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Imperial/TP/93-94/54 UCSBTH-94-31 hep-th/9409021

arXiv:hep-th/9409021v2 29 Sep 1994

A new class of exact solutions in string theory

Gary T. Horowitz? Physics Department University of California, Santa Barbara, CA 93106, USA and A.A. Tseytlin? Theoretical Physics Group, Blackett Laboratory Imperial College, London SW7 2BZ, U.K. Abstract We prove that a large class of leading order string solutions which generalize both the plane-wave and fundamental string backgrounds are, in fact, exact solutions to all orders in α′ . These include, in particular, the traveling waves along the fundamental string. The key features of these solutions are a null symmetry and a chiral coupling of the string to the background. Using dimensional reduction, one ?nds that the extremal electric dilatonic black holes and their recently discovered generalizations with NUT charge and rotation are also exact solutions. We show that our bosonic solutions are also exact solutions of the heterotic string theory with no extra gauge ?eld background.

September 1994

? ?

e-mail address: gary@cosmic.physics.ucsb.edu On leave from Lebedev Physics Institute, Moscow, Russia.

1. Introduction To address strong ?eld e?ects in string theory, it is necessary to obtain exact classical solutions and study their properties. As in other ?eld theories, symmetries have been used to help ?nd these solutions. It is easy to show that every Killing vector on spacetime gives rise to a conserved current on the string world sheet. If the antisymmetric tensor ?eld is related to the spacetime metric in a certain way, these currents are chiral. The existence of such chiral currents turns out to simplify the search for exact solutions. One example is the WZW model which describes string propagation on a group manifold. This background has a large symmetry group, and all the associated currents are chiral. (Since the gauged WZW models can be represented in terms of the di?erence between two WZW models for a group and a subgroup, a similar statement applies there.) Another example is provided by the F -models discussed in [1,2] which have two null Killing vectors and two associated chiral currents. In addition to these two examples, the only other known exact solutions to (bosonic) string theory are the plane waves and their generalizations [3,4], which are characterized by the existence of a covariantly constant null Killing vector. We will show that the F -models and generalized plane waves are both special cases of a larger class of exact solutions which have a null Killing vector and an associated conserved chiral current. Backgrounds of this type are described by σ-models which we will refer to as “chiral null models”. We will see that they include a number of interesting examples. The presence of a null chiral current is associated with an in?nite-dimensional a?ne symmetry of the σ-model action. This implies special properties of the spacetime ?elds. The generalized connection with torsion equal to the antisymmetric ?eld strength plays an important role since it is the one that appears in the classical string equations of motion. We will see that this connection has reduced holonomy. A certain balance between the metric and the antisymmetric tensor resulting in chirality of the action is the crucial property of our models which is in the core of their exact conformal invariance. There are several levels of describing solutions to string theory. The string equation is usually expressed in terms of a power series in α′ . If one keeps only the leading order terms, one obtains an equation analogous to Einstein’s equation and a large number of solutions have been found. The form of the higher order terms is somewhat ambiguous due to the freedom of choosing di?erent renormalization schemes (or ?eld rede?nitions). For the plane - wave type solutions and the F -models, it has been shown that there exists a scheme in which the leading order solution does not receive α′ corrections, and thus corresponds to an exact solution as well. We will see that the same is true for the more general chiral null models. To explore the properties of a given solution, one would like to know not only that a given background is an exact solution to the ?eld equations, but also what the string states and interactions are in this background. In other words, one would like to know the corresponding conformal ?eld theory explicitly. This is known only for gauged WZW models. But some chiral null models can be realized as gauged WZW models [5,1] so in these cases, one has more information about the solution. Many of the chiral null model backgrounds have unbroken space-time supersymmetry and some models admit extended world sheet supersymmetry. For example, the F -models in even dimensions always have at least (2,0) world sheet supersymmetry. However, our argument that they are exact string solutions is not based on this fact. We will show that these backgrounds are solutions in the bosonic as well as the superstring and heterotic string theories. What types of solutions belong to this class? To begin, all of the plane 1

wave type solutions are included, as well as all of the F -models [1] which contain the fundamental string solution [6] as a special case. In addition, several generalizations of these solutions are in this class, including the traveling waves along the fundamental string [7]. Although the bosonic string does not have fundamental gauge ?elds, e?ective gauge ?elds can arise from dimensional reduction. In this way, we will show that the charged fundamental string solutions [8,9] are exact. Perhaps of most importance is the fact that four dimensional extremal electrically charged black holes [10,11,12] can be obtained from the dimensional reduction of a chiral null model, and hence are exact. Similarly, we will see that the generalizations of the extremal black holes which include NUT charge and rotation [13,14,15] are also exact. Finally, the chiral null models also describe some backgrounds with magnetic (and no electric) ?elds, as well as other solutions which appear to be new. If one considers only the leading order string equations, many of these solutions arise as the extremal limit of a family of solutions with a regular event horizon. The nonextremal solutions are not of the chiral null form and are likely to receive α′ corrections in all renormalization schemes. Finding the exact analogs of these solutions (which include the Schwarzschild metric as a special case) remains an outstanding open problem. The fact that we only obtain a particular charge to mass ratio from a chiral null model can be understood roughly as follows. To have chiral currents, one needs a balance between the spacetime metric and antisymmetric tensor ?eld, which upon dimensional reduction results in a relation between the charge and the mass. This paper is organized as follows. In the next section we introduce the chiral null models, and discuss their properties as well as some special cases and examples of solutions. In Section 3 we describe a general scheme of Kaluza-Klein type dimensional reduction working directly at the level of the string world sheet action. Unlike the more traditional approach which uses the leading order terms of the spacetime e?ective action, our approach applies to all orders in α′ . Section 4 will be devoted to solutions obtained from the dimensional reduction of a chiral null model. These include the charged fundamental string, extremal electric black holes and their generalizations. Section 5 contains our main result: we prove that for a chiral null model, the leading order solutions do not receive any α′ corrections (in a particular scheme). In Section 6 we extend this argument to the case of superstring and heterotic string theory. We show that the (1, 0) supersymmetric extensions of our bosonic models are conformally invariant without any extra gauge-?eld background. We also discuss the world sheet supersymmetry properties of these models. Section 7 is devoted to some concluding remarks. In Appendix A we summarize the geometrical properties of the string backgrounds described by the chiral null model (the generalized connection with torsion, its holonomy and curvature tensor, parallelizable spaces, etc.). In Appendix B we elaborate on the discussion of D = 3 models in [1] and show that the general chiral null model in three dimensions is actually a gauged WZW model. 2. Chiral null models: general properties and examples 2.1. Review of previous work A bosonic string in a general ‘massless’ background is described (in the conformal gauge) by the σ-model I= 1 πα′ d2 z L , ? L = (GM N + BM N )(X) ?X M ?X N + α′ Rφ(X) , 2 (2.1)

where GM N is the metric, BM N is the antisymmetric tensor and φ is the dilaton [16] (R 1√ ? is related to the world sheet metric γ and its scalar curvature by R ≡ 4 γR(2) ; ? and ? stand for ?+ and ?? when the world sheet signature is Minkowskian). In [1], two types of models were studied, which were called the K-model and the F model. In terms of the coordinates X M = (u, v, xi), the simplest (?at transverse xi -space) K- and F -model Lagrangians are ? ? ? LK = ?u?v + K(x) ?u?u + ?xi ?xi + α′ Rφ0 , φ0 = const , (2.2) (2.3)

These two models are dual in the sense that applying a spacetime duality transformation 1 [17] with respect to u turns the K-model into the F -model with F = K ?1 , φ = φ0 + 2 lnF . The general K-model includes arbitrary u dependence and describes the standard plane fronted waves. It is conformal to all orders if it is conformal at leading order, i.e. ? 2 K = 0. There exists a special scheme [1] in which a similar statement is true for the F -model, i.e. it is conformal to all orders if ? 2 F ?1 = 0 , 1 φ = φ0 + lnF (x) . 2 (2.4)

? ? LF = F (x) ?u?v + ?xi ?xi + α′ Rφ(x) .

Perhaps the most important solution in this class is the one describing the ?elds outside of a fundamental string (FS) [6] which is given by F ?1 = 1 + M r D?4 , D>4; F ?1 = 1 ? M ln r , r0 D=4, (2.5)

where r 2 = xi xi and D is the total number of space-time dimensions. The key property of the K-model is that it has a covariantly constant null vector ?/?v. The main features of the F -model are that there are two null Killing vectors corresponding to translations of u and v, and that the coupling to u, v is chiral (since Guv = Buv ). This means that the F -model is invariant under the in?nite dimensional symmetry u′ = u + f (τ ? σ) and v ′ = v + h(τ + σ). Associated with this symmetry are two conserved ? ? world sheet chiral currents: Ju = F ?v, Jv = F ?u. These properties are preserved if the i transverse x -space is modi?ed. In fact, the two models (2.2) and (2.3) can be generalized [1] to the case when the transverse space corresponds to an arbitrary conformal σ-model. The simplest generalization is to keep the transverse metric ?at but include an extra linear term in the dilaton. 2.2. The general chiral null model The fact that a leading order solution turns out to be exact applies to a larger class of backgrounds than represented by the K-model and F -model. We will consider the following Lagrangian which will be called the chiral null model: ? ? ? ? ? ? L = F (x)?u?v + K(x, u)?u?u + 2Ai (x, u)?u?xi + ?xi ?xi + α′ Rφ(x, u) . (2.6)

We need to assume that F does not depend on u since otherwise the argument for conformal invariance given in Section 5 does not go through. As in the case of the K-model and F model, it is possible to replace the ?at transverse space by an arbitrary conformal σ-model, but we will not consider that generalization here. 3

This model has roughly half the symmetries of the F -model. There is one null Killing vector generating shifts of v, and the action is invariant under the a?ne symmetry v ′ = v + h(τ + σ) which is related to the existence of the conserved chiral current Jv = F (x)?u. This in turn implies the special geometrical (holonomy) properties of the corresponding string backgrounds (see Appendix A). Like the F -term, the vector coupling has a special chiral structure: the Gui and Bui components of the metric and the antisymmetric tensor are equal to each other. The action (2.6) can be represented in the form ? ? ? ? L = F (x)?u ?v + K(x, u)?u + 2Ai (x, u)?xi + ?xi ?xi + α′ Rφ(x, u) , ? K ≡ F ?1 K , ? Ai ≡ F ?1 Ai , (2.7)

and thus is invariant under the subgroup of coordinate transformations v ′ = v ? 2η(x, u) combined with a ‘gauge transformation’ K ′ = K + 2?u η , A ′ = A i + ?i η . i (2.8)

It is clear that using this freedom one can always choose a gauge in which K = 0. However, we will often consider the special case when K, Ai and φ do not depend on u, i.e. when ?/?u is a Killing vector. In this case, K cannot be set to zero without loss of generality. When the ?elds do not depend on u, one can perform a leading-order duality transformation along any non-null direction in the (u, v)-plane. Setting v = v + au (a=const) ? in (2.7) and dualizing with respect to u yields a σ-model of exactly the same form with F, K, Ai and φ replaced by F ′ = (K + a)?1 , K ′ = F ?1 , A′ = Ai , i 1 φ′ = φ ? ln[F (K + a)] . 2 (2.9)

In other words, chiral null models are ‘self-dual’: the null translational symmetry and chiral couplings are preserved under duality. In Section 5 we shall determine the conditions on the functions F, K, Ai and φ under which these models are conformal to all orders in α′ . As in the case of the simplest F model (2.3) there exists a scheme in which these conditions turn out to be equivalent to the leading-order equations (derived in Appendix A) 1 ? ? 2 F ?1 + bi ?i F ?1 = 0 , 2 1 ? ?i F ij + bi F ij = 0 , 2 (2.10) (2.11) (2.12)

1 2 ? ? 2 K + bi ?i K + ? i ?u Ai ? 2bi ?u Ai + 2F ?1 ?u φ = 0 , 2 1 φ(u, x) = φ(u) + bi xi + lnF (x) , 2 where Fij ≡ ?i Aj ? ?j Ai , ? 2 ≡ ? i ?i .

