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Some Exact Solutions of String Theory

arXiv:hep-th/9110067v1 24 Oct 1991EFI-91-57Some Exact Solutions of String Theoryin Four and Five DimensionsPetr Hoˇrava∗Enrico Fermi InstituteUniversity of Chicago5640 South Ellis AvenueChicago, IL 60637, USAe-mail: horava@yukawa.uchicago.eduWe find several classes of exact classical solutions of critical bosonic string theory, con-structed as twisted products of one Euclidean and one Minkowskian 2D black hole coset.One class of these solutions leads (after tensoring with free scalars and supersymmetrizing)to a rotating version of the recently discovered exact black fivebrane. Another class rep-resents a one-parameter family of axisymmetric stationary four-dimensional targets withhorizons.

Global properties and target duality of the 4D solutions are briefly analyzed.October 1991∗Robert R. McCormick Fellow; work also supported by the NSF under Grant No. PHY90-00386; the DOE under Grant No.

DEFG02-90ER40560; the Czechoslovak Chart 77 Foundation;and the ˇCSAV under Grant No. 91-11045.

1. IntroductionExact solutions of (perturbative) string theory that do not represent strings in staticspacetimes with flat time coordinate (plus possible internal degrees of freedom), but ratherdescribe spacetimes of nontrivial metric properties, have attracted much interest recently.One of the most excellent examples are Witten’s black holes in 2D string theory [1], aswell as their generalizations to charged black holes in 2D [2], black strings in 3D [3], andexact fivebranes in 10D superstring theory [4].

With these solutions at hand, one naturallywonders whether the techniques can be used to construct exact solutions of string theoryin four dimensions.To answer this question in the affirmative, let us start with Witten’s black hole cosetin 2D spacetime with Minkowskian signature. We might obtain a model which describesstrings in a 4D manifold with Minkowskian signature by tensoring the 2D black hole withanother conformal field theory, describing strings in a 2D manifold of Euclidean signature.Obviously, we have one excellent candidate for such a manifold: The 2D black hole itself,now in the Euclidean regime.To get a nontrivial 4D spacetime, we would like to allow a ‘twist’ in the productof the two conformal field theories.

Technically, the basic idea of this paper is to startwith a direct product of two WZW models, and gauge a group that acts nontrivially onboth of them, thus producing a conformal field theory that is no longer a direct product(compare [3]). In the case of two 2D black holes, a naive way of doing so might be asfollows.

Starting with the direct product of two SL(2, R) WZW models (referred to asSL(2, R)M,E throughout),LWZW = ikM4πZd2z Tr (g−1M ∂gM g−1M ∂gM) −ikMΓ(gM)+ ikE4πZd2z Tr (g−1E ∂gE g−1E ∂gE) −ikEΓ(gE)(1.1)where gM,E ∈SL(2, R)M,E, we will gauge two Abelian symmetry groups of the model.First, we will gauge the compact Abelian groupgE →hE gE hE(1.2)with hE generated by01−10. At this stage, we get a direct product of the 2D blackhole coset with Euclidean signature in sector E and the ungauged WZW model in sectorM.

In the second step, we will gauge the noncompact groupgM →hM gM hM,gE →hαE gE h−αE(1.3)1

with hM generated by100−1. Here α ∈R is a ‘distortion parameter’: For α = 0, weget a direct product of one Euclidean and one Minkowskian 2D black hole.The total conformal anomaly of the gauged model isc =3kMkM −2 +3kEkE −2 −2.

(1.4)To get a critical theory in four dimensions, we set c = 26. This condition restricts thevalues of kM,E tokM,E = k ± ˜k,(1.5)with˜k = ±r(k −2811)(k −2).

