We discuss the duality between two type I compactifications to four
dimensions and an heterotic construction with spontaneous breaking of the N=4
supersymmetry to N=2. This duality allows us to gain insight into the
non-perturbative properties of these models. Through the analysis of the
gravitational corrections, we then investigate the connections between
four-dimensional, N=2 M-theory vacua constructed as orbifolds of type II,
heterotic, and type I strings.Comment: 7 pages, latex, to appear in the proceedings of the TMR meeting on
  Quantum aspects of gauge theories, supersymmetry and unification, Paris, 1--7
  September 199

Gregori, Andrea

English

arXiv

We discuss the duality between two type I compactifications to fourdimensions and an heterotic construction with spontaneous breaking of the N=4supersymmetry to N=2. This duality allows us to gain insight into thenon-perturbative properties of these models. Through the analysis of thegravitational corrections, we then investigate the connections betweenfour-dimensional, N=2 M-theory vacua constructed as orbifolds of type II,heterotic, and type I strings

Gregori, A

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TMR9: Quantum aspects of gauge theories, supersymmetry and unicationPROCEEDINGSNon-perturbative string connectionsAndrea GregoriUniversita di Milano{Bicoccavia Celoria 16, 20133 MilanoAbstract: We discuss the duality between two type I compactications to four dimensions and anheterotic construction with spontaneous breaking of theN = 4 supersymmetry to N = 2. This dualityallows us to gain insight into the non-perturbative properties of these models. Through the analysisof the gravitational corrections, we then investigate the connections between four-dimensional, N = 2M-theory vacua constructed as orbifolds of type II, heterotic, and type I strings.After the \second string revolution", thattook place after 1995, it is a common belief thatall the string constructions are manifestations ofa unique underlying theory. In most of the cases,the dierent string models correspond to dier-ent regions in the moduli space of the underly-ing theory. There are however several cases inwhich apparently disconnected vacua turn out tobe indeed equivalent, being simply related by a\change of parametrization".In some cases, this reparametrization mapsperturbative moduli into non-perturbative ones.In these cases, knowing this map allows to com-pute in an easy way quantities that would be be-yond a perturbative, often short, computation.This is the case for instance of the duality be-tween the heterotic string compactied to fourdimensions and the type IIA string compactiedon a K3 bration [1]. The heterotic dilaton{axion eld is mapped into a perturbative mod-ulus, associated to the volume form of the baseof the bration, on the type IIA side. There-fore, what is non-perturbative on the heteroticside is perturbative on the type IIA side. Thisrelation, that recently received a further conr-mation [2], has been used in order to computethe non-perturbative correction to the eectivecoupling of the R2 term in some specic exam-ples [3]{[6] (An analogous relation exists also be-tween some type II asymmetric orbifold com-pactications and the type IIA string [6]{[8]).However, the cases in which such duality existsare far from covering the main part of the het-erotic constructions. It seems indeed that someof the most interesting cases don’t fall in thisclass. The question is therefore whether it isnevertheless possible, in some of these cases, toobtain (at least partial) information on the non-perturbative behavior. We present here an anal-ysis of a four-dimensional, N = 2 heterotic con-struction which, although without type IIA dual,possesses nevertheless a pair of type I string du-als. Putting together the informations comingfrom all these constructions, it is possible to ob-tain partial but not irrelevant informations aboutthe non-perturbative behavior of this model.Through the analysis of the gravitational cor-rections, we discuss then the connections of theseconstructions with other type II and heteroticconstructions.  