Notice that the leading order equations allow a linear term bi xi in the dilaton. Eq. (2.12) implies that the central charge of the model is given by c = D +6bi bi . One can easily verify 4

that these equations are invariant under the ‘gauge’ transformations (2.8) (and, when the ?elds do not depend on u, under the duality transformations (2.9)). When F, K, Ai, φ are independent of u and bi = 0, these equations take the simple form ? 2 F ?1 = 0 , ? 2K = 0 , ?i F ij = 0 , 1 φ = φ0 + lnF (x) . 2 (2.13)

A crucial feature of these equations is that they are linear. Thus all solutions satisfy a solitonic no-force condition and can be superposed (this is also true for the more general equations (2.10) - (2.12) provided bi is held ?xed). Since these equations are exact conformal invariance conditions, changing F , K or Ai while preserving (2.10)–(2.12) can be viewed as ‘marginal deformations’ of the corresponding conformal ?eld theory. 2.3. Some special cases We now discuss some special cases of the general chiral null model (2.6). If F = 1, we obtain a class of plane fronted wave backgrounds which have a covariantly constant null vector. The general background with a covariantly constant null vector contains another vector coupling [3] ? ? ? ? ? L = ?u?v + K(x, u)?u?u + 2Ai (x, u)?u?xi + 2Ai (x, u)?xi?u ? + ?xi ?xi + α′ Rφ(x, u) . (2.14)

The conditions of conformal invariance of this model turn out to take the form [18] (for simplicity we set bi = 0) ?i F ij = 0 , ? ?i F ij = 0 , φ = φ(u) , (2.15)

1 2 ? ? ? (2.16) ? ? 2 K + ? i ?u (Ai + Ai ) + F ij Fij + 2?u φ + O α′s+k ? s F ? k F = 0 . 2 ? Thus, if one breaks the chiral structure by introducing the Ai -coupling, then, in general, there are corrections to the uu-component of the metric conformal anomaly coe?cient (2.16) to all orders in α′ . The higher-loop corrections still vanish in one special case: when ? Ai and Ai have ?eld strengths constant in x (in general, the ?eld strengths may still depend on u) 1 1? ? (2.17) Ai = ? Fij xj , Ai = ? Fij xj . 2 2 Such a model represents a simple and interesting conformal theory in its own right.1 When the ?elds do not depend on u one may de?ne the dual σ-model which is also conformal to all orders and will be discussed at the end of Section 5.

One particular case corresponds to the D = 4 non-semisimple WZW model of ref. [19], 1 ? namely, K = ?xi xi , Ai = ?Ai = ? 2 ?ij xj , φ = const, which is obviously a solution of ? (2.15),(2.16). Since Ai = ?Ai , Ai represents the antisymmetric tensor part of the action (2.14).

1

Another equivalent (related by a u-dependent coordinate transformation of xi ) representation of ? the model of [19] is K = 0, Ai = ? 1 ?ij xj , Ai = 0 which will be useful at the end of Section 4 2 (see also Appendix A).

5

? The special property of the model with Ai = 0 or Ai = 0 (i.e. with Gui = ±Bui ) resulting in cancellation of the vector-dependent contributions to the β-function for K was noted at the one-loop level in [18] and extended to the two-loop level in [20].2 It was further shown [22] that such backgrounds are (‘half’) supersymmetric when embedded in D = 10 supergravity theory and it was conjectured that these ‘supersymmetric string waves’ remain exact heterotic string solutions to all orders in α′ when supplemented with ? some gauge ?eld background. As we shall demonstrate, (2.14) with Ai = 0 is, in fact, an exact solution of the bosonic string theory. In Section 6 we shall prove that, furthermore, it can be promoted to an exact superstring and heterotic string solution with no need to introduce an extra gauge ?eld background. It is the chiral structure of this solution which is behind this fact. ? If K = 0, and Ai (x), φ are independent of u, the chiral null model (2.6) reduces to ? ? ? ? L = F (x)?u?v + 2Ai (x)?u?xi + ?xi ?xi + α′ Rφ(x) . (2.18)

This background is also supersymmetric [23] when embedded in D = 10 supergravity theory (and was also conjectured [23] to correspond to an exact heterotic string solution when suplemented by a gauge ?eld). As above, we will prove in Section 6 that it is an exact solution of the heterotic string theory by itself, i.e. that the (1, 0) supersymmetric extension of (2.18) is a conformally invariant model without extra gauge ?eld terms added. 2.4. Examples of solutions We now discuss some examples of solutions which are described by chiral null models. These solutions can be viewed as di?erent generalizations of the fundamental string solution (2.5). It is straightforward to describe the general solution for the conformal D = 5 chiral null model which is independent of u (and has bi = 0). It is given by ? ? ? ? L = F (x)?u ?v + K(x)?u + 2Ai (x)?xi + ?xi ?xi + α′ Rφ(x) , (2.19)

where the functions F, K, Ai and φ satisfy (2.13). Since the transverse space is now three dimensional, every solution Ai to Maxwell’s equation can be written in terms of a scalar3 ?ijk ?j Ak = ? i T (x) ,

2

?2T = 0 .

(2.20)

It was observed in [20] that introducing the generalized connection with the antisymmetric ? tensor ?eld strength as torsion, one ?nds that if Ai = 0 the generalized curvature (see Appendix A) ?v ?v is nearly ?at: the only non-trivial components of it are R?ijk = 2?i Fjk , R?iuj = 2?i ?u Aj ??i ?j K. λρσ ? Then assuming that all terms in the β?ν -function have the structure Y? R?λρσν , where Y depends on H?νλ and R?νλρ (in a special renormalization scheme this is true at the 2-loop order [21]) one can argue [20] that all higher-order corrections vanish. This argument is not completely rigorous and, in fact, unnecessary, since a simpler direct proof of conformal invariance of this model can be given (see Section 5).

3

Another simple case is D = 4 since in two transverse dimensions Ai = q?ij xj .

6

With F ?1 and K also satisfying Laplace’s equation in the transverse space, the general solution is characterized by three harmonic functions. It is clear from (2.11) that the model remains conformal if we let K have an arbitrary u dependence. If we set Ai = 0, take F and φ given by the FS solution (2.5), and keep K general, the solutions describe traveling waves along the fundamental string and were ?rst discussed in [7]. Consider now spherically symmetric solutions with Ai = 0 and no u dependence. Since all spherically symmetric solutions to Laplace’s equation take the form a + br 4?D , the function K can always be represented as K(x) = c + nF ?1 (x). After a shift of v the ? model then takes the form (2.6) with K = n. In view of the freedom to rescale u and v the only non-trivial values of the constant n are 0 and 1. n = 0 corresponds the standard FS while n = 1 yields the following simple generalization ? ? ? L = F (x)?u?v + ?u?u + ?xi ?xi + α′ Rφ(x) , (2.21)

where F and φ are given by (2.5) and (2.4). This solution was ?rst found in [9] and further discussed in [24]. It is known [25] that the fundamental string is the extremal limit of a family of charged black string solutions to the leading order equations. The generalization (2.21) can similarly be viewed as the extremal limit of a black string as follows (we consider D = 5 for simplicity). The charged black string can be obtained by boosting the direct product of the Schwarzschild background with a line, and applying a duality transformation [26]. The result is (S ≡ sinh α, C ≡ cosh α, α is the original boost parameter) ds =

2

2mS 2 1+ r Byt

?1

2m ? 1? r 2mS 2 1+ r

dt + dy

?1

2

2

2m + 1? r e?2φ = 1 +

?1

dr 2 + r 2 d? ,

(2.22)

C = S

,

2mS 2 . r

The extremal limit corresponds to sending m → 0, α → ∞ in such a way that M ≡ 2me2α is held ?xed. In this limit the horizon at r = 2m shrinks down to zero size and becomes singular. The charged black string solution (2.22) approaches the fundamental string (2.3). ? If we add linear momentum to (2.22) by applying a boost t = t cosh β + y sinh β, y = ? ?sinh β + y cosh β, and then take the extremal limit m → 0, α, β → ∞ with M ≡ 2me2α = t ? 2me2β ?xed, we obtain the generalized fundamental string (2.21). So this solution can also be viewed as the extremal limit of a charged black string, but now with a non-zero linear momentum. 3. Dimensional reduction To consider further applications of the chiral null models to, for example, extremal dilatonic black holes in D = 4 and charged FS solutions, we need to discuss ?rst the Kaluza-Klein re-interpretation of higher dimensional bosonic string solutions (heterotic string solutions will be discussed in Section 6). To have extremal black holes we need gauge ?elds. There are no fundamental gauge ?elds in bosonic string theory but they appear once the theory is compacti?ed on a torus or a group manifold and is expressed in terms of ‘lower-dimensional’ geometrical objects. 7

The usual treatment of dimensional reduction in ?eld theory starts with a spacetime action. This is possible also in string theory, but di?cult to do exactly. One would have to start with the full massless string e?ective action in, say, ?ve dimensions containing terms of all orders in α′ . Assuming the ?fth direction x5 is periodic we can expand the metric, antisymmetric tensor and dilaton in Fourier series in x5 and explicitly integrate over x5 . The result will be the e?ective action in D = 4 containing massless ?elds as well as an in?nite tower of massive modes with masses proportional to a compacti?cation scale. Any exact solution of the D = 5 theory which does not depend on x5 can then be directly interpreted as a solution of the equations of the D = 4 ‘compacti?ed’ theory with all massive modes set equal to zero (but all ‘massless’ α′ -terms included). Fortunately, in string theory there is a simpler alternative – to perform the dimensional reduction directly at the more fundamental level of the string action itself. Let us start with the general string σ-model (2.1), split the coordinates X M into ‘external’ x? and ‘internal’ y a and assume that the couplings do not depend on y a , ? ? ? L = (G?ν + B?ν )(x)?x? ?xν + (A?a + B?a )(x)?x? ?y a + (A?a ? B?a )(x)?x? ?y a (3.1) ? + (Gab + Bab )(x)?y a?y b + α′ Rφ(x) , where A?a ≡ G?a , B?a ≡ B?a . (3.2) Assuming for simplicity that Bab = 0, it is easy to represent the action in a form which is manifestly invariant under the space-time gauge transformations of the vector ?elds Aa ≡ Gab A?b and B?a ? ? ? ? ? L = (G?ν + B?ν )(x)?x? ?xν + B?a (x)(?x? ?y a ? ?x? ?y a ) ? ? + Gab (x) ?y a + Aa (x)?x? ?y b + Ab (x)?xν + α′ Rφ(x) ,

? ν ? ν

(3.3)

Like all σ-model Lagrangians, (3.3) changes by a total derivative if one adds the curl of a vector to the antisymmetric tensor ?eld. Since we are assuming no dependence on y a , the (?, a)-component of this transformation is simply B?a → B?a + ?? λa , i.e. the standard gauge transformation for the vector ?elds B?a . The action (3.3) is also invariant under shifting y a → y a ? η a (x) together with A a → A a + ?? η a , B?ν → B?ν ? 2?[? η a Bν]a . (3.5) ? ? The ?rst transformation is the usual one for the vector ?elds Aa while the second implies ? that the gauge-invariant antisymmetric tensor ?eld strength is given by4 ? Hλ?ν = 3?[λ B?ν] ? 3Aa B?ν]a , B?νa ≡ 2?[? Bν]a . (3.6)

[λ

? where the gauge-invariant ‘Kaluza-Klein’ metric G?ν is de?ned by ? G?ν ≡ G?ν ? Gab Aa Ab .

(3.4)

From the world sheet point of view we are using there seems to be no reason to rede?ne the antisymmetric tensor B?ν in (3.3) by the term Aa Bν]a as it is sometimes done in the e?ective [? ? action approach to dimensional reduction. If one does such a rede?nition, the new B?ν also ? transforms under the B?a gauge transformations and the generalized ?eld strength tensor Hλ?ν takes a more ‘symmetric’ form with respect to the two vector ?elds Aa and B?a . It should be ? ? λ?ν that has an invariant meaning, and it remains the same noted, however, that it is the full H ? irrespective of the de?nition of B?ν .

