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Evidence for Heterotic/Heterotic Duality

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Evidence for Heterotic/Heterotic Duality CTP-TAMU-54/95 hep-th/9601036 EVIDENCE FOR HETEROTIC/HETEROTIC DUALITY 1 M. J. Du Center for Theoretical Physics, Texas A&M University, Col lege Station, Texas 77843, U. S. A. 2 R. Minasian Theory Division, CERN, CH 1211 Geneva 23, Switzerland and 3 Edward Witten School of Natural Sciences, Institute for Advanced Study, Olden Lane, Princeton, NJ 08540, U. S. A. ABSTRACT We re-examine the question of heterotic - heterotic string dualityinsix dimensions and argue that the E E heterotic string, compacti ed on K 3 8 8 with equal instanton numb ers in the two E 's, has a self-duality that inverts 8 the coupling, dualizes the antisymmetric tensor, acts non-trivially on the hyp ermultiplets, and exchanges gauge elds that can b e seen in p erturbation theory with gauge elds of a non-p erturbative origin. The sp ecial role of the symmetric emb edding of the anomaly in the two E 's can b e seen from 8 eld theory considerations or from an eleven-dimensional p oint of view. The duality can b e deduced by lo oking in two di erentways at eleven-dimensional 1 M -theory compacti ed on K 3 S =Z . 2 1 Research supp orted in part by NSF Grant PHY-9411543. 2 World Lab oratory Fellow. 3 Research supp orted in part by NSF Grant PHY92-45317. 1 Intro duction Prior to the recent surge of interest in a dualitybetween heterotic and Typ e I IA strings [1, 2, 3 , 4, 5], it was conjectured (on the basis of D = 10 heterotic string/ vebrane duality [6, 7]) that in D 6 dimensions there ought to exist a dualitybetween one heterotic string and another [8, 9, 10 , 11, 12 , 13 , 14 ]. A comparison of the fundamental string solution [15] and the dual solitonic string solution [10, 11 ] suggests the following D = 6 duality dictionary: the e e f dilaton , the string -mo del metric G and 3-form eld strength H of MN the dual string are related to those of the fundamental string, , G and MN H by the replacements e ! = e G ! G = e G MN MN MN f H ! H = e H (1) In going from the fundamental string to the dual string, one also inter- changes the roles of worldsheet and spacetime lo op expansions. Moreover, since the dilaton enters the dual string equations with the opp osite sign to the fundamental string, it was argued in [8, 10, 11] that in D = 6 the strong coupling regime of the string should corresp ond to the weak coupling regime of the dual string: =2 e =<e >=1= (2) 6 6 e where and are the fundamental string and dual string coupling con- 6 6 stants. Because this dualityinterchanges worldsheet and spacetime lo op expansions { or b ecause it acts by dualityonH { the duality exchanges the tree level Chern-Simons contributions to the Bianchi identity 0 2 dH = (2 ) X 4 1 2 2 X = [trR v trF ] (3) 4 2 4(2 ) with the one-lo op Green-Schwarz corrections to the eld equations 0 2 f f dH = (2 ) X 4 1 2 2 f e X = [trR v trF ] (4) 4 2 4(2 ) 2 th Here F is the eld strength of the comp onent of the gauge group, tr e denotes the trace in the fundamental representation, and v ; v are constants. (As explained in App endix A, wemay, without loss of generality,cho ose the string tension measured in the string metric and the dual string tension mesured in the dual string metric to b e equal.) In fact, the Green-Schwarz anomaly cancellation mechanism in six dimensions requires that the anomaly eight-form I factorize as a pro duct of four-forms, 8 f I = X X ; (5) 8 4 4 and a six-dimensional string-string duality with the general features summa- rized ab ovewould exchange the two factors. Until now, there has not b een a really convincing example of heterotic- heterotic duality in six dimensions. In [11], it was prop osed that the D =10 SO(32) heterotic string compacti ed to D =6onK3 might b e dual to the D =10 SO(32) heterotic vebrane wrapp ed around K 3. However, this candidate for a heterotic/heterotic dual string pair su ered from the following drawbacks: 1) The existence of a vebrane carrying the requisite SO(32) quantum numb ers is still unclear. Even if it exists, its prop erties are not well-understo o d. 2) The anomaly eight-form