PHYSICAL REVIEW D VOLUME 51, NUMBER 6 15 MARCH 1995

Duality versus and compactification

Eric BergshoeF" Institute for , University of Groningen, Nijenborgh g, g7$7AG Groningen, The Netherlands

Renata Kallosht Department of Physics, , Stanford, California 94905

Tomas Ortin~ Department of Physics, Queen Mary and Westfield College, Mile End Road, London El gNS, United Kingdom (Received 4 November 1994)

We study the interplay between T duality, compactification, and supersymmetry. We prove that when the original configuration has unbroken space-time , the dual configuration also does if a special condition is met: the Killing spinors of the original configuration have to be independent of the coordinate which corresponds to the isometry direction of the bosonic fields used for duality. Examples of "losers" (T duals are not supersymmetric) and "winners" (T duals are supersymmetric) are given. PACS number(s): 11.30.Pb, 04.65.+e, 04.70.Dy, 04.30.—w

I. INTRODUCTION condition that guarantees the preservation of unbroken supersymmetries that it is not satisfied by these coun- Target-space duality (T duality) is a powerful tool for terexamples. We will perform this analysis in the con- generating new classical solutions of theory. It can text of N = 1, d = 10 without vector Gelds. 0. be used in the model context to generate new exact More general results involving Abelian and non-Abelian solutions but also in the context of the leading order in vector fields and order o' corrections will be re- o' higher effective action to generate new solutions to the low- ported elsewhere [8]. Some of the results presented in energy equations of motion. Some of these solutions have this paper were announced in [9]. unbroken supersymmetries. The purpose of this paper is The first counterexample known to us appears in a to study the generic relation between the supersymmetric very simple case. We have found some time ago that properties of the original configuration and the dual one if one starts with ten-dimensional Hat space (which has in the context of the low-energy efI'ective action. all supersymmetries unbroken) in polar coordinates and It has been observed that in some cases T duality pre- performs a T-duality transformation with respect to the serves unbroken supersymmetry. Well-known examples angular coordinate p, the resulting conGguration has no are the supersymmetric string wave solutions (SSW's) unbroken supersymmetries whatsoever. [1] and their partners, dual waves [2], which in partic- The explanation of this apparently inconsistent sit- ular include fundamental string solutions [3]. Another uation will be found in a Kaluza-Klein-type analysis example of T-dual partners with unbroken supersymme- of the fermionic supersymmetry transformation rules of tries is given by a special class of fivebrane solutions [4], N = 1, d = 10 supergravity. In the conventional di- multimonopoles [5] and their duals, a special class of mensional reduction of this theory by compactiGcation stringy asymptotically locally Euclidean (ALE) instan- of one dimension (with coordinate x, say) one only con- tons [6] which has a multicenter metric. siders those Geld conGgurations that do not depend on The preservation of unbroken supersymmetries by du- the compact coordinate, and one only considers those su- ality is related in principle to the fact that T duality is persymmetry transformations generated by parameters e just one of the hidden symmetries of the supergravity that are independent of x as well, projecting the rest theory that arises after dimensional reduction [7]. These out of the resulting N = 1, d = 9 theory which is the hidden symmetries are indeed consistent with the super- case n = 1 of Ref. [10]. If the Killing spinor of the symmetry of the dimensionally reduced theory. However, ten-dimensional configuration depends on x, the config- some recently discovered counterexamples seem to con- uration will not be supersymmetric in nine dimensions. tradict this preliminary understanding. Therefore, one The efI'ect of the nine-dimensional hidden symmetries in of the main goals of our analysis is to Gnd the general the ten-dimensional supersymmetry is unknown, while

*Electronic address: bergshoeOth. rug. nl t Electronic address: kalloshphysics. stanford. edu It was suggested to us by Tseytlin to check whether super- ~ Electronic address: ortinqmchep. . ch symmetry is preserved by duality in this case.