4

8

Although the world sheet approach to dimensional reduction in string theory is the most straightforward and simplest, it is useful to recall what the corresponding procedure looks like from the point of view of the space-time e?ective action. For example, if we start with just the leading-order term in the D = 5 bosonic string action √ 1 (3.7) S5 = κ0 d5 x G e?2φ { R + 4(?M φ)2 ? (HM NK )2 + O(α′ )} , 12 and assume that all the ?elds are independent of x5 , we obtain the four dimensional reduced action (for the general case, see e.g. [27] and refs. there) S4 = κ0 ? ? d4 x ? ? G e?2φ+σ { R + 4(?? φ)2 ? 4?? φ? ? σ (3.8)

1 1 1 ? (H?νλ )2 ? e2σ (F?ν )2 ? e?2σ (B?ν )2 + O(α′ )} , 12 4 4 where we have de?ned G55 ≡ e2σ , F?ν = 2?[? Aν] , B?ν = 2?[? Bν] , Setting the action (3.8) becomes S4 = κ0 ? ? d4 x ? = 2φ ? σ ? ? G e?? { R + (?? ?)2 ? (?? σ)2 (3.10) A? ≡ A5 , B? ≡ B?5 . ? (3.9)

(3.11)

1 1 1 ? (H?νλ )2 ? e2σ (F?ν )2 ? e?2σ (B?ν )2 + O(α′ )} . 12 4 4 In the Einstein frame (3.11) takes the form S4 = κ0 ? ? 1 ? ? d4 x GE { RE ? (?? ?)2 ? (?? σ)2 2 (3.12)

1 ?2? ? 1 1 e (H?νλ )2 ? e??+2σ (F?ν )2 ? e???2σ (B?ν )2 + O(α′ )} . 12 4 4 Thus, in general, the four dimensional theory contains two scalars, two vectors, and the antisymmetric tensor, as well as the metric. In certain special cases, the nontrivial part of the action (3.12) can be expressed in terms of only one scalar and one vector, so that it takes the familiar form5 1 1 ? ? S4 = κ0 d4 x GE { RE ? (?? ψ)2 ? e?aψ (F?ν )2 + O(α′ )} . ? (3.13) 2 4 For example, if one sets φ = 0 and HM NK = 0 in the D = 5 action, or equivalently ? ? =√ ?σ, H?νλ = 0 = B?ν directly in (3.12), one obtains (3.13) with ψ = ?aσ and a = 3. This is, of course, the standard Kaluza-Klein reduction of the Einstein action. ? Another possibility is to set σ = 0 (G55 = 1), H?νλ = 0 and either the two vector ?elds proportional to each other, or let one of them vanish. This case corresponds to (3.13) with ψ = ? and a = 1.

5

Such ansatzes must, of course, be consistent with D = 5 equations of motion.

9

4. Solutions involving dimensional reduction In this section we discuss the dimensional reduction of some of the exact solutions described by chiral null models (2.6). We will see that several previously found solutions of the leading order string e?ective equations can be easily obtained in this way. In addition, we ?nd some solutions which appear to be new. 4.1. Charged fundamental string solutions Our ?rst example is the charged FS solution found at the leading order level in [8,9].6 This solution is obtained by starting with the general chiral null model in D+N dimensions, and requiring that all ?elds be independent of u and N of the transverse dimensions labeled by y a . If we further assume that the vector coupling has only y a -components, we obtain ? ? ? ? ? ? ? L = F (x)?u?v + K(x)?u?u + ?xi ?xi + 2Aa (x)?u?y a + ?ya ?y a + α′ Rφ(x) , (4.1)

? ? which is conformal to all orders provided F, K ≡ F ?1 K, Aa ≡ F ?1 Aa and φ satisfy (2.13). If we are looking for FS-type solutions which are rotationally symmetric in D ? 2 coordinates xi , then solving the Laplace equations we can put the functions F, K, Aa in the form7 M 1 F ?1 = 1 + D?4 , φ = φ0 + lnF (r) , r 2 = xi xi , r 2 Qa P Aa = D?4 . (4.2) K = c + D?4 , r r ? Shifting v we can thus in general replace K in (4.1) by a constant. To re-interpret (4.1) as a D-dimensional model coupled to N internal coordinates we rewrite it in the form (3.3) ? ? ? ? L = F (r)?u?v + K ′ (r)?u?u + ?xi ?xi + α′ Rφ(r) ? ? ? ? ? ? ? + Aa (r)(?u?y a ? ?y a ?u) + [?y a + Aa (r)?u][?ya + Aa (r)?u] , ? ? ? K ′ (r) ≡ K ? (Aa )2 . The ?rst four terms give the D-dimensional space-time metric, antisymmetric tensor and dilaton while the last two identify (see (3.3)) the presence of two equal vector ?eld backgrounds (two equal components Gua and Bua conspire as one D-dimensional Kaluza-Klein vector ?eld, cf. (3.11)). Note that since Gab = δab , the modulus ?eld is constant and the lower dimensional dilaton is the same as the higher dimensional one. In the case of just one internal dimension we get one abelian vector ?eld u-component and the resulting background becomes that of the charged FS in [8,9].

6

(4.3)

The method of [8] was to start with the neutral solution and to make the most general

leading order duality rotation in all available isometric directions (including the internal ones). Since the duality transformation has, in general, α′ -corrections, this procedure does not guarantee the exactness of the resulting solution.

7

In the zero charge Qa = 0 limit we get not just the FS solution of [6] but its modi?cation

(2.21) which corresponds to momentum running along the string.

10

4.2. D = 4 solutions with electromagnetic ?elds To obtain four dimensional solutions with electromagnetic ?elds, we can reduce a D = 5 chiral null model. It was recently shown [2] that extremal electrically charged black holes can be obtained in this way. If one starts with the standard D = 5 √ (2.3),(2.5) FS one gets [28] the extremal electric black hole solution to (3.13) with a = 3 which was discussed in [10], while starting with the generalized FS (2.21) one obtains the extremal electric black hole solution to (3.13) with a = 1 discussed in [11,12].8 Here we shall consider the most general D = 5 chiral null model which is independent of u. It will yield a large class of D = 4 solutions. Some of these backgrounds were recently found [13,14,15] as leading-order string solutions, i.e. solutions of the dilatonaxion generalization of the D = 4 Einstein-Maxwell theory. They are the analogs of the IWP (Israel-Wilson-Perj?s [29]) solution of the pure Einstein-Maxwell theory.9 Special e cases of this generalized IWP solution describe a collection of extremal electric dilatonic black holes (Majumdar-Papapetrou-type solution) and an extremal electric Taub-NUTtype solution. The D = 5 chiral null model which is independent of u ? ? ? ? L5 = F (x)?u ?v + K(x)?u + 2Ai (x)?xi + ?xi ?xi + α′ Rφ(x) , (4.4)

was discussed in section 2.4 where it was noted that the general solution depends on the three harmonic functions F ?1 , K and T (see (2.20)) of the three coordinates xi . This model can be reduced to D = 4 along any space-like direction in the u, v plane. Shifting v by a multiple of u changes, of course, the direction of ?/?u, but this transformation is equivalent to a shift of K by a constant. Shifting u by a multiple of v can be undone by a particular case of the gauge transformation (3.5) (which gives an equivalent background, ? in particular, leaves H?νλ invariant). Thus it su?ces to use u as the internal coordinate y (which is possible, provided F K > 0) and to identify v with 2t. Then we can put (4.4) in the “four-dimensional” form (3.3) as follows L5 = ?K(x)?1 F (x) ?t + Ai (x)?xi ? ? ? ?t + Ai (x)?xi + ?xi ?xi + α′ Rφ(x) (4.5)

? ? ? ? + F (x)(?y ?t ? ?t?y) + F (x)Ai (x)(?y ?xi ? ?xi ?y) + K(x)F (x) ?y + K ?1 (x)?t + K ?1 (x)Ai (x)?xi ? ? ? × ?y + K ?1 (x)?t + K ?1 (x)Ai (x)?xi .

The corresponding four-dimensional background is thus represented by the following met1 ric, two abelian gauge ?elds A5 ≡ A? , B?5 ≡ B? , two scalars (the ‘modulus’ σ = 2 lnG55 ? ? and the dilaton) and the antisymmetric tensor ?eld strength H (cf. (3.3), (3.11)) ds2 = ?F (x)K ?1 (x) dt + Ai (x)dxi

2

+ dxi dxi ,

(4.6)

√ The a = 3 black hole can also be obtained [10] from the D = 5 plane-wave-type background (2.2) which is dual to FS. Similarly, one can get the a = 1 electric dilatonic D = 4 black hole from a duality-rotated (2.9) version of the generalized FS (2.21). Such model is, however, essentially

8

equivalent to (2.21), since it is ‘self-dual’. It was shown also that these backgrounds are supersymmetric when embedded in a supergravity [13].

9

11

Bt = ?F (x) , Bi = ?F (x)Ai (x) , At = K ?1 (x) , Ai = K ?1 (x)Ai (x) , 1 1 ? σ = ln[F (x)K(x)] , φ = φ0 + lnF (x) , Hλ?ν = ?6A[λ ?? Bν] . 2 2 Notice that even though the D = 4 antisymmetric tensor B?ν vanishes, the gauge invariant ? ?eld strength Hλ?ν is nonzero due to the contribution from the gauge ?elds in (3.6). This background represents a solution of the equations following from the D = 4 e?ective action (3.11) since Ai satis?es ?ijk ?j Ak = ? i T (x), and F ?1 , K and T are solutions of the three dimensional Laplace equation. Let us now consider some special cases. If K = 1 and Ai = 0, the gauge ?eld A? becomes trivial and the two scalars coincide (up to a constant). Since the gauge ?elds have ? only time components being nonzero, the antisymmetric tensor H vanishes. If we now set ?1 F = 1 + M/r, the original D = 5 theory (4.4) describes the fundamental string and the D = 4 reduction is the ‘Kaluza-Klein’ extremal black hole, √ i.e. the extreme electrically charged black hole solution corresponding to (3.13) with a = 3. We see that this solution has a straightforward generalization to the case of Ai = 0. The case K = F ?1 is of particular interest. The D = 5 model (4.4) is the Ai generalization of (2.21) while the corresponding D = 4 background is ds2 = ?F 2 (x) dt + Ai (x)dxi

2

+ dxi dxi ,

(4.7)

At = F (x) , Ai = F (x)Ai (x) , B? = ?A? , 1 ? φ = φ0 + lnF (x) , Hλ?ν = 6A[λ ?? Aν] , σ=0. 2 Since σ = 0 and the two gauge ?elds di?er only by a sign, these backgrounds are solutions to (3.13) with a = 1 provided the antisymmetric tensor term of (3.12) is included. These are precisely the D = 4 dilatonic IWP solutions [13,14,15]. If we restrict further to Ai = 0 ? and F ?1 = 1 + M/r, then Hλ?ν = 0 and we obtain the ‘standard’ extremal dilatonic black 10 hole [11,12] ds2 = ?F 2 (r)dt2 + dxi dxi , (4.8)

Let us note that the D = 4 extremal electric dilatonic black hole background can also be ? ? ? related to a D = 6 chiral null model with K = 0, L6 = F (x)?u ?v + 2A(x)?y′ ? ?y′ ?y′ + i ′ ? ?xi ?x , where the internal coordinate y has the ‘wrong’ (time-like) signature. Introducing the new coordinate y′ = y+u and choosing A = F ?1 (which is consistent with the conformal invariance conditions) we ?nd that this model takes the form of (2.21) plus an extra free time-like direction, ? ? ? ? L6 = F (x)?u?v + ?u?u + ?xi ?xi ? ?y?y, and thus can also be related to the D = 4 extremal electric black hole. An equivalent observation was made at the level of the leading-order terms in the e?ective action in [30] (ref. [13] also discussed a similar higher (six) dimensional interpretation of the IWP solution). It should be emphasized that it is our D = 5 model (4.4) that provides the correct higher-dimensional embedding of these D = 4 black-hole type solutions: though the presence of an extra time-like ‘internal’ coordinate in the above D = 6 model is irrelevant from the point of view of the proof of exactness of the D = 4 solution, it is unphysical, since complex coordinate transformations are needed if one wants to keep the physical signature of the full higher-dimensional space.

10

12

At = ?Bt = F (r) ,

1 φ(x) = φ0 + lnF (r) , 2

? Ai = Bi = σ = Hλ?ν = 0 .

