PHYSICAL REVIEW D VOLUME 51, NUMBER 6 15 MARCH 1995 Duality versus supersymmetry and compactification Eric BergshoeF" Institute for Theoretical Physics, University of Groningen, Nijenborgh g, g7$7AG Groningen, The Netherlands Renata Kallosht Department of Physics, Stanford University, 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 supersymmetries, 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 string theory. It can text of N = 1, d = 10 supergravity 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. cern. 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 dilaton 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 string theory 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.