0556-2821/95/51(6)/3009(8)/$06. 00 51 3009 1995 The American Physical Society 3010 ERIC BERGSHOEFF, RENATA KALLOSH, AND TOMAS ORTIN 51 in the nine-dimensional theory is just an O(l, 1) group where the fields are the metric, the axion, and the completely consistent with supersymmetry [10]. This is (g~„-, B„-„-,P) and our conventions are those of Ref. [2]. exactly what happens in the counterexample above: the In particular the axion field strength H is given by Killing spinor depends on y when a p-independent frame is used. Hp, vp —[pBv p] exam- Recently Bakas [11]has found a more interesting All the D-dimensional entities carry a caret and the (D— ple of the loss of unbroken supersymmetries after a series 1)-dimensional ones do not. Then the indices take the of T and S-duality transformations, T duality being the values responsible of this loss. In his scheme supersymmetry is lost if the Killing vector with respect to which one du- p, = (0, . . . , D —2, D —1) = (p, D —1) . alizes has not self-dual covariant derivatives. We believe that his example also satisfies our criterion: if one calcu- We call the coordinate x = x. To distinguish be- lated explicitly the Killing spinors of such configurations tween curved and fiat indices when confusion may arise, in an x-independent frame, they would depend on the we underline the curved ones ((—,for instance). Now we isometry direction x. We hope these different criteria assume that the fields are independent of the coordinate can be shown to be equivalent for these configurations. x, i.e., there exists a Killing vector k" such that This work is structured as follows. In Sec. II we set k"8„- = t9 up the general problem of dimensionally reducing one ab- dimension in the low-energy string effective action in Then, in this coordinate system, the components of the sence of gauge fields, Inainly for fixing the conventions Killing vector are and notation. We describe the effect of T duality on the compactified dimension from the point of view of k„- =gp k = k"kp —g the lower-dimensional theory. In Sec. III we study the effect of T duality on the supersymmetry properties of and the metric can be rewritten as purely bosonic configurations of the zero-slope limit of ds = k (kpdx") k )dx"dx". heterotic in ten dimensions. Accordingly + (g„„— k„k (6) we investigate the behavior under T duality of the su- The Killing vector can be either timelike or spacelike, persymmetry rules of pure N = 1, d = 10 supergravity. but not null. We will keep our expressions valid for Using a Kaluza-Klein basis of zehnbeins we rewrite the both cases because from the point of view of T duality ten-dimensional supersymmetry transformation rules in both are equally interesting [12] and the compactifica- a manifestly T-duality-invariant form for configurations tion of a timelike coordinate is not usually considered which have unbroken supersymmetries with the Killing in the literature because it gives rise to an inconsistent spinor independent on the isometry direction. In Sec. IV lower-dimensional theory. We consider here the lower- we present examples of configurations with (broken) un- dimensional theory just as a tool. broken supersymmetry after duality in accordance with The above action enjoys invariance under the following (dependence) independence of the Killing spinor on isom- Buscher's [13] T-duality transformations etry direction. Section V contains our conclusions. Fi- nally, the Appendix contains some additional results of — g = 1/g~~, B~~ —g~~/g~~, g~„= B~„/g~~ ) our work: we dimensionally reduce N = 1, d = 10 su- to d = 4 and study the truncation of the pergravity B„=B„„+(g „B —g „B„)/g lower-dimensional theory consisting in setting to zero all the fields which are matter from the point of view of 9@v (9x~gxv BxpBxv)/gee 1 N = 4, d = 4 supergravity. The remaining fields are found to be duality invariant. Therefore, when a su- — persymmetric compactification is done and the resulting P = gb — 1n/g theory is truncated to pure supergravity, T duality in the compactified directions has no effect whatsoever on the Checking directly the invariance of the action Eq. (1) theory. under the above transformations is a very involved cal- culation but if we compactify the redundant coordinate II. PROM D TO D —1 DIMENSIONS x, checking duality will be very easy. Now we are going to dimensionally reduce the above The D-dimensional action we start from is action to D —1 dimensions by compactifying the redun- dant dimension x. We use the standard techniques of — S = d xQ ge ~[ R+ 4(cjoy—) —sH—], (1) Scherk and Schwarz [14]. First we parametrize the D 2 bein as follows

P 0 kJ' okl where To apply our criterion one has to find the Killing spinors explicitly. k = /krak" [~, DUALITY VERSUS SUPERSYMMETRY AND COMPACTIFICATION 3011 and A = e "A„. The functions e„- do not depend on x. transformation rules derived in Ref. [2] Note that krak" = g k . With our conventions (mostly Q 1-- minuses signature) rI is positive if x is a timelike coor- e e g~x dinate and k a timelike vector, and g is negative if x and k" are both spacelike. A Q P Q 1 e~ = +e~ (g „kB„)e With this parametrization, the D-dimensional fields g~~ decompose as follows: and it is this property of the Kaluza-Klein basis, Eq. k,2 B„=B„, (8), which simplifies the transformation rules is justifies its use here. k A„, B„=B„+A]I„B„), Of course, the transformation one sees in the lower- — dimensional theory is part of the O(d, d) symmetry ex- gp, v g p+q k AA„, P=P+ ink, 2 hibited in Ref. [7] when one compactifies d dimensions. Now we have made this relation very explicit and it is go- where gp„, A k, are the (D —1)-dimen- ( B„, pB„, g) to be extremely useful for the study of the unbroken sional fields. They are given in terms of the D- ing supersymmetries of the dual conGgurations. dimensional Gelds by = J gpu gp, v gee, gxv/gxx ) B„=B„, III. DUALITY VERSUS SUPERSYMMETRY'