Adding a nonzero Ai to this solution by setting T = q/r has the e?ect of adding a NUT charge. The result is the extremal electrically charged dilatonic Taub-NUT solution. Linear superposition of an arbitrary number of solutions of this type is possible by setting

N

F

?1

=1+

k=1

Mk , |x ? xk |

N

T =

k=1

qk . |x ? xk |

(4.9)

To add angular momentum, one takes solutions to Laplace’s equation which are singular on circles, rather than points as in (4.9). Finally, if we set K = F = 1 in (4.7), the dilaton becomes constant. This solution depends only on Ai and describes a spacetime with a magnetic ?eld Fij = 2?[i Aj] and ? antisymmetric tensor Htij = Fij . The corresponding D = 5 exact conformal σ-model (4.4) can be put (by a shift of v) in the following simple form ? ? ? L = ?u?v + 2Ai (x)?u?xi + ?xi ?xi , ?i F ij = 0 , (4.10)

and deserves further study. Some special choices of Ai are particularly interesting. One is the monopole background, Fij = q?ijk xk /|x|3 . Another is the case of a uniform magnetic 1 ?eld, Fij = const, i.e. Ai = ? 2 Fij xj . This model is equivalent (see Appendix A.3) to a product of the non-semisimple D = 4 WZW model of [19] and an extra free spatial direction and thus has a CFT interpretation. One can choose coordinates so that the D = 4 metric for the uniform magnetic ?eld solution is simply 1 ds = ? dt + Hr 2 dθ 2

2 2

+ dz 2 + dr 2 + r 2 dθ 2 ,

(4.11)

? and describes a rotating universe (while the antisymmetric tensor H is constant). This uniform magnetic ?eld solution may be contrasted with the dilatonic Melvin solution [11,31] in which the magnetic ?eld decreases with transverse distance. The latter solution contains a nonconstant dilaton (but no antisymmetric tensor or rotation) and is expected to have higher order α′ corrections. The solutions (4.6) with generic K and thus di?erent gauge ?elds Ai and Bi appear to be new. 5. Conformal invariance of the chiral null models The aim of this section is to demonstrate that the general chiral null model (2.6) is conformal to all orders in α′ provided the couplings satisfy the conditions (2.10) - (2.12) and one choses a special renormalization scheme. Our discussion will be based on the approach of [1] where more details about the special choice of the scheme can be found. In [1] it was shown that the F -model (2.3) (i.e. (2.6) with K = Ai = 0) which has two null Killing vectors and two associated chiral currents, is exact. It turns out that a single chiral current associated with a null symmetry is, in fact, su?cient to establish the exact conformal invariance of the more general backgrounds (2.6). 13

To ?nd the conditions for conformal invariance of a σ-model we must de?ne it on a curved two dimensional surface, introduce sources for the σ-model ?elds and determine when the resulting generating functional (or its Legendre transform) does not depend on the conformal factor of the 2-metric. There are two reasons why the models (2.6) are special. First, the null symmetry and chiral coupling to v imply that the path integral over v is readily computable giving a δ-function constraint on u which expresses u in terms of xi and a source. Second, chirality of the ?u?x-coupling implies that the resulting e?ective x-theory has only tadpole divergences (or conformal anomalies) in a properly chosen scheme. We shall ?rst give the proof of conformal invariance in a few special cases mentioned in section 2 (when some of the functions in (2.6) are trivial) and then give the general argument. 5.1. F =1 The argument is simplest when F = 1. To ?nd the exact conditions of conformal invariance we follow [1] by adding the source terms (z denotes the two world sheet coordinates) ? ? ? Lsource = V (z)? ?u + U (z)? ?v + Xi (z)? ?xi , (5.1) to (2.6) and performing the path integral over v. The resulting δ-function sets u to its classical value U (up to a zero mode which we absorb in U ). Thus u is ‘frozen’ and the e?ective x-theory is ? ? ? Lef f = ?xi ?xi + K(x, U )?U ?U + 2Ai (x, U )?U ?xi + α′ Rφ(x, U ) ? ? + Xi ? ?xi + V ? ?U . ? Computing the classical dilaton contribution (? ? ?φ) to the trace of the stress energy ? ?xi ) quantum contributions (in view of tensor and observing that there cannot be O(?U ? the absence of the O(?U ) vector coupling and simple dimensional considerations) one ?nds that the necessary conditions for this theory to be conformal are ?i ?u φ = 0, ?i ?j φ = 0, so that φ(x, u) = φ(u) + bi xi , bi = const . (5.3) ? One also learns that (in the minimal subtraction scheme) the renormalization of the ?U ?U ? i may come only from the one-loop tadpole diagrams. The conclusion is that this and ?U ?X model is conformal to all orders once the leading-order conditions of conformal invariance are satis?ed (see also [18]) 1 2 ? ? 2 K + bi ?i K + ? i ?u Ai ? 2bi ?u Ai + 2?u φ = 0 , 2 1 ? ?i F ij + bi F ij = 0 . 2 (5.4) (5.2)

These relations follow from a direct computation of the tadpole graphs and use of classical σ-model equations to transform the dilaton contribution (for simplicity, one may gauge away K by using the freedom (2.8)). They agree, of course, with the standard general expression for the one loop Weyl anomaly coe?cients given in Appendix A. 14

5.2. Ai = 0 ? Let us now set Ai = 0 and assume that K = F ?1 K and φ do not depend on u, i.e. consider ? ? ? ? L = F (x)?u?v + K(x)?u?u + ?xi ?xi + α′ Rφ(x) . (5.5) Introducing the source terms (5.1) and integrating over v one ?nds the constraint ?u = F ?1 (x)?U . (5.6)

Integrating then over u and taking into account the determinant contribution that shifts the dilaton as well as ?xing the same special ‘leading-order’ scheme (related to the standard one by an α′ -rede?nition of the ij-component of the metric) as in the F -model [1] one ?nds that the e?ective x-theory takes the form11 ? ? ? Lef f = ?xi ?xi ? F ?1 (x)?U ?V + K(x)?U ?? ?1 [F ?1 (x)?U ] ? + α′ Rφ′ (x) + Xi ? ?xi , 1 φ′ ≡ φ ? lnF (x) . 2 (5.8) (5.7)

The conditions of exact conformal invariance include the linearity of the dilaton φ′ in x φ′ = φ0 + bi xi , 1 φ = φ0 + bi xi + lnF , 2 (5.9)

and the standard scalar (‘tachyonic’) equation for F ?1 1 ? ? 2 F ?1 + bi ?i F ?1 = 0 . 2 (5.10)

The conformal anomaly must be local, so it is only the local part of the quantum average of the non-local O(?U ?U ) term that may contribute to it. Since this non-local term already contains two factors of ?U it cannot produce ?x-dependent counterterms. That means we may expand the functions K(x) and F ?1 (x) in it near a constant, xi (z) = xi + η i (z), 0 ? d2 zd2 z ′ [K(x)?U ](z)? 2??1 (z, z ′ )[F ?1 (x)?U ](z ′ ) 1 ?i1 ...?im K(x0 )?j1 ...?jn F ?1 (x0 ) n!m! n,m=0

∞

(5.11)

=

? d2 zd2 z ′ (η i1 ...η im )(z)?U (z)? 2 ??1 (z, z ′ )(η j1 ...η jn )(z ′ )?U (z ′ ) ,

11

Note that if F were u-dependent the integral over u would not be easily computable and the

argument below would not apply.

15

? where we de?ned ??1 by ? ???1 = δ (2) (z, z ′ ). Then the only contractions of the quantum i ? ?elds η that can produce local O(?U ?U ) divergences are the one-loop tadpoles on the left and right side of the non-local propagator ??1 (z, z ′ ). Any contraction between η n (z) and η m (z ′ ) gives additional ??1 (z, z ′ )-factor and thus contributes only to the non-local part of the corresponding 2d e?ective action. As a result, we ?nd the following conformal invariance condition F ?1 ? 2 K + K? 2 F ?1 = 2bi F ?1 ?i K + 2bi K?i F ?1 , or, combined with (5.10), ? 2 F ?1 = 2bi ?i F ?1 , 5.3. General chiral null model For the general chiral null model (with u dependence), one can set K = 0 by the gauge transformation (2.8). Adding sources and integrating over v and u as above we arrive at the following e?ective x-theory ? ? ? Lef f = ?xi ?xi ? F ?1 (x)?U ?V + 2Ai x, ? ?1 [F ?1 (x)?U ] ?U ?xi (5.14) where φ′ is as in (5.8) and we again use a special scheme to keep the free kinetic term ? of xi unchanged (see [1]). The condition of conformal invariance in the ?x?x direction is straightforward generalization of (5.3) and the condition in the model with Ai = 0 (5.9), ? i.e. φ′ = φ(u) + bi xi . The ?U ?V term is conformally invariant, provided one imposes ? (5.10) as in the Ai = 0 model. The conditions of conformal invariance in the ?u?u and ? ?u?x directions are similar to (5.4) with K = 0, 1 2 ? ?i F ij + bi F ij = 0 . ? i ?u Ai ? 2bi ?u Ai + 2F ?1 ?u φ = 0 , (5.15) 2 The reason why there are no extra terms involving F is that the locality of the conformal anomaly implies that the only contributions depending on derivatives of F are tadpole ones which thus vanish due to (5.10). This is easy to see by expanding the argument xi (z) of F ?1 and Ai near its ‘classical’ value. Contractions of the quantum ?elds on the opposite sides of the ? ?1 -operator produce only non-local contributions to the corresponding e?ective action. Equation (5.15) is valid in the gauge K = 0. The general form of this conformal invariance condition can be obtained by doing the gauge transformation (2.8). Combining all the conditions together we obtain12 1 1 φ = φ(u) + bi xi + lnF (x) , (5.16) ? ? 2 F ?1 + bi ?i F ?1 = 0 , 2 2 1 1 2 ? ? 2 K + bi ?i K + ? i ?u Ai ? 2bi ?u Ai + 2F ?1 ?u φ = 0 , ? ?i F ij + bi F ij = 0 . (5.17) 2 2

Let us note that the fact that the model (2.7) is Weyl invariant means also that when considered on a ?at world sheet this σ-model is ultra-violet ?nite to all loop orders on the mass shell. The latter clari?cation means that the standard β-functions vanish only modulo a di?eomorphism term (which is related to the presence of a non-trivial dilaton in the corresponding Weyl-invariant model).

12

(5.12)

? 2 K = 2bi ?i K ,

1 φ = φ0 + bi xi + lnF . 2

(5.13)

? + α′ Rφ′ x, ? ?1 [F ?1 (x)?U ] + Xi ? ?xi ,

16

5.4. Further generalizations? Can one extend the chiral null model (2.6) to include a larger class of backgrounds and maintain their conformal invariance? As we have already remarked, one possible generalization is to replace the transverse space with a nontrivial conformal ?eld theory. Another possibility would appear to be the addition of a second vector coupling ? ? ? ? ? L = F (x)?u?v + K(x, u)?u?u + 2Ai (x, u)?u?xi ? ? ? + 2Si (x, u)?xi?v + ?xi ?xi + α′ Rφ(x, u) . This σ-model shares with the chiral null model the following three properties: (i) conformal invariance of the transverse part of the model; (ii) existence of an a?ne symmetry v ′ = v + h(τ + σ) in a null direction; (iii) chirality of all vector couplings. The second condition implies the existence of the associated conserved chiral current. At the ‘point-particle’ (zero mode) level this a?ne stringy symmetry reduces to the null Killing symmetry v ′ = v + h, h = const. However, the model (5.18) is not, in general, conformal to all orders if only the leadingorder equations are satis?ed. As before, we can still explicitly integrate out v and then u. But the result is a complicated x-theory for which the conditions of conformal invariance seem di?cult to formulate and solve explicitly to all orders.13 To illustrate this point, let us consider a particular example of (5.18) with F = 1, ?i = 0 and u-independent couplings, A ? ? ? ? ? ? L = ?u?v + K(x)?u?u + 2Si (x)?xi ?v + ?xi ?xi + α′ Rφ(x) . (5.19) (5.18)

The corresponding target space metric has a null Killing vector, but in contrast to the case of the model (2.6) with F = 1 this vector is not covariantly constant. Making the coordinate transformation u → u + p(x) we get ? ? ? ? ? ? L = ?u?v + K?u?u + K?i p(?u?xi + ?xi ?u) ? ? ? ? + (2Si + ?i p)?xi ?v + (δij + K?i p?j p)?xi ?xj + α′ Rφ(x) . ? ? If we now choose Si = ? 1 ?i p, the new ?x?v-coupling disappears. We learn that in this 2 case the model (5.20) is equivalent to a modi?cation of (2.6) with a non-trivial transverse ? ? metric and non-chiral ?u?x and ?x?u - couplings (cf. (2.14)). Integrating over v it is ? ? easy to see that the the resulting conformal invariance conditions (both in ?u?u and ?x?x ′ directions) contain non-trivial corrections to all orders in α . This example makes it clear that the above three conditions are not su?cient to ensure that leading order solutions will be exact. One needs an additional condition which can be taken to be (iv) the null Killing vector should be orthogonal to the transverse subspace.

