the shift t →t + π/2 gives the resultsfor region uv > 1.

There we see that the dilaton field, which gives the string coupling20

constant, blows up as expected only at the singularity uv = 1 + ρ2. In the D →26 limitof these solutions, the torsion vanishes and the solution collapses to the vector gaugingcase (making it self-dual!).

For D > 26 analytic continuation gives the same picture as forD < 26 (as in the vector case), so we see that the critical dimension plays an interestingrole in our solutions. Note that we are unable to obtain solutions for all allowed values ofthe pi parameters via a WZW construction, but expect that exactly marginal deformationsof the present CFT’s will access those parameters to complete the class of solutions.4.

DualityThe two different spacetime geometries, corresponding to the vector and axial gaugingsof the G/H WZW model, can be viewed as different modular invariant combinations ofrepresentations of the same holomorphic and anti-holomorphic chiral algebras. There aregeneral arguments [10] that show that the vector and axial gaugings are dual, in the senseof having equal partition functions.

The duality is similar to the familiar r →1/r dualityin c = 1 conformal field theory where two seemingly different theories are as well relatedby a changing the sign of the left (or holomorphic) currents J = JL with respect to theright (or anti-holomorphic) currents J = JR. The lorentzian D = 2 case is special sincethe same geometry (2D black hole) is obtained by either the vector or axial gauging, so wesay that the model is self-dual.

We now point out the sense in which geometries for D ≥2are dual, placing the vector/axial duality in a more generalized context.In ref. [6], following previous developments in supergravity, the r →1/r duality ofcompactified string theories was generalized to any string background for which the world-sheet action has at least one isometry.

For completeness, we review this analysis and treatexplicitly the case of N commuting isometries in bosonic string theory. The worldsheetaction for the bosonic string isS =14πα′Zd2zGMN(X) + BMN(X)∂XM∂XN + α′R(2) Φ(X),(4.1)where M, N = 1, ...D; GMN, BMN and Φ are the metric, antisymmetric tensor and dila-ton backgrounds respectively, and R(2) is the 2D curvature.

For a background with Ncommuting isometries, we write the action in the formS =14πα′Zd2zQµν(Xα) ∂Xµ∂Xν + Qµn(Xα)∂Xµ∂Xn + Qnµ(Xα)∂Xn∂Xµ+Qmn(Xα)∂Xm∂Xn + α′R(2)Φ(Xα),(4.2)21

where QMN ≡GMN +BMN and lower case latin indices m, n label the isometry directions.Since the Lagrangian (4.2) depends on Xm only through their derivatives, we can describeit in terms of the first order variables V m = ∂Xm,S =14πα′Zd2zQµν(Xα) ∂Xµ∂Xν + Qµn(Xα) ∂XµVn + Qnµ(Xα) V n∂Xµ+Qmn(Xα) V mVn + bXm(∂Vm −∂V m) + α′R(2) Φ(Xα). (4.3)This can be alternatively interpreted as gauging the isometry with the constraint of van-ishing gauge field strength [16].

Integrating the Lagrange multipliers bXm in the abovegives back (4.2). After partial integration and solving for V m and Vm, we find the dualactionS′ =14πα′Zd2zQ′µν(Xα) ∂Xµ∂Xν + Q′µn(Xα) ∂Xµ∂bXn+ Q′nµ(Xα) ∂bXn∂Xµ + Q′mn(Xα) ∂bXm∂bXn + α′R(2) Φ′(Xα).

(4.4)The dual backgrounds are given in terms of the original ones byQ′mn = Q−1mnQ′µν = Qµν −Q−1mn Qnν QµmQ′nµ = Q−1nm QmµQ′µn = −Q−1mn Qµm . (4.5)To preserve conformal invariance, it can be seen by a careful consideration of the pathintegral [6] (and from other approaches [17]) thatΦ′ = Φ −logpdet Gmn(4.6)Notice that equations (4.5) reduce to the usual duality transformations for the toroidalcompactifications of [18] in the limit Qmµ = Qµm = 0.

