Generalised chiral null models. and rotating string backgrounds. A.A. Tseytlin 1. Theoretical Physics Group, Blackett Laboratory.

Transcription

1 Imperial/TP/95-96/3 hep-th/ March 996 Generalised chiral null models and rotating string backgrounds A.A. Tseytlin Theoretical Physics Group, Blackett Laboratory Imperial College, London SW7 BZ, U.K. Abstract We consider an extension of a special class of conformal sigma models (`chiral null models') which describe extreme supersymmetric string solutions. The new models contain both `left' and `right' vector couplings and should correspond to non-bps backgrounds. In particular, we discuss a conformal sixdimensional model which is a combination of fundamental string and 5-brane models with the two extra couplings representing rotations in the orthogonal planes. If the two rotation parameters are independent the resulting background is found to be either singular or not asymptotically at. The non asymptotically at solution has a regular short distance limit described by a `twisted' product of SL(;R) and SU() WZW theories with two twist parameters mixing the isometric Euler angles of SU() with a null direction of SL(;R). tseytlin@ic.ac.uk. On leave from Lebedev Institute, Moscow.

2 Introduction Superstring solutions with vanishing R-R background elds are described by conformal -models [, ]. Related solutions with non-vanishing R-R elds can be obtained, e.g., by applying SL(;Z) duality of D= 0 type IIB superstring theory [3]. Knowing a -model description of a given NS-NS string solution is important since many of its features can be (at least, qualitatively) understood using conformal eld theory considerations (see, e.g., [4, 5, 6]). This may complement the picture obtained for the corresponding R-R solution using D-brane approach (see, e.g, [7, 8, 9, 0]). An important class of exact (supersymmetric, BPS saturated) string solutions is described by a special -model { `chiral null model' (CNM) []. Here the equations on background elds obtained by imposing the conformal invariance condition are decoupled Laplace-type equations. The solutions are expressed in terms of harmonic functions, i.e. can be freely superposed (in agreement with the BPS nature of resulting eld congurations). Remarkably, the `pure-electric' CNM with at transverse space [] admits a `dyonic' generalisation [, 5, 6] where the transverse space is represented by a non-trivial N = 4 supersymmetric -model. CNM is characterised by the presence of only certain special chiral couplings in the action. It is of interest to study what happens when one introduces extra integrable marginal perturbations getting as a result a mixture of `left' and `right' couplings (which, in general, may break supersymmetry). The corresponding conformal invariance conditions will no longer factorize into a simple set of Laplace equations, i.e. one will no longer have a simple BPS-type superposition principle. These models may describe certain `non-bps' solutions which may still have some special properties (e.g. may still be exact to all orders). An example of such model will be discussed below. It presumably still corresponds to extremal eld congurations but with less (than in standard CNM case) or no supersymmetry. Our investigation of such models was partly motivated by an attempt to construct `rotational' generalisations of D = 6 CNM's describing D = 4 [, 5] and D =5[6] extreme dyonic black holes (thus providing exact string-theory generalisations of the solutions of leading-order eective equations in [3] and [7]). These D = 6 conformal models have a short-distance (horizon or `throat' [4, 5]) r! 0 region described by (a factor of) SL(;R)SU() WZW theory. The latter is remarkable in that in the supersymmetric case it has the free value c ef f = 6 of the central charge and thus the dilaton is constant at the `throat'. Adding one rotational parameter to a D = 5 solution parametrised by 3 independent charges (two electric and one `magnetic') can be understood [6] as adding a perturbation (@u J 3 ) which `mixes' the Cartan J 3 curent ofsu() with an isometric (Gauss decomposition) coordinate u of SL(;R). It can be induced by shifting one of the isometric Euler angles of SU(),! + q u. The `integrated' CNM which extrapolates such perturbed throat region model to nite r describes the 4-parameter Closely related `magnetic' models were discussed in [6, 7, 8].

