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Generalized Duality in Curved String-Backgrounds

arXiv:hep-th/9112070v1 23 Dec 1991IASSNS-HEP-91/84ITP-SB-91-67December 1991Generalized Duality in Curved String-BackgroundsAmit Giveon∗and Martin Roˇcek†School of Natural SciencesInstitute for Advanced StudyPrinceton, NJ 08540AbstractThe elements of O(d, d, Z) are shown to be discrete symmetries of the space of curvedstring backgrounds that are independent of d coordinates. The explicit action of thesymmetries on the backgrounds is described.

Particular attention is paid to thedilaton transformation. Such symmetries identify different cosmological solutionsand other (possibly) singular backgrounds; for example, it is shown that a compactblack string is dual to a charged black hole.

The extension to the heterotic string isdiscussed.∗Email: giveon@iassns.bitnet† Permanent address: ITP, SUNY at Stony Brook, Stony Brook NY 11794-3840.Email: rocek@dirac.physics.sunysb.edu

1. IntroductionIn string theory, conformal field theories (CFT’s) correspond to classical vacua (fora review, see [1]).

In general, a CFT can be deformed by truly marginal operators.The space of couplings to these operators is a connected subspace of the space ofstring vacua. The space of all string vacua has many such components (which mayapproach the same boundaries).

Many tractable cases involve CFT’s that can bedescribed by a sigma-model action, which can be thought of as a string moving ina metric, antisymmetric tensor, and dilaton background. In these theories, somemarginal operators generate deformations of the background.Different string theory backgrounds may correspond to the same CFT.

The sim-plest example of this phenomenon is the R →1/R duality for a free boson compact-ified to a circle [2]. For d-dimensional toroidal backgrounds, this duality generalizesto a discrete symmetry group isomorphic to O(d, d, Z) (or O(d, d + 16, Z) for theheterotic string) [3, 4, 5].In this paper, we discuss the discrete symmetry group acting on the space ofcurved D-dimensional backgrounds that are independent of d coordinates (d < D).Such backgrounds include many explicitly known string vacua: black holes [6], p-branes [7], cosmological solutions [8, 9], etc.The discrete symmetries identify vacua with geometries that in general are radi-cally different.

This characteristic stringy property is a consequence of the possibilitythat strings can wind around compact coordinates. The geometries related by thediscrete symmetries differ in their identification of the (local) momentum modes andthe (non-local) winding modes.

Thus, to an observer composed of, e.g., momentummodes, geometries mathematically equivalent as CFT’s will appear physically dis-tinct.The discrete symmetries in general act on the dilaton; in the flat case, the con-stant dilaton is transformed to a new constant such that the string coupling remainsinvariant [10, 5]. In the curved case, the dilaton transforms analogously [11].

Thedilaton transformation plays an important role in the understanding of the propa-gation of a string in the D = 2 black hole geometry [12], and in the understandingof duality invariant cosmological solutions [9].The structure of the moduli space that the discrete symmetries act on is notknown in general.In the flat (D-dimensional) case, the moduli space is locally2

isomorphic to O(D, D, R)/(O(D, R))2 [13]. Studies of low energy effective actionssuggest that there is a moduli subspace of backgrounds independent of d coordinatesisomorphic to O(d, d, R)/G [14, 15, 16], where G is at least the diagonal subgroupO(d, R) of (O(d, R))2 (the maximal compact subgroup of O(d, d, R)).

This is consis-tent with results from string field theory [17]. This moduli subspace is only correctwhen the d coordinates have the topology of a torus.

Though in general the localstructure of the moduli space is unknown, we can still identify a discrete symmetrygroup that acts on it.The main result of this paper is that there is a discrete symmetry group trans-forming curved D-dimensional backgrounds independent of d coordinates (d < D),which is isomorphic to O(d, d, Z). This group is naturally embedded in O(D, D, Z),and acts on the D-dimensional (metric + antisymmetric tensor) background by frac-tional linear transformations, together with a dilaton transformation that preservesthe string coupling.

Though the explicit transformations are valid only to leadingorder in the inverse string tension α′, we generalize the result of [18] to argue thatthe symmetry survives to all orders.The basic example of a nontrivial CFT that has two curved spacetime interpre-tations related by duality is the D = 2 black hole [12]. More general discussions ofsuch duality have appeared in [11, 19].The paper is organized as follows: In section 2, we generalize the result of [18] andconstruct general (conformal) curved D-dimensional backgrounds that are indepen-dent of d coordinates as abelian quotients [20] of CFT’s with (D + d)-dimensionalbackgrounds.In section 3, we use the construction of section 2 to find the dis-crete symmetries of the space of these D-dimensional backgrounds.