For the case of a single isometry(m = n = 0), we recover the explicit expressions of [6]. This is to our knowledge the mostgeneral statement of duality in string theory .

In particular we see that a space with notorsion (Qmµ = Qµm) can be dual to a space with torsion (Q′mµ = −Q′µm), as found inthe previous section. To prove the duality we should identify the particular isometry (ofthe D + 1 total) that relates them.

Notice that for every isometry we do not have to goto the first order formalism, i.e. we can integrate the Lagrange multipliers bXm for some ofthe fields and instead the V m for the remaining fields with isometries.

This is the most22

general form of these duality transformations, and eqs. (4.5) and (4.6) should be read withindices m, n running over only the variables with isometries that have been dualized.Before applying this formalism to the solutions we found in the previous section, wecompare this approach to duality with others in the literature.

In string theory, dualitysymmetry was originally discovered in toroidal compactifications and found to interchangewinding states with momentum (Kaluza–Klein) states in the compactified theory. We haveseen that the toroidal compactification is a particular case of a σ–model with isometriesand thus has this symmetry manifest.

The interchange of winding and momenta statesrealizes the duality symmetry in this particular background but is not necessarily a genericfeature of duality, so we might expect duality even in backgrounds where winding modesare not present. A particular example is given by the 2D Lorentzian black hole reviewedhere in section 2.The existence of duality as well implies a continuous noncompact global symmetry(3.6) at the classical level that relates the field equations of the theory (4.2) with theBianchi identities of the dual theory (4.4).

For the case of two dimensional σ–models inthe context of string theory, this symmetry was found in [19] to be SO(N, N). Duality isof course a discrete subgroup of this continuous symmetry.

The noncompact continuoussymmetries have been very useful to identify the moduli space in certain string compact-ifications and more recently have been used to find new nonstatic solutions from knownones [20,21]. We wish to emphasize that they are not true string symmetries since, just asin static backgrounds, they are broken by nonperturbative effects on the worldsheet.

Thesurviving symmetry is a discrete symmetry which includes duality and integer shifts of theantisymmetric tensor, and as shown in [22] generalizes to SO(N, N, ZZ).In order to make a connection between duality in this formulation and the vector–axial duality in G/H WZW models, we recall the discussion of duality for the latter[10].If H is abelian, group elements g ∈G can be parametrized as g = eiσφbg where σ is thegenerator of H (considered to be U(1) for concreteness).Substituting into the vectorgauged WZW action, it turns out that the action depends only on ∂φ, and proceedingwith the standard duality transformations (4.5) the axial gauged action is obtained. Theisometry in this case is then φ →φ + δφ which is generated by the right transformationg →gh.After gauging the g →hgh−1 transformation, we see that there will alwaysbe a remaining isometry generated by the other independent transformation g →hg, ormore symmetrically by the axial transformation g →hgh.

The same occurs for the axialgauging, where the remaining isometry is given by the vector action g →hgh−1. (In the23

case of gauged non-abelian symmetries, it followed from the analysis of section 2 that dueto quantum effects the ungauged symmetry does not remain even as a global symmetry. )Now we are ready to analyze duality in the models of the previous section.

Startingwith the vector gauging, we have to see how the gauged fixed parameters transform underg →hg, and go to a basis where only one of the coordinates transforms. In that basis themetric is not necessarily diagonal so the duality transformations (4.5) will be non-trivial.3Let us consider the 0 < ab < 1 region for concreteness.

In that case we see that underg →hg the original parameters transform asδa = εq0a ,δb = −εq0b ,δri = εqi . (4.7)We can see that the coordinates which diagonalize the metric transform asδt = 0δX0 = ε(ρ2 + 1)δXi = 0 ,i = 1, .

. ., D −3δXD−2 = ε ,(4.8)where we have used (3.17).

Note that if we exchange XD−2 for Y ≡X0 −ΩXD−2, withΩ≡ρ2 + 1, the independent system of coordinates t, X0, Y and Xi (i = 1, . .