fh B„„=B„„+g[„B]/g In this section we investigate the general relation be- tween unbroken supersymmetries before and after a T- duality transformation using the results of the previous Then the D-dimensional action Eq. (1) is identically section with D = 10. Specifically we are going to analyze equal to the effect of a T-duality transformation on N = 1, d = 10 supergravity Killing spinors. To do this one needs to 1 know how the zehnbeins transform under duality. As we S = — d[ ']xgq ge ~[ R+ 4—(8$) — H- 2 explained in the previous section, the zehnbein duality transformation laws were found in Ref. and reduce —(c[lnk) —g 4k F (A) —g 4k F (B)], [2] to Eqs. for the 2:-independent Kaluza-Klein basis of (12) (14) zehnbeins, Eq. (8), where a clear distinction between the where cases in which unbroken supersymmetry is preserved and those in which it is not arises naturally. F„„(A)= 28[„A ], F„„(B)= 28[„B], We consider here the zero slope limit of the heterotic 1 1 string theory without gauge fields, which is given by N = Hpvp = ~[pBvp] + A[pFvp](B) + B[pFvp](A) l 2 2 1, d = 10 supergravity. The bosonic part of the action of % = 1, d = 10 supergravity in absence of vector fields tensor field are the vectors and antisymmetric strengths. is given by Eq. (1) with D = 10: Equation (12) can be interpreted as a (D —1)- dimensional action for the above (D —1)-dimensional S = — d"xQ ge '~[ R—+ 4(0$)2——4sH2], fields. Observe that, when x is a timelike coordinate, 2 the vector fields kinetic terms have the wrong signs in the above action. with H given by Eq. (2). The corresponding fermionic Now, using first the definitions of the (D —1)- supersymmetry transformation rules are dimensional fields in terms of the D-dimensional ones, Eqs. (11),and Buscher's duality rules, Eqs. (7), it is very easy to check that the duals of the (D —1)-dimensional Gelds are Now we assume that some spinor e makes these equa- g„=g„, A„= B„, tions vanish (i.e., e is a Killing spinor ) for some spe- B„=B„„, B„=A„, (14) cific x-independent Geld configuration and we want to investigate whether this e is also a Killing spinor of the k=k T-dual field configuration or whether it is related to an- other spinor of the dual Geld as that is, in the (D —1)-dimensional theory the only effect Killing conGguration, of T duality is to interchange the vector fields A„and in the S-duality case [15]. To investigate this problem B~ and to invert k. This is an obvious symmetry of the (D —1)-dimensional action, Eq. (12), which, on the other hand is identically equal to the D-dimensional one, Eq. (1). Prom the lower-dimensional point of view, the invariance of the action under T duality is manifest. Actually a spinor that makes Hqs. (17) vanish needs to have Observe that, in particular, the (D —1)-beins are du- a speci6c asymptotic behavior in order to be a Killing spinor, ality invariant. This is completely consistent with the but these details will not concern us in this discussion. 3012 ERIC BERGSHOEFF, RENATA KALLOSH, AND TOMAS ORTIN 51 we rewrite the above equations in terms of the nine- are the nine-dimensional ones) dimensional fields. It is perhaps worth stressing that we are not reducing the ten-dimensional matrices nor the p Pa = 0 +aha ='7 Habc ~ ten-dimensional spinors, which are the objects we are in- 0 terested in. Again, here, dimensional reduction can be (18) understood as a tool for having under control the duality transformations. All the indices used below are Bat. We We get for the x component (flat) and the a compo- also use the slightly unusual notation (observe that the p nent of the gravitino transformation and for the dilatino matrices are ten-dimensional but the indices contracted transformation