The heterotic model we consider is constructedas a Z2, \freely acting" orbifold of the string com-pactied on T 6 = T 2  T 4. The Z2 projectionacts as a reflection, xi ! −xi, on T 4 and asa half-circumference translation in one circle ofT 2. Imposing modular invariance and requiringthe shift of the momenta of the T 2-lattice pro-duced by this translation to be left-right sym-metric, leads to a specic embedding of the spinconnection into the gauge group, such that, atthe orbifold point, the massless spectrum origi-TMR9: Quantum aspects of gauge theories, supersymmetry and unication Andrea Gregorinating from the c = (0; 16) currents always con-tains an equal number of vector and hypermulti-plets. This equality is however not due to a \leveltwo" realization of the gauge group: the rankis sixteen, i.e. the maximal allowed in pertur-bative heterotic constructions. The free actionof Z2 produces then a spontaneous breaking ofthe N = 4 supersymmetry, that can be restoredwhen the translated coordinated is decompacti-ed. This corresponds to a special limit in thespace of the moduli of the two-torus, T and U ,associated respectively to the Ka¨hler class andthe complex structure. At the U(1)16 point, thismodel has sixteen vector multiplets and sixteenhypermultiplets from the currents, plus the threevectors and four hypermultiplets from the com-pact space. Due to the free action of Z2, there areno massless states originating from the twistedsector. Although it exists a type IIA orbifoldconstruction with the same massless spectrum,obtained by compactication on an orbifold limitof a Calabi Yau manifold with Hodge numbers(19,19), this is not dual to the heterotic orbifold;the CY19;19 manifold is not in fact a K3 bration.The heterotic orbifold possesses on the otherhand a type I dual, constructed as an orientifoldof theN = 4, type IIB string, in which theN = 8supersymmetry is spontaneously broken by a Z2freely acting projection, whose action of T 6 is thesame as that of the heterotic model. In the caseof the heterotic string, T-duality makes irrelevantthe choice of the translation, that can equiva-lently be performed on the momenta, projection(−1)m, or on the windings, projection (−1)n; inthe case of type I orientifolds on the other handthe two projections lead to rather dierent mod-els [9]. In the rst case, the model obtained is ex-actly equivalent to the heterotic one, with a per-turbative, spontaneous breaking of theN = 4 su-persymmetry and a gauge group of rank sixteen,originating entirely from D9-branes: there areno D5-branes. By introducing appropriate Wil-son lines it is possible to obtain exactly the samespectrum of the heterotic construction. The typeI orientifold obtained from the type IIB stringwith translation (−1)n, on the other hand, hasstill a gauge group of rank sixteen, but that noworiginates half from the D9-branes sector and halffrom the D5-branes sector. Moreover, the spec-trum on the branes doesn’t feel the breaking ofsupersymmetry. This model doesn’t look there-fore dual to the previous constructions.In order to investigate the conjectured du-ality between the heterotic and the rst of thetwo type I constructions, we consider the stringcorrections to the eective coupling of the R2term. As it was discussed in Refs. [5, 6, 10], inorder to compare string constructions, it is nec-essary to project out the non-universal contribu-tion coming from the coupling of the \bulk" sec-tor with the currents. This can be consistentlydone because the contribution of the \currents"to the gravitational and gauge couplings are pro-portional, and can be subtracted by properly re-dening the amplitude we compute. On the het-erotic side, a further contribution must be sub-tracted, namely that coming at the singularitiesin the space of the moduli T and U : at thesepoints, new vector and/or hypermultiplets ap-pear. This phenomenon doesn’t happen on thetype I side. Once this projection has been per-formed, the renormalization of the gravitationalamplitudes read, in the two models:Het: :16 2g2= 16 2=S − 2 log=T j#4(T )j4−2 log=U j#4(U)j4+O(logM) ; (1)Type I :16 2g2= 16 2=S − 2 log=U j#4(U)j4+O(logM) : (2)Apparently, a term is missing in Eq. (2). The so-lution to this puzzle comes from the observationthat the second term in the r.h.s. of Eq.