., D −3) issuch that only X0 transforms, so this defines the isometry.In these coordinates, the metric takes the formds2 = −dt2 +tan2 t + 1Ω2dX20 + 1Ω2 dY 2 −1Ω2 dX0 dY +D−3Xi=1dX2i(4.9)From equations (4.5), we can find the dual metric with the single isometry X0 →X0+δX0.It is straightforward to verify that it coincides with the one given in equation (3.24) comingfrom the hgh gauging. This proves that regions III of both geometries are mapped to eachother under duality.

An identical analysis can be carried out for the other regions obtained3 Duality for the diagonal metric was explicitly proved in [23] to lowest order in α′, andextended to next order in [24] where either a field redefinition or equivalently a change in thedilaton transformation was required. This had the interesting consequence that large to smallradius duality could be explicitly realized in cosmology, as treated in [25], but requiring only weakcoupling information from string theory, as recently analyzed in [26,27].

Duality for nondiagonalmetrics relates not only large to small radius, but as well relates different cosmological models.24

by analytic continuation of this region. It is straightforward to see that region V of fig.

1is mapped to region I of fig. 6 and in particular the singularity of the first is mapped toone horizon in the second.

Also, region I of the vector gauging black hole gets mappedto regions V and VII together of the axial gauging black hole. This has the interestingimplication that a surface which in one geometry is perfectly regular (ab = ρ2) is mappedto the singularity in the other geometry (uv = 1 + ρ2).This goes even further thanthe black hole singularity/horizon duality of the 2D black holes [11], since in that casethe horizon is a better behaved region than the singularity but there remains nontrivialbehavior such as the exchange of spacelike and timelike coordinates.

In the present case itcan be seen explicitly that string theory can deal with spacetimes that have singularitiesat the classical level, in the sense that there still exists a description of interactions, etc.for that region of spacetime. The situation is not so different from situations have beenencountered in compactified cases where singular spaces (orbifolds) are dual to nonsingularones (tori).

(For a review in the simplest c = 1 case, see [28]; in a more general context thathas recently arisen, see e.g. [29].) It would be interesting to study the present geometriesat the singularity in more detail to start probing string theory in those regimes.We have therefore established the duality between the (2D black hole)⊗IRD−2 and(3D black string)⊗IRD−3 geometries.

A more general analysis may be performed for thecombination of all the isometries using (4.5) and for the more general solutions discussedearlier. Also we can easily check that the solutions (3.42) are related to those of (3.35)for three dimensions by duality — we rotate the spacelike coordinates in (3.35) to get anondiagonal metric and apply (4.5).Similarly, starting from (3.35) for any number of dimensions we can find new (cosmo-logical) string solutions with torsion by applying (4.5) after going to a nondiagonal basis.For the boosted variables,XM ≡ΛMNYN ,(4.10)we see that dualizing equations (3.35) gives the metric (using the same index conventionmentioned after (4.5))Gmn =XMα2M tan2pM γt ΛMm ΛMn−1Gµν = GmnXM,Nα2M α2N tan2pM+2pN γt· ΛNµΛMn(ΛNνΛMm −ΛMνΛNm) ,(4.11)25

and torsionBnµ = GnmXMαM tan2pM γt ΛMm ΛMµ ,(4.12)with a corresponding expression for the dilaton determined from (4.6). These expressionsprovide new explicit cosmological solutions with torsion for any value of m, n ≤N (and Nvarying from one to the total number of isometries).

Notice that they depend on D2−D+1arbitrary parameters (ΓMN ≡αMΛMN, pM, and t0) which equals the number of boundaryconditions allowed for GMN, BMN and Φ and their first derivatives (minus the constraintprovided by equation (4.6)). Eq.

(3.42) is the D = 3 case of these general solutions. It iscertainly interesting to explore the possible cosmological consequences of these solutionsas well as the implications of their duality to the known solutions of [15].