' = (k 'ct —I'[r) k P'(A) + k P'(B)] + 2j (Pink))e = 0, — ' — — — = ([8 4(~ 2H ')jb, ] —1'j [f' (A) q f' (B)] A 0 )e = 0, = — &s& (PQ+ 4 P 4r) k 'j P'(B) + 2 Pink)e = 0, (19) respectively. As they stand, none of these equations is separately manifestly duality invariant. Unless we assume in what follows that the Killing spinor of the original configuration does not depend on the isometry direction x, no further progress can be made in relating the supersymmetry of the original configurations with that of the Anal one. Thus we require that 0 a=0. (20) Using this assumption we have the Killing spinor equations in the form (I~[r) k P'(A) + k P'(B)] —zj (Pink))e = 0,

([o~ 4 (~(7, 2 H ) ybg] s~p [yF (A) g P (B)])&— (PP+ —P —'rl k j -P'(B) + — Pink)e = 0.

Still, after assuming 8 e = 0, not all of the Killing not straightforward. It is clear, however, that the corre- spinor equations are manifestly separately duality invari- spondence disappears if we do not impose Eq. (20) to the ant. To be precise (and here we take q = —1) using supersymmetry parameters. the nine-dimensional version of Buscher s duality rules, We would like to stress that we have derived the condi- Eq. (14), the first and the second are duality invariant tion of preservation of unbroken supersymmetry Eq. (20) but the third is clearly not. However, since by assump- using heavily a zehnbein basis of the form Eq. (8). How- tion all of them are satisfied by e, we are allowed to com- ever, after deriving this condition in that special frame we bine them. If we substitute the erst into the third, we may ask ourselves to which extent this condition is frame get the duality-invariant equation dependent. The answer is that the same criterion is valid ' ' in any x-independent frame. Indeed, if one changed from (PP+ —, P + —,'~. [k P'(A) + k P'(B)])e = 0. (22) the x-independent Kaluza-Klein frame discussed above This proves that i is a Killing spinor of the dual con- to any other x-independent frame, the Lorentz rotation figuration if it is so for the original configuration, that involved would not change the fact that the spinor is or is is not x dependent since the same x-independent param- eter u appears in the spinors and frames transforma- (23) A tion laws e' = exp (4cu p b)e and e„- = exp ( wM b)e—„ If we take g = +1 (timelike duality) it is easy to see where the M bs are the generators of the ten-dimensional that Lorentz group in the vector representation. P, A A (24) IV. EXAMPLES Examples of these results will be discussed in Sec. III. The set of T-duality-invariant supersymmetry equa- In this section we are going to study examples of su- tions that we have generated by dimensional reduction persymmetric con6.gurations and duality transformations should be nothing but the explicitly O(1,1)-invariant which illustrate the results of Sec. III. N = 1, d = 9 supergravity theory of Ref. [10] for the case (1) Losers: configurations that lose their unbroken su- n = 1 and in stringy frame, although the dimensional re- persymmetries after T duality. duction of the supersymmetry parameters, p matrices, (i) Our first example is flat ten-dimensional space-time e4' etc. , still has to be done. There are factors of relating in polar coordinates pz = (xi)2 + (x2)2, tang = xz/xi: the Einstein-frame and string-frame spinors too and the comparison between our results and those or Ref. [10] is ds = dt —dp —p dp —dx dx, I = 3, . . . , 9. (25) DUALITY VERSUS SUPERSYMMETRY AND COMPACTIFICATION 3013

