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Topological Defects in the Moduli Sector of String Theory⋆

arXiv:hep-th/9109044v2 25 Sep 1991UPR–485–TSeptember 1991Topological Defects in the Moduli Sector of String Theory⋆Mirjam CvetiˇcDepartment of PhysicsUniversity of PennsylvaniaPhiladelphia, PA 19104–6396ABSTRACTWe point out that the moduli sector of the (2, 2) string compactification withits nonperturbatively preserved non-compact symmetries is a fertile framework tostudy global topological defects, thus providing a natural source for the large scalestructure formation. Based on the target space modular invariance of the non-perturbative superpotential of the four-dimensional N = 1 supersymmetric stringvacua, topologically stable stringy domain walls are found.

They are supersym-metric solutions, thus saturating the Bogomolnyi bound. It is also shown thatthere are moduli sectors that allow for the global monopole-type and texture-typeconfigurations whose radial stability is ensured by higher derivative terms.⋆Talk Presented at Strings ‘91 Workshop, May 20–25, 1991, Stony Brook, N.Y.

Topological defects occur during the spontaneous break-down of gauge sym-metries, as a consequence of the nontrivial homotopy group Πn of the vacuummanifolds. Their existence has important cosmological consequences.

In particularglobal topological defects, like textures1,2 and more recently, global monopoles3,4as well as global Π2 textures5,6 were proposed as a source of large scale struc-ture formation. On the other hand, in the framework of grand-unified theories(or theories beyond the standard model) it is often unnatural to impose globalnon-Abelian symmetries that would ensure the existence of such global topologicaldefects.

Here we would like to point out that in the string theory, the moduli sec-tor of (2, 2) string compactification provides a natural framework for such globaldefects, with its potentially important physical implications.In (2, 2) string compactifications, where (2, 2) stands for N = 2 left-moving aswell as N = 2 right-moving world-sheet supersymmetry, there are massless fields –moduli M – which have no potential, i.e. V (M) ≡0, to all orders in string loops.7Thus perturbatively there is a large degeneracy of string vacua, since any vacuumexpectation value of moduli corresponds to the vacuum solution.

On the otherhand it is known that nonperturbative stringy effects like gaugino condensation8and axionic string instantons9 give rise to the nonperturbative superpotential.In the case of the modulus T associated with the internal size of the com-pactified space for the so-called flat background compactifications (e.g., orbifolds,self-dual lattice constructions, fermionic constructions) the generalized target spaceduality is characterized by noncompact discrete group PSL(2, Z) = SL(2, Z)/Z2specified by T →aT −ibicT +d with a, b, c, d ∈Z and ad −bc = 1. If one assumes thatthe generalized target space duality is preserved even nonperturbatively,10,11 theform of the nonperturbative superpotential is very restrictive.11 The fact that thisis an exact symmetry of string theory even at the level of nonperturbative effects issupported by genus-one threshold calculations,12,13 which in turn specify the formof the gaugino condensate.14,15This phenomenon has intriguing physical implications leading to the stable su-persymmetric domain walls.16 This physics of modulus T is actually a generalizationof the well known axion physics17 introduced to solve the strong CP problem inQCD.

Spontaneously broken global U(1) Peccei-Quinn symmetry is non-linearlyrealized through a pseudo-Goldstone boson, the invisible axion θ. Nonperturba-tive QCD effects through the axial anomaly break explicitly U(1) symmetry downto ZNf, by generating an effective potential proportional to 1 −cos Nfθ.Thispotential leads to domain wall solutions18 with Nf walls meeting at the axionicstrings.17As an instructive example let’s first consider a global supersymmetric theory2

by withL = GT ¯T|∇T|2 + GT ¯T|∂TW(T)|2(1)Here, GT ¯T ≡∂T ∂¯T K(T, ¯T) is the positive definite metric on the complex mod-ulus space and the superpotential, W, is a rational polynomial P(j(T)) of themodular-invariant function j(T)19 i.e. a modular invariant form of PSL(2, Z).The potential V ≡GT ¯T|∂T W(T)|2 = GT ¯T|∂jP(j)∂Tj(T)|2 has at least two iso-lated zeros at T = 1 and T = ρ ≡eiπ/6 in the fundamental domain D for T, i.e.when |∂T j(T)|2 = 0.19 Other isolated degenerate minima might as well arise when|∂jP(j)|2 = 0.Then, the mass per unit area of the domain wall can be written as:20µ =∞Z−∞dz GT ¯T|∂zT −eiθGT ¯T∂¯T ¯W( ¯T)|2 + 2Re(e−iθ∆W)(2)where ∆W ≡W(T(z = ∞)) −W(T(z = −∞)).