Similar remarksapply for the generalization of the other black hole regions (by analytic continuation) andit would be interesting to determine if together they could lead to a geodesically completesystem of coordinates for a black hole type of geometry with (4.11) as the cosmologicalregion. This is certainly true for the cases we constructed in section 3 (p1 = 1, all otherpi = 0), where (4.11) and (4.12) generate new dual black branes (although they are notdirectly obtained from the WZW construction).5.

ConclusionsWe have presented a general discussion of the class of geometries that can be obtainedin string theory from noncompact coset conformal field theories in the large k limit. TheWZW approach plays a crucial role in allowing identification the background fields, inparticular the metric of the target spacetime.

This is not necessary for the CFT describ-ing the internal degrees of freedom in superstring compactifications, since in that case ageometrical interpretation for that sector of the string vacua is unnecessary (and does noteven necessarily exist in general).We have found that the number of possible geometries obtained in this way is veryrestricted due to the constraint of having a single timelike coordinate.4In the case ofsuperstring compactifications, however, the coset G/H describing the internal degrees offreedom actually provides many string vacua. These arise both from the different possible4 Euclidean cosets, corresponding to the cases listed in table 1, remain useful for describ-ing instanton-like configurations such as euclidean black holes, and for describing compactifieddimensions.26

choices of boundary conditions (orbifoldizing) and also because they are only special pointsin a degenerate space of vacua parametrized by the exactly marginal deformations ofthe CFT. In the case of (2,2) compactifications, for example, the coset models describeparticular points of a Calabi-Yau manifold.The study of the spectrum of the noncompact coset models, especially the marginaldeformations, will thus generate new classes of geometries which would be interesting toinvestigate and still represent exact CFT’s.

A motivation for considering the models ofsection 3 was to find a conformal field theory realization of the cosmological solutions foundin [15]. We have only partially succeeded since in the vector gauging our solutions in thecosmological region correspond to those of [15] only for fixed values of the parameters piof equations (3.35).

A natural expectation is that exactly marginal deformations of ourmodels will turn on those parameters pi and generate the whole class of cosmological modelsof reference [15], as well as the new class of models with torsion we give in (3.42) for the axialgauging. This could be an interesting extension of the results here.

Another possibility isthat by marginal deformations, the black hole–like singularities of the noncompact cosetscould be “blown-up”, reminiscent of the way orbifold singularities can be blown-up toobtain smooth string compactifications.This would illustrate the ameliorating controlthat string theory seems to have over singularities, already exemplified in section 4 bythe duality between singular and regular regions of the different black hole geometries ofsection 3.Our discussion of duality in section 4, following [6], is based entirely on the existenceof isometries of the σ–model. Although it is the most general statement of that symmetryto date (including the known toroidal compactifications as particular cases), it cannot bethe final statement because we know that there are geometries, such as Calabi-Yau spaces,that have no continuous isometries and still are known to have duality-like symmetries,for example the mirror symmetry of [30].

It would be very interesting to have a unifiedunderstanding of these dualities from a σ–model point of view.Finally, since the subject of the present article has been evolving faster than our abilityto write it up, we briefly discuss some of those recent results that have partial overlap withour work. First, the list of single-time cosets for the case of simple groups (table 2) wasindependently given in [31] using the known list of supersymmetric (K¨ahler) cosets (withthe exception of the SO(D −1, 2)/SO(D −1, 1) given earlier in [1]) but with no claim tocompleteness.

The particular case D = 3 of the axial gauging geometry of the models ofsection 3, was discussed in [32] where a complete discussion of the black string geometry27

can be found. The expressions for the background fields given therein are in agreementwith those given here after a simple change of variables.

Duality of that geometry wasdiscussed in [33] and the relation with the vector gauging was briefly mentioned, althoughthe proof of the duality of both geometries was not explicitly presented.5 More recently,duality for several commuting symmetries was discussed in [34] and an SO(N, N, ZZ) wasidentified as the modular group in agreement with our comments about the general resultsof [19] and [22]. The periodic coordinates and the winding mode / momentum dualityinsisted upon by these authors, however, is not considered essential here.