In section 4,we explore the group structure of these symmetries, and find a group isomorphic toO(d, d, Z), as well as a simple expression for its action on the backgrounds. In section5, we focus on the dilaton and its transformations.

In section 6, based on analogieswith the flat case, we conjecture how these results extend to the heterotic string.In section 7, we study two examples: duality between compact black strings andcharged black holes in the bosonic string, and duality between neutral and chargedblack holes in the heterotic string. In section 8, we close with a few comments anddiscuss related open problems.3

2. Curved target spaces independent of d coordinates as quotientsIn this section we construct a general (conformal) curved background in D dimen-sions that is independent of d coordinates as an abelian quotient of a CFT witha (D + d)-dimensional background.

Generalizing the result of [18], we start witha CFT with d abelian left handed currents Ji and right handed currents ¯Ji. Theaction isSD+d = S1 + Sa + S[x] ,S1 = 12πZd2zh∂θi1 ¯∂θi1 + ∂θi2 ¯∂θi2 + 2Σij(x)∂θi2 ¯∂θj1 + Γ1ai(x)∂xa ¯∂θi1 + Γ2ia(x)∂θi2 ¯∂xaiSa = 12πZd2zh∂θi1 ¯∂θi2 −∂θi2 ¯∂θi1iS[x] = 12πZd2zhΓab(x)∂xa ¯∂xb −14Φ(x)R(2)i,(2.1)where i, j = 1, .

. .

, d and a, b = d + 1, . .

. , D, and Σij, Γ1ai, Γ2ia, Γab are components ofarbitrary x-dependent matrices, such that, together with the dilaton Φ, the theorydescribed by the action SD+d is conformal.The antisymmetric term Sa is (locally) a total derivative, and therefore may giveonly topological contributions, depending on the periodicity of the coordinates θ.∗To specify the periodicity, we defineθi = θi2 −θi1 ,˜θi = θi1 + θi2 ,(2.2)such thatθi ≡θi + 2π ,˜θi ≡˜θi + 2π .

(2.3)In these coordinates, Sa becomesSa = 12πZd2z 12h∂˜θi ¯∂θi −∂θi ¯∂˜θii,(2.4)which takes half-integer values, and therefore contributes to the path-integral.The action SD+d (2.1) is invariant under the U(1)dL × U(1)dR affine symmetry∗This term was omitted in [18]; it is needed for gauge invariance of the gauged action, see below.4

generated by currentsJi = ∂θi1 + Σji∂θj2 + 12Γ1ai∂xa= 12h−(I −Σ)ji∂θj + (I + Σ)ji∂˜θj + Γ1ai∂xai,¯Ji = ¯∂θi2 + Σij ¯∂θj1 + 12Γ2ia ¯∂xa= 12h(I −Σ)ij ¯∂θj + (I + Σ)ij ¯∂˜θj + Γ2ia ¯∂xai. (2.5)We choose to gauge the d anomaly-free axial combinations of the symmetries (2.5);†other options are generated by discrete symmetries discussed later.The gaugedaction is [20]Sgauged = SD+d + 12πZd2zhAi ¯Ji + ¯AiJi + 12Ai ¯Aj(I + Σ)iji.

(2.6)The antisymmetric term Sa (2.4) is needed to insure gauge invariance under largegauge transformations.‡ Integrating out the gauge fields Ai, ¯Ai gives:SD = 12πZd2zhEIJ(x)∂XI ¯∂XJ −14φ(x)R(2)i= 12πZd2zhEij(x)∂θi ¯∂θj + F 2ia(x)∂θi ¯∂xa + F 1ai(x)∂xa ¯∂θi+ Fab(x)∂xa ¯∂xb −14φ(x)R(2)i,(2.7)where{XI}I=1...D = {θi, xa}i=1...d,a=d+1...DandEIJ = GIJ + BIJ = EijF 2ibF 1ajFab. (2.8)We have split E into its symmetric and antisymmetric parts G and B.