(1.6)We will further restrict ourselves tok ≥2811,(1.7)to avoid complex values of the levels.1Let us parametrize the group manifolds by their Euler angles:gM =etL/200e−tL/2 cosh rM2sinh rM2sinh rM2cosh rM2 e−tR/200etR/2,gE = cos θL2sin θL2−sin θL2cos θL2 cosh rE2sinh rE2sinh rE2cosh rE2 cos θR2−sin θR2sin θR2cos θR2,(1.8)with rM ∈R, rE ∈[0, ∞), tL,R ∈R, and θL,R ∈[0, 2π],2 and denote the gauge fieldsthat correspond to (1.2) and (1.3) by AE, AE and AM, AM respectively. Upon choosing aunitary gauge by settingtL = tR ≡t,θL = θR ≡θ,(1.9)1 We have omitted here the other region with real kM,E, namely k ≤85.

This region wouldcorrespond to the analytic continuation of one of the SL(2, R)’s to SU(2) (see below).2 Two facts seem worth stressing. First, the Euler angle parametrization of gM we have useddoes not cover the whole SL(2, R)M manifold.

Second, the ranges for θL,R, rE cover SO(2, 1)rather than its double cover SL(2, R).2

we arrive at the following Lagrangian:L4D = LWZW + ikMπZd2z sinh2 rM2 (AM∂t −AM∂t)+ 2ikMπZd2z cosh2 rM2AMAM+ ikEπZd2z sinh2 rE2 (AE∂θ −AE∂θ + αAM∂θ + αAM∂θ)+ 2ikEπZd2z (α2 sinh2 rE2 AMAM −cosh2 rE2 AEAE)+ ikEπZd2z α(AEAM −AMAE)(cosh2 rE2 + sinh2 rE2 ). (1.10)As the gauge fields enter quadratically, they can be integrated out by solving their equationsof motion.

The final (lowest order) Lagrangian readsL4D = i2πZd2z kE4 ∂rE∂rE + kE sinh2 rM2 (cosh2 rE2 −K)cosh2 rM2 cosh2 rE2 −K∂θ∂θ+ kM4 ∂rM∂rM −kMsinh2 rE2 (cosh2 rM2 −K)cosh2 rM2 cosh2 rE2 −K ∂t∂t+αkE sinh2 rM2 sinh2 rE22(cosh2 rM2 cosh2 rE2 −K)(∂θ∂t −∂θ∂t)!,(1.11)with K a shorthand for 4α2kE/kM. Obviously, it describes strings in a 4D target, withthe following metric and antisymmetric tensor background:ds2 = kE4 dr2E + kM4 dr2M + kEsinh2 rM2 (cosh2 rE2 −K)cosh2 rM2 cosh2 rE2 −K dθ2−kMsinh2 rE2 (cosh2 rM2 −K)cosh2 rM2 cosh2 rE2 −K dt2,B =αkE sinh2 rM2 sinh2 rE22(cosh2 rM2 cosh2 rE2 −K)dθ ∧dt.

(1.12)The nontrivial determinant coming from the integration over the gauge fields leads to adilaton background,Φ = ln (cosh2 rM2 cosh2 rE2 −K) + const. (1.13)Throughout the paper, we will only consider sigma-model metrics.

The so-called canon-ical metrics can be obtained from the sigma-model metrics by a proper rescaling by anexponential of Φ.3

A priori we might have expected that the 4D theory would have an exact U(1)×U(1)symmetry. Indeed, two abelian symmetries survive our gauging of the WZW model.

With-out any additional arguments, our conformal field theory seems to represent a class of exactstationary and axisymmetric solutions of string theory in four dimensions. Unfortunately,this is not true, the reason being that the gauging we have attempted to do is in factanomalous.

Indeed [5], we can easily check that the anomaly-cancellation condition,Tr (Ta,LTb,L −Ta,RTb,R) = 0,(1.14)is not met. (Here we have used the notation of [5], i.e.

a, b are gauge group indicesand Ta,L, Ta,R generate the gauge group action on the WZW fields, δg = ǫa{Ta,L · g + g ·Ta,R}.) We thus cannot hope that (1.12) is more than a solution of the low-energy effectiveapproximation to string theory.To obtain genuine solutions of string theory, we will modify our basic strategy in twodirections.