? ? ? We assume that K or Ai do not vanish at the same time. In the special case when K = 0 ? and Ai = 0 the model (5.18) is equivalent to the special case (2.18) of (2.6) with u → v, v → u.

13

(5.20)

17

One can further generalize (5.18) by introducing a non-trivial transverse space metric. Then there may exist some special cases in which such a model may remain conformal to all orders once it is conformal to the leading order. An example is provided by L = F (x) ?u + 2Si (x)?xi ? ? ? ?v + 2Ai (x)?xi + ?xi ?xi + α′ Rφ(x) . (5.21)

This model is related by u-duality to the u-independent case of the ‘non-chiral’ generalization of the K-model (2.14) with two non-vanishing vector couplings (the relation of the 1 ? functions is F = K ?1 (x), Si = Ai (x), Ai = Ai (x), φ = φ0 + 2 lnF (x)). In the case when Si and Ai have constant ?eld strengths (2.17), the theory (5.21), like (2.14), can be shown to be conformally invariant to all loop orders, provided (cf. (2.16)) 1 ? ? ? 2 F ?1 + F ij Fij = 0 , 2 1 φ = φ0 + lnF , 2 ? Fij = 2?[i Sj] . (5.22)

The proof is a simple version of the arguments used in the previous subsections (in the special case of Si = ?Ai it was given already in the Appendix B of [1]). Introducing the sources and integrating out u and v one obtains the following e?ective x-theory (cf. (5.2), (5.7), (5.8)) ? ? ? ? Lef f = ?xi ?xi ? F ?1 (x)?U ?V + 2Ai (x)?U ?xi + 2Si (x)?xi ?V ? + α′ Rφ′ (x) + Xi ? ?xi , (5.23) so that if Ai and Si are linear in x all conformal anomaly contributions come only from one-loop diagrams. 6. Superstring and heterotic string solutions So far we have discussed exact classical solutions of the bosonic theory. A generalization to the case of the closed superstring theory is straightforward. The superstring action is given by the (1, 1) supersymmetric extension of the bosonic σ-model ? ? (2.6) (with x? = (u, v, xi) in (2.6) replaced by (1, 1) super?elds X ? (z, θ, θ) ). Repeating the arguments of section 5 starting with the (1, 1) supersymmetric extension of (2.6) ? ? ?? I(1,1) = d2 zd2 θ(G?ν + B?ν )(X)D X ? D X ν and using that the one-loop conformal invariance conditions are the same as in the bosonic case one ?nds that our exact bosonic backgrounds also represent superstring solutions. One can also start with the component 1 representation (here ω±n? = ω m ± 2 H m ) ?m n? n? I(1,1) =

m? ? ? d2 z[(G?ν + B?ν )(x)?x? ?xν + λRm (δn ? + ω?n? (x)?x? )λn ?m R

(6.1)

1? m +λLm (δn ? + ω+n? (x)?x? )λn ? R+mnpq λm λn λp λq ] , ?m L L R R L 2 write down the fermionic part of the action explicitly with the help of (A.9),(A.16) and directly integrate over the left and right fermions. One then ?nds that the only e?ect of the ? fermionic contributions on the e?ective bosonic xi -theory is to cancel the local ?lnF ?lnF 18

term coming from the bosonic u, v-determinant.14 Thus there is no need for a special adjustment of a scheme compared to the pure bosonic case (see also [1]). As for the heterotic string solutions, one approach is to start with a closed superstring solution and embed it into a heterotic string theory by identifying the generalized Lorentz connection ω+n? (or ω?n? ) with a Yang-Mills background, i.e. by rewriting the (1, 1) ?m ?m supersymmetric σ-model in the (1, 0) (or (0, 1)) supersymmetric heterotic σ-model form [33,34,35,36]. For this to be possible, the holonomy group of the generalized connection ω+ (or ω? ) should be a subgroup of the heterotic string gauge group. In general, such em? ? bedding is problematic for solutions with a curved space-time (i.e. with a non-trivial timelike direction) since the holonomy is then (a subgroup of) a non-compact Lorentz group SO(1, D ? 1) while the heterotic gauge group should be compact on unitarity grounds.15 In fact, as shown in Appendix A.2, the holonomy groups of ω+ and ω? for generic chiral ? ? null models are non-compact (except for the case of the plane wave background (4.10) when the holonomy of ω+ is SO(D ? 2)) and thus cannot be embedded into SO(32) or ? E8 × E8 . 6.1. Exact heterotic string solutions One should thus try a more direct approach. As indicated above, given a bosonic string theory, there exist, in principle, two possible ways to construct a heterotic string theory depending on whether the “right” or “left” parts of the bosonic coordinates are supersymmetrized, i.e. on whether one considers a (1, 0) or (0, 1) supersymmetric world sheet theory. The two heterotic theories are related by interchanging left- and right- movers in the vertex operators, and, in general, are inequivalent. The fermionic parts of the heterotic σ-models corresponding to the two theories depend on ω? and ω+ respectively.16 ? ? In what follows we shall concentrate on the standard (1, 0) (or “right”) theory since it turns out that the (0, 1) (or “left”) theory does not have chiral null models as exact solutions. The action of the (1, 0) heterotic σ-model is given by (we ignore the “internal”

14

A simple test that this cancellation does take place is provided by the observation that the

two-loop β-function must vanish (in a “supersymmetric” scheme) in the (1, 1) supersymmetric ? σ-model [32], while the one-loop induced term ?lnF ?lnF term would contribute to the two-loop conformal anomaly.

15

A special case of this was pointed out in [37]. Notice that if the gauge group is non-compact, at

least one of the internal fermions has a negative norm but (compared to the (1, 1) supersymmetric superstring case) there is no extra local world sheet superconformal symmetry to gauge it away [38].

16

In particular, the σ-model β-functions and low-energy e?ective actions corresponding to the

two theories are related by simply changing the sign of B?ν (the e?ective actions of bosonic or supersymmetric string theories are invariant under B?ν → ?B?ν since these theories are invariant extensions of a bosonic background which is chiral (i.e. which distinguishes between left and right, e.g., having B?ν = 0) will be inequivalent. under world sheet parity transformation). That implies, e.g., that the “right” and “left” heterotic

19

fermionic part with a possible gauge ?eld background) I(1,0) = ? ? ?? d2 zdθ(G?ν + B?ν )(X)D X ? ? X ν (6.2)

=

m? ? ? d2 z (G?ν + B?ν )(x)?x? ?xν + λRm (δn ? + ω?n? (x)?x? )λn . ?m R

The (1,1) superstring σ-model action (6.1) can be formally obtained from (6.2) by adding the internal left fermionic part coupled to the gauge ?eld background equal to ω+ . ? Thus ω? appears in the fermionic part of the σ-model action (6.2) (and also in the ? leading-order space-time supersymmetry transformation laws). The β-functions and the e?ective action S of this theory will depend on ω? but also explicitly on the curvature R of ? G?ν and the antisymmetric tensor ?eld strength H. The σ-model anomaly will also naturally involve ω? . However, since the form of the anomaly is ambiguous (scheme dependent) ? [39,40] it can be arranged so that it will be ω+ that will enter the anomaly relation as well ? as the “anomaly-related” terms in the e?ective action (this, in fact, is a common assumption, see e.g. [41,42]). It should be emphasized that there is no unambiguous de?nition of such “anomaly-related” terms since S is scheme dependent and, in general, cannot be represented only in terms of ω+ . There are always other H-dependent terms which are not ? expressed in terms of the generalized curvature of ω+ (so that one can equally well use ω? ? ? 17 in place of ω+ at the expense of modifying the rest of the terms). ? Let us now show that our bosonic solutions are exact solutions of the heterotic string theory without any extra gauge-?eld background: the direct (1, 0) supersymmetric extension of the bosonic σ-model (2.6) is conformally invariant if the bosonic model is conformal. The fermionic part of the action (6.2) does not actually contribute to the conformal anomaly. This follows from the special “null” holonomy property of ω? : according to Appendix ? A (see (A.16)) the only non-vanishing component of the generalized Lorentz connection ω? is ω??? (?, v, ? are tangent space indices).18 The non-trivial fermionic terms in (6.2) ? ? ui? u ? i

17

O(α′ )-terms in the heterotic string e?ective action were computed in [43] and [21] starting

from the string S-matrix. As was shown in [21], there exists a scheme in which the α′ -term (its part which is not related to Chern-Simons modi?cation of the leading-order H 2 -term) in the heterotic string action is the same as in the bosonic string one up to an extra overall factor of 1/2. The same result was obtained from the analysis of the 3-loop conformal anomaly of the heterotic σ-model [44].

18

This property of ω? is also responsible for the “one-half” extended space-time supersymmetry ?

of our bosonic backgrounds when they are embedded into D = 10 supergravity as shown for the special cases of the (generalized) FS in [6,9] and for the F = 1 and K = F ?1 cases in [22,23] (our notation for ω? and ω+ are opposite to that of [22,23]). The general chiral null model also ? ? has unbroken spacetime supersymmetry, at least to leading order in α′ . It should be possible to address higher order corrections to the spacetime supersymmetry transformations for this model in the worldsheet approach using Green-Schwarz superstring action in a light-cone type gauge (cf. [9]).

20

are thus given by

? ? ? ? ? ? ? ? ? ? ui? L(1,0) (λR ) = λu ?λv + λi ?λi + ω??? (x)?x? λu λi . R R R R R R

(6.3)

The “null” structure of the coupling implies that integrating out fermions does not produce a non-trivial contribution to the x? -theory which remains conformally invariant. There is an obvious similarity with integrating out u and v in the bosonic theory (cf. Section 5). Thus we do not need a non-trivial gauge ?eld background to promote our bosonic solutions to heterotic ones. We conclude that, for example, the exact D = 5 bosonic solutions (4.4) are also heterotic string solutions and so are their four dimensional ‘images’ (4.6). In particular, the D = 4 extremal electric black holes discussed in Section 4.2 are thus exact heterotic string solutions [2] without any extra gauge ?eld background. Let us compare the above conclusion with the perturbative result for the two-loop β-function of the heterotic σ-model. Let us consider the “non-anomalous” α′ contribuG tion to the metric β-function β?ν (i.e. we shall ignore other non-covariant α′ -corrections which modify the one-loop H 2 -term by the Chern-Simons terms). The contribution of the fermions λR is essentially the same form as the standard two-loop “F 2 ”-term that comes from the internal fermionic sector ψL [34] except for the fact that the gauge ?eld is represented by the connection ω? [45]. Thus ? 1 ? G(2) G(2) ? mnλ (β?ν )(1,0) = (β?ν )0 ? α′ R?mnλ? R? ν , 4

G(2)

(6.4)

where (β?ν )0 is the bosonic contribution. There exists a special chiral “right” scheme in which the latter is given by [21]

G(2) (β?ν )0 =

1 ′ ? ? βλα ? ? αβλ ? α 2R? (? R?ν)αβλ ? R? (? R?ν)αβλ + R?α(?ν)β H αρσ H β ρσ 4

.

(6.5)

? ? mnλ As follows from (A.9) R?mnλ? R? ν (i.e. the fermionic contribution) indeed vanishes for our backgrounds. As for (6.5), it also vanishes when F = 1 but in general one needs to choose a di?erent scheme to avoid α′ -corrections (see [1]). Given the scheme dependence of the β-function, in the heterotic σ-model context there may exist a scheme in which the bosonic contribution to the σ-model β-function (6.4) can be put in the following “left-right symmetric” form

G(2) (β?ν )0 =

1 ′ ? ? mnλ ? ? mnλ α R+mnλ? R+ ν + R?mnλ? R? ν 4

.