The compo-nents areEij = (I −Σ)ik(I + Σ)−1kj ,(2.9)†In [18], the designation of axial vs. vector was interchanged.‡Indeed, with the choice of Sa above, minimal coupling ∂α˜θi →∂α˜θi + Aiα gives the correctgauged model.5

andF 2ia = (I +Σ)−1ij Γ2ja ,F 1ai = −Γ1aj(I +Σ)−1ji ,Fab = Γab−12Γ1ai(I +Σ)−1ij Γ2jb . (2.10)This D-dimensional target space background is independent of the d coordinates θi.Since the relations (2.9,2.10) can be inverted to solve for (Σ, Γ) in terms of (E, F), itis the most general such background.

Following the reasoning of [18], if the originalmodel SD+d (2.1) is conformal, then SD is conformal to one loop order with§φ = Φ + ln det(I + Σ) . (2.11)This relation is also invertible, and thus a D-dimensional field theory with a back-ground that is independent of d coordinates can be described as a quotient of a(D +d)-dimensional field theory with d chiral currents.

If the D-dimensional theoryis conformally invariant, then the (D + d)-dimensional theory is as well [18], andhence any CFT with a background that is independent of d coordinates can bedescribed as a quotient of a CFT with a (D + d)-dimensional background.This construction will allow us to understand the discrete symmetries of themoduli space of string vacua in backgrounds that are independent of d coordinates.3. Discrete SymmetriesDifferent string theory backgrounds may describe the same conformal field theory.Here we study discrete symmetries that relate different but equivalent backgrounds(E(x), F(x), φ(x)) (2.7).

We first discuss transformations of (E(x), F(x), φ(x)) thatfollow from manifest symmetries of the action SD+d (2.1). We then combine themwith transformations that are manifest symmetries of SD (2.7) to find a discretesymmetry group isomorphic to O(d, d, Z).The action SD+d is invariant under the transformationsθ1 →O1θ1 ,θ2 →O2θ2 ,O1,2 ∈O(d, Z) ,(3.1)§However, as noted in [18], higher order corrections to the background that give an exact CFTwith a D-dimensional background exist; these corrections come from the integration measure.6

together withΣ →O2ΣOt1 ,Γ1 →Γ1Ot1 ,Γ2 →O2Γ2 ,Γ →Γ(3.2)such that12(O1 ± O2)ij ∈Z . (3.3)Here O(d, Z) is the group of matrices O with integer entries satisfying OOt = I.These symmetries can be found as follows: The action SD+d is invariant up tototal derivatives under O(d, R)×O(d, R) acting on (θ1, θ2) as in (3.1), together with(3.2) for the backgrounds.

The periodic coordinates θ, ˜θ (2.2) transform as: θ˜θ→12 O1 + O2O1 −O2O1 −O2O1 + O2 θ˜θ(3.4)To preserve the periodicities of θ, ˜θ (2.3), the condition (3.3) must be satisfied. Inparticular, this implies O1,2 ∈O(d, Z).The total derivative comes from the transformation of Sa (2.4), and isSa[O1θ1, O2θ2] −Sa[θ1, θ2] = 12πZd2zhMij(∂˜θi ¯∂˜θj −∂θi ¯∂θj) + Nij(∂θi ¯∂˜θj −∂˜θi ¯∂θj)i,(3.5)whereM = 14(Ot1 −Ot2)(O1 + O2) ,N = 14(Ot1 −Ot2)(O1 −O2) .

(3.6)The condition (3.3) implies that Mij, Nij ∈Z, and hence the total derivative is aninteger, and does not contribute to the path integral. This concludes the proof that(3.1,3.2) are symmetries of the action SD+d.The transformations of the background (3.2) induce transformations of the back-ground (E(x), F(x), φ(x)) (2.9,2.10,2.11) in SD (2.7):E→E′=(I −O2ΣOt1)(I + O2ΣOt1)−1(3.7)F 1→F 1′=−Γ1(O1 + O2Σ)−1F 2→F 2′=(Ot2 + ΣOt1)−1Γ2F→F ′=Γ −12Γ1(Ot2O1 + Σ)−1Γ2(3.8)7

φ →φ′ = Φ + ln det(I + O2ΣOt1) . (3.9)These transformations can be rewritten asE′ =h(O1 + O2)E + (O1 −O2)ih(O1 −O2)E + (O1 + O2)i−1(3.10)F 1′ = 2F 1h(O1 −O2)E + (O1 + O2)i−1F 2′ = 12h(O1 + O2) −E′(O1 −O2)iF 2F ′ = F −F 1h(O1 −O2)E + (O1 + O2)i−1(O1 −O2)F 2(3.11)φ′ = φ + 12 lnh detGdetG′i= φ + 12 lnh detGdetG′i(3.12)where G is the background metric as defined in (2.8) and G is the symmetric partof E (2.9).