3 (3 charges and two equal angular momenta in two orthogonal planes) extreme D = 5 black hole solution [6] which generalises the solution of the leading-order eective string equations originally found in [0] (where the 3 non-trivial charge parameters were all equal). A natural idea is to generalise the throat region model further shifting also the second Euler angle of SU(), '! ' + q u,thus inducing the 3 which should correspond to the second independent rotation parameter in the resulting D = 5 background. For conformal invariance one needs also to add non-trivial O(q term. Similar string model was considered in [8] in the context of continuous supersymmetry breaking by `magnetic' backgrounds. The -model which extends such(q ;q )-perturbed SL(;R)SU() WZW model to nite r is a simple generalisation of CNM discussed below. We shall nd that for q 6= 0 the conformal invariance conditions do not have solutions which describe asymptotically at at r! and non-singular at r! 0 backgrounds. This implies that there are no non-singular extremal D = 5 black holes with two independent rotational parameters (in agreement with similar remarks in [0]). The solution that reduces to the regular (q ;q )-perturbed throat region model is not asymptotically at and describes a rotating magnetic universe. Analogous solutions were discussed in [, 9]. 3 Given that the background with q = q =0 has a simple `fundamental string + 5-brane' interpretation it may beofinterest to study the D-brane description of the direct analog of this q q 6= 0 solution which has non-vanishing R-R elds. The D-brane picture in [0] based on two U() currents of SU()SU() KM algebra of underlying d conformal model has a striking similarity with the above description of the throat region model with the two (`left' and `right') Cartan current perturbations. Review The `standard' CNM with curved transverse part is dened by the Lagrangian [] 4 h L = F + +Ai (x)@x ii + Rln F (x)+l? ; () L? =(G ij + B ij )(x)@x j + R(x) : () There exists a renormalisation scheme in which () is conformal to all orders in 0 provided the `transverse' -model () is conformal and the functions F ;K;A i ; satisfy certain conformal invariance conditions. The simplest tractable case is when 3 There is also a similarity to Kaluza-Klein Melvin model [0, 6] which can be obtained by shifting an angular coordinate by a Kaluza-Klein one. 4 We shall ignore possible u-dependence of couplings. When both K and A i depend on u one can redene v to absorb K into A i [].

4 the transverse theory is either at or has at least (4; 0) extended world-sheet supersymmetry so that the conformal invariance conditions essentially preserve their -loop form, i.e. are the `Laplace' equations in the `transverse' background [, i (e p GG j F i (e p GG j K)=0; (3) ^r +i (e F ij )=0; i (e p GF ij ) e p GH kij F ki =0; (4) where ^i jk = i jk + Hi jk ; F i A j A i ;H ijk =3@ [i B jk] : For example, in the case when the transverse space is described by the `5-brane' model [4] (i; j; ::: =;;3;4) L? = f(x)@x i + B ij (x)@x j + R(x) ; (5) G ij = f(x) ij ; H ijk = pg p ; = ln i f=0; the equations for F; K; f become the free 4-d Laplace i F =0;@ i K= 0;while the equation for A i can be re-written as follows i (e p GF ij + )=0; i (f F ij + )=0; (6) F ij + F ij + F ij ; F ij = p G ijkl F kl : This equation can be solved, e.g., by imposing F ij + = 0 which is again a linear atspace equation (the scale factor of conformally at G ij drops out) and should describe supersymmetric BPS-type backgrounds. A particular solution is the 4-parameter rotating dyonic D = 5 black hole [6] which generalises the solution of [0]. In general the above equation for A i depends on the harmonic function f of the transverse theory and does not reduce to a at-space Maxwell equation. This is illustrated by a related example with an extra `left' electric charge added to a dyonic model with two electric and two magnetic functions which was discussed in [5]. The resulting equation for the new electric charge function A(x) (a component ofa i )was not a free at-space Laplace equation. Generically there is an interaction between electric (longitudinal or time-like) and magnetic (transverse or space-like) parts of the model which cannot be switched o. 5 In addition to the `longitudinal'{`transverse' coupling one can also introduce an interaction between `left-moving' and `right-moving' coupling functions. The simplest example is provided by the `plane-wave' model [] + +A i i + A i (x)@x + L? ; (7) 5 This raises the following question: given a type II or heterotic theory compactied to D =4,are there `dyonic' solutions parametrised by more than two general (not necessarily one-center) harmonic functions of 3 spatial coordinates? 3