First, we will obtain in section 2 a class of models in five dimensions, simplyby gauging the U(1) group that acts on both of the SL(2, R)’s, and forgetting aboutthe other U(1). As we will see, this leads (after continuing analytically, tensoring withfree bosons, and supersymmetrizing) to rotating versions of the recently discovered [6][4]fivebrane solitons of superstring theory.

Second, in section 3 we will modify the action ofU(1) × U(1) so as to avoid the violation of Witten’s condition of non-anomalousness. Thiswill result in a class of exact, stationary and axisymmetric solutions in four dimensions.2.

Exact Rotating Black Fivebranes from 2D Black HolesLet us now start with the tensor product Lagrangian (1.1), and gauge the U(1) groupacting bygE →hE gE hE,gM →hβM gM hβM(2.1)4

with β a distortion parameter. Upon gauging this group, the Lagrangian becomesL = LWZW + ikE2πZd2z A Tr01−10∂gEg−1E+ ikE2πZd2z A Tr01−10g−1E ∂gE+ ikE2πZd2z AA−2 + Tr01−10gE01−10g−1E+ ikM2πZd2z βA Tr100−1∂gMg−1M+ ikM2πZd2z βA Tr100−1g−1M ∂gM+ ikM2πZd2z β2AA2 + Tr100−1gM100−1g−1M(2.2)where we have denoted by A, A the gauge field associated with (2.1).

Upon parametrizingthe group manifolds as in (1.8) and fixing the gauge byθL = θR ≡θ,(2.3)we observe that the model describes the following five-dimensional background (to lowestorder):ds25D = kM4 dr2M + kE4 dr2E + kE sinh2 rE2 (1 −L cosh2 rM2 )cosh2 rE2 −L cosh2 rM2dθ2+ kM sinh2 rM2 (L −cosh2 rE2 )cosh2 rE2 −L cosh2 rM2dt2 + 2kMβ sinh2 rE2 sinh2 rM2cosh2 rE2 −L cosh2 rM2dt dθ+ kM cosh2 rM2 cosh2 rE2cosh2 rE2 −L cosh2 rM2d˜t2,B = kM sinh2 rM2 cosh2 rE2cosh2 rE2 −L cosh2 rM2dt ∧d˜t −βkM sinh2 rE2 cosh2 rM2cosh2 rE2 −L cosh2 rM2dθ ∧d˜t,Φ = ln (cosh2 rE2 −L cosh2 rM2 ) + const(2.4)where t, ˜t = 12(tL ± tR) and L = β2kM/kE. As (2.3) does not fix the gauge completely, ˜tis orbifoldized, and ˜t ≡˜t + 2πβ.While this geometry is interesting in itself, we can find connections to some resultsobtained recently [6],[4],[7]– [11] by continuing it analytically to a gauged SL(2, R)×SU(2)WZW model.

Upon parametrizing the SU(2) group manifold by its Euler angles,g =eiθL/200e−iθL/2 cos φ2i sin φ2i sin φ2cos φ2 e−iθR/200eiθR/2,(2.5)5

with φ ∈[0, π) and the ranges for θL,R as before, we can see that the corresponding gaugedmodel is related to the one constructed previously, by the analytic continuation of rE to φvia rE = iφ. In addition, this analytic continuation has to be supplemented with the signreversal of the level, in order to preserve the relative metric signature of the two groupmanifolds.

We will thus assume k ≡−kE ≥0 henceforth, as well as reverse the sign of Lso as to ensure L ≥0. The central charge of the model isc =3kMkM −2 +3kk + 2 −1(2.6)and k is restricted by unitarity to a discrete set of values, as usual.