The arbitrary phase θ has tobe chosen such that eiθ = ∆W/|∆W|, thus maximizing the cross term in Eq. (2).Then, we find µ ≥K ≡2|∆W|, where K denotes the kink number.

Since ∂T Wis analytic in T, the line integral over T is path independent as for a conserva-tive force.The minimum is obtained only if the Bogomolnyi bound ∂zT(z) =GT ¯Teiθ∂¯T ¯W( ¯T(z)) is saturated. In this case ∂zW(T(z)) = GT ¯Teiθ|∂T W(T(z))|2,which implies that the phase of ∂zW does not change with z.

Thus, the super-symmetric domain wall is a mapping from the z-axis [−∞, ∞] to a straight lineconnecting between two degenerate vacua in the W-plane. We would like to em-phasize that this result is general; it applies to any globally supersymmetric theorywith disconnected degenerate minima that preserve supersymmetry.For the superpotential, e.g.

W(T) = (α′)−3/2j(T) the potential has two iso-lated degenerate minima at T = 1 and T = ρ ≡eiπ/6. At these fixed points,j(T = ρ) = 0 and j(T = 1) = 1728.Therefore, the mass per unit area isµ = 2 × 1728(α′)−3/2.

Other cases can be worked out analogously.21The case with gravity restored has a K¨ahler potential K = −3 log(T + ¯T)and the superpotential should transform as a weight −3 modular function undermodular transformations.22,11 The most general choice, nonsingular everywhere inthe fundamental domain D, isWm,n(T) = Hm,n(T)η(T)6 ,Hm,n ≡(j(T) −1728)m/2 · jn/3(T)P(j(T)), m, n = R+(3)Here, η(T) is the Dedekind eta function, a modular form of weight 1/2 and P(j(T))3

is an arbitrary polynomial of j(T). The potential is of the following form:Vm,n(T, ¯T) =3|H|2(T + ¯T)3|η|12(|(T + ¯T)3(∂T HH+ 32πˆG2)|2 −1)(4)where ˆG2 = −4π∂T η/η −2π/(T + ¯T).

In general the scalar potential (4) has ananti-de Sitter minimum with broken supersymmetry.11 However, one can see thatfor m ≥2, n ≥2 and P(j) = 1, the potential is semi-positive definite with thetwo isolated minima at T = 1 and T = ρ with it unbroken local supersymmetryjust like in the global supersymmetric case.We now minimize the domain wall mass density. By the planar symmetry,the most general static Ansatz for the metric24 is ds2 = A(|z|)(−dt2 + dz2) +B(|z|)(dx2 + dy2) in which the domain wall is oriented parallel to (x, y) plane.Using the supersymmetry transformation lawsδψµα = [∇µ(ω) −i2Im(GT∇µT)]ǫα + 12(σµ¯ǫ)αeG2 ,δχα = 12(σµ¯ǫ)α∇µT −eG2 GT ¯TG ¯T ǫα(5)with commuting, covariantly constant, chiral spinors ǫ±, the ADM mass density µcan be expressed as25µ ∓K =Zdz √g[gijδψ†iδψj + 12GT ¯Tδχ†δχ] ≥0.

(6)The i, j indices are for spatial directions. The minimum of the Bogomolnyi boundis achieved if Eq.

(5) vanish. Again, the stringy domain wall is stabilized by thetopological kink number.Unfortunately, the nice holomorphic structure of the scalar potential is lost.In other words, there is now a holomorphic anomaly in the scalar potential due tothe supergravity coupling.

This implies that the path connecting two degeneratevacua in superpotential space is not a straight line. In fact, one can understandthe motion as a geodesic path in a nontrivial K¨ahler metric, thus in G(T, ¯T).

Onecan show (numerically) that in our example the path along the circle T = exp iθ(z)with θ = (0, π/6), i.e. , the self-dual line of T →1/T modular transformation, isthe geodesic path connecting between T = 1 and T = ρ in the scalar potentialspace.

Thus, we have again established an existence of stable domain walls. Thesuperpotential is quite complicated, however, the numerical solution in can beobtained.214

It is interesting to note that stringy cosmic strings#1 can be viewed as bound-aries of our domain walls. Because the domain wall number is two, the intersectionof two such domain walls is precisely the line of stringy cosmic strings.