This solution of % = 1, d = 10 supergravity has all shows that all the cases found in Ref. [11] to violate su- supersymmetries unbroken. In the zehnbein basis persymmetry su8'er from the problem of dependence of the Killing spinor on the coordinate associated with the 0 — 1 2 J — J eq —1, e~ =1, e~ = p, eI —~I isometry. Observe that one of his examples with V = 1 and z. is case above. which is of the type of that in Eq. (8), the Killing spinors p;~. g h, provided by (i) Winners: configurations with unbroken supersym- are given by (2) metries that are preserved by T duality. Alternatively we 1 4 91&2 could refer to them as those configurations with unbro- 0 ) (27) ken supersymmetries and x-independent Killing spinors where e0 is a completely arbitrary constant spinor. since the results of Sec. III guarantee, without the need After a duality transformation in the direction p, we of further proof, the supersymmetry of the dual configu- get the solution rations. The first example is provided the SSW solutions ds' = dt' —dp' — 'dp' ——dx'dx', = — (i) by p y ln p. and the generalized fundamental string (GFS) solutions which are both supersymmetric and are known to be re- lated by duality in the direction x [1—3]. Let us describe The dilatino supersymmetry rule implies that the Killing briefly these two classes of solutions. The SSW solutions spinors of this solution have to satisfy the constraint are V'& = 0) (29) ds = 2dudv + 2A„du + 2A, dx' du —dx-'dx- *A - which can only be satisfied by e = 0. Therefore, all the B = 2A, dx du, supersymmetries of the dual solution of Minkowski space g=0, are broken. As we saw in Sec. III this is related to the de- and the GFS solutions are pendence of the Killing spinors on p when we use adapted coordinates and a y-independent frame. ds = 2e ~(dudv + A, dx' du) —dx-'dx- Our second example is the one recently found (ii) by - Bakas in Ref. [11].He studied self-dual Euclidean metrics B = —2e ~((1 —e ~)du A dv + A, du A dx '), admitting a Killing vector associated to the coordinate A 1 P = ——ln(l —A„) . v, which generally can be written in the form 2 ds = V (d~ + (u, dx') + V p, dx*dx' . . (3o) — Here i = 1, . . . , 8, u = ~(t+ x), v = ~(t x), and the Self-duality of the metric is an integrability condition for fields do not depend on x = x and on t = x . the existence of unbroken supersymmetries. What was To use the machinery developed in the main body of actually observed in [11] was the breaking of the self- the paper we have to identify the nine-dimensional fields. duality condition of the configuration after the T-S-T For our purposes it is enough to do it for just the SSW chain of duality transformations. solutions. First of all we need a zehnbein basis of the The violation of supersymmetry in this example could form of Eq. (8). Fields k and A„ that appear in it are be attributed to T-duality, since, as we have said, S- readily identified: duality is perfectly consistent with supersymmetry. Fur- k —1 —— thermore, the violation of supersymmetry by T duality k = (1 —A„)2 A~ 2 A; . (33) was related to the nature of the Killing vector: T dual- ity with respect to "translational" Killing vectors would A neunbein basis, necessary to complete the zehnbein not violate supersymmetry while T duality with respect basis, is provided by to "rotational" Killing vectors would. In particular, this criterion was sufhcient to show that for configurations (k-' o) k o~ ()=I kA. g' I ()=„( with flat three-dimensional metrics p;~ = b;~ no violation 2 2 2 of supersymmetry happened. However, for some special and the rest of the nine-dimensional fields are (with choices of nonflat . the self-duality of the final configu- p, ~ curved indices) ration was violated. From our point of view, this gives an interesting exam- 1 1+ k2 Bg,. —— Bg —0, ple of our general statement that unless the Killing spinor k in Kaluza-Klein basis is shown to be independent on du- —, ality direction there is no reason to expect the preserva- B~=0, B, = — A, =k A, , 2 tion of supersymmetry by T duality. A preliminary study =0.

In Cartesian coordinates and in the most obvious frame In order to avoid ambiguities we will always assume that e~ = b„- the Killing spinors are just arbitrary constant A —1 ( 0 so the solution will always have the same signature spinors and so have the right asymptotic behavior. as the asymptotic infinity (when the fields vanish). 3014 ERIC BERGSHOEFF, RENATA KALLOSH, AND TOMAS ORTIN

The field strengths of the nine-dimensional vector fields that the Killing spinors change after timelike duality ac- A, B are given by cording to Eq. (24). There seems to be a contradiction — between these two facts, but, actually, they are consistent Fp; (A) = 2k 8;k, Fp; (B) = 0, with each other because the above constraint is invariant (36) under multiplication by pp(= p ) and the Killing spinor F;~(A) = —4k A(;8~)k+ F~(A), F;~(B) = k F;~(A) . is simply transformed into another Killing spinor. It would be interesting to apply our results to super- If we write the Killing spinor equation h;@ = 0 in the symmetric configurations which are not timelike invari- form ant since in general timelike duality seems to change the sign of the energy of the configurations and interchanges (8 +M)e=0, (37) singularities and horizons [12, 17] while supersymmetry (as we have shown) is preserved. we have

