I discuss the role of spacetime supersymmetry in the interplay between
strong/weak coupling duality and target space duality in string theory which
arises in string/string duality. This can be seen via the construction of
string soliton solutions which in $N=4$ compactifications of heterotic string
theory break more than $1/2$ of the spacetime supersymmetries but whose analogs
in $N=2$ and $N=1$ compactifications break precisely $1/2$ of the spacetime
supersymmetries. As a result, these solutions may be interpreted as stable
solitons in the latter two cases, and correspond to Bogomol'nyi-saturated
states in their respective spectra.Comment: 6 pages, harvma

Khuri, Ramzi R.

English

arXiv

I discuss the role of spacetime supersymmetry in the interplay between strong/weak coupling duality and target space duality in string theory which arises in string/string duality. This can be seen via the construction of string soliton solutions which in N=4 compactifications of heterotic string theory break more than 1/2 of the spacetime supersymmetries but whose analogs in N=2 and N=1 compactifications break precisely 1/2 of the spacetime supersymmetries. As a result, these solutions may be interpreted as stable solitons in the latter two cases, and correspond to Bogomol'nyi-saturated states in their respective spectra

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arXiv:hep-th/9506086v1  13 Jun 1995McGill/95-24CERN-TH/95-156hep-th/9506086Spacetime Supersymmetry and Duality in String TheoryRamzi R. KhuriPhysics Department, McGill UniversityMontreal, PQ, H3A 2T8 CanadaandCERN, CH-1211, Geneva 23, SwitzerlandI discuss the role of spacetime supersymmetry in the interplay between strong/weakcoupling duality and target space duality in string theory which arises in string/stringduality. This can be seen via the construction of string soliton solutions which in N = 4compactifications of heterotic string theory break more than 1/2 of the spacetime super-symmetries but whose analogs in N = 2 and N = 1 compactifications break precisely1/2 of the spacetime supersymmetries. As a result, these solutions may be interpreted asstable solitons in the latter two cases, and correspond to Bogomol’nyi-saturated states intheir respective spectra.McGill/95-24CERN-TH/95-156June 1995Talk given at MRST’95 Meeting, May 8-9, 1995, Rochester University1. IntroductionThe construction of soliton solutions in string theory is intimately connected with thepresence of various dualities in string theory (for recent reviews of string solitons, see [1,2]).Most of the solitonic solutions found so far break half of the spacetime supersymmetriesof the theory in which they arise. Examples of string-like solitons (i.e. with one Killingdirection) in this class are the fundamental string solution of [3] and the dual string solutionof [4], which are interchanged once the roles of the strong/weak coupling S-duality andtarget space T -duality are interchanged.In this talk I will first summarize the basic features of S duality, T duality andstring/string duality in heterotic string theory. Then I will discuss newly constructed [5]classes of string-like soliton solutions, making connections between the solution-generatingsubgroup of the T -duality group and the number of spacetime supersymmetries broken inN = 4, D = 4 compactifications of N = 1, D = 10 heterotic string theory, as well as thenatural realization of these solutions in N = 1 and N = 2 four-dimensional compactifica-tions. For simplicity, I will restrict myself to solutions in the gravitational sector of thestring (i.e. all Yang-Mills fields will be set to zero).For an interesting discussion of six-dimensional string/string duality see [6]. New andexciting connections between the various dualities in heterotic string theory and type IIstring theory can be found in [7].2. S DualityWe adopt the following conventions for N = 1, D = 10 heterotic string theory com-pactified toN = 4, D = 4 heterotic string theory: (0123) is the four-dimensional spacetime,z = x2 + ix3 = reiθ, (456789) are the compactified directions, S = e−2Φ + ia = S1 + iS2,where Φ and a are the four-dimensional dilaton and axion. S duality generalizes strong-weak coupling duality, since g = eΦ is the string loop coupling parameter. In N = 4, D = 4heterotic string theory S duality corresponds to the group SL(2, Z). In other words, thefour-dimensional theory exhibits an invariance underS → aS + bcS + d, (2.1)where a, b, c, d are integers and ad− bc = 1.There is now considerable evidence [8–15,4,16] in favor of S duality also being an exactsymmetry of the full string theory. One obvious attraction to demonstrating S dualityexactly in string theory is that it would