On theother hand such stable domain walls are disastrous from the cosmological point ofview. One possible solution to this problem is that after supersymmetry breaking,the degeneracy of the two minima is lifted.

In that case, the domain wall becomesunstable via the false vacuum decay.26We would now like to point out27 the existence of other global topologicaldefects, like global monopole-type and texture-type defects in the moduli sector ofstring theory. Such defects could exist in the study of the symmmetry structureof the effective theory when there are more than one modulus (which of course isa generic situation).

We shall illustrate the idea using examples based on the socalled flat backgrounds, i.e. generalization of SL(2, Z).For that purpose we shall study the simplest example of Z4 manifold withcontinuous symmetry SU(2, 2) on the four moduliT ≡"T11T12T21T22#(7)of compactified space.

Note that the moduli T live on the coset SU(2, 2)/SU(2) ×SU(2) × U(1). The continuous non-compact symmetry SU(2, 2) is an exact sym-metry28 at least at the string tree-level.

Note that this continuous symmetry in themodulus could be broken down to the discrete subgroup SU(2, 2, Z) due to non-perturbative effects, e.g. gaugino condensation and/or axionic instanton effects.However, at this point we shall stick to the continuous symmetry.

For the timebeing we shall keep in mind that SU(2, 2, Z) is the vacuum symmetry and thusthe T fields should live in the fundamental domain of SU(2, 2, Z).The maximal comapct symmetry of SU(2, 2) is SU(2)A × SU(2)B × U(1) ⊂SU(2)A+B. Note also that in projective coordinates:29 Z = (1 −T)/(1 + T).

Ztransforms as 1 + 3 under SU(2)A+B. The ansatz Z = P3a=1 σaVa with Va =f(r)xa/r ensures the map of Z on the S2.Let us concentrate now on the Lagrangian for the Z field and thus a specificsolution for f(r).

Note that the Z fields have no potential to all orders in stringloops. Thus the kinetic energy term28 shrinks f →0 due to Derrick’s theoremand thus should be stabilized by higher derivative terms.

Such higher derivativeterms arise even at the tree level of the string theory. They should respect thenoncompact SU(2, 2) symmetry.

Also, if one sticks to terms with at most two#1 It is intriguing that the present kink solitons also appear in integrable, supersymmetrictwo-dimensional N = 2 Landau-Ginzburg models205

time derivatives, one has a unique form for the terms that involve four derivatives,which is very similar in nature to the Skyrme term30 in the Skyrmion model andcan serve the same role as the stabilizing term. In this case f = Cr as r →0and f = D/r2 as r →∞The energy stored in such a configuration is finite.

Thisis different from the standard global monopole configuration,3 which has f →f0as r →∞which has linearly divergent energy and thus long range interactionrelevant for large scale formation.Another interesting observation would be to study the texture-type configu-rations, which have a chance to occur within this sector. Namely, the Z fieldstransform as 4 under the compact symmetry SU(2)A × SU(2)B ∼SO(4) andthus the ansatz: Z = a(r) + b(r) P3a=1 σaxa/r is mapped onto S3.

The potentialproblem in this case is an impossibility to ensure a2(r) + b2(r) = f2 where f is aconstant. Interestingly, {a(r), b(r)} →0 as r →∞and thus the knot configurationdisappears at large distances.The above studied configurations are much milder defects than strings anddomain walls and they have finite range and thus finite energy.

Further study ofcosmological implications of such global defects is under consideration.I would like to thank my collaborators S. Griffies, F. Quevedo, and S.-J. Rey, formany fruitful discussions and enjoyable collaborations.

I would also like to thankthe Aspen Center for Physics, the International Centre for Theoretical Physics,Trieste, and CERN for their hospitality. The work is supported in part by theU.S.

DOE Grant DE–22418–281 and by a grant from University of PennsylvaniaResearch Foundation and by the NATO Research Grant #900–700.REFERENCES1. R. Davis, Gen. Rel.

Grav. 19, 331 (1987).2.

N. Turok, Phys. Rev.

Lett. 63, 2625 (1989).3.

M. Barriola and A. Vilenkin, Phys. Rev.

Lett. 63, 341 (1989).4.

D. Bennett and S. Rhie, Phys. Rev.

Lett. 65, 1709 (1990).5.