(1) be-haves, for large-T , as: log=T j#4(T )j4  log=T .There is no linear divergence in T , and the log-arithmic behavior can be interpreted as due tonon-perturbative phenomena. Indeed, it can beremoved by switching on an appropriate infraredcut-o (see Ref. [11]). The corrections (1) and(2) are therefore consistent with the duality ofthese constructions. We may however ask whatis indeed the meaning of the heterotic eld T ,whose contribution is apparently lost when go-2TMR9: Quantum aspects of gauge theories, supersymmetry and unication Andrea Gregoriing to the type I dual. In the opposite limit,T ! 0, the heterotic correction (1) diverges lin-early in the inverse eld, ~T  −1T . By reden-ing ~T  S0, we see that in the large ~T limit, (1)behaves as:16 2g2 16 2=S + 16 2=S0−2 log=U j#4(U)j4 : (3)This is indeed the behavior of a type I modelwith both D9- and D5-branes sectors. Actually,this is the correction as it would be computed onthe second type I construction. We guess thatindeed the second type I construction is dual tothe heterotic and the other type I model. Thishypothesis is supported by a look at the pathfollowed by the D5-branes when their coupling,parametrized by S0, becomes very weak (S0 ! 0)[10]. The crucial point is that this model can beviewed as obtained via a freely acting Z2 projec-tion performed along the eleventh coordinate ofthe M-theory [9]. In this model, owing to the freeaction of this projection, S(S0)-duality is broken.Indeed, in the limit S0 ! 0, that correspondsto a decompactication of the T 4, the D5-branesmassless elds are still present in the spectrum.However, being the D5-branes coupling missing,these states have to be interpreted as originatingfrom D9-branes. Supersymmetry of the branesspectrum is indeed an artifact, due to the sep-aration into closed and open sectors, typical oftype I constructions. It seems therefore possi-ble to imagine to break supersymmetry only onthe bulk, represented by the closed string sector.However, through interactions of the bulk eldswith the elds living on the branes, the break-ing of supersymmetry is then communicated tothe whole theory. It is indeed from the heteroticdual, where there is no such a fake separation,that we learn how the supersymmetry breakingis actually communicated from one sector to theother.The relation to the M-theory is better un-derstood by going to the dual, type I0 picture,where D9- and D5-branes appear as D4- and D8-branes, and the eleventh coordinate correspondsto the tenth coordinate of the string. From theM-theory point of view, the freely acting, Scherk{Schwarz supersymmetry breaking projection onthe eleventh coordinate leaves untouched, at leastin a rst approximation, the two \Horava{Wittenwalls", whose massless elds correspond respec-tively to the D4- and D8-branes elds [9]. Sincethe system is symmetric under reflection of theeleventh coordinate, we argue that the breakingof the S0-duality on the D4 sector has a mirrorin an analogous breaking of S-duality on the D8-branes sector. Under this hypothesis, the correc-tion given in Eq. (1) should be promoted to:16 2g2= −2 log= ~Sj#4( ~S)j4 − 2 log= ~S0j#4( ~S0)j4−2 log=U j#4(U)j4 +Oe−( ~S; ~S′;U);(4)where ~S  −1S, ~S0  −1S0, and the last termis a series of exponentials, symmetric in the threeelds, suppressed in the large and small eldslimits. This is the typical behavior of a theorywith spontaneous breaking of the N = 8 super-symmetry. Expression (4) diverges linearly forlarge-S, large-S0 (and small U), but only loga-rithmically for large ~S, S0, U . This is the limitof restoration of the N = 8 supersymmetry.  