(6.6)

Including the gauge ?eld contribution of the internal left fermions the heterotic σ-model β-function corresponding to this “symmetric” scheme then is given by

G(2) (β?ν )(1,0) =

1 ′? 1 ? mnλ α R+mnλ? R+ ν ? α′ FIJλ? (V )F IJλ (V ) . ν 4 4

(6.7)

This expression is consistent with the expectation that the two-loop β-function should IJ vanish once we identify the gauge ?eld V? with ω+ since then the heterotic σ-model ? becomes identical to the (1,1) supersymmetric model (6.1). The two-loop contribution 21

(6.7) with V? = 0 does not vanish for our backgrounds even in the simplest plane-wave case F = 1. As already mentioned above, in general, we cannot make it vanish by the identi?cation V = ω+ since the holonomy group of ω+ is non-compact. Thus in this scheme ? ? our solutions will be modi?ed by higher-order α′ corrections. In the special case of F = 1, K = 0, Ai = Ai (x), the only non-vanishing component ? ? mnλ of ω+ is ω+?? = ?Fij and one ?nds that R+mnλ? R+ ν in (6.7) has non-vanishing uu? ? iju component equal to (?k Fij )2 . If we set V? = 0 and start with the leading-order solution 1 K = 0, ?i F ij = 0 then K receives the α′ -correction K1 satisfying (cf. (A.27)) ? 2 ? i ?i K1 + 1 ′ α (?k Fij )2 = 0, i.e. K1 = 1 α′ (Fij )2 . Such modi?cation can be thought of as a local ?eld 4 4 rede?nition corresponding to the transformation from the chiral “right” scheme (6.4),(6.5) where K1 = 0 to the “left-right symmetric” scheme (6.7).19 Since in this exceptional case the holonomy of ω+ is compact (SO(D ? 2)), there is ? IJ also an alternative option to introduce the gauge ?eld background V? equal to ω+ and ? in this way cancel the higher order correction. This was suggested in [22] where (6.7) was assumed to be the form of the α′ -correction in the heterotic string equation of motion.20 As we have mentioned, the idea of embedding of ω+ into the gauge group does not have ? consistent generalizations to other cases except the one with F = 1, K = 0, Ai = Ai (x) so that we disagree with the suggestion of [22,23] that F = 1 and F = K ?1 models are exact heterotic string solutions only when supplemented by a gauge ?eld background. The need to introduce a non-trivial gauge ?eld background in [30,23] was caused by having implicitly taken the α′ term in the e?ective action in a speci?c “symmetric scheme” (in which ω+ appears in the “anomaly-related” terms). As we have explained above, the form ? ′ of α -corrections is scheme dependent and in the natural chiral scheme there is no need for an extra gauge ?eld background at all. The plane wave model (4.10) with F = 1, K = 0, Ai = Ai (x) and the gauge ?eld ij background Vu = ω+?? = ?Fij is equivalent to the (1, 1) supersymmetric superstring σ? iju model and thus represents an exact solution according to the discussion at the beginning of this section. It is instructive to see explicitly why the resulting model remains conformally m invariant: the fermionic terms now are (ψL are internal fermions; see also (A.9),(A.16))

? ? ? ? ? ? ? ? ? L(1,1) (λR , ψL ) = λu ?λv + λi ?λi + Fij (x)?xj λu λi R R R R R R

(6.8)

1 ? ? ? ? ? j ? i j i i u ? v ? ? ? ? k +ψL ?ψL + ψL ?ψL ? Fij (x)?uψL ψL ? ?i Fjk (x)λu λi ψL ψL . R R 2

19

This rede?nition of Guu could be thought of as induced by G′ = G?ν + 1 α′ H?λρ Hνλρ . It ?ν 4

IJ 1 may also be related to the non-covariant rede?nition G′ = G?ν + 4 α′ (? + ? ω+mnν ? V? VIJ ν ) ω mn ? ?ν

used in [40] in order to preserve world sheet supersymmetry (there is only ω+ ω+ -term if V = 0 ? ? and the whole rede?nition is trivial if V = ω+ ). ?

20

While the e?ective action considerations in [22,23] are not su?cient to demonstrate the

exactness of the solutions to all orders in α′ since they were ignoring “anomaly-unrelated” terms (in particular, no explanation was given why these backgrounds are superstring solutions, i.e. why the corresponding (1, 1) supersymmetric σ-model should be conformally invariant), this is possible within our direct world sheet approach. Although the approach of [22,23,37] is incomplete, our present work was much motivated and in?uenced by these interesting papers.

22

? ? i j Integration over λv ‘freezes’ out λu , while the term Fij (x)?uψL ψL does not produce new R R ? divergencies in the uu-direction since the total action does not contain local ?u-couplings (cf. (5.11)). Finally, let us consider the (0,1) (“left”) heterotic theory. Here the superstring fermions are coupled to ω+ . Since according to (A.17) ω+ has general holonomy, one ? ? should expect non-trivial fermionic contributions to the conformal anomaly. The gauge ?eld background cannot be consistently introduced since the (abelian) holonomy group of ω? is “null” (non-compact). The corresponding leading-order solutions thus should have ? corrections to all orders in α′ . Given that ω+ (which in this theory appears also in the ? space-time supersymmetry transformation laws) is of generic form, one should not also expect to ?nd Killing spinors, i.e. a residual supersymmetry.

?

?

6.2. Extended world sheet supersymmetry It is clear that the abelian gauge ?elds of the four dimensional solutions (4.6) have a Kaluza-Klein and not a heterotic Yang-Mills origin. In general, given a D = 4 leadingorder bosonic background, its embedding into the heterotic string theory is not unique. The embeddings of extremal D = 4 dilatonic black holes in which the U (1) gauge ?elds have a Kaluza-Klein (i.e. N = 4 supergravity) and not a heterotic Yang-Mills origin have extended (e.g. N = 2, D = 4) space-time supersymmetry [46]. Since our general bosonic D ≤ 10 backgrounds have extended space-time supersymmetry when embedded into D = 10 supergravity theory [6,9,22,23,13] one may also try to envoke supersymmetry to argue that they are exact superstring solutions. In fact, the presence of extended space-time supersymmetry suggests (cf. [47,42]) that the corresponding (1, 0) supersymmetric σ-models may have extended world sheet supersymmetry. If the latter supersymmetry is large enough, one may use the fact that there exists a scheme in which the (4, n) supersymmetric σ-models are conformal to all orders [48]. In contrast to our approach described in Section 5 and in the previous subsection, any argument based on extended world sheet supersymmetry is bound to have a limited applicability. The standard discussions of extended world sheet supersymmetry apply to the case of Euclidean target space signature. To have (2, n) supersymmetry the dimension D must be even; the (4, n) supersymmetry is possible only when D is multiple of 4. Most of our models (e.g. all with odd spacetime dimension) do not admit extended world sheet supersymmetry since they do not admit a complex structure when analytically continued to Euclidean signature. The generic chiral null model (2.6) does not have a natural analytic continuation with a real Euclidean target space metric. For example, if one analytically continues u + v keeping u ? v real, so that u and v become complex conjugates (v = u), then the metric ? is no longer real unless K and Ai in (2.6) are taken to be zero. There may exist a wellde?ned Euclidean analog of (2.6) for some special choice of Ai but we shall ignore this possibility for simplicity. In the special case of the F -model (2.3) one gets a real action on the Euclidean world sheet (but thus a complex string action in the Minkowski world sheet signature case) ?? ? L = F (x)?u? u + ?xi ?xi + α′ Rφ(x) . (6.9) The corresponding Euclidean metric ds2 = F (x)dud? + dxi dxi is real but the antisymu metric tensor is imaginary. If the dimension is even, D = 2N , the metric is hermitian. 23

Replacing xi by a set of complex coordinates ws (s = 1, ..., N ? 1) the metric and the antisymmetric tensor are ds2 = F (w, w)dud? + dws dws , ? u ? Bu? = u 1 F , 2 Hsu? = u 1 ?s F . 2 (6.10)

The corresponding (1, 0) supersymmetric σ-model admits (2, 0) extended supersymmetry. This is clear from the comparison with the conditions on geometry implied by (2, 0) supersymmetry [36], as reviewed, e.g., in [49] (for some earlier discussions of related complex geometries, see [50,51]). Provided ? 2 F ?1 = 0 the generalized connection with torsion has special (not U (N ) but SU (N )) holonomy and satis?es the generalized quasi Ricci ?atness condition (see Appendix A) ? ? R?ν = D? Vν , V? = ??? lnF . (6.11)

If the dimension D is a multiple of four, i.e. N = 2N ′ , a (2, 0) supersymmetric σ-model may admit (4, 0) extended supersymmetry. In fact, the Euclidean F -model (6.9) does have it, as is clear again from the comparison with the expressions in [49]. In particular, the holonomy of the generalized connection is an Sp(N ′ ) subgroup of SU (N ).21 Given that (4, 0) supersymmetric σ-models are conformally invariant to all orders (in a properly chosen scheme) [48] we get (for D = 4N ′ ) an independent proof of the fact that the F -model corresponds to an exact solution of heterotic string theory.22 It should be stressed again that our explicit proof given in [1] and in the present paper is more direct and applies for any D as well as to a more general class of models (2.6). In general, a relevant property which is important for the proof of exactness is the special holonomy of the generalized connection with torsion and not an extended supersymmetry (which is just a consequence of the special holonomy under certain additional conditions like existence of a complex structure).23 6.3. Relation to other D = 4 heterotic solutions What about non-supersymmetric solutions of D = 4 heterotic string theory? For example, the charged dilatonic black hole may be considered as a non-supersymmetric leading-order solution [12] of the D = 4 heterotic string theory with the charge being that

21

In the simplest case of D = 4 and F = F (|w|) the metric becomes conformal to a K¨hler a The conclusion about extended supersymmetry of the F -models is consistent with the fact

metric, cf. [42,49].

22

that some of them correspond to special nilpotently gauged WZW models [5]. The latter are formulated essentially in terms of the WZW model on a (maximally non-compact) group G and thus their Euclidean versions should admit (2, 0) or (2, 2) supersymmetry [51,52].

23

A somewhat related remark was made in [53], where it was pointed out that since the σ-model

on a Calabi-Yau space has a special holonomy it thus has an extra in?nite-dimensional non-linear classical symmetry. That symmetry (if it were not anomalous at the quantum level) would rule out all higher-loop corrections to the β-function [53]. In our case, the analogous symmetry is linear and is the a?ne symmetry generated by the null chiral current.

24

of the U (1) subgroup of the Yang-Mills gauge group. This solution must have an extension to higher orders in α′ which, in general, may not be the same as the above supersymmetric ‘Kaluza-Klein’ solution obtained by dimensional reduction from D = 5. Even though the leading-order terms in the compacti?ed (from D = 5 to D = 4) bosonic string theory and D = 4 heterotic string theory with a U (1) gauge ?eld background look the same, the α′ -corrections are di?erent, so that our bosonic result does not automatically imply that the extremal electric black hole considered as a D = 4 heterotic string solution is also exact. In fact, it is known that the non-supersymmetric extremal magnetic solution of the D = 4 heterotic string has α′ -corrections [54]. The same is likely to be true for the non-supersymmetric extremal electric solution. To explain this di?erence between “supersymmetric” and “nonsupersymmetric” solutions it is useful to consider the space-time e?ective action approach. Our exact D = 4 solutions (4.7) obtained by dimensional reduction are actually D = 5 bosonic or heterotic string solutions. This means that there exists a choice of (?ve dimensional) ?eld rede?nitions in which the D = 5 e?ective equations evaluated on our background contain no α′ corrections. As shown in section 3, the dimensional reduction of the D = 5 action includes two gauge ?elds (as well as an extra scalar modulus ?eld). Even though these two gauge ?elds are equal for our solution (4.7), the general ?eld rede?nition treats them independently. In contrast, the D = 4 non-supersymmetric heterotic action contains a single gauge ?eld and thus a smaller group of ?eld rede?nitions. Thus the fact that nontrivial α′ corrections inevitably arise in this case (for the magnetically charged black hole [54] and, most likely, for the electrically charged case as well) does not contradict our claim that the supersymmetric electrically charged solution obtained from dimensional reduction is exact. In general, the starting point is the D = 10 heterotic string with the leading-order term in the e?ective action being represented by the N = 1, D = 10 supergravity coupled to D = 10 super Yang-Mills theory. Compacti?ed on a 6-torus, this e?ective action becomes that of N = 4, D = 4 supergravity coupled to a number of abelian N = 4 vector multiplets and N = 4 super Yang-Mills. The simplest charged dilatonic black hole solution may be embedded in this theory in several inequivalent ways, depending on which vector ?eld(s) is kept non-vanishing. The dependence of higher-order α′ -terms in the full e?ective action on di?erent vector ?elds is di?erent, so it should not be surprising that the solutions that happen to coincide at the leading-order level may turn out to receive di?erent α′ -corrections. Finally, let us note that it may be possible to utilize some of our D > 4 exact bosonic solutions to construct other D = 4 heterotic string solutions.24 The idea is to start with an exact higher dimensional bosonic solution and then fermionize the ‘internal’ coordinates in an appropriate way to obtain a heterotic σ-model. A similar method was used in [55] to ?nd the heterotic solution representing a D = 2 monopole theory (which was related to the throat limit of the D = 4 extreme magnetically charged black hole) and in [56] to describe a non-trivial throat limit of the D = 4 dilatonic Taub-NUT solution [13,14].