L. Perivolaropoulos, BROWN–HET–775 (November 1990).6. D. Bennett and S. Rhie, UCRL–IC–104061 (June 1990).7.

M. Dine and N. Seiberg, Nucl. Phys.

B301, 357 (1988).8. J. P. Derendinger, L. E. Ib´a˜nez, and H. P. Nilles, Phys.

Lett. 155B, 65(1985); M. Dine, R. Rohm, N. Seiberg, and E. Witten, Phys.

Lett. 156B,55 (1985).6

9. S.-J.

Rey, Axionic String Instantons and Their Low-Energy Implications,Invited Talk at Tuscaloosa Workshop on Particle Theory and Superstrings,ed. L. Clavelli and B.

Harm, World Scientific Pub., (November, 1989); Phys.Rev. D 43, 526 (1991).10.

S. Ferrara, D. L¨ust, A. Shapere, and S. Theisen, Phys. Lett 225B, 363(1989).11.

M. Cvetiˇc, A. Font, L. E. Ib´a˜nez, D. L¨ust, and F. Quevedo, Nucl. Phys.B361 (1991) 194.12.

V. Kaplunovsky, Nucl. Phys.

B307, 145 (1988).13. L. Dixon, V. Kaplunovsky, and J. Louis, Nucl.

Phys. B355, 649 (1991);J. Louis, SLAC–PUB–5527 (April 1991).14.

A. Font, L. E. Ib´a˜nez, D. L¨ust, and F. Quevedo, Phys. Lett.

245B, 401(1990); S. Ferrara, N. Magnoli, T. R. Taylor, and G. Veneziano, Phys. Lett.245B, 409 (1990); P. Binetruy and M. K. Gaillard, Phys.

Lett. 253B, 119(1991).15.

H. P. Nilles and M. Olechowski, Phys. Lett.

248B, 268 (1990).16. M. Cvetiˇc, S. J. Rey and F. Quevedo,“Stringy Domain Walls and TargetSpace Duality”, UPR–0445-T (April 1991) to appear in Phys.

Rev. Lett.17.

J. E. Kim, Phys. Rev.

Lett. 43, 103 (1979); M. Dine, W. Fischler, andM.

Srednicki, Phys. Lett.

104B, 199 (1981); and Nucl. Phys.

B189, 575(1981); M. B. Wise, H. Georgi, and S. L. Glashow, Phys.

Rev. Lett.

47, 402(1981); A good review is by J. E. Kim, Phys. Rep. 150, 1 (1987).18.

P. Sikivie, Phys. Rev.

Lett. 48, 1156 (1982); G. Lazarides and Q. Shafi,Phys.

Lett.115B (1982) 21; For a review, see A. Vilenkin, Phys. Rep. 121,263 (1985).19.

B. Schoeneberg, Elliptic Modular Functions, Springer, Berlin-Heidelberg(1970); J. Lehner, Discontinuous Groups and Automorphic Functions, ed.by the American Mathematical Society, (1964).20. P. Fendley, S. Mathur, C. Vafa and N.P.

Warner, Phys. Lett.

B243(1990)257.21. M. Cvetiˇc, S. Griffies, and S.-J.

Rey, Physical Implications of Stringy DomainWalls, UPR–471.22. S. Ferrara, D. L¨ust, A. Shapere and S. Theisen, Phys.

Lett. B225, (1989)363.23.

M. Cvetiˇc, A. Font, L.E. Ib´a˜nez, D. L¨ust and F. Quevedo, Target SpaceDuality, Supersymmetry Breaking and the Stability of Classical StringVacua, Nucl.

Phys. B361 (1991) 194.7

24. A. Vilenkin, Phys.

Lett. 133B (1983) 177: J. Ipser and P. Sikivie, Phys.Rev.

D30 (1984) 712.25. E. Witten, Comm.

Math. Phys.

80 (1981) 381.26. S. Coleman, Phys.

Rev. D15 (1977) 2929: C. Callan and S. Coleman, Phys.Rev.

D16 (1977) 1762.27. M.Cvetiˇc, F. Quevedo, and S.-J.

Rey, in preparation.28. M. Cvetiˇc, J. Louis, and B. Ovrut, Phys.

Lett. 206B, 227 (1988) and Phys.Rev.

D 40, 684 (1989).29. R. Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications, JohnWiley and Sons, (1974).30.

Skyrme, G. Adkins, C. Nappi, and E. Witten, Nucl. Phys.

B228, 5521(1983).8

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