The solution of3D cosmological backgrounds discussed in [35] are particular cases of our solutions (3.42)(up to analytic continuations). Other models have been recently explored [36] and togetherwith the models considered here, most of the possible cases of our table 3 for D ≤4 havebeen investigated.AcknowledgementsWe thank I.

Bars, J. Distler, and E. Verlinde for discussions at the Aspen Center forPhysics. We are also grateful for helpful conversations with P. Candelas, X. de la Ossa,J.-P. Derendinger, L. Ib´a˜nez, M. Crescimanno, and as well as J. Horne and G. Horowitzfor information on their related work.

This work was supported in part by DOE contractW-7405-ENG-36.5 In [32], duality is found to map the singularity of the axial gauged geometry to a region insidethe singularity of the vector gauged geometry, instead of to the asymptotically flat region as foundhere. The discrepancy is due to the presence of two isometries, and hence more than one possibleduality.

The two results are easily reconciled by applying a second duality transformation.28

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GH#H GeneratorsDim(G/H)SL(p,C)SU(p)p2 −1p2 −1SL(p, IR)SO(p)12p(p −1)12p(p + 1) −1SU ∗(2p)USp(2p)p(2p + 1)(p −1)(2p + 1)SU(p, q)SU(p) × SU(q) × U(1)p2 + q2 −12pqSO(p,C)SO(p, IR)12p(p −1)12p(p −1)SO(p, q)SO(p) × SO(q)12p(p −1) + q(q −1)pqSO∗(2p)SU(p) × U(1)p2p(p −1)Sp(2p,C)USp(2p)p(2p + 1)p(2p + 1)Sp(2p, IR)SU(p) × U(1)p2p(p + 1)USp(2p, 2q)USp(2p) × USp(2q)p(2p + 1) + q(2q + 1)4pqGc2G2(−14)1414G2(+2)SU(2) × SU(2)68F c4F4(−52)5252F4(+4)USp(6) × SU(2)2428F4(−26)SO(9)3616Ec6E6(−78)7878E6(+6)USp(8)3642E6(+2)SU(6) × SU(2)3840E6(−14)SO(10) × SO(2)4632E6(−26)F4(−52)5226Ec7E7(−133)133133E7(+7)SU(8)6370E7(−5)SO(12) × SO(3)6964E7(−25)E6(−78) × SO(2)7954Ec8E8(−248)248248E8(+8)SO(16)120128E8(−24)E7(−133)136112Table 1: Noncompact coset spaces G/H with no timelike coordinates. G is simple and His the maximal compact subgroup.

GH# G Generators# H GeneratorsSignaturecompactnoncompactcompactnoncompactSU(p, q) SU(p) × SU(q)p2 + q2 −12pqp2 + q2 −20(1, 2pq)SO(p, 2)SO(p, 1)12p(p −1) + 12p12p(p −1)p(1, p)SO(p, 2)SO(p)12p(p −1) + 12p12p(p −1)0(1, 2p)Sp(2p, IR)SU(p)p2p(p + 1)p2 −10(1, p(p + 1))SO∗(2p)SU(p)p2p(p −1)p2 −10(1, p(p −1))E6(−14)SO(10)4632450(1, 32)E7(−25)E6(−78)7954780(1, 54)Table 2: Coset spaces G/H with only one time coordinate (for simple groups G)