M = —(k' P"(A) —P'(B) 4j* P—k) = —(p —j*)(j*O,k) . V. CONCLUSION 8 2 (38) Bosonic configurations may have Killing vectors and, when embedded in a supergravity theory, also Killing This implies that the Killing spinor is x independent 0 e spinors. We have studied the case in which both are if it is constrained by present and one performs a T duality transformation in the direction associated to a Killing vector. (39) Usually, the existence of a Killing vector means that This is just the constraint found in Ref. [1], using a dif- there exist a system of coordinates (adapted coordinates) ferent (but also x-independent) zehnbein basis, though. in which the fields [here the metric (or zehnbeins), the As it was explained in the end of the previous section, the dilaton, and the two-form field] do not depend on the independence of the Killing spinor on x in a basis of the coordinate associated. to the Killing vector. One of our form of Eqs. (8) and (34) follows &om its x independence main conclusions is that if a bosonic configuration ad- on any other x-independent frame, in particular that of mits a Killing vector and a Killing spinor and one uses Ref. [1]. adapted coordinates, even if the bosonic fields do not de- (ii) A second example is provided by the dual relation pend on the coordinate associated to the isometry it is between a special class of fivebrane solutions [4] called not guaranteed that the Killing spinor will not depend on well. have multiinonopoles in Ref. [5] and the stringy ALE instan- it as We exhibited diferent examples of this Our second tons [6] which have the multicenter Gibbons-Hawking situation. main conclusion is that in this one metric. It was observed in Ref. [6] that these two so- situation, if performs a T-duality transformation in lutions are related by T duality. The reason why only the direction associated to the Killing vector, the dual the multimonopole configurations are dual to the stringy configuration will not admit Killing spinors. ALE is simple. The characteristic property of The main result of our paper is that T duality does those class of fivebranes is the independence on the di- preserve the unbroken supersymmetries of those configu- co- rection x = w which is the one used for duality. Generic rations whose Killing spinors are independent of the ordinate associated with the used for fivebrane [4] as well as generic self-dual metrics [6] do isometry duality not have such an isometry. The fivebrane solutions, in- and the Killing spinors transform in a very simple way. cluding the multimonopoles, have unbroken supersym- It is interesting to compare this situation with the case metries with constant chiral (in four-dimensional Eu- of S duality. S duality always preserves the unbroken su- clidean space) Killing spinors in a 7.-independent zehn- persymmetries of the configurations at the classical level bein basis. According to the results of the previous sec- [15]. However S duality and T duality are on equal foot- tion this would be sufBcient to claim that the dual solu- ing in some contexts [18]: when the effective action of the tions (the stringy ALE instantons) have unbroken super- type-II superstring is compactified on a six torus, the hid- symmetries with the same Killing spinors. den symmetry of the resulting four-dimensional theory (iii) Our last example illustrates our results for timelike (% = 8 supergravity) is , which contains the SO(6,6) duality, although it cannot be said it is a natural born T-duality group and the SL(2, R) S-duality group. Ob- "winner. " It is easy to show that the extreme magnetic viously, &om the four-dimensional point of view, both T dilaton , uplifted to ten dimensions in [16], is and S duality must be consistent with supersymmetry. invariant under timelike duality. We also know that it has However, in the case of T duality, we are not interested unbroken supersymmetries with constant Killing spinors in four-dimensional configurations for which T duality restricted by the same condition as the fivebrane Killing amounts to a rotation of vector and scalar fields but, of- spinors of Ref. [4]: the Killing spinors are chiral in the ten, we are interested in the nontrivial 'ects induced four-dimensional Euclidean space spanned by the coordi- by T duality in the ten-dimensional metric. From the ten-dimensional point of view (the one we adopt here) T nates x, . . . , x, that is duality will not be consistent with supersymmetry in the (1 + pi2s4)&+ = 0. (40) cases explained above. Since the configuration is invariant, the Killing spinors The investigation of n' corrections with respect to T are invariant too. On the other hand, in Sec. III we found duality may also lead to the discovery of some new fea- DUALITY VERSUS SUPERSYMMETRY AND COMPACTIFICATION 3015