5 or its u-dual [] ( ~ F = K ; ~A i = A i, v 0 = v +~u) L= ~ F(x)[@~u ~ A i (x)@x i 0 +A j j ]+ Rln ~ F (x)+l? : (8) which are conformal i [e p GF ij (A)] e p GH kij F ki (A) =0; i [e p GF ij ( A)] + e p GH kij F ki ( A) i (e p GG j K) e p GF ij (A)F ij ( A) =0: The equation for K (or F ~ ) is no longer a free Laplace equation, i.e. there is no simple superposition principle. In contrast to the standard plane-wave case [] one may also expect higher 0 -corrections (unless there exists a special `exact' scheme). The special cases in which the simplicity is restored correspond to taking the two vector eld strengths to be constant orchoosing an anti-self-dual F ij (A) and selfdual F ij ( A) so that F ij (A)F ij ( A) = 0, i.e. the equation for K becomes again the homogeneous Laplace one. 3 Generalised CNM Below we shall consider the following generalisation of the above models (),(7) L = F i i + A i (x)@x Rln F (x)+l? : (0) The u-dual of this model is a generalisation of (8) ~L = ~ F(x)[@~u ~A i (x)@x i + ~ +Aj j ]+ Rln ~ F(x)+L? ; () ~F = K ; ~ K = F ; ~ Ai = F A i : () In contrast to the original CNM () the generalised model (0) is no longer self-dual. This is also an indication of its `non-bps' nature. The -loop conditions of conformal invariance of (0) can be derived from the standard equation ^R +^r ^r = 0 for a general -model (x =(u; v; x i )). A more illuminating way of obtaining them is to add source terms and to integrate out u and v as in []. 6 As a result, we get L 0 = @V (3) +A +F A : 6 The fact that the action is still linear in v so that the null coordinates u and v can be integrated out strongly indicates that there still exist a scheme in which the -loop conformal invariance conditions are exact to all orders (provided this is also true for the transverse part of the model). 4

6 The O(ln F ) dilaton term cancels out. Since U and V are external elds it is then straightforward to study conditions of conformal invariance with respect to the transverse coordinates only. One nds the following set of equations which generalises i ( p Ge G j F )=0; i ( p Ge G j K) p Ge F ij ( ~A)[F ij (C) F F ij ( ~A)] = 0 ; i [ p Ge F ij ( A)] ~ p + Ge H kij F ki ( A)=0; ~ i f p Ge [F ij (C) F F ij ( A)]g ~ p Ge H kij F ki (C) =0: (7) Here the gauge-invariant eld strengths correspond to the following potentials 7 ~A i F A i ; C i A i + A i : (8) 3. Flat transverse space: charged fundamental string Let us rst discuss an example of generalised CNM (0) with at transverse space but A i 6= 0 which corresponds to a generalisation of the charged D = 5 fundamental string solution [3]. Splitting the coordinates x i into N = 5 compact toroidal ones y n and 3 non-compact spatial ones x s wemay set A = A n (x)dy n ; A = A n (x)dy n ; ~A n = F A n ; C n = A n + A n ; (9) and assume that all the functions will depend only on x s. Then the equations (4){ (7) s F s K 4@ s ~A n (@ s C n s ~A n )=0; s (@ s C n s ~A n s ~A n =0: () Though for A n 6= 0 these equations no longer reduce to free Laplace equations, their general solution can still be expressed in terms of N + = independent harmonic functions. 8 Choosing all harmonic functions to be -center ones we nd F =+Qr ; ~ An = p n r ; C n =(p n +q n )r +p n Qr ; () A n = q n r ; An = p n r + p n Qr ; K =+ ~ Qr +q n p n r + 3 p np n Qr 3 ; r = x s x s : (3) 7 As it is clear from () the dual action is invariant under ~u! ~u + h(x); v! v + s(x); A m!a m s ~ K@m h; ~ Am! ~ A m m h; so that the gauge-invariant vector eld strengths are d ~ A and da + df ^ ~A = dc F d ~ A or just d ~ A and dc. 8 In the heterotic string case one may introduce extra 6 `right' vector couplings leading to extra 6 harmonic functions. 5