(We don’t set c = 26here, as it is more interesting to tensor the model with five free scalars and supersymmetrizeit. Imposing ctot = 15 afterwards, we obtain a solution of N = 1 superstring theory.) Withthese conventions, we getds25D = kM4 dr2M + k4dφ2 + k sin2 φ2 (1 + L cosh2 rM2 )cos2 φ2 + L cosh2 rM2dθ2−kM sinh2 rM2 (L + cos2 φ2 )cos2 φ2 + L cosh2 rM2dt2 −2kMβ sin2 φ2 sinh2 rM2cos2 φ2 + L cosh2 rM2dt dθ+ kM cosh2 rM2 cos2 φ2cos2 φ2 + L cosh2 rM2d˜t2,B = kM sinh2 rM2 cos2 φ2cos2 φ2 + L cosh2 rM2dt ∧d˜t + βkM sin2 φ2 cosh2 rM2cos2 φ2 + L cosh2 rM2dθ ∧d˜t,Φ = ln (cos2 φ2 + L cosh2 rM2 ) + const.

(2.7)This model is a rotating analog of the fivebrane soliton discovered in [6][4]. The core ofthe fivebrane is surrounded by a spherical horizon3 localized at rM = 0.

At fixed t, thehorizon inherits the following metric:ds2horizon = k4dφ2 + k sin2 φ2 (1 + L)cos2 φ2 + Ldθ2 + kM cos2 φ2cos2 φ2 + Ld˜t2. (2.8)The exact black fivebrane of [4] has the structure of the direct product of SU(2) andSL(2, R)/U(1), which we recover in the limit of L →∞.3 Our coordinates φ, θ, ˜t do parametrize a 3-sphere, albeit in an unusual manner.

More standardcoordinates on the sphere would result from an alternative gauge choice in the gauged WZWmodel, namely tL = tR ≡0.6

It is worth noting that the model is indeed not asymptotically flat, rather it is asymp-totic to S3 × R, as can be seen in the rM →∞limit of the metric:4ds25D →kM4 dr2M + k4dφ2 + k sin2 φ2 dθ2 −(kM + kβ2 cos2 φ2 )dt2−2kβ sin2 φ2 dt dθ + kβ2 cos2 φ2 d˜t2. (2.9)We can also observe dragging of inertial frames, a typical effect of rotating bodies in generalrelativity.As we have argued that our class of conformal field theories represents essentially arotating deformation of the fivebrane solution constructed by Giddings and Strominger, itis natural to look for the exact marginal vertex operator that governs this deformation.

Inthe approximation of (2.7), the vertex operator can be easily identified asVmarg ∼ik2πZd2z sin2 φ2htanh2 rM2 (∂t∂θ + ∂t∂θ) −(∂θ∂˜t −∂θ∂˜t)i. (2.10)This indeed represents a lowest order approximation to an exactly margninal operator.

Theexact form of the operator can be identified by looking at the full-fledged coset Lagrangian,leading toVmarg = ik2πZd2zA Tr01−10∂gEg−1E+ A Tr01−10g−1E ∂gE. (2.11)One can easily check that after integrating out the gauge field in (2.11), one arrives at(2.10).

Note the interesting fact that the exactly marginal vertex operator (2.11) acts onthe conformal field theory of the non-rotating fivebrane (which corresponds to the limitof L →∞in our parametrization) by redefining the BRST charge, thus leading to aone-parametric class of deformations of the BRST cohomology of the model.Recalling that φ ∈[0, π), we can see that the set of coordinates we have used todescribe the rotating fivebrane covers just one half of the external spacetime. Obviously,the metric can be continued analytically to φ ∈[0, 2π); nevertheless, another way to the4 Actually, one might expect the model to represent a limiting, exactly solvable case of a classof solutions to the low-energy action of string theory, quite analogously as in [6].

These low-energy solutions can be expected to open the throat at infinity. I am indebted to JeffHarvey forilluminating discussions on this point.7

analytic continuation exists. To show this, let us again start with the Lagrangian (1.1),but now gaugegE →hE gE h−1E ,gM →hβM gM hβM.