' tures. We know that T duality gets o. corrections and 0, .. . , 9, the 6ve-dimensional indices will carry a caret this means that the hidden symmetries of the conven- p, v = 0, . . . , 4, and the compactified dimensions will be tional supergravity theories (and the theories themselves) denoted by capital I's and J's, I, J = 5, . . . , 9. We take will be modi6ed in a form unknown at present time. We the d = 10 fields to be related to the D = 5 ones by have some relevant results on T duality which includes (io) . (io) g "v gPv ~ "a ccv non-Abelian vector fields and n' corrections which ex- (xo) (io) plain the fact that the SSW [1] solutions as well as the gI- —0, BI- —0, dual wave solutions [2] have unbroken supersymmetry (A1) ' (io) with account of o. corrections. These results will be pub- gII —'9IJ = ~II, BII = 0, lished elsewhere [8). y(10) Note added in proof. Note that for p = 0 the pp com- ponent of the metric given in Eq. (25) vanishes. This We get leads to a singular point in the dual metric given in Eq. — — @ — (28). Nevertheless, one can perform a duality transfor- S= d x ge R+4g (A2) mation in the y direction as has been discussed in Ref. As a second step we reduce from = 5 to d = —1 = 4 [23]. D D using the results and notation of the previous section. We get ACKNO%'LEDC MENTS —1 S = d xe ~v' g[ R+ 4—(c)$—) —4H We are grateful to E. Alvarez, G. W. Gibbons, G. — T. Horowitz, and A. Tseytlin for most fruitful discus- (o)ink) + 4k E (A) + 4k E (B)]. (A3) sions. One of us (T.O.) is extremely grateful to the hos- Now, if we look to the gravitino supersymmetry rule in pitality, friendly environment and 6nancial support of d = 10, the Institute for Theoretical Physics of the University of Groningen and the Physics Department of Stanford University, where part of this work was done. The work and observe that, setting k = 1 of E.B. has been made possible by the financial support of the Royal Netherlands Academy of Arts and Sciences ——H, = — E, (A+B—), (A5) (KNAW). The work of R.K. was supported by NSF Grant No. PHY-8612280. The work of E.B. and R.K. has also it is clear that the identification of the matter vector fields been partially supported by a NATO Collaboration Re- D„and the supergravity vector fields V„ is the same as search Grant. The work of T.O. was supported by a in Chamseddine's paper up to factors of 1/2: European Union Human Capital and Mobility program grant. D„= —(A„—B„), 1 APPENDIX: FROM d = 10 TO d = 4: V„= —(A„+B„), (A6) THE SUPERSYMMETRIC TRUNCATION respectively. We also have to put k = 1, because there Now we want to make connection with the action is no such a scalar in the N = 4, d = 4 supergravity of N = 4, d = 4 supergravity. Therefore we have multiplet. Now we want to truncate the theory keeping to compactify six spacelike coordinates and we substi- only the supergravity vector field V„. We have then tute everywhere g = —1. The compactification from A: (A7) N = 1, d = 10 to N = 4, d = 4 was done by Chamsed- =1, V„=A„=B„,D„=O. dine in Ref. [19]. However Chamseddine worked in the The truncated action is Einstein frame and that makes it very difBcult to study 1 the efFect of T duality on the resultant theory. Our goal S = — d xQ ge ~[ R+4—(c)$) ——4H + 2E (V)), here will be to obtain pure N = 4, d = 4 supergravity (or part of it) in string frame, identifying which fields (AS) belong to the matter multiplet and which 6elds belong where to the how supergravity multiplet and the dimensionally E„„(V)= 2c)(„V„I, reduced action has to be truncated in order to get rid of — the matter GeMs. H„„p —B(„B„p)+ V(„E„p)(V) . (A9) We perform the dimensional reduction of the theory The embedding of the four-dimensional 6elds in this from d = 10 to d = 4 for a simpli6ed model in which action in d = 10 is mast of the d = 10 fields are trivial. This simplified {xo) (») model is enough to discuss the important features of the g„=g„—V„V, B„=B„ dimensional reduction versus duality. We da it in two steps. First we reduce from d = 10 to D = 5. We denote the ten-dimensional Gelds an (A10) by (10) 1 y(10) upper index 10 and the five-dimensional fields by a caret. (») The ten-dimensional indices are capital letters M, N = gIJ —'gI J — ~IJ. 3016 ERIC BERGSHOEFF, RENATA KALLOSH, AND TOMAS ORTIN

These formulas can be used to uplift any four- tion satisfying Eq. (All) is invariant under T duality in dimensional field configuration with one , one the direction x. That is also obviously true for the rest axion, one vector, and a dilaton to a ten-dimensional field of the compactified directions. configuration in a way consistent with supersymmetry, as We can state this result as follows: if in the four- in Refs. [20, 16]. dimensional action we interpret the vector field V„as One obvious but important observation is that this belonging to the supergravity multiplet coming from the action is not just invariant under x duality (here x combination A„+B„,then the lifting to ten dimensions x ), but all the fields that appear in it are individually of any four-dimensional configuration will be an x du- invariant. ality invariant configuration if z is one of the compact But there is more. If we rewrite the truncation, dimensions. Eq. (A7), in terms of the original ten-dimensional fields, One example is provided by the SSW [1] and the GFS it looks like this: [3, 2] solutions. These solutions are described in Sec. IV, = (io) Eqs. (31) and (32). If x x is the nontrivial compact- ified dimension (what we called before x ), then, impos- (io) ing the conditions Eq. (All) means for these solutions (io) A = P = 0. This subset of SSW and GFS are identical, ~~1 (All) are duality invariant in the x = x direction and give rise Now one can check that a ten-dimensional configura- to the same supersymmetric solutions of N = 4, d = 4 supergravity [20, 16, 21, 22].