7 When both q n and p n are non-vanishing the o-diagonal components of the D =0 metric and the antisymmetric tensor are dierent, i.e. one nds the fundamental string background with both `left' and `right' charges. Separating the `internal' y n - part ((@y n + FC + :::) the rest of the action becomes the neutral fundamental string CNM with redened function K! K 0 = K FCn. The reduction to D =4 then leads to a family of electrically charged black holes parametrised by the charges Q; Q;p ~ n ;q n ( in type II or +6 in the heterotic string case). The corresponding D = 4 Einstein-frame metric is given by ds E = (r)dt + (r)(dr + r d) ; (4) = F K C n =+(Q+ ~ Q)r +(Q ~ Q q n p n)r 4 3 p n Qr 3 3 p Q r 4 : For p n 6= 0 the background is more singular than in the usual p n = 0 case. This seems to be a general pattern: solutions with non-vanishing A i coupling have stronger shortdistance singularities (see also below) brane model as transverse theory In what follows we shall consider an example with a curved transverse theory represented by the 5-brane model (5). Since here p Ge H kij = l e one can put the equations (4){(7) into the form (repeated indices are now contracted using at space F K f F ij ( ~ A)[F ij (C) F F ij ( ~ A)] = 0 ; i (f [F ij ( ~A) F ij ( ~A)]) = 0 ; i (f [F ij (C)+F ij(c) F F ij ( ~ A)]) = 0 : (7) Note the last two equations have the following special solution F ij ( ~ A)=F ij( ~ A) ; F ij ( ~ A)= F[F ij(c)+f ij(c)] ; (8) for which the equation for K becomes again the free Laplace K = f F ij ( ~ A)[F ij (C) F ij(c)] = 0 : (9) For F = this solution reduces to the case of self-dual A i and anti-self-dual A i mentioned at the end of Section. 4 D =6conformal model with `rotations' in transverse planes Let us now solve the equations (5){(7) using the SO(4) symmetric choice for the harmonic functions f and F f =+Pr ; F =+Qr ; r = x i x i ; (30) 6

8 and the following ansatz for A i ; A i with rotational symmetry in the two orthogonal planes A i dx i = h (r)sin d'+h (r)cos d ; Ai dx i = h (r)sin d'+ h (r)cos d : (3) Here the four spatial transverse coordinates x i are chosen as x + ix = rsine i' ; x 3 + ix 4 = rcose i ; so that the 5-brane model Lagrangian L? in (5) takes the form L? = + r )] (3) + )+ Rln f(r) : It then follows from (5) that in general K should depend also on K(r;)=K (r)+k (r)cos : (33) For the gauge potential of the form in (3) one nds the following non-vanishing components of the eld strengths (prime denotes the derivative over r and F ij (A) F ij (A) F ij (A)) F r' = r 3 (rh 0 h ); F r = r 3 (rh 0 h ); (34) F ' = r 4 cot(rh 0 +h ); F = r 4 tan(rh 0 +h ): Let us rst consider the special case of Ai = 0 discussed in [6]. Then the general solution of the equation (7) for C i (i.e. for A i ) is found to be (k a ;n a = const) h + h + h = (k + k )r + k P + (n + n )r + 3 n Pr 4 ; (35) h + h + h = (k k )r + k P + (n n )r 3 n Pr 4 : The resulting -model (0) is regular at r! 0 only if there is no r 4 pole, i.e. if n = 0. Moreover, the background is asymptotically at (r!)ifk =k =0. The resulting special solution A i dx i = n r (sin d' + cos d ) ; A i =0; (36) describes [6] the D = 5 rotating black hole [0] with equal angular momenta in the two orthogonal planes J ' = J = J = n =4G N. The general solution of the equation (6) for ~ A i is similar to (35) ~h F h = (q + q )r + q P + (p + p )r + 3 p Pr 4 ; (37) ~h F h = (q q )r q P + (p p )r + 3 p Pr 4 : The self-dual solution (8) corresponds to the case of p =0; q = 0. It is easy to see that (independently of the form of the associated solutions for A i and K) the 7

