A. Font, L.E. Ib'a~nez, D. Lust and F. Quevedo, Phys. Lett. B245 (1990) 401; B249 (1990) 35.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... effective dilation potential suggesting the need of strong coupling [7] and of the implied presence of those interactions in the superstring, do not hold once S Gammaduality symmetry is imposed in the effective theory through demanding modular invariance of the N = 1 supergravity superpotential [8]. It is worth mentioning that the idea of dropping the Lorentz invariance has been previously considered. Indeed, a background or constant cosmological vector field throughout space has been suggested as a way to introduce into the physical description our velocity with respect to a preferred ....

A. Font, L.E. Ib'a~nez, D. Lust and F. Quevedo, Phys. Lett. B245 (1990) 401; B249 (1990) 35.

....manifestly modular covariant form with the correct modular weight F (g 1) T ) 1 a 2 g B 2g 2g(2g Gamma 2) 2g Gamma 2) D 2(g Gamma1) T log(j(T ) Gamma j(1) 4. 27) where the holomorphic modular covariant derivative D T of a weight modular form P (T ) is given by [35] D T P (T ) i T Gamma G 2 (T ) j P (T ) 4.28) and where G 2 (T ) is related to the Dedekind function j(T ) by G 2 (T ) T log j 2 (T ) For later use we define the non holomorphic modular function G 2 (T ) of weight 2, G 2 (T; T ) 1 T T G 2 (T ) 4.29) 22 ....

....e Gamma2(kT lU) 1 Gamma e Gamma2(kT lU) 2 ; C.5) where the constants c(n) are determined by j(T ) Gamma 744 = 1 X n= Gamma1 c(n) q n ; C.6) and where the integers k and l can take the same values as indicated above. The third line of (C.2) can be expanded as follows. Using [35] G 2 (T ) Gamma 6 1 Gamma 24 1 X n=1 oe 1 (n) q n ; C.7) where oe 1 (n) denotes the sum of the divisors of n, one finds that G 2 (T ) G 2 (U) 2 36 i 1 Gamma 24 1 X m=1 oe 1 (m) q m 1 (1 q m 3 ) 576 1 X n;m=1 oe 1 (m) oe 1 (n) q n m 1 q n 3 j ; C.8) ....

M. Cvetic, A. Font, L. E. Ib'a~nez, D. Lust and F. Quevedo, Nucl. Phys. B 361 (1991) 194.

....theory so that the VEV of T remains undetermined. However, one expects that non perturbative effects will generate a superpotential for T. The simplest expression for W (T ) compatible with modular invariance is [11] W (T ) j(T ) Gamma6 (5) and the related scalar potential is given by [13]: V (T ) 1 T 3 R jj(T )j 12 T 2 R 4 2 j G 2 (T )j 2 Gamma 1 ; 6) where TR = 2Re T . The function j(T ) q 1=24 Q n (1 Gamma q n ) is the well known Dedekind function, q j exp( Gamma2T ) G 2 = G 2 Gamma 2=TR is the weight two Eisenstein function and G 2 = 1 3 ....

A. Font, L.E. Ib'a~nez, D. Lust and F. Quevedo, Phys. Lett. B249 (1990) 35.

.... related with the Polonyi problem associated with scalar fields that couple only gravitationally and which may dominate the energy density of the universe at present [5, 6] In the present letter, we show that the abovementioned difficulties can be avoided through a new mechanism introduced in [7] for fixing the dilaton. Indeed, it was shown that the dilaton potential develops a suitable minimum once the requirement of S duality invariance (see below) is imposed. Already in Ref. 8] this mechanism was used to stabilise the dilaton while inflation was accomplished by chiral fields. Our ....

....are satisfied by domain walls that separate degenerate minima in S dual superstring potentials, implying that topological domain wall inflation is, as antecipated in Ref. 6] through general arguments, an attractive cosmological scenario in string inspired models. 2. S duality was conjectured [7] in analogy with a well established symmetry of string compactification T duality. Indeed, it was shown that the effective supergravity action following from string compactification on orbifolds or even Calabi Yau manifolds is severely constrained by an underlying string symmetry, the so called ....

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A. Font, L.E. Ib'a~nez, D. Lust and F. Quevedo, Phys. Lett. B245 (1990) 401.