(2.12)Repeating the same story as above, we arrive atds25D = kM4 dr2M + kE4 dr2E + kE cosh2 rE2 (1 + L cosh2 rM2 )sinh2 rE2 + L cosh2 rM2dθ2−kM sinh2 rM2 (L + sinh2 rE2 )sinh2 rE2 + L cosh2 rM2dt2 + 2kMβ cosh2 rE2 sinh2 rM2sinh2 rE2 + L cosh2 rM2dt dθ+ kM cosh2 rM2 sinh2 rE2sinh2 rE2 + L cosh2 rM2d˜t2,B = kM sinh2 rE2 sinh2 rM2sinh2 rE2 + L cosh2 rM2dt ∧d˜t −βkM cosh2 rE2 cosh2 rM2sinh2 rE2 + L cosh2 rM2dθ ∧d˜t,Φ = ln (sinh2 rE2 + L cosh2 rM2 ) + const(2.13)where we have used the notation of (2.4). Continuing analytically to the SL(2, R)×SU(2)gauged model, we obtain (in the notation of (2.7))ds25D = kM4 dr2M + k4dφ2 + k cos2 φ2 (1 + L cosh2 rM2 )sin2 φ2 + L cosh2 rM2dθ2−kM sinh2 rM2 (L + sin2 φ2 )sin2 φ2 + L cosh2 rM2dt2 −2kMβ cos2 φ2 sinh2 rM2sin2 φ2 + L cosh2 rM2dt dθ+ kM cosh2 rM2 sin2 φ2sin2 φ2 + L cosh2 rM2d˜t2,B = kM sin2 φ2 sinh2 rM2sin2 φ2 + L cosh2 rM2dt ∧d˜t + βkM cos2 φ2 cosh2 rM2sin2 φ2 + L cosh2 rM2dθ ∧d˜t,Φ = ln (sin2 φ2 + L cosh2 rM2 ) + const,(2.14)which is exactly the analytic continuation of (2.7) from φ ∈[0, π) to φ ∈[π, 2π).3.

Exact Axisymmetric Stationary Solutions in Four DimensionsNow let us return to our attempt at constructing a 4D exact solution of bosonicstring theory by combining two 2D black holes. The problem we have arrived at is theviolation of Witten’s non-anomalousness condition (1.14) by the proposed gauge group8

action (1.2). In [5], Witten has shown that even in such anomalous cases, it is possibleto choose a gauged Lagrangian in such a way that the gauge non-invariant terms don’tdepend on the WZW group variable.

This leads us to suspect that a U(1)×U(1) action onSL(2, R)M×SL(2, R)E exists which is still anomalous on each of the SL(2, R)’s separately,but the gauge non-invariances cancel between sectors M and E. This is indeed the case,as we are now going to see.Let us start with (1.1) once more, and set kM = kE ≡k for simplicity. The U(1)×U(1)group to be gauged acts byU(1)E :gE →hE gE hE,gM →hαM gM h−αM ,U(1)M :gE →hαE gE h−αE ,gM →hM gM hM.

(3.1)It is easy to show that (3.1) satisfies condition (1.14). The shift from (1.2) to (3.1) addsnew α-dependent terms to (1.10),L4D →L4D −ikMπZd2z α sinh2 rM2 (AE∂t + AE∂t)−2ikMπZd2z α2 sinh2 rM2 AEAE+ ikMπZd2z α(cosh2 rM2 + sinh2 rM2 )(AEAM −AMAE),(3.2)thus modifying the (lowest order) background fields tods24D = k4dr2E + k4dr2M + k cosh2 rM2 sinh2 rE2∆dθ2 −k sinh2 rM2 cosh2 rE2∆dt2,B = kα sinh2 rE2 sinh2 rM2∆dt ∧dθ,Φ = ln (∆) + const,(3.3)where we have shortened eΦ ≡∆, with∆≡cosh2 rM2 cosh2 rE2 −α2 sinh2 rM2 sinh2 rE2 .

(3.4)(3.3) is (the lowest order approximation to) the class of exact, stationary and axisymmetricsolutions of four-dimensional string theory advertised above.5 To avoid naked singularities,5 This construction can be obviously generalized to N = 1 superstrings. As the cosets will infact carry N = 2 supersymmetry, they will correspond to solutions of N = 1 superstring theorywith N = 1 supersymmetry in the target.9

we will restrict ourselves to |α| < 1. (Note the singular behavior of the full Lagrangian(3.2) at the limiting values of α, α = ±1.