DG/H2SL(2,IR)SO(1,1)3SO(2,2)SO(2,1); {(D = 2 case);SL(2,IR)U(1)} × IR4SO(3,2)SO(3,1);SO(2,2)SO(1,1)×SO(2);SO(3,C)SO(3,IR);SO(3,1)SO(3) ; (D = 3)} × IR5SU(2,1)SU(2) ; SO(4,2)SO(4,1); SO(2,2)SO(2) ; SL(2,IR)×SL(2,C)SO(1,1)×SU(2){SU(2,1)SU(2)×U(1);SO(2,2)SO(2)2 ;SO(4,1)SO(4) ; (D = 4)} × IR6SO(5,2)SO(5,1); { SL(3,IR)SO(3) ; SO(5,1)SO(5) ; (D = 5)} × IR; (D = 4) × SL(2,IR)SO(2)(D = 3) × SL(2,C)SU(2) ; (D = 2) × {SU(2,1)SU(2)×U(1); SO(4,1)SO(4) ; SO(2,2)SO(2)2 }7{SU(3,1)SU(3)×U(1);SO(4,C)SO(4,IR);SO(6,1)SO(6) ;SO(3,2)SO(3)×SO(2); (D = 6)} × IRSU(3,1)SU(3) ;SO(6,2)SO(6,1);SO(3,2)SO(3) ; (D = 5) × SL(2,IR)SO(2) ; (D = 4) × SL(2,C)SU(2)(D = 3) × {SU(2,1)SU(2)×U(1);SO(4,1)SO(4) ;SO(2,2)SO(2)2 }; (D = 2) × { SL(3,IR)SO(3) ;SO(5,1)SO(5) }8SO(7,2)SO(7,1); { SO(7,1)SO(7) ; (D = 7)} × IR; (D = 6) × SL(2,IR)SO(2) ; (D = 5) × SL(2,C)SU(2)(D = 4) × {SU(2,1)SU(2)×U(1);SO(4,1)SO(4) ;SO(2,2)SO(2)2 }; (D = 3) × { SL(3,IR)SO(3) ;SO(5,1)SO(5) }(D = 2) × {SU(3,1)SU(3)×U(1);SO(4,C)SO(4,IR);SO(6,1)SO(6) ;SO(3,2)SO(3)×SO(2)}9SU(2,2)SU(2)×SU(2);SU(4,1)SU(4) ;SO(8,2)SO(8,1);SO(4,2)SO(4) ; (D = 7) × SL(2,IR)SO(2) ; (D = 6) × SL(2,C)SU(2)(D = 5) × {SU(2,1)SU(2)×U(1);SO(4,1)SO(4) ;SO(2,2)SO(2)2 }; (D = 4) × { SL(3,IR)SO(3) ;SO(5,1)SO(5) }(D = 3) × {SU(3,1)SU(3)×U(1);SO(4,C)SO(4,IR);SO(6,1)SO(6) ;SO(3,2)SO(3)×SO(2)}; (D = 2) × SO(7,1)SO(7) ; IR×{ SL(3,C)SU(3) ;SU(2,2)SU(2)2×U(1);SU(4,1)SU(4)×U(1);SO(8,1)SO(8) ;SO(4,2)SO(4)×SO(2);USp(4,2)USp(4)×USp(2); (D = 8)}10(D = 8) × SL(2,IR)SO(2) ; (D = 7) × SL(2,C)SU(2) ; (D = 6) × {SU(2,1)SU(2)×U(1);SO(4,1)SO(4) ;SO(2,2)SO(2)2 }(D = 3) × SO(7,1)SO(7) ; (D = 4) × {SU(3,1)SU(3)×U(1);SO(4,C)SO(4,IR);SO(6,1)SO(6) ;SO(3,2)SO(3)×SO(2)}(D = 2) × { SL(3,C)SU(3) ;SU(2,2)SU(2)×U(1);SU(4,1)SU(4)×U(1);SO(8,1)SO(8) ;SO(4,2)SO(4)×SO(2);USp(4,2)USp(4)×USp(2)}(D = 5) × { SL(3,IR)SO(3) ;SO(5,1)SO(5) }; { SL(4,IR)SO(4) ;SO(9,1)SO(9) ; (D = 9)} × IR;SO(9,2)SO(9,1)Table 3: Noncompact coset spaces G/H with one time coordinate with dim(G/H) ≤10,where G is a product of simple noncompact groups.

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