allow us to use perturbative string techniques inthe strong-coupling regime.1In the absence of nontrivial moduli and Yang-Mills fields, the low-energy four-dimensional bosonic effective action in the gravitational sector of the heterotic string hasthe formS4 =∫d4x√−g(R − gµν2S21∂µS∂νS¯). (2.2)A solution of this action is given by [3]ds2 = −dt2 + dx21 +ReS(dx22 + dx23)S = − 12piN∑i=1ni ln(z − ai)ri0,(2.3)where N is an arbitrary number of string-like solitons each with arbitrary location ai in thecomplex z-plane and arbitrary winding number ni respectively. One can replace z by z¯ inS, thereby changing the orientation of the windings. There is also an SL(2, R) symmetrymanifest in the low-energy action, which is broken down to SL(2, Z) in string theory viaaxion quantization and from which the above solution can be generalized further. Notethat the x1 Killing direction gives the above solution the structure of a parallel multi-string configuration. Each string is interpreted as a macroscopic fundamental string [3].For dynamical evidence for this identification see [17].3. T DualityT duality in string theory is the target space duality, and generalizes the R → α′/Rduality in compactified string theory. For N = 4, D = 4 compactifications of heteroticstring theory, T -duality corresponds to the discrete group O(6, 22;Z) and is known to bean exact symmetry of the full string theory [18–25].Let us consider a simple special compactification, in which the only nontrivial moduliare given by T = T1 + iT2 =√det gmn − iB45, where m,n = 4, 5. For trivial S field,the low-energy four-dimensional bosonic effective action in the gravitational sector has theformS4 =∫d4x√−g(R − gµν2T 21∂µT∂ν T¯). (3.1)A solution of this action is given by [4]ds2 = −dt2 + dx21 +ReT (dx22 + dx23)T = − 12piM∑j=1mi ln(z − bj)rj0,(3.2)2where M is an arbitrary number of string-like solitons each with arbitrary location bjin the complex z-plane and arbitrary winding number mj respectively. Again, one canreplace z by z¯ in T and reverse the windings, and there is an SL(2, R) symmetry manifestin the low-energy action which is broken down to SL(2, Z), this time due to the presenceof instantons, and from which the above solution can be generalized further. Note thatthe x1 Killing direction gives the above solution the structure of a parallel multi-stringconfiguration as well, but in this case each string is interpreted as a dual string [4].4. String/String DualityNote that in interchanging the S field in the action (2.2) with the T field in theaction (3.1), one is interchanging the S (fundamental) string with the T (dual) stringand effectively interchanging their respective couplings. In this form, the string/stringduality conjecture postulates the existence of a dual string theory, in which the roles ofthe strong/weak coupling duality and target space duality are interchanged. It followsthat the string/string duality conjecture requires the interchange of worldsheet couplingassociated with T duality and spacetime coupling associated with S duality.Of course the full T duality group O(6, 22;Z) is much larger than the S duality groupSL(2, Z), but from the six-dimensional viewpoint, the strong/weak coupling duality ofthe fundamental string can be seen to emerge as a subgroup of the target space dualitygroup of the dual string. From the ten-dimensional viewpoint, the dual theory is a theoryof fundamental fivebranes (5 + 1-dimensional objects) [26]. However, given the variousdifficulties in working with fundamental fivebranes (see discussion in [6]) and the fact thatthe technology of fundamental string theory is reasonably well-developed, it seems naturalto prefer string/string duality (in D = 6 or D = 4) over string/fivebrane duality in D = 10.Both the fundamental (S) and dual (T ) string break 1/2 the spacetime supersym-metries, which can be seen either from the N = 1, D = 10 uncompactified theory orfrom the N = 4, D = 4 compactified theory. They also both arise in a larger O(8, 24;Z)solution generating group (for an explicit O(8, 24;Z) transformation that takes one fromthe fundamental string to the dual string see [5]). As a consequence, they both saturateBogomol’nyi bounds and correspond to states in the spectrum of the theory [12,16].5. Generalized Solutions and Supersymmetry BreakingNow consider the following ansatz, in which the solution-generating subgroup of the3O(6, 22;Z) T duality group is contained in SL(2, Z)3 = SL(2, Z)× SL(2, Z)× SL(2, Z):T (1) = T(1)1 + iT(1)2 =√detgmn − iB45, m, n = 4, 5,T (2) = T(2)1 + iT(2)2 =√detgpq − iB67, p, q = 6, 7,T (3) = T(3)1 + iT(3)2 =√detgrs − iB89, r, s = 8, 9(5.1)are the moduli. We assume dependence only on the coordinates x2 and x3 (i.e. x1 remainsa Killing direction), and that no other moduli than the ones above are nontrivial.The canonical four-dimensional bosonic action for the above compactification ansatzin the gravitational sector can be written in terms of gµν (µ, ν = 0, 1, 2, 3), S and T(a), a =1, 2, 3 asS4 =∫d4x√−g(R− gµν2S21∂µS∂νS¯− gµν2T(1)21∂µT(1)∂ν T¯(1) − gµν2T(2)21∂µT(2)∂ν T¯(2) − gµν2T(3)21∂µT(3)∂ν T¯(3)).