9 -model (0) has singular short distance (r! 0) region unless p = p =0. The resulting background is asymptotically at only if q = q =0. We conclude that there are no regular asymptotically at solutions with A i 6=0. In particular, there does not exist a non-singular `extremal' generalisation of the special solution (36) to the case of two independent rotational parameters. There is still a non asymptotically at solution with a regular r! 0 region which is a natural -parameter generalisation of the horizon region [6] of the solution (36). Relaxing the condition of asymptotic atness, i.e. choosing ~A i dx i =[ (q +q )r +q P]sin d' +[ (q q )r q P]cos d ; (38) we nd that (7) has the following general solution which is regular at r = 0 (cf.(35)) h + = (k + k )r + k P + n r + ff (q + q )r ; (39) h + = (k k )r + k P + n r + ff (q q )r : Then A i is given by (3) with h = (k0 + k 0 )r +(k 0 P +q Q+q Q)+ n0 r ; (40) h = (k0 k 0 )r +(k 0 P +q Q q Q)+ n0 r ; k 0 =k +q ; k 0 =k +q ; n 0 =n +QP q : The function K (33) can be determined from (5), i.e. from r (r 3 K 0 ) 0 =f [r ~ h 0 h0 + + r ~ h 0 h ~ h h + +8 ~ h h + (4) F (r ~ h 0 ~ h 0 + r ~ h 0 ~ h 0 +8 ~ h ~ h +8 ~ h ~ h )] ; (r 3 K 0 ) 0 8rK = r 4 [r 5 (r 4 K ) 0 ] 0 = rf ( ~ h 0 h0 + ~h 0 h0 +) : (4) A particular solution corresponding to the special case of k = k = n =0is K=c 0 + ~ Qr + F fr (q + q q q cos) : (43) 5 Short-distance region: relation to SL(;R)SU() WZW theory The r! 0 limit of the D = 6 conformal model discussed in the previous section is described by the following Lagrangian L = +[Q ~ Q P(q +q q q + )+Pq ) 8

10 + P @ )] ; z ln(q r ) : The total dilaton = ln(ff) is constant inther!0 limit. This model is related to the SL(;R)SU() WZW model 9 L = + Q ~ + 4 (45) + P )] ; by the formal coordinate shift 0 '! ' +(q +q )u;! +(q q )u: (46) Thus the throat region model (44) is locally a direct product (but is globally nontrivial for generic values of q ;q ). It has broken supersymmetry [8] unless the twists q ;q take quantised values (for which the resulting string model becomes equivalent to the model with q = q = 0). Note that the positivity of the coecient of@u@u term in both (44) and (45) implies a bound on jq q j: (q q ) <Q P Q. ~ Though this model has two natural `twist' parameters, we have found above that for q 6= 0 it does not have an extension to nite r which is asymptotically at. It may beofinterest to study related non asymptotically at models from the point of view of their possible `magnetic' interpretation, given that the corresponding backgrounds can be transformed into solutions with non-vanishing R-R elds which have straightforward D-brane interpretation. Non-extremal 5-dimensional black hole solutions with two rotational parameters were recently constructed in [4, 5]. Acknowledgments I am grateful to M. Cvetic and J. Maldacena for useful discussions. This work was supported by PPARC, ECC grant SC*-CT and NATO grant CGR The standard SL(;R) WZW Lagrangian written in the Gauss decomposition parametrisation can be put in the `non-standard' form which appears here by the following eld redenition [] e = u 0 = eu ; v 0 = v e u ; x 0 = x + u: The levels of SU() and SL(;R) WZW models are both equal to P= 0. 0 The standard SU() Euler angles ( 0 ;' 0 ; 0 ) are related to the angular coordinates used above by = 0 ; ' = ('0 + 0 ); = ( 0 ' 0 ); 0 0 ; 0 ' 0 ; 0 0 4: 9