....algebra of charge densities on the p brane. This is of potential interest for particle physics because it has been speculated that the p = 5 case is relevant if one wants to incorporate non perturbative effects in superstring theory by appealing to a dual formulation in terms of the five brane [11]. Such a duality was conjectured in [12] before the five brane was proven to exist [13] The consequences of the existence of such a Montonen Olive [14, 15, 16, 17] type duality in string theory might be physically extremely important [11] involving phenomena like supersymmetry breaking and the ....

.... appealing to a dual formulation in terms of the five brane [11] Such a duality was conjectured in [12] before the five brane was proven to exist [13] The consequences of the existence of such a Montonen Olive [14, 15, 16, 17] type duality in string theory might be physically extremely important [11], involving phenomena like supersymmetry breaking and the occurrence of a non zero potential for the dilaton. Recently [18, 19, 20, 21, 22] an attempt has been made to use a straightforward generalization of the MF algebra to higher dimensions in the context of the p branes. In this attempt, the ....

A. Font, L.E. Ib'a~nez, D. Lust and F. Quevedo, Phys. Lett. B249 (1990) 35.

....interesting exact properties [1 4] One of the most intriguing features that can now be investigated is the duality between strong and weak coupling. The possibility in field theory of such a property was first discussed by Montonen and Olive [5] and in the context of string theory by Font et al. [6]. For N = 1; 2, duality relates different low energy descriptions of the same theory at different points in the quantum moduli space. While in these cases duality is not a symmetry of the theory, the situation changes drastically if we instead consider theories with vanishing fi function, e.g. N ....

A. Font, L.E. Iba~nez, D. Lust and F. Quevedo, Phys.Lett. 249B (1990) 35.

....values of the moduli. Moreover, topological inflation solves the problem of initial conditions for the onset of inflation as, due to the fact that inflation is driven by a topological defect, the field sits necessarily at the top of the potential. In N = 1 supergravity, S duality was conjectured [14] in analogy with T duality, a wellestablished symmetry of string compactification [15] Indeed, the target space modular invariance of string effective actions contains a duality transformation as well as discrete shifts of the axionic background. The conjecture is that there would be a further ....

....(this was an implicit assumption in [11] Notice that, at this stage, S duality (and also supersymmetry) is broken since a particular non vanishing vacuum state has been chosen. 2 Another realization of S duality is f 1=f , but this requires the presence of the so called magnetic condensate [14, 16]. In Ref. 11] it has been shown that the conditions for topological inflation, a scenario first put forward by Linde [10] and Vilenkin [18] to occur at the core of the domain walls separating degenerate minima of the potential can be met for some range of parameters. In this scenario, ....

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A. Font, L.E. Ib'a~nez, D. Lust and F. Quevedo, Phys. Lett. B245 (1990) 401; B249 (1990) 35.

....S S 1=2 (25) where f and G are the gauge kinetic function and S field Kahler potential. This relationship is related to the SU(1; 1) structure of the field S in the N = 4 supergravity Lagrangian [44] So the dilaton dominated soft terms are intimately connected to the S Gamma duality symmetry [45, 46] underlying these theories. Recently Seiberg and others [47] have discussed the existence of certain duality properties between different classes of N = 1 theories with different particle content and with or without Yukawa couplings. It would be desirable to see to what extent their results ....

....possibilities. Indeed, the well known T duality symmetries would suggest that the most natural values for T should be around the selfdual point, T 1 and that kind of result is obtained in T Gamma duality invariant versions of gaugino condensation [50] If some sort of S Gamma duality [45] is correct in N = 1 theories, one should also expect S 1. I would argue that this is not necesarily unreasonable if the massless sector of the theory contains particles beyond the ones in the MSSM, which is in fact the generic case in explicit string models. In this case the gauge couplings ....

A. Font, L.E. Ib'a~nez, D. Lust and F. Quevedo, Phys.Lett.B249(1990) 35.

....model itself. We do not know yet how SU(2) Theta U(1) breaking takes place. But, assuming that somehow a (composite or elementary) operator with the quantum numbers of a doublet gets a vev, we get a lot of information. The idea is to apply a similar philosophy for SUSY breaking in string theory [32, 8] . We have to try and identify possible chiral fields OE i such that their auxiliary fields F i could get non vanishing vev and break SUSY. In string models there are some natural candidates to do the job: the complex dilaton S = 4 =g 2 i and the moduli fields T i whose vevs determine the ....