We discuss the dilaton transformation (3.12) in detail in section 5.The transformations (3.1,3.2) induce a non-trivial action on the currents (2.5),and hence S′D (SD with a transformed background (E′, F ′, φ′)) is derived from SD+d(2.1) by a different quotient. For example, some symmetries simply change the signof a Ji without changing ¯Ji; this corresponds to a vector gauging (as opposed to anaxial gauging) of the i’th U(1).We now consider the additional transformations that are manifest symmetries ofthe action SD itself.

The first are integer “Θ”-parameters that shift E:Eij →Eij + Θij ,Θij = −Θji ∈ZF 1 →F 1 ,F 2 →F 2 ,F →F . (3.13)We refer to the group generated by these transformations as Θ(Z); these are obvi-ously symmetries, as they shift the action SD (2.7) by an integer and thus do notcontribute to the path integral.The second type of transformations are given by homogeneous transformationsof E, F 1, F 2 under A ∈Gl(d, Z):E →AtEA ,F 2 →AtF 2 ,F 1 →F 1A ,F →F .

(3.14)8

These are obviously symmetries of the theory described by SD, as they generate achange of basis in the space of θ’s that preserves their periodicities.Neither the Θ(Z) nor the Gl(d, Z) transformations affect the dilaton, since theydo not change the integration measure of the path integral; consequently, they aresymmetries of the exact CFT’s, and receive no higher order corrections.The group generated by all the symmetries discussed is isomorphic to O(d, d, Z).A natural embedding of O(d, d, Z) in O(D, D, Z) acts on the background E by frac-tional linear transformations, as explained in the next section.4. The action of O(d, d, Z)We begin by establishing our notation following [14].

The group O(d, d, R) can berepresented as 2d × 2d-dimensional matrices g preserving the bilinear form J:g = abcd,J = 0II0,(4.1)where a, b, c, d, I are d × d-dimensional matrices, andgtJg = J⇒atc + cta = 0 ,btd + dtb = 0 ,atd + ctb = I . (4.2)This has an obvious embedding in O(D, D, R) asˆg = ˆaˆbˆcˆd(4.3)where ˆa,ˆb, ˆc, ˆd are D × D-dimensional matrices of the formˆa = a00I,ˆb = b000,ˆc = c000,ˆd = d00I(4.4)(here I is the (D −d) × (D −d)-dimensional identity matrix).9

We define the action of ˆg on E by fractional linear transformations:ˆg(E) =E′ = (ˆaE + ˆb)(ˆcE + ˆd)−1=E′(a −E′c)F 2F 1(cE + d)−1F −F 1(cE + d)−1cF 2(4.5)whereE′ = (aE + b)(cE + d)−1(4.6)is a fractional linear transformation of E under O(d, d).The group O(d, d) is generated by [14]:Gl(d): abcd= At00A−1s.t.A ∈Gl(d) . (4.7)Θ: abcd= IΘ0Is.t.Θ = −Θt .

(4.8)Factorized duality: abcd= I −e1e1e1I −e1s.t.e1 = diag(1, 0, . .

., 0) . (4.9)The maximal compact subgroup of O(d, d) is O(d) × O(d) embedded as abcd= 12 o1 + o2o1 −o2o1 −o2o1 + o2s.t.o1,2 ∈O(d) .

(4.10)This subgroup includes factorized duality (4.9).We now turn from definitions to the actual symmetries of the CFT. These forman O(d, d, Z) discrete subgroup of O(d, d, R) that acts on the background as above.The elements of the subgroup O(d, d, Z) are given by matrices g of the form (4.1,4.2)with integer entries.Just as in the continuous case, the discrete group is generated by Gl(d, Z), Θ(Z),and factorized duality; these are given by (4.7,4.8,4.9) with integer entries.