of this mo del is given by (5) with [19] 1 2 2 2 X = [trR trF 2trF ] 4 SO(28) SU (2) 2 4(2 ) 1 2 2 2 f X = [trR + 2trF 44tr F ]; (6) 4 SO(28) SU (2) 2 4(2 ) and one of the gauge co ecients in the second factor enters with the wrong sign. The structure of this equation actually presents a problem that is indep en- dentofany sp eculation ab out string-string duality.Itwas shown by Sagnotti [20] that corrections to the Bianchi identities of the typ e (3) and to the eld equations of the typ e (4) are entirely consistent with sup ersymmetry, with e no restrictions on the constants v and v . Moreover, sup ersymmetry relates these co ecients to the gauge eld kinetic energy. In the Einstein metric c =2 G = e G , the exact dilaton dep endence of the kinetic energy of MN MN the gauge eld F ,is M N 3 p (2 ) =2 =2 MN c e L = G v e + v e trF F : (7) g aug e M N 0 8 3 e Positivity of the kinetic energy for all values of thus implies that v and v should b oth b e non-negative, and at least one should b e p ositive. This fails for the SO(32) heterotic string, as we see from the formula for the anomaly eight-form. Some interesting new \phase transition" must o ccur at the value of at which the SO(28) coupling constant app ears to change sign, and at least until this phase transition is understo o d, its o ccurrence mightwell obstruct simple attempts to extrap olate from a string description at large negative to a dual string description at large p ositive. In this pap er, we shall attempt to remedy these problems as follows. 0 1 ) It has recently b een recognised that the ten-dimensional E E het- 8 8 10 1 erotic string is related to eleven-dimensional M -theory on R S =Z [16 ], 2 10 1 just as the ten-dimensional Typ e I IA string is related to M -theory on R S 6 1 [17]. By lo oking in two di erentways at M -theory on R K 3 S =Z ,we 2 get a de nite framework for deducing string-string duality. This framework shows that the gauge group should b e E E , the vacuum gauge bundle 8 8 should have equal instanton numb ers in each E (a situation we will refer to 8 4 as symmetric emb edding ), and the duality acts in a non-trivial fashion on the hyp ermultiplets. From this eleven-dimensional p oint of view, one heterotic string comes by 1 wrapping the D = 11 membrane around S =Z and the dual heterotic string 2 1 is obtained by reducing the D =11 vebrane on S =Z and then wrapping 2 around K 3. This is quite similar to the eleven-dimensional derivation of 1 1 heterotic - Typ e I IA duality, which is recovered if we replace S =Z by S in 2 5 the ab ove scenario [18 ]. 0 2 )Now let us discuss the anomaly p olynomial. Picking a vacuum on K 3 with equal instanton numb ers in each E will break E E to a sub- 8 8 8 group. Generically E E is completely broken, so there are no questions of 8 8 whether the gauge contributions to the anomaly eight-form are compatible with duality. But we also want to understand how the duality acts on vacua with non-trivial unbroken gauge groups. For instance, in a vacuum in which the gauge bundle breaks E E to E E (a maximal p ossible unbroken 8 8 7 7 4 Symmetric emb edding entered naturally in [21] in constructing simple examples of heterotic/Typ e I I duality in four dimensions. 5 The interpretation of the heterotic string as a wrapping of a vebrane around K 3, or around a K 3 sub-manifold of a Calabi-Yau manifold or Joyce manifold, is presumably the explanation for the ubiquity in string/string dualityofK3 itself and Calabi-Yau and Joyce manifolds corresp onding to brations of K 3. 4 subgroup of E E ) the anomaly eight-form is 8 8 1 1 1 1 2 2 2 2 I = [trR trF trF ] [trR ] (8) 8 E E 7 7 2 2 4(2 ) 6 6 4(2 ) e We see that v = 0, so (i) there is no wrong sign problem and one can p ossibly extrap olate to strong coupling without meeting a phase transition, e but (ii) since v 6= v , there is no manifest self-duality. Qualitatively similar e results hold (that is, v = 0) for any other unbroken subgroup of E E . 8 8 This qualitative picture dep ends on having equal instanton numb ers in the e two E 's; in any other case, v < 0 for some subgroups of E E , and phase 8 8 8 transitions of some kind are unavoidable.
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