....theory and it is expected that non perturbative effects will i) induce a non trivial scalar potential for those fields yielding S 6= 0, T i 6= 0 and ii) break supersymmetry spontaneously. The crucial assumption here is to locate the origin of SUSY breaking in the dilaton moduli sector [32, 8, 33, 34] . It is perfectly conceivable that other fields in the theory, like charged matter fields, could contribute in a leading manner to supersymmetry breaking. If that is the case the structure of soft SUSY breaking terms will be totally model dependent and we would be able to make no modelindependent ....

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M. Cvetic, A. Font, L.E. Ib'a~nez, D. Lust and F. Quevedo, Nucl.Phys. B361 (1991) 194.

.... one hidden gauge sector [30] Then G S = 0 is rather generic, however several fi function coefficients have to be tuned in a careful way in order to get m 3=2 O(1TeV) A different, very interesting possibility is that the non perturbative dilaton dynamics is governed by the socalled S duality [31, 32]. This means that the true non perturbative string partition function is actually PSL(2;Z) invariant resp. covariant with respect to the S field due to non perturbative monopollike configuration in target space. The simplest possibility within this context is that the partition function looks ....

....This means that the true non perturbative string partition function is actually PSL(2;Z) invariant resp. covariant with respect to the S field due to non perturbative monopollike configuration in target space. The simplest possibility within this context is that the partition function looks like [31] Z 1 (S S)jj(S)j 4 : 3.23) In the effective field theory this could mean that the effective superpotential contains a term j(S) Gamma2 instead of the standard e S dependence. Such types of superpotentials possibly lead to G S = 0. Finally one has to remark in this context that ....

A. Font, L.E. Ib'a~nez, D. Lust and F. Quevedo, Phys. Lett. B 249 (1990) 35.

....fixed) vev s of the moduli fields. It is quite evident that in general these SSBP are nonuniversal [3] The non universality arises due to the non universal moduli dependence of the gauge couplings and the matter kinetic energies. Similar expression can be also obtained for the trilinear couplings [17, 4, 5] Finally let us investigate the possible apperance of a mass mixing term for the two standard model Higgs fields H 1 , H 2 which is necessary for the correct radiative breaking of the electro weak gauge symmetry. Clearly, a tree level mixing due to a quadratic term in the superpotential, W tree = ....

....complete Green Schwarz cancellation, i.e. b 0 = 3ffi i GS , there is no T i dependence in the potential (as well as in m 3=2 ) and T i still remains as a undetermined parameter. On the other hand, for 3ffi i GS 6= b 0 , the modulus T i gets dynamically fixed. A specific analysis was performed in [15, 17] for the case ffi i GS = 0 with the result that at the minimum T i 1:2 supersymmetry gets spontaneously broken in the T i sector since at that point G T i 6= 0. However there is an important caveat witin this analysis since it used the assumption that at the minimum G S = 0. In fact, the ....

M. Cvetic, A. Font, L.E. Ib'a~nez, D. Lust and F. Quevedo, Nucl. Phys. B 361 (1991) 194.

....to the spontaneous breaking of supersymmetry. In particular, one can show that non perturbative gaugino condensation in the hidden gauge sector potentially breaks supersymmetry [14] Integrating out the dynamical degrees of freedom corresponding to the gaugino bound states, the duality invariant [15, 16] gaugino condensation can be described by an effective non perturbative superpotential, which depends holomorphically on the moduli fields: W SUSY Gammabreaking (M) e 24 2 ba fa(M) 2.6) b a is the N = 1 fi function coefficient) It is remarkable that this expression is in a sense exact ....

....complete Green Schwarz cancellation, i.e. b 0 = 3ffi i GS , there is no T i dependence in the potential (as well as in m 3=2 ) and T i still remains as a undetermined parameter. On the other hand, for 3ffi i GS 6= b 0 , the modulus T i gets dynamically fixed. A specific analysis was performed in [15, 17] for the case ffi i GS = 0 with the result that at the minimum T i 1:2 supersymmetry gets spontaneously broken in the T i sector since at that point G T i 6= 0. However there is an important caveat witin this analysis since it used the assumption that at the minimum G S = 0. In fact, the ....

A. Font, L.E. Ib'anez, D. Lust and F. Quevedo, Phys. Lett. B 245 (1990) 401.

....quantum level of the effective field theory since, for example, in supergravity they entail chiral transformations on fermions. In the case of modular invariance counterterms [4, 5] must be added to the effective field theory so as to restore the discrete modular symmetry. It has been conjectured [6] that a similar situation might hold with respect to S duality. Since the vev of the complex scalar field s that is the scalar member of the dilaton supermultiplet S determines the gauge coupling constant: hsi = g Gamma2 Gamma i =8 2 , this corresponds to strong weak coupling duality. ....