(5.2)A solution of this action is given by [5]ds2 = −dt2 + dx21 +ReSReT (1)ReT (2)ReT (3)(dx22 + dx23)S = − 12piN∑i=1ni ln(z − ai)ri0,T (1) = − 12piM∑j=1mj ln(z − bj)rj0,T (2) = − 12piP∑k=1pk ln(z − ck)rk0,T (3) = − 12piQ∑l=1ql ln(z − dl)rl0,(5.3)where N,M, P and Q are arbitrary numbers of string-like solitons in S, T (1), T (2) and T (3)respectively each with arbitrary location ai, bj, ck and dl in the complex z-plane and arbi-trary winding number ni, mj , pk and ql respectively. One can replace z by z¯ independentlyin S and in each of the T moduli, and in each of S and the T moduli there is an SL(2, Z)symmetry manifest in the action in each of the moduli, and from which the above solutionscan be generalized further. Thus one has an overall effective solution-generating group ofSL(2, Z)4.It can be shown [5] that the solutions with trivial S and 1, 2 and 3 nontrivial Tfields preserve 1/2, 1/4 and 1/8 of the spacetime supersymmetries respectively, while the4solutions with nontrivial S and 0, 1 and 2 nontrivial T fields preserve 1/2, 1/4 and 1/8spacetime supersymmetries respectively. The solution with nontrivial S and 3 nontriv-ial T fields preserves 1/8 of the spacetime supersymmetries for one chirality choice of S,and none of the spacetime supersymmetries for the other, although the ansatz remains asolution to the bosonic action in this latter case. In short, the maximum number of space-time supersymmetries preserved in the N = 4 theory for a solution generating subgroupSL(2, Z)n of O(8, 24;Z) is given by (1/2)n [5].6. DiscussionSo what is the interpretation of these new solutions which break more than half thesupersymmetries, since they are not expected to arise within the spectrum of Bogomol’nyi-saturated states in the N = 4 theory? It turns out that most of the above solutionsthat break 1/2, 3/4 or 7/8 of the spacetime supersymmetries in N = 4 have analogsin N = 1 or N = 2 compactifications of heterotic stirng theory that break only 1/2the spacetime supersymmetries∗. Of course no solution actually preserves a higher totalnumber of supersymmetries in the lower supersymmetric theory (N = 1 or N = 2) thanin N = 4, but the relative number of supersymmetries preserved may be increased intruncating the N = 4 theory to N = 1 or N = 2 by the removal of non-supersymmetricmodes. Thus a solution that preserves 1/8 of the spacetime supersymmetreis in N = 4 and1/2 of the spacetime supersymmetries in N = 1 actually preserves the same total amountof supersymmetry in both theories. The only difference is that in the N = 4 case one isstarting with four times as many supersymmetries, so that a greater number of those arebroken than in the N = 1 case.These solutions are therefore in some sense realized naturally as stable solitons only inthe context of eitherN = 1 orN = 2 compactifications, and should lead to the constructionof the Bogomol’nyi spectrum of these theories. In these two case, however, the situationis complicated by the absence of non-renormalization theorems present in the N = 4 casewhich guarantee the absence of quantum corrections. An exception to this scenario arisesfor N = 2 compactifications with vanishing β-function. The construction of these spectraremains a problem for future research.∗ However, when at least one of the fields, either S or one of the T fields, has a differentanalyticity behaviour from the rest, no supersymmetries are preserved in N = 1 [5].5References[1] R. R. Khuri, Helv. Phys. Acta 67 (1994) 884.[2] M. J. Duff, R. R. Khuri and J. X. Lu, NI-94-017, CTP/TAMU-67/92, McGill/94/53,CERN-TH.7542/94, hepth/9412184 (to appear in Physics Reports).[3] A. Dabholkar, G. W. Gibbons, J. A. Harvey and F. Ruiz Ruiz, Nucl. Phys. B340(1990) 33.[4] M. J. Duff and R. R. Khuri, Nucl. Phys. B411 (1994) 473.[5] M. J. Duff, S. Ferrara, R. R. Khuri and J. Rahmfeld, hep-th/9506057.[6] M. J. Duff, NI-94-033, CTP-TAMU-49/94, hepth/9501030.[7] C. M. Hull and P. K. Townsend, Nucl. Phys. B438 (1995) 109.[8] A. Font, L. Iba´n˜ez, D. Lust and F. Quevedo, Phys. Lett. B249 (1990) 35.[9] S. J. Rey, Phys. Rev. D43 (1991) 526.[10] A. Sen, Nucl. Phys. B404 (1993) 109.[11] A. Sen, Phys. Lett. B303 (1993) 22.[12] A. 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Spacetime Supersymmetry and Duality in String Theory

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