)Now let us analyze shortly the global structure of the solution. The surface at rM = 0is an event horizon, which inherits at fixed t the geometry of the Euclidean 2D black hole:ds2horizon = k4dr2E + k tanh2 rE2 dθ2.

(3.5)The structure of the horizon suggests that it might be reasonable to interpret the solutionas a black string.The external geometry (3.3) can be continued behind the horizon as follows. As wehave remarked above, the Euler angle parametrization we have used does not cover theSL(2, R)M group manifold completely.

In another region, the following Euler angles areuseful:gM =etL/200e−tL/2 cos rM2−sin rM2sin rM2cos rM2 e−tR/200etR/2(3.6)with rM ∈(−π, π). In this parametrization of the Lagrangian, we are led tods24D = k4dr2E −k4dr2M + k cos2 rM2 sinh2 rE2∆intdθ2 + k sin2 rM2 cosh2 rE2∆intdt2,B = −kα sinh2 rE2 sin2 rM2∆intdt ∧dθ,Φ = ln (∆int) + const,(3.7)now with∆int ≡cos2 rM2 cosh2 rE2 + α2 sin2 rM2 sinh2 rE2 .

(3.8)This describes the geometry of the solution behind the horizon. Note that rM has becometimelike while t is now spacelike, as might have been expected.At rM = π we encounter a singularity.6 However, an interesting effect occurs here:While in the direct product geometry of α = 0 any observer behind the horizon must fallinto the singularity (which exists at rM = π for any value of rE), with α nonzero thesingularity is localized at rE = 0.

What then happens to the observer at fixed nonzero rEwith increasing timelike coordinate rM? It is easy to see that the internal geometry (3.7)6 Strictly speaking, this is a future singularity.

The same analysis can be carried out for thepast singularity at rM = −π.10

can be continued further to another region. This region corresponds to the remaining partof the SL(2, R)M group manifold, parametrized by7gM =etL/200e−tL/2 sinh rM2cosh rM2−cosh rM2−sinh rM2 e−tR/200etR/2.

(3.9)After crossing the event horizon at rM = 0 in (3.3), the observer can avoid the singularity,cross a new, inner horizon at rM = π with rE ̸= 0, and enter the portion of the universecoming from (3.9). At the inner horizon, θ becomes timelike, while rM turns spacelikeagain.

The former time coordinate t remains spacelike, and plays the role of an angularvariable. The inner horizon carries the geometry of the dual Euclidean 2D black hole:ds2horizon = k4dr2E + kα2 coth2 rE2 dt2.

(3.10)Hence, the roles of t and θ have been completely interchanged in the region behind theinner horizon, when compared to the geometry of (3.3). The lowest order background inthis region isds24D = k4dr2E + k4dr2M −k sinh2 rM2 sinh2 rE2˜∆dθ2 + k cosh2 rM2 cosh2 rE2˜∆dt2,B = −kα sinh2 rE2 cosh2 rM2˜∆dt ∧dθ,Φ = ln ( ˜∆) + const,˜∆≡α2 cosh2 rM2 sinh2 rE2 −sinh2 rM2 cosh2 rE2 .

(3.11)This geometry describes, for rM close enough to zero, a throat with a naked singularity.The throat can be continued through its future horizon, and the analysis can be repeatedinfinitely many times, leading to an infinite strip of geometries and horizons.Without any computation, the residual (Killing) global symmetry of the coset is(at least) U(1) × U(1), as precisely these symmetries survive the gauging of (3.1) onSL(2, R)M × SL(2, R)E. It is worth stressing that the U(1) × U(1) is an exact Killingsymmetry of the full-fledged, exact CFT, not just an accidental symmetry of the lowestorder background.In this sense, we are guaranteed to have obtained a class of exact,stationary and axisymmetric classical solutions of string theory in four dimensions.7 Here rM > 0. The region with rM < 0 corresponds to the continuation through the analogoushorizon in the past, and/or to a ‘mirror geometry,’ analogously as in, say, the Reissner-Nordstrømblack hole.11

The solution we have just constructed is a twisted product of two 2D black holes. Asthe 2D black hole cosets enjoy an interesting property of target duality [12], one mightwonder whether there is an analogy of this stringy symmetry for the 4D cosets.