24

To establish a relation between heterotic and bosonic models one can use the following strat-

egy: start with a leading-order heterotic string solution, write down the corresponding heterotic σ-model and then try to bosonize it to put it in a form of a bosonic σ-model for which it may be possible to prove the conformal invariance directly.

25

7. Concluding remarks To obtain exact solutions in string theory, it is rather hopeless to start with the ?eld equations expressed as a power series in α′ , and try to solve them explicitly. One must ?rst make an ansatz which simpli?es the form of these equations. We have studied such an ansatz, the chiral null models (2.6), and shown that they have the property that there exists a scheme in which the leading order string solutions are exact. This generalizes a number of previous results. The chiral null models include the plane wave type solutions and the fundamental string background which were previously shown to be exact. But as we have seen, they also include, e.g., the solution describing traveling waves along the fundamental string, and, after a dimensional reduction, the extremal electrically charged dilaton black holes and the dilaton IWP solutions. Moreover, there are interesting solutions describing magnetic ?eld con?gurations. It is rather surprising that such a large class of leading-order solutions turn out to be exact in bosonic, superstring and heterotic string theories. One can, in fact, turn the argument around. Even the leading order string equations (analogous to Einstein’s equations) can be rather complicated when the dilaton and antisymmetric tensor are nontrivial. By choosing an ansatz at the level of the string world sheet action which yields simple equations for the σ-model β-functions, one can easily ?nd new solutions of even the leading order equations. The chiral null models are an example of this. Some of the solutions we have discussed, e.g. (4.6) with a general K, appear to be new. However, it is clear that not all of the solutions of the leading order equations can be obtained from chiral null models. The chiral coupling, which is an important feature of these models, leads to a no-force condition on the solutions, and the possibility of linear superposition. This happens only for a certain charge to mass ratio which typically characterizes extreme black holes or black strings. Furthermore, we have obtained only four dimensional black-hole type solutions with electric charge. Extreme magnetically charged black holes do not appear to be described by chiral null models. We have considered examples of chiral null models with a ?at transverse space. As we have remarked, it is straightforward to extend this class of models to any transverse space which is itself an exact conformal ?eld theory. It may be interesting to explore the new solutions (with non-trivial mixing of “space-time” and “internal” directions) which can be obtained in this way. An important open problem is to study string propagation in the backgrounds discussed here. This will improve our understanding of the physical properties of these solutions in string theory.

8. Acknowledgements We would like to thank G. Gibbons, R. Kallosh and A. Strominger for useful discussions. G.H. was supported in part by NSF Grant PHY-9008502. A.A.T. is grateful to CERN Theory Division for hospitality while this paper was completed and acknowledges also the support of PPARC. 26

Appendix A. Geometrical quantities for the chiral null model A.1. Generalized connection The classical string equations for a σ-model ? L = C?ν (x)?x? ?xν , C?ν ≡ G?ν + B?ν , (A.1) (A.2) (A.3)

1 1 ? ? ? Γλ = Γλ ± H λ ?ν , Γλ = Γλ = Gλρ (?? Cρν + ?ν C?ρ ? ?ρ C?ν ) . ??ν +ν? ±?ν ?ν 2 2 In the case of our model (2.7) x? = (u, v, xi ) and 1 Guv = F , Gui = F Ai , Guu = F K , Gvi = 0 , Gvv = 0 , Gij = δij , 2 Gui = Guu = 0, Gvi = ?2Ai , Guv = 2F ?1 , Gvv = 4(Ai Ai ? F ?1 K) , Gij = δ ij , Cuv = F , Cvu = 0 , Cui = 2F Ai , Ciu = 0 , Cuu = F K , Cvi = Civ = 0 , Cvv = 0 , Cij = δij . We shall use the following de?nitions 1 h(x) = lnF (x) , Fij = ?i Aj ? ?j Ai , K = K(x, u) , Ai = Ai (x, u) . 2 ? +?ν ? ?ν? The corresponding components of the connection are (Γλ = Γλ ) ? Γu = 2?i h , ?ui ? Γi ?uj

i ? Γi ?uv = ?F ? h ,

are naturally expressed in terms of the generalized connection with torsion ? ? ? ? ? ? ? ?xλ + Γλ (x)?x? ?xν = 0 or ? ?xλ + Γλ (x)?xν ?x? = 0 , ??ν +?ν

(A.4)

(A.5)

(A.6)

We get

? It is straightforward to compute the curvature tensor corresponding to Γλ ±?ν (note that the torsion here is a closed form) ?λ ? ? ? ? R±?νρ = Rλ?νρ (Γλ ) , R?λ?νρ = R+νρλ? . (A.8) ±?ν ?u R??νρ = 0 , ?i R?jνρ = 0 , ?λ R??uv = 0 , ?? R?vνρ = 0 , (A.9)

? Γv = ?i K ? 2K?i h + 2Ai Aj ?j F ? 2F Aj Fij , ?ui ? Γv = ?u K ? 2F Ai ?u Ai + Ai ?i (F K) ,

?uu

? ? Γv = 2?i h , Γv = 2?i Aj + 4Aj ?i h , ?iv ?ij

1 i i ? Γi ?uu = F ?u A ? ? (F K) , 2 i i ?i ?i = ?Aj ? F ? F F j , Γ?v? = Γ?j? = 0 ,

? ? ? ? Γu = Γu = Γu = Γu = 0 , ?ij ?vj ??u ??v

(A.7)

? Γv = ?i K + 2K?i h , ?iu ? Γv = Ai ?i F , ?uv v ? Γ =0.

?v?

?i ?v R?iuj = 2F ?1 R?uju = 2?i ?u Aj + 4?i h?u Aj ? 2K?i ?j h ? ?i ?j K ? 2?i h?j K , ? mn ? Note that product of the curvatures R? λρ R?mn?ν vanishes. 27 ?v ?i R?ijk = ?2F ?1 R?ujk = 2?i Fjk + 4?i hFjk + 4Ak ?j ?i h ? 4Aj ?k ?i h .

?v ?i R?ivj = 2F ?1 R?ujv = ?2?i ?j h ,

?v R?uvj = ?2F Ai ?i ?j h ,

A.2. Special holonomy The expressions for the curvature (A.9) re?ect holonomy properties of the generalized ? ? connections Γλ . It turns out that the holonomy group of Γλ is an abelian (D ? 2) ±?ν ??ν dimensional “null” subgroup of the Lorentz group SO(1, D ? 1). The holonomy group of ? Γλ is not special for generic functions F, K, Ai It becomes the Euclidean group in D ? 2 +?ν dimensions when F = 1 and reduces further to its rotational subgroup SO(D ? 2) when F = K = 1, Ai = Ai (x). ? It is easy to argue that a special holonomy of the generalized connection Γλ ??ν in (A.2) is a direct consequence of the presence of a chiral current in the σ-model (A.1) (for a related more general discussion see [53] and refs. there). If one introduces the vierbeins and de?nes the following di?erentials (or ‘currents’) θ m = em ?x? , ? ? ? θ m = em ?x? , ? G?ν = ηmn em en , ? ν (A.10)

where ηmn is the tangent space metric, then the string equation (A.2) can be written in the form ? ? ? ?θ m + ω?n? ?x? θ n = 0 or ? θ m + ω+n? ?x? θ n = 0 , ?m ? ?m (A.11) where ω±nν are the generalized Lorentz connections ?m ? ω±nν = em Γλ e? + em ?ν eλ . ?m λ ±?ν n λ n

(A.12)

In the case of (2.7) one may choose (the tangent space indices take the following values: m = (?, v, ? u ? i)) ? ? ? (A.13) θ u = F ?u , θ v = ?v + K?u + 2Ai ?xi , θ i = ?xi , so the Lagrangian (2.7) takes the form

? ?? ? L = θ u θ v + θ i θ i + α′ Rφ(x) . ? ?

(A.14)

? ? ? Then the existence of the null v-isometry implying ?θ u = 0 tells us that ω?n? = 0, i.e. ?u that the connection ω? has a reduced holonomy.25 ? 1 De?ning the connection 1-forms (ηuv = 2 , η?? = δ?? ) ?? ij ij

ω ω±mn = ηmp ω±n? dx? = ?? ±nm , ? ?p we ?nd ω?m? = ω??? = 0, and ? v ? ij 1 ω??? = ?i hdv + ( ?i K ? ?u Ai + K?i h)du + (Fij + 2Aj ?i h)dxj , ? ui 2

25

(A.15)

(A.16)

In the case of the F -model [1] one has two null chiral currents (u and v are on an equal footing)

and so both ‘left’ and ‘right’ connections should have the same properties. Note, however, that our choice of vierbein in (A.13) is not symmetric in u and v so an extra Lorentz transformation will be needed to relate ω? to ω+ . ? ?

28

ω+?v = ?i hdxi , ? u?

ω+?? = ?F Fij du , ? ij ω+?? = F ?i hdu . ? vi

(A.17)

1 ω+?? = ( ?i K ? ?u Ai )du , ? ui 2

Since the algebra of the Lorentz group SO(1, D ? 1) is generated by M ≡ Muv , L? ≡ ?? i Mu?, R? ≡ Mv?, M?? satisfying, in particular, ?i i ?i ij [M, M?? ] = 0 , [M, L?] = L? , [M, R?] = ?R? , [L?, R?] = 2δ?? M + M?? , ij i i i i i j ij ij [M?? , Lk ] = 4L[?η? k , ? ij j i] ? [M?? , Rk ] = 4R[?η? k , [L?, L?] = [R?, R?] = 0 , ? i j i j ij j i] ? (A.18)

we conclude that the holonomy group of ω? is equivalent to the non-compact abelian ? subgroup of the Lorentz group generated by Mu? (it is “null”, having zero norm associated ?i with it). The holonomy of ω+ is not special in general.26 ? Let us now consider some particular cases. When F = const we ?nd that ω+?v = ? u? ω+?? = 0 and thus the holonomy algebra of ω+ reduces the Euclidean algebra generated ? vi ? by M?? and Mu?. It reduces further to the algebra of SO(D ? 2) when K = 1, Ai = Ai (x) ij ?i (i.e. in the case of the model (4.10)). In the case of the generalized FS solution related to the black hole type solutions (4.7) we have K = F ?1 , Ai = Ai (x) in (4.4) so that the non-vanishing components of the connections are ω??? = ?i hdv + (Fij + 2Aj ?i h)dxj , ? ui (A.19) ω+?v = ?i hdxi , ? u? ω+?? = ?F Fij du , ? ij ω+?? = F ?i hdu . ? vi ω+?? = ?F ?1 ?i hdu , ? ui

When Ai = 0 the holonomy algebra of ω+ becomes the 2D ? 3 dimensional non-semisimple ? subalgebra of the Lorentz algebra generated by M, L? and R?. i i A special holonomy is known [50,51,48] to be related to the presence of extended world sheet supersymmetry in the supersymmetric extensions of the σ-models. In fact, some of the models (2.7) (which, in particular, admit a complex structure) have extended supersymmetry (see Section 6). Let us note also that special holonomy does not guarantee, by itself, conformal invariance since for that the dilaton is crucial as well. Still, it is related (in a proper renormalization scheme) to the on-shell ?niteness of our models on a ?at world sheet. A.3. Parallelizable spaces and connection to WZW models based on non-semisimple groups One may be interested which of our spaces are parallelizable with respect to the ?λ ?λ generalized connection, i.e. have R??νρ = 0 (and thus R+?νρ = 0, see (A.8)). One expects parallelizable spaces to be related to group spaces and indeed this is what we ?nd.

26

In the absence of torsion the irreducible holonomy groups (or “special geometries”) on non-

symmetric spaces have been classi?ed [57]. No systematic classi?cation seems to be known in the torsionful case. We thank J. De Boer and G. Papadopoulos for helpful comments on this subject.