11 References [] C. Callan, D. Friedan, E. Martinec and M. Perry, Nucl. Phys. B6 (985) 593. [] A.A. Tseytlin, Class. Quant. Grav. (995) 365, hep-th/ [3] C.M. Hull and P.K. Townsend, Nucl. Phys. B438 (995) 09; J.H. Schwarz, Phys. Lett. B360 (995) 3; E. Bergshoe, C.M. Hull and T. Ortin, Nucl. Phys. B45 (995) 547, hep-th/ [4] F. Larsen and F. Wilczek, PUPT-576, hep-th/ [5] M. Cvetic and A.A. Tseytlin, IASSNS-HEP-95-0, hep-th/9503. [6] A.A. Tseytlin, Imperial/TP/95-96/, hep-th/96077 (revised), Mod. Phys. Lett. (to appear). [7] A. Strominger and C. Vafa, HUTP-96-A00, hep-th/ [8] C.G. Callan and J.M. Maldacena, PUPT-59, hep-th/ [9] G. Horowitz and A. Strominger, hep-th/ [0] J.C. Breckenridge, R.C. Myers, A.W. Peet and C. Vafa, HUTP-96-A005, hepth/ [] G.T. Horowitz and A.A. Tseytlin, Phys. Rev. D5 (995) 896, hep-th/94090; Phys. Rev. D50 (995) 504, hep-th/ [] M. Cvetic and A.A. Tseytlin, Phys. Lett. B366 (996) 95, hep-th/ [3] M. Cvetic and D. Youm, Phys. Rev. D53 (996) 584, hep-th/ ; UPR T, hep-th/ ; Phys. Lett. B359 (995) 87, hep-th/ [4] C.G. Callan, J.A. Harvey and A. Strominger, Nucl. Phys. B359 (99) 6; in Proceedings of the 99 Trieste Spring School on String Theory and Quantum Gravity, J.A. Harvey et al., eds. (World Scientic, Singapore 99). [5] D.A. Lowe and A. Strominger, Phys. Rev. Lett. 73 (994) 468, hep-th/ [6] J.G. Russo and A.A. Tseytlin, CERN-TH/95-5, hep-th/ [7] E. Kiritsis and C. Kounnas, Nucl. Phys. B456 (995) 699, hep-th/ [8] A.A. Tseytlin, Imperial/TP/94-95/6, hep-th/ [9] J.G. Russo and A.A. Tseytlin, Nucl. Phys. B448 (995) 93, hep-th/

12 [0] F. Dowker, J.P. Gauntlett, G.W. Gibbons and G.T. Horowitz, hep-th/950743; F. Dowker, J.P. Gauntlett, S.B. Giddings and G.T. Horowitz, Phys. Rev. D50 (994) 66. [] A.A. Tseytlin, Nucl. Phys. B390 (993) 53. [] D. Amati and C. Klimcik, Phys. Lett. B9 (989) 443; G. Horowitz and A.R. Steif, Phys. Rev. Lett. 64 (990) 60; Phys. Rev. D4 (990) 950; G. Horowitz, in: Proceedings of Strings '90, College Station, Texas, March 990 (World Scientic,99). [3] A. Sen, Nucl. Phys. B388 (99) 457; D. Waldram, Phys. Rev. D47 (993) 58. [4] J.C. Breckenridge, D.A. Lowe, R.C. Myers, A.W. Peet, A. Strominger and C. Vafa, hep-th/ [5] M. Cvetic and D. Youm, to appear.