Thesubgroup O(d, Z)×O(d, Z) is given by the matrices (4.10), again with integer entries.10

Clearly, the O(d, Z) × O(d, Z) symmetries (3.10,3.11), the Θ(Z) symmetries (3.13),and the Gl(d, Z) symmetries (3.14) that we found in the previous section act on thebackground by the O(d, d, Z) ⊂O(D, D, Z) fractional linear transformations (4.5)with the matrices a, b, c, d given by (4.10, with o1,2 = O1,2), (4.8, with Θ ∈Z), and(4.7, with A ∈Gl(d, Z)), respectively.These results are compatible with the known discrete symmetries of the space offlat D-dimensional toroidal backgrounds [3, 4, 5]. In that case, the O(d, d, Z) symme-tries described above are simply a subgroup of the full O(D, D, Z) symmetry groupacting as in (4.5), for any ˆg ∈O(D, D, Z).

For curved D-dimensional backgrounds,we expect that some large symmetry group acts (analogous to O(D, D, Z)); herewe have described the O(d, d, Z) subgroup that is associated with a d-dimensionaltoroidal isometry of the background.In the flat case, the fractional linear transformation is an exact map betweenequivalent backgrounds; in the curved case, in general one expects higher ordercorrections to the transformed background [11]. For Θ(Z) and Gl(d, Z), the trans-formations are exact; however, the factorized duality receives corrections from thepath-integral measure.Nevertheless, because the transformation is exact in the(D + d)-dimensional model, we know that non-perturbative correction must existsuch that factorized duality is exact.∗We close this section with a general remark.

The group O(d, d) has two discon-nected components (of the generators given in (4.7,4.8,4.9), only factorized duality(4.9) has det = −1, and hence is not connected to the identity). Therefore one ex-pects that in general the submoduli space generated by O(d, d) can be disconnected.In the flat case, it happens that the O(d, d) acts on the moduli space by a doublecovering, and as a result, the moduli space is connected.

However, in general onegets two disconnected components of backgrounds with different topologies: for ex-ample, in the D = 2, d = 1 curved case, the two-dimensional black-hole and its dualare in two disconnected components mapped into each other by O(1, 1). The dualitytransformation identifies the two components as conformal field theories.

This is ageneral feature: factorized duality (4.9) maps one component of the moduli spaceof backgrounds to the other.†∗N = 4 supersymmetry can protect duality transformations, and there are examples where theone loop transformations are exact even in curved backgrounds.†Factorized duality is similar to mirror symmetry [21], which also identifies two (possibly) dis-connected components of moduli space corresponding to backgrounds with different topologies. ForN = 2 toroidal (orbifold) backgrounds, mirror symmetry and factorized duality are identical [22].11

5. The dilatonTo complete the previous discussion of the transformation of the background underthe O(d, d, Z) symmetries, we consider the transformation of the dilaton.

This issummarized in eq. (3.12), which we derive by proving a theorem: The quantity˜φ = φ + 12 ln detG(5.1)is invariant under O(d, d, Z) transformations.

This impliesφ′ = φ + 12 ln detGdetG′. (5.2)The second equality in (3.12) follows from the proof of (5.1) given below.We begin with the identity GijGibGajGab= Gik0GakIac Ikj(G−1)klGlb0Gcb −Gck(G−1)klGlb(5.3)which impliesdet(G) = det(Gij)det(Gab −Gak(G−1)klGlb) .

(5.4)We next prove that the following two quantities are separately invariant underO(d, d, Z) transformations:the invariant fiber dilaton :ˆφ = φ + 12 ln det(Gij) ,(5.5)the quotient metric :Gab −Gak(G−1)klGlb . (5.6)Geometrically, a D-dimensional space whose metric is independent of d coordinatesθi can be thought of as a bundle M with fiber coordinates θ.

The metric on thefiber is Gij, and hence we refer to ˆφ as the “invariant fiber dilaton”. The inducedmetric on the quotient space M/{θi} is the quotient metric (5.6).To prove the invariance of (5.5), we consider the action of the generators ofO(d, d, Z) separately.

The Gl(d, Z) and Θ(Z) transformations trivially leave φ and12

det(Gij) invariant, and hence ˆφ as well. To show invariance under factorized duality,we write ˆφ explicitly in terms of Σ:ˆφ = Φ + ln det(I + Σ) + 12 ln det12h(I −Σ)(I + Σ)−1 + (I + Σt)−1(I −Σt)i= Φ + 12 ln det(I −ΣtΣ) .