Note that the truncation itself is duality invariant, i.e., k = I =S, D„=D„=O. "Which is necessary to have supersymmetry.

[1] E. Bergshoeif, R. Kallosh, and T. Ortin, Phys. Rev. D [12] D.L. Welch, Phys. Rev. D 50, 6404 (1994). 4'7, 5444 (1993). [13] T. Buscher, Phys. Lett. 159B, 127 (1985); Phys. Lett. B [2] E. BergshoeiF, I. Entrop, and R. Kallosh, Phys. Rev. D 194, 59 (1987); 201, 466 (1988). 49, 6663 (1994). [14] J. Scherk and J. Schwarz, Nucl. Phys. B153, 61 (1979); [3] A. Dabholkar, G. Gibbons, J. Harvey, and F. Ruiz, Nucl. E. Cremmer, in Supergravity '81, edited by S. Ferrara and Phys. B340, 33 (1990). J.G. Taylor (Cambridge University Press, Cambridge, [4] C.G. Callan, Jr., J.A. Harvey, and A. Strominger, Nucl. England, 1982). Phys. B359, 611 (1991). [15] T. Ortin, Phys. Rev. D 4'7, 3136 (1993); 51, 790 (1995). [5] R. Khuri, Nucl. Phys. B387, 315 (1992); J. Gauntlett, J. [16] R. Kallosh and T. Ortin, Phys. Rev. D 50, 7123 (1994). Harvey, and J. Liu, ibid. B409, 363 (1993). C.P. Burgess, R.C. Myers, and F. Quevedo, "On spher- [17] " [6] M. Bianchi, F. Fucito, G.C. Rossi, and M. Martellini, pre- ically symmetric string solutions in four dimensions, sented at the 7th Marcel Grossmann Meeting on General McGill University Report No. McGill-94/47, Universite Relativity, Stanford, California, 1994 (unpublished). de Neuchatel Report No. NEIP-94-011, hepth/9410142 [7] J. Maharana and J.H. Schwarz, Nucl. Phys. B390, 3 (unpublished). (1993); J.H. Schwarz, in String Theory, Quantum Grav [18] C.M. Hull and P.K. Townsend, "Unity of superstring ity and the Unification of Fundamental Interactions, Pro- dualities, " Queen Mary and Westfield College Report ceedings of the International Workshop, Rome, Italy, No. QMW-PH-94-30, DAMTP Report No. R/94/33, 1992, edited by M. Bianchi et al. (World Scientific, Sin- hepth/9410167 (unpublished). gapore, 1993). [19] A. Chamseddine, Nucl. Phys. B185, 403 (1981). [8] E. BergshoefF, R. Kallosh, and T. Ortin (unpublished). [20] E. BergshoefF, R. Kallosh, and T. Ortin, Phys. Rev. D [9] T. Ortin, talk given at the 7th Marcel Grossman Meeting 50, 5188 (1994). on General Relativity, Stanford, California, 1994 (unpub- [21] G.W. Gibbons, Nucl. Phys. B207, 337 (1982); G.W. Gib- lished). bons and K. Maeda, ibid. B298, 741 (1988); D. Garfin- [10] S.J. Gates, Jr., H. Nishino, and E. Sezgin, Class. Quan- kle, G.T. Horowitz, and A. Strominger, Phys. Rev. D 43, tum Grav. 3, 21 (1986). 3140 (1991). [ll] I. Bakas, "Space-time interpretation of S duality and su- [22] R. Kallosh, A. Linde, T. Ortin, A. Peet, and A. Van persymmetry violations of T duality, " CERN Report No. Proeyen, Phys. Rev. D 46, 5278 (1992). CERN- TH.7473/94, hep-th/9410104 (unpublished). [23] M. Rocek and E. Verlinde, Nucl. Phys. B373, 630 (1992).