Actually,for general sigma models with a Killing vector, there is a duality transformation [13], givenbyˆG00 =1G00,ˆG0i = B0iG00,ˆGij = Gij −(G0iG0j −B0iB0j)/G00,ˆB0i = G0iG00,ˆBij = Bij + (G0iB0j −B0iG0j)/G00,ˆΦ = Φ + ln (−G00). (3.12)Note that in the case of our 4D solution, we obtain ˆB = 0, but instead of nonzero valuesof the antisymmetric tensor field, nonzero off-diagonal components of the metric tensoroccur.

When applied to the region represented by (3.3), this duality transformation wouldgive a metric with a naked singularity at rM = 0, as it is easy to see that (3.12) maps thehorizon to a singularity. To obtain a metric without naked singularities, it is a better ideato apply (3.12) to the region behind the naked singularity in (3.11).

Indeed, upon doingthis we obtain a remarkably simple geometry,bds24D = k4dr2E + k4dr2M+ k tanh2 rE2hdθ2 −2α dt dθ −(tanh2 rM2 coth2 rE2 −α2)dt2i,ˆB ≡0,ˆΦ = ln (cosh2 rE2 cosh2 rM2 ) + const. (3.13)Quite surprisingly, this is a direct product of two 2D black holes!

Indeed, upon changingcoordinates to Θ = θ −αt, T = t, we obtain the standard direct product metric of oneEuclidean and one Minkowskian black hole, parametrized by rE, Θ and rM, T respectively.How does it come about that the full class of highly nontrivial spacetimes is dual to asimple, tensor product structure? The crucial point is that (3.12) assumes a preferredKilling vector with respect to which the duality transformation is performed.8 We havetacitly assumed that this Killing vector coincides with ∂/∂t.

Nevetheless, it is a peculiarity8 Compare the recent discussion of duality by Roˇcek and E. Verlinde in [14]. Note also thatthere is a similarity between the duality found above, and the twisting procedure studied by Senin [15].12

of our geometry that there are two commuting Killing vectors in the target, and the dualitytransformation now requires a choice of preferred basis in the space of Killing vectors.Naively, we can apply (3.12) to any particular choice of the basis, thus obtaining a classof a priori different geometries. The same situation emerges for the tensor product of twoblack holes, which explains the duality observed above.We have constructed a class of U(1) × U(1) symmetric solutions of 4D string theory,and have found indications of a remarkable duality of the models.In general relativ-ity, U(1) × U(1) symmetric metrics have attracted much interest, and appealing resultshave been achieved: not only many physically important exact solutions of Einstein’sequations belong to this class, but Einstein’s equations become exactly solvable in thislimit by inverse scattering methods, in particular multi-monopole solutions can be found,infinite-dimensional solution-generating groups of symmetries exist, explainable via inti-mate relations to dimensional reduction to 2D, to mention at least some of the crucialaspects of axisymmetric stationary metrics in Einstein gravity.

On the other hand, the 2Dblack hole cosets, which serve as basic building blocks for the constructions of our paper,have been shown recently to enjoy a rich internal structure [16], related in particular toW∞algebras [17]. It would indeed be desireable to study possible interplays between thedeep results of general relativity of axisymmetric stationary geometries on one hand, andstring theory with its extremely rich mathematical structure on the other.

We hope thatthe exact solutions we have constructed above might serve as a starting point for furtherinvestigation in this direction.Acknowledgement. It is a pleasure to thank Peter Bowcock, Tohru Eguchi, JeffHarvey, Elias Kiritsis and Emil Martinec for valuable discussions.13

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