29

Since the string naturally ‘feels’ the generalized connection with torsion, the vanishing of the generalized curvature is the analogue of the ?atness condition in the point-particle n ? ? theory. In particular, R± = 0 means that locally Γλ = f ?1λ ?ν f±? . Then (A.2) implies ±?ν ±n n n ? the existence of D chiral and D antichiral conserved currents f?? (x)?x? and f+? (x)?x? . As follows from (A.9), a necessary condition for parallelizability is ?i ?j h = 0, i.e. h = h0 + pi xi . Then the two remaining conditions take the form ?v R?ijk = ?i Fjk + 2pi Fjk = 0 , ?v R?iuj = 2?i ?u Aj + 4pi ?u Aj ? ?i ?j K ? 2pi ?j K = 0 . In view of the gauge freedom (2.8) we may set K = 0. If pi = 0 the solution is Ai = Ci (u) exp(?2pj xj ). By rede?ning the coordinates v ′ = v + exp(?2pi xi )g(u), x′i = xi + wi (u) the corresponding model can be transformed into the product of the SL(2, R) WZW model (cf. (B.7)) and RD?3 . The case of pi = 0, i.e. F = const is more subtle. The solution is Ai = Ci (u) ? 1 j ′ 2 Fij x , Fij = const. One can further eliminate Ci by a coordinate transformation v = i ′i i i v +q(u)+si (u)x , x = x +w (u). We are ?nally left with the following model (cf. (4.10)) F =1, K=0, 1 Ai = ? Fij xj . 2 (A.21) (A.20)

These spaces can be interpreted as boosted products of group spaces, or, equivalently, as spaces corresponding to WZW models for non-semisimple groups. To show this one should ?rst put Fij into the block-diagonal form by a coordinate xi rotation, so that its elements are represented by constants H1 , ..., H[D/2?1] and the corresponding Lagrangian (4.10) is (we split xi into pairs representing 2-planes; a, b = 1, 2)

[D/2?1]

? L = ?u?v +

s=1

? ? Hs ?ab xa ?u?xb + ?xa ?xas s s s

.

(A.22)

The ?rst non-trivial case is that of D = 4, i.e. Fab = H?ab . The corresponding model (x1 = r cos θ, x2 = r sin θ) ? ? ? L = ?u?v + H?ab xa ?xb ?u + ?xa ?xa ? ? ? ? = ?u?v + Hr 2 ?θ?u + ?r ?r + r 2 ?θ ?θ , (A.23)

c is equivalent to the E2 WZW model of ref. [19] (note that H can be set equal to –1 by a rescaling of u, v). In fact, the coordinate transformation [58] x1 = y1 + y2 cos u, x2 = y2 sin u, v = v ′ + y1 y2 sin u puts (A.23) in the form

? ? ? ? L = ?u?v ′ + ?y1 ?y1 + ?y2 ?y2 + 2 cos u ?y1 ?y2 ,

(A.24)

which is obtained from the R × SU (2) WZW action by a singular boost and rescaling of the level k or α′ (see [59]). If s is a time-like coordinate of the R-factor and ψ is an angle of SU (2) one should set s = u, ψ = u + ?v, rescale k and yi by powers of ? and 30

? ? where θ is the dual coordinate and A, A are components of the 2d gauge ?eld. Shifting A ? by ?H?u and v by Hθ we get a model which is equivalent to the ?at space one. The same transformation can be done independently for each plane. The original σ-model (A.21) is thus related to a ?at space one by a combination of duality, coordinate transformation and “inverse” duality. If, however, the true starting point is the “doubled” or “gauged’ model (A.25), then the transformation to the model corresponding to the ?at space is just ? ? a coordinate transformation on the extended con?guration space of (u, v, r, θ, θ, A, A). A.4. Leading-order conformal invariance equations The standard leading-order conformal invariance conditions are ? ? ? R??ν + 2D?? D?ν φ = 0 , (A.26)

take ? to zero. The D = 5 model (A.23) is equivalent to the product of the D = 4 model and a free space-like direction. The D = 6 model (which contains two sets of planar coordinates xa , s = 1, 2) is equivalent to the non-semisimple or boosted version of the s SL(2, R)?k1 × SU (2)k2 WZW model (see eq. (4.16) in [59]). The required coordinate transformation is ψ1 = u, ψ2 = u + ?v, etc. The non-trivial parameter H1 /H2 is equal to the ratio of the levels k1 /k2 . The next non-trivial model is with D = 8. It can be obtained by boosting SL(2, R)?k1 × SU (2)k2 × SU (2)k3 WZW model (ψ1 = u, ψ2 = u + ?v ? ?λ, ψ3 = u + ?λ) with the direction λ decoupling in the limit ? → 0. All higher D models are related to similar WZW models based on direct products of SL(2, R)?k , SU (2)k and R factors, or, equivalently, on corresponding non-semisimple groups. The parameters Hs are essentially equivalent to the rescaled levels kn of the factors. Finally, it is interesting to note that all the models (A.21), like the D = 4 model (A.23), can be related to the ?at space model in the same way as this was shown [60,58] for the D = 4 model of [19]. In fact, let us consider one pair of planar coordinates xa and gauge the rotational symmetry in the plane. We get the following model ? ? ? ? ?? ? ? ? ? L = ?u?v + ?r ?r + r 2 (?θ + A)(?θ + A) + Hr 2 ?u(?θ + A) + A? θ ? A? θ , (A.25)

? ? ?λ ? where R±?ν = R?ν? = R±?λν and D±? are the Ricci tensor and covariant derivative for ? ±?ν the connection Γλ (the symmetric and antisymmetric parts of (A.26) give equations for G?ν and B?ν ). Computing the Ricci tensor from (A.9) one ?nds ? ? ? ? ? R?uv = ?F ? i ?i h , R?ij = ?2?i ?j h , R?iu = R?iv = R?v? = 0 , 1 ? R?uu = ?F ( ? i ?i K + ? i h?i K + K? i ?i h ? ? i ?u Ai ? 2? i h?u Ai ) , 2 ? R?ui = ?F (?j F j i + 2?j hF j i + 2Ai ? j ?j h) . Then (A.26) implies and ?nally we get the same relations as in (5.16),(5.17) 1 1 ? ? 2 F ?1 + bi ?i F ?1 = 0 , ? ?i F ij + bi F ij = 0 , 2 2 1 2 ? ? 2 K + bi ?i K + ? i ?u Ai ? 2bi ?u Ai + 2F ?1 ?u φ = 0 . 2 31 ??i ?j h + ?i ?j φ = 0 , ?i ? u φ = 0 , φ(x, u) = φ(u) + bi xi + h(x) , (A.27)

(A.28)

(A.29) (A.30)

Appendix B. General D = 3 chiral null model As was shown in [1] the generic D = 3 F -model (2.3) is equivalent to a special [SL(2, R) × R]/R gauged WZW model and can also be identi?ed with the extremal limit of the charged black string solution of [61]. Here we shall consider the generic D = 3 model belonging to the class (2.7), ? ? ? ? L3 = F (x)?u ?v + K(x, u)?u + 2A(x, u)?x + ?x?x + α′ Rφ(x, u) . (B.1)

Since the transverse space here is one-dimensional, one can set A = 0 by a transformation of v (see (2.8)). The functions F, K and φ are then subject to (see (2.10)–(2.12))

2 ?x F ?1 = 2b?x F ?1 , 2 2 ?x K = 2b?x K ? 4F ?1 ?u φ ,

1 φ = φ(u) + bx + lnF (x) . 2

(B.2)

Assuming for simplicity that K and φ do not depend on u we get the following solutions F ?1 = a + me2bx , K = a′ + m′ e2bx = c + nF ?1 (x) , (B.3)

so that by shifting v we ?nish with the following conformal D = 3 model ? ? ? L3 = F (x)?u?v + n?u?u + ?x?x + α′ Rφ(x) , F ?1 = a + me2bx , 1 φ(x) = φ0 ? ln(ae?2bx + m) . 2 (B.4) (B.5)

a, n, m are arbitrary constants which take only two non-trivial values: 0 and 1 (–1 case is related to the +1 one by an analytic continuation). In what follows we shall set m = 1. The n = 0 model is the F -model discussed in [1]. In what follows we shall keep n general thus treating both n = 0 and n = 1 cases at the same time. The solution (B.3) with a = 0 has a constant dilaton and thus the corresponding model must be equivalent to the SL(2, R) WZW model (since there are no other φ = const solutions in D = 3 in a properly chosen scheme [1]). In fact, the a = 0 model ? ? ? L3 = e?2bx ?u?v + n?u?u + ?x?x + α′ Rφ0 , (B.6)

is related to the SL(2, R) WZW Lagrangian written in the Gauss decomposition parametrization (we follow the notation of [1] and set α′ = 1) ? ? Lwzw = k e?2r ?u?v + ?r ?r , (B.7)

by the following coordinate transformation (u′ , v ′ stand for the coordinates in (B.7))

√ 1 u′ = √ e2b nu , 2 n

v ′ = bv ?

√

ne2bx ,

√ r = bx + b nu ,

b2 = 1/k .

(B.8)

B.1. Gauged WZW model interpretation Like the n = 0 model, the n = 1 one (B.6) can be related to a special [SL(2, R)×R]/R gauged WZW model. This provides an explicit illustration of our claim that the chiral 32

null backgrounds are exact conformal models.27 The SL(2, R) × R WZW model written ? in the Gauss decomposition parametrization, i.e. (B.7) with an additional R-term ?y ?y, has the following obvious global symmetries: independent shifts of u, v, y and shifts of r combined with rescalings of u and (or) v. Gauging the translational subgroup u→u+? , v →v+? , y → y + ρ? , ρ = const , (B.9)

?xing the gauge y = 0 and integrating out the two dimensional gauge ?eld, one gets the n = 0 model (B.4) with a = ρ?2 [1]. The subgroup which is to be gauged to get the n = 1 model is28 √ √ (B.10) u → e2 n? u , v → v + ? , r → r + n? , y → y + ρ? .

In view of (B.8) this is just the translational symmetry (B.9) (with ? → b?1 ?) of the action ? (B.6) (with ?y ?y added). Since (B.6) is a coordinate transformation of the WZW action (B.7) we can start directly with (B.6) in the gauging procedure, ? ? ? ? ? ? ? Lgwzw = e?2bx (?u+A)(?v + A)+n(?u+A)(?u+ A)+?x?x+(?y +ρA)(?y +ρA) . (B.11) ? Fixing y = 0 as a gauge and integrating out A, A we get Lgwzw = nρ2 ρ2 e?2bx ? ? ?u?v + 2 ?u?u ρ2 + n + e?2bx ρ + n + e?2bx (B.12)

1 ? + ?x?x + α′ R[φ′ + ln(ρ2 + n + e?2bx )] . 0 2 The rede?nition u′ = (1 + nρ?2 )1/2 u , v ′ = (1 + nρ?2 )1/2 (v + ρ2 n u) , +n (B.13)

puts this action into the desired form (B.4),(B.5) with a = (ρ2 + n)?1 . B.2. Extremal black string interpretation The generic D = 3 F -model (i.e. (B.4) with n = 0) can be considered as an extremal limit of the charged black string solution of [61]. Here we point out that a similar statement is true for the n = 1 model (B.4). This is a particular case of the relation between the model (2.21) and the charged black string solution discussed in Section 2.4 (see (2.22)). Starting with the non-extremal charged black string σ-model which has the metric ds2 = ?f1 (r ′ )dt′2 + f2 (r ′ )dy ′2 + h(r ′ )dr ′2 ,

27

(B.14)

It is very likely that there exists a generalization of the nilpotent gauging procedure of ref.

d i=1

[5] which makes it possible to identify not just one D = 3 model but a whole subclass of the chiral null models with F ?1 =

28

eαi ·x (αi are the simple roots of the algebra of a maximally

non-compact Lie group of rank d = D ? 2) with the gauged WZW models. [1]: u → e? u, v → e? v, r → r + ?, y → y + ρ?.

It may be useful to recall that the subgroup that leads to the charged black string of [61] is

33

M1 M2 ) , f2 = (1 ? ′ ) , ′ r r boosting the solution f1 = (1 ?

h = (4r ′2 f1 f2 )?1 ,

M1 = M , M2 =

Q2 , (B.15) M

1 t = λv + ( λ ? λ?1 )u , 2

y = λ?1 u ,

λ≡(

M1 ? 1)1/2 , M2

(B.16)

and then taking the extremal limit M → Q, i.e. M1 → M2 or λ → 0 in the resulting σ-model one ?nishes with the model (B.4) with the metric ds2 = 2(1 ? M )dudv + du2 + h(r ′ )dr ′2 = F (x)dudv + du2 + dx2 . ′ r (B.17)

So the generic u-independent D = 3 chiral null model can be obtained as an extremal limit of a black string solution.

34

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