(5.7)To get the second equality, we need to split ln det(I+Σ) as 12 ln det(I+Σ)+ 12 ln det(I+Σt). Under Σ →O2ΣOt1 (3.2), (I −ΣtΣ) →O1(I −ΣtΣ)Ot1, and hence ˆφ is invariant.This completes the proof of the invariance of the fiber dilaton ˆφ.The proof of the invariance of (5.6) is entirely parallel; again, the Gl(d, Z) andΘ(Z) transformations trivially leave (5.6) unchanged.

To prove invariance underfactorized duality, we express (5.6) explicitly in terms of Σ, Γ1, Γ2, Γ (see 2.9,2.10):Gab −Gak(G−1)klGlb=h12(Γ + Γt) −14{Γ1(I + Σ)−1Γ2 + Γ2t(I + Σt)−1Γ1t}+ 12{Γ1(I + Σ)−1 + Γ2t(I + Σt)−1}{(I −Σ)(I + Σ)−1 + (I + Σt)−1(I −Σt)}−1× {(I + Σ)−1Γ2 + (I + Σt)−1Γ1t}iab=h12(Γ + Γt) + 14{Γ1(I −ΣtΣ)−1(Γ1t + ΣtΓ2) + Γ2t(I −ΣΣt)−1(Γ2 + ΣΓ1t)}iab . (5.8)The final expression is manifestly invariant under (3.2), which completes the proofof the invariance of the quotient metric (5.6).The theorem (5.1) that ˜φ is invariant follows from the invariance of (5.5,5.6)together with the identity (5.4).

This is compatible with results in low-energy ef-fective field theories [15, 16] and string field theory [17, 23], and physically impliesthat the string coupling constant g−1string =< e˜φ >=<√detG eφ > is invariant underO(d, d, Z). The invariance of (5.5,5.6) actually prove thatqdet(Gij) eφ and the quo-tient metric (5.6) are separately invariant; of course, this holds only for O(d, d, Z),and not the full O(D, D, Z) that it is embedded in.13

6. The Heterotic StringIn this section, we make a conjecture about the discrete symmetries of the heteroticstring by requiring compatibility with the flat limit [3, 4] and with the bosonic case.We start with a curved heterotic background, which we assume is a consistent,conformally invariant, heterotic string theory, with an action:Shet = 12πZd2zhEIJ(x)∂XI ¯∂XJ + AIA(x)∂XI ¯∂Y A+ EAB∂Y A ¯∂Y B −14φ(x)R(2) + (fermionic terms)i,(6.1)and with the second-class constraints that the Y A are chiral bosons: ∂Y A = 0.

Asbefore, {XI} = {θi, xa}, with i = 1 . .

. d, and a = d + 1 .

. .

D. In addition, we havedint internal chiral bosons Y A: A = 1 . .

. dint.

In flat space, we have dint = 16, butmore generally, we may find other solutions [24, 25]. The spacetime background isgiven by (E,φ), as in the bosonic case (2.8,2.11), and, in addition, the gauge fieldA.

We assume that this curved background is independent of the d coordinates θi.The constant internal background isEAB = GAB + BAB ,(6.2)where GAB is the metric on the internal lattice (one half the Cartan matrix ofthe internal symmetry group when the lattice is the root lattice of a group) andBAB is its antisymmetrization [26], i.e., EAB is upper triangular. In the spacetimesupersymmetric flat case, the symmetry group is E8 × E8.∗In curved space, thisgroup is in general different [24, 25].Following [3], the expected symmetry group is isomorphic to O(d, d + dint, Z) ⊂O(D, D + dint, Z) as in section 4, acting by fractional linear transformations on the(D + dint) × (D + dint) dimensional matrixΞ(x) = EIJ + 14AIA(G−1)ABAJBAIA0EAB(6.3)where G−1 is the inverse of GAB in (6.2).

The matrix Ξ is the embedding of theheterotic background into a bosonic (D + dint)-dimensional background, and the∗The Spin(32)/Z2 string can be described as the E8 × E8 string with a particular gauge fieldbackground [27].14

group O(D, D + dint) is the subgroup of O(D + dint, D + dint) that preserves theform (6.3). Note that the spacetime metric GIJ (the symmetric part of EIJ in (6.3))is the quotient metric of the (D + dint)-dimensional space modulo {Y A} (here, GABis the fiber metric).

This leads to a simple expression for the transformation of thedilaton:φ′ = φ + 12 ln detGdetG′;(6.4)note that this is independent of the gauge fields.7. ApplicationsIn this section we explore a number of consequences of the discrete symmetries.