A. Font, L. Ib'a~nez, D. Lust and F. Quevedo, Phys Lett. 249B: 35 (1990).

....contain the states OE 0 H . Thus the logarithmic singularity in the moduli T i is nothing else than the threshold effect of OE 0 H with (intermediate) mass scale M OE 0 H (T i ) T i Gamma P c ) This discussion was entirely based on field theoretical arguments. As already discussed in [28], in string theory these threshold functions are given in terms of automorphic functions of the underlying target space duality group with the appropriate singularity structure. In fact, since in string theory there is generically an infinite number of states which may become massless at duality ....

....can moreover be extended to the point at infinity, which is a fixed point of order 1 under the translation f M e T . Since the limit U; T 1 describes the decompactification of the torus, infinitely many Kaluza Klein states become massless there, as predicted by the order of the fixed point [28]. In the Z 2 orbifold the critical points are the same, but since one must form twist invariant combinations the numbers of extra massless gauge bosons is divided by the order of the twist. Therefore equation (3.10) holds as well. The case of the Z 3 orbifold will be discussed below. Let us now ....

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M. Cvetic, A. Font, L. E. Ib'a~nez, D. Lust and F. Quevedo, Nucl. Phys. B361 (1991) 194.

....Z) T Omega SL(2; Z)U ) Theta Z T GammaU 2 Theta Z S GammaT 2 Theta Z S GammaU 2 : 2. 7) It contains the tree level T duality group O(2; 2; Z) T;U (SL(2; Z) T Omega SL(2; Z)U ) Theta Z T GammaU 2 , which is a classical symmetry, together with the S duality group SL(2; Z) S [16] and the exchange transformations S T and S U , which are non perturbative transformations. 4 As a concrete example let us specify the symplectic matrices realizing the S duality transformations S aS Gammaib icS d , T T , U U : U = d1; V = a1; W = bH; Z = cH ; 2.8) where H = ....

A. Font, L. E. Ib'a~nez, D. Lust and F. Quevedo, Phys. Lett. B 249 (1990) 35.

....superpotential W includes a SUSY mass term (S; T i )H 1 H 2 induced by a non perturbative effect, then a B term is automatically generated and it is called B . Also it was pointed out [22] that the presence of a non renormalizable term 3 Several kinds of modular functions are shown in Ref. [18]. in the superpotential WH 1 H 2 yields dynamically a parameter when W acquires VEV (S; T i ) T i )W (S; T i ) and the corresponding B term is denoted by B . In general, there could be an admixture of the above three possibilities. Thus we will treat B as a free parameter whose value can ....

M. Cvetic, A. Font, L.E. Ib'a~nez, D. Lust and F. Quevedo, Nucl. Phys. B361 (1991) 194.

....FIELD AND SUSY BREAKING In the absence of non perturbative effects, the dilaton field interacts with all scalar fields with an 1=S interaction, and the scalar potential does not have a stable solution. There are several possibilities to stabilize the dilaton. Firstly, one can impose an S duality[4] (analogous to the T dual symmetry) invariance to the potential. Another possibility is to consider gaugino condensation. Gaugino condensation [1] offers a very plausible origin for SUSY breaking for it is very reasonable to expect such a condensate to form at a scale between the Planck scale ....

A. Font, L. E. Ibanez, D. Lust and F. Quevedo, Phys. Lett. B249 (1990) 35.

.... V 0 = V 1 g 0 Gamma b 1 b ; V 00 = V 2 2 g 00 Gamma 1 b 2 2 (1 b ) 2 V 0 1 g 0 Gamma b 1 b : 29) 11 The other fixed point in the fundamental domain, namely t I = 1, is a saddle point of potential (28) see e.g. [33]. We require V 0 = 0; V 00 0 for stabilization, which means g 0 1 = b 1 b ; 2 g 00 1 Gamma b 2 2 (1 b ) 2 : 30) The function f( is related to g( by a differential equation (which assures a canonical form for the Einstein term in the Lagrangian) g 0 ....

A. Font, L. Iban~nez, D. Lust and F. Quevedo, Phys. Lett. B245, 401 (1990).

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