The analogous gravitational correction in thetype IIA orbifold with Hodge numbers (19,19), is(see Ref. [8]):16 2g2= −2 log=T (1)j#4(T (1))j4−6 log=T (2)j(T (2))j4−6 log=T (3)j(T (3))j4 ; (5)where T (1), T (2), T (3) are the moduli associatedto the Ka¨hler classes of the three tori of T 6 =T 2T 2T 2, and we omit for simplicity the termlogarithmically dependent on the infrared cut-o.It is clear that this correction doesn’t match theheterotic one. Indeed, this type IIA orbifold isdual to a type II asymmetric orbifold, in whichthe modulus T (1) plays the role of the dilaton{axion eld. It is also clear that, if the heteroticmodel has a spontaneously broken N = 8 super-symmetry, it cannot possess a type IIA dual, inwhich this super-Higgs phenomenon would ap-pear as perturbative. Indeed, a type IIA orbifold3TMR9: Quantum aspects of gauge theories, supersymmetry and unication Andrea Gregoriwith the spontaneous breaking of N = 8 exists:it is a Z2Z2 orbifold in which both the projec-tions act freely [7, 8]. All the states of the twistedsectors are massive, and the massless spectrumcontains only, besides the gravity multiplet, threevector multiplets and four hypermultiplets. Thegravitational correction reads:16 2g2= −2 log=T (1)j#4(T (1))j4−2 log=T (2)j#4(T (2))j4−2 log=T (3)j#4(T (3))j4 : (6)In a particular limit in the moduli space (T (1) !0), this orbifold behaves as a K3 bration, andcan be compared to an heterotic dual with thesame massless spectrum [6]. The map of mod-uli is in that case: T (1) ! −1S, T (2) ! T ,T (3) ! U . In the opposite limit, T (1) ! 1,the model matches instead a type II asymmet-ric, freely acting orbifold, with map of moduli:T (1) ! S, T (2) ! T , T (3) ! U . Heterotic andtype II asymmetric orbifolds turn out thereforeto be S-dual the one to the other.Although not coinciding in the \bulk" of themoduli space, the corrections (4) and (5) matchat the corner of moduli space, precisely at thelimit in which the supersymmetry breaking Z2projection of the heterotic/type I side ends to actfreely: in this limit, corresponding to S0 ! 1,U ! 0, (4) matches (5) in the limit T (2) ! 1,T (3) ! 1. This is however not evidence thatat the limit the two theories really match: in-deed, when we recover a genuine orbifold limit,we expect new massless states to appear, associ-ated to the orbifold xed points1. On the otherhand, it is possible to interpolate between thefreely acting type IIA/heterotic/type II asym-metric orbifolds and the heterotic/type I con-structions with rank 16 by switching on an ap-propriate Wilson line that, already at the N = 4level, lifts the mass of all the elds originatingfrom the currents. It turns out therefore thatsome of these constructions can be connected bypassing through appropriate limits in the modulispaces.1In the light of recent investigations, this part of theconclusions presented in [10] needs a slight modication.AcknowledgmentsThis talk is based on a work done in collabora-tion with C. Kounnas; I take this opportunity toaddress my thanks to him, and to acknowledgethe EEC, under the contract TMR-ERBFMRX-CT96-0045, the Swiss National Science Founda-tion and the Swiss Oce for Education and Sci-ence, for nancial support.References[1] C.M. Hull and P.K. Townsend, Nucl. Phys. B438 (1995) 109;S. Kachru and C. Vafa, Nucl. Phys. B 450(1995) 69;A. Klemm, W. Lerche and P. Mayr, Nucl. Phys.B 357 (1995) 313;P. S. Aspinwall and J. Louis, Phys. Lett. B 369(1996) 233.[2] E. Kiritsis, N. A. Obers and B. Pioline, J. HighEnergy Phys. 0001 (2000) 029.[3] J. A. Harvey and G. Moore, Phys. Rev. D 57(1998) 2323; Phys. Rev. D 57 (1998) 2329.[4] A. Gregori, E. Kiritsis, C. Kounnas, N.A. Obers,P.M. Petropoulos and B. Pioline, Nucl. Phys. B510 (1998) 423.[5] A. Gregori, C. Kounnas and P.M. Petropoulos,Nucl. Phys. B 537 (1999) 317;A. Gregori, Talk given at 6th HellenicSchool and Workshop on Elementary ParticlePhysics:, Corfu, Greece, 6-26 Sep 1998. 3InCorfu 1998, Quantum aspects of gauge theo-ries, supersymmetry and unication, 348-355,hep-th/9811096.[6] A. Gregori, C. Kounnas and P.M. Petropoulos,Nucl. Phys. B 553 (1999) 108.[7] A. Sen and C. Vafa, Nucl. Phys. B 455 (1995)165.[8] A. Gregori, C. Kounnas and J. Rizos, Nucl.Phys. B 549 (1999) 16.[9] I. Antoniadis, G. D'Appollonio, E. Dudas andA. Sagnotti, Nucl. Phys. B 553 (1999) 133.[10] A. Gregori and C. Kounnas, Nucl. Phys. B 560(1999) 135.[11] E. Kiritsis, C. Kounnas, P.M. Petropoulos andJ. Rizos, Phys. Lett. B 385 (1996) 87.4

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