Wefirst discuss an exact D = 3 closed string background that is independent of d = 2coordinates [28]. We then turn to D = 2 heterotic backgrounds [24, 25].

In bothcases, we find that uncharged black compact objects (strings or holes) are equivalentto charged D = 2 black holes.∗7.1. The closed string exampleThe simplest nontrivial example after the D = 2 black hole duality [12] is a compactblack string given by attaching a circle to every point of the D = 2 black holespacetime (Sl(2, R)k/U(1) × U(1)).

To leading order, the action isSBlackString = 12πZd2zhk(∂x¯∂x + tanh2x∂θ1 ¯∂θ1) + α∂θ2 ¯∂θ2 −14φ(x)R(2)i,(7.1)whereφ(x) = φ0 + ln(cosh2x) . (7.2)The first term in SBlackString is the euclidean black hole metric [6], and the secondterm describes a circle of radius √α attached to each point.

This D = 3 backgroundis independent of d = 2 coordinates θi, and is described by the matrix (2.8)E = E00F,(7.3)∗While writing up our results, we found similar observations in [29].15

whereE = k tanh2x00α,F = k . (7.4)The group of generalized duality tranformations O(2, 2, Z) maps this backgroundinto other backgrounds that (in general) have different spacetime interpretations.Since F1 = F2 = 0 in (7.3, cf.

2.8), O(2, 2) acts on the background by transformingonly E and φ as given in (4.6) and (5.2). A particularly interesting point on thetrajectory of O(2, 2, Z) is reached by acting with the elementg = IΘ0I p00p 0II0 I−Θ0I= 100−1 0000 0110 100−1,(7.5)whereI = 1001,Θ = 01−10,p = 0110.

(7.6)This transforms E and φ toE′ = g(E) =11 + αk tanh2xk tanh2xαk tanh2x−αk tanh2xα=1cosh2x −λ k(1 −λ)sinh2xλsinh2x−λsinh2xλcosh2x/k(7.7)φ′(x) = φ0 −ln(1 −λ) + ln(cosh2x −λ) ,(7.8)whereλ =kα1 + kα . (7.9)This gives an actionSCharge = 12πZd2zhk(∂x¯∂x + (1 −λ)sinh2xcosh2x −λ ∂θ1 ¯∂θ1)+λsinh2xcosh2x −λ(∂θ1 ¯∂θ2 −∂θ2 ¯∂θ1) +λcosh2xk(cosh2x −λ)∂θ2 ¯∂θ2 −14φ′(x)R(2)i,(7.10)16

and, after wick rotating θ1 →it, corresponds to a charged black hole of the typefound in [28] with mass MM = (1 −λ)M0 =s2keφ0 ,M0 =s2keφ′0 ,(7.11)and chargeQ =qλ(1 −λ)2M0k=sλ1 −λ2Mk ,(7.12)where φ′0 = φ0 −ln(1 −λ) is the constant part of the dual dilaton (7.8). The action(7.10) is related to the precise action for the coset (Sl(2, R)k × U(1))/U(1)) [28] byrescaling θ2 →kθ2.†This shows that the charged black hole is equivalent to the compact black stringas a conformal field theory.

We now consider some particular limits of this solution.The limits α →0, ∞in (7.1) is the 2D black hole × a degenerate circle [6], withthe two limits related by R →1/R duality. These limits correspond to the limitsλ →0, 1 in (7.9).

In SCharge (7.10), λ →0 is precisely the 2D black hole × the samedegenerate circle; however, the λ →1 limit gives the action (modulo an integer totalderivative term)Sλ→1 = 12πZd2zhk∂x¯∂x + 1kcoth2x∂θ2 ¯∂θ2 + (1 −λ)∂θ1 ¯∂θ1 −14φ′(x)R(2)i, (7.13)φ′(x) = φ0 −ln(1 −λ) + ln(sinh2x) ,which corresponds to the dual 2D black hole × a degenerate circle [12]. In bothcases, the degenerate limits are equivalent as CFT’s to a noncompact black string.7.2.

The heterotic string exampleWe focus on the example of [25], which is a D = 2 heterotic string with internaldegrees of freedom taking values in a standard 12-dimensional lattice (the vectorweights of SO(24)).‡We find that a family of charged black holes (and naked†Our k matches [6], which is 2k of [28].‡More precisely, only spacetime bosons have internal quantum numbers in the vector representa-tions of SO(24); spacetime fermions have internal quantum numbers in the spinor representationsof SO(24).17

singularities) are dual to a neutral one, which is the exact CFT given by the heteroticD = 2 black hole [25].We start with a heterotic D = 2 action:Shet = 12πZd2zhk(∂x¯∂x + tanh2x∂θ¯∂θ) + ∂Y A ¯∂Y A −14φ(x)R(2)+(fermionic terms)i,(7.14)where A = 1, . .

. , 12, k = 5/2 (for criticality), and φ = φ0 + ln(cosh2x).

This actiondescribes a neutral heterotic D = 2 black hole. The conformal field theory (7.14)corresponds to a background (6.3)Ξ =k000k tanh2x00I,(7.15)where the internal background I is the 12 × 12 identity matrix corresponding to thevector weights of SO(24).

Only a 2 × 2 block E of the matrix Ξ is affected by thediscrete transformations we discuss in this example:E = k tanh2x001. (7.16)By transforming E and φ with a group element gn ∈O(1, 2, Z) ⊂O(1, 13, Z)(where n is an arbitrary integer):gn = 0II0 InΘ0I Atn00A−1n= 0000 10−n1 1n01 −n2n−n0,(7.17)whereI = 1001,Θ = 01−10,An = 10n1,(7.18)one findsgn(E) = E′n = (n2 + k tanh2x)−1−2n(n2 + k tanh2x)−101,(7.19)18

φ′(x) = φ0 + ln(n2 + k) + ln(cosh2x −kn2 + k) . (7.20)After rescalingθ →k + n2√kt ,(7.21)and defining r to be a linear function of the dilaton φ′ (7.20),Qr = ln(cosh2x −kn2 + k) ,(7.22)where Q is a constant determined below, the background (7.19) gives rise to anaction§SCharge = 12πZd2zhf(r)∂t¯∂t + f(r)−1∂r ¯∂r −A(r)∂t¯∂Y 1 + ∂Y A ¯∂Y A−14φ′(r)R(2) + (fermionic terms)i,(7.23)withf(r) = 1 −2me−Qr −q2e−2Qr,(7.24)A(r) = nQ + 2qe−Qr ,φ′(r) = Qr + φ0 + ln(n2 + k) ,where Q = 2/√k′ is determined by the normalization of Grr in (7.23), and2m = n2 −k′n2 + k′ ,q = n√k′n2 + k′ .

(7.25)Following [6] we have replaced k with k′ = k −2 = 1/2 in (7.25). Wick rotatingt →it, along with q →−iq (necessary to maintain hermiticity of the action), thetheory (7.23 with |n| > 1) describes a D = 2 charged black hole with mass andcharge [25]M = Q(n2 −k′)eφ0 ,Q = n√8eφ0 .

(7.26)For n = −1, 0, 1, the theory (7.23) describes a naked singularity.§Recall that Gtt = Ξ′tt −14A2 (see 6.3).19

We emphasize that these backgrounds, for all n, are different spacetime inter-pretations of the same CFT: the exact CFT given by the neutral heterotic D = 2black hole.8. Concluding Remarks and Open ProblemsWe have shown that O(d, d, Z) acts on the space of backgrounds that are independentof d coordinates.

We expect that in general the full symmetry group acting on thespace of D-dimensional curved backgrounds is larger. Some of these extra symmetrygenerators can be found by considering quotients of (D + d)-dimensional actionsthat are more general than (2.1); we hope to discuss this somewhere else.

Ideally,one would like to find the complete symmetry group for the space of all curvedbackgrounds.The O(d, d, Z) subgroup already leads to interesting relations between differentgeometries.We have illustrated this with charged black hole examples; similarstudies in the context of string cosmology may lead to surprising consequences.Elements of O(d, d, Z) with det = −1 relate backgrounds with (possibly) differenttopologies. In the flat case, such transformations coincide [22] with mirror symmetryfor N = 2 superconformal backgrounds [21].

It would be interesting to understandthe relation between the two in the general case.Another open problem is the issue of higher order corrections; this is a prob-lem for the spacetime interpretation of quotients as well as for discrete symmetrytransformations.Acknowledgments: We would like to thank R. Dijkgraaf, C. Nappi, E. Verlinde,E. Witten, and B. Zwiebach for discussions.

The work of AG is supported in partby DOE grant No. DE-FG02-90ER40542, and that of MR is supported by the JohnSimon Guggenheim Foundation.20

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