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Domain Walls in N = 1 Supergravity

arXiv:hep-th/9209117v2 8 Oct 1992UPR–527–TSeptember 1992Domain Walls in N = 1 Supergravity⋆Mirjam Cvetiˇc† and Stephen Griffies‡Department of PhysicsUniversity of Pennsylvania209 So. 33rd StreetPhiladelphia, PA 19104–6396ABSTRACTWe discuss a study of domain walls in N = 1, d = 4 supergravity.

The wallssaturate the Bogomol’nyi bound of wall energy per unit area thus proving stabilityof the classical solution. They interpolate between two vacua whose cosmologicalconstant is non-positive and in general different.

The matter configuration andinduced geometry are static. We discuss the field theoretic realization of thesewalls and classify three canonical configurations with examples.

The space-timeinduced by a wall interpolating between the Minkowski (topology ℜ4) and anti-de Sitter (topology S1(time) × ℜ3(space)) vacua is discussed.⋆Talk given at the International Symposium on Black Holes, Membranes, Wormholes, andSuperstrings, The Woodlands, Texas, January 1992.† email CVETIC@cvetic.hep.upenn.edu‡ email GRIFFIES@cvetic.hep.upenn.edu

1. IntroductionGlobal and local topological defects are known to arise during symmetry break-ing phase transitions if the vacuum manifold is not simply connected.

Textures,monopoles, strings, domain walls and combinations thereof are examples. Theseobjects may have important physical implications, especially in the cosmologicalcontext.The inclusion of gravity in the study of topological defects is straightforwardand usually leads to insignificant modifications to the otherwise stable topologicaldefects.

However, in superstring theories, for example, gravity and other moduliand matter fields are on an equal footing so the effects of gravity can yield qualita-tively different features. With the advent of deeper understanding of semi-classicalsuperstring theories in a topologically nontrivial sector, various stringy topologicaldefects were discovered: stringy cosmic strings[1 ,2], axionic instantons[3 ,4 ] as wellas related heterotic five-branes and other solitons[5 ,6 ,7 ,8].The above solutions were known to exist for free moduli fields, i.e.vanishingsuperpotential.Additionally, there exist supersymmetric domain walls when anontrivial superpotential for the moduli fields exists[9 ,10].

These domain walls areinteresting by themselves as well as in connection to the dynamical supersymmetrybreaking mechanism in superstring theory[11 ,12]. Additionally, they serve as a classof stringy topological defects in which a nonzero superpotential is essential to theirexistence.The present discussion centers on the construction and properties of domainwalls in N = 1, d = 4 supergravity.

There are three major results of our analysis.The first is a proof of a positive energy density theorem for a topologically nontrivialextended object in which the matter part of the theory has a generic nonzerosuperpotential. To the best of our knowledge, the proof has not been addressedpreviously.The second result is an existence proof for static domain wall solutions forboth the space-time metric and the matter field interpolating between two super-symmetric vacua.

It is known that the inclusion of gravity to reflection symmetricdomain walls of infinite extent and infinitesimal thickness generically admits onlytime-dependent solutions to Einstein’s equations[13]. We show that by allowingfor a reflection asymmetric solution interpolating between either a Minkowski andanti-de Sitter space-time or anti-de Sitter and anti-de Sitter space-time, the metricand matter field can both be time-independent.The last result is that supersymmetric domain walls can interpolate betweentwo vacua of different scalar potential energy: for example, between a supersym-metric vacuum with zero cosmological constant (Minkowski space-time) and an-2

other with a negative cosmological constant (anti-de Sitter space-time). This resultis at first counter to the notion of a domain wall interpolating between degeneratevacua.

The point is that in defining degenerate energy solutions, one must includeall the relevant energy in the theory; in this case both matter and gravity. It turnsout that when the vacua of the theory preserve supersymmetry, their energy, whichis defined in the appropriate way according to the ADM prescription[14], are thesame regardless of the particular matter vacuum energy.

This result is consistentwith there being no semi-classical tunnelling bubble causing the decay of one su-persymmetric vacua into another with a lower matter vacuum energy. In[15] thisresult has been proven by showing the minimum energy bubble which one couldconceivably form separating two supersymmetric vacua has an infinite radius andthus will never form.

This result is complementary to positive energy theorems(see, for example[16] and references therein) derived to show the stability of super-gravity theories with a matter potential unbounded below. Indeed, without thisresult, one could never expect to find the domain wall solutions we wish to describe.This paper is organized as follows.

We start in chapter 2 with a discussionof the formal aspects of the realization of these walls in the supergravity theory.Chapter 3 gives a classification of the walls in terms of the various combinationof vacua that they interpolate between. Chapter 4 presents examples of the threecanonical wall configurations and chapter 5 gives the geodesic structure for thespace-time induced by these walls.

We finish with some further remarks on thewall interpolating between Minkowski and anti-de Sitter space-times.Most of the work presented here is developed in the following references. Thefield theoretic results can be found in reference[18].

Additional work addressingthe problem of the stability of supersymmetric vacua can be found in reference[15].Discussion of the classification of the types of walls and their geodesic structurecan be found in reference[19]. Finally, the causal structure of the Minkowski-AdSwall as well as some phenomena related to quantum fields on this background iswork in progress[20].3

2. Supergravity realization of the wallsConsider an N = 1 locally supersymmetric theory with one chiral mattersuperfield T .

The bosonic part of the N = 1 supergravity Lagrangian is[21] ⋆e−1L = −12κR + KT ¯Tgµν∂µ ¯T∂νT −eκK(KT ¯T|DT W|2 −3κ|W|2)(2.1)where e = |detgµν|12, K(T, ¯T) = K¨ahler potential and DTW ≡e−κK(∂T eκKW).†In order to have stable domain wall solutions, topological arguments implythat the degenerate vacua be disconnected. Thus one must have isolated vacua ofthe matter potential.

However, the inclusion of gravity will turn out to play animportant role in removing the constraint that the isolated minima of the matterpotential have to be degenerate. We shall see that with the inclusion of gravita-tional energy, the notion of degenerate vacua will be defined as supersymmetrypreserving vacua just as in globally supersymmetric theories.

Indeed, formal ar-guments for the stability follow from the existence of local supersymmetry chargeswhich satisfy an algebra which is a generalization of the global algebra. Thus,the inclusion of gravity, when the dust settles, merely adds to the technology nec-essary to formulate the existence and stability criteria of these extended objects.Therefore, in a formal sense, the arguments are analogous to those in the globalcase[9 ,10 ,18].Supersymmetry preserving minimum of the potential in (2.1) satisfy DTW = 0.This in turn implies that the supersymmetry preserving vacua have either zero cos-mological constant (Minkowski space-time) when W = 0, or negative cosmologicalconstant −3eκK|κW|2 (anti-de Sitter space-time) when W ̸= 0.Note that wedefine the cosmological constant as follows.

The energy momentum tensor whenT is at its vacuum value (DT W = 0) is Tµν = −3κ|WeκK2 |2gµν. Therefore, Ein-stein’s equation Rµν −12gµνR = κTµν can be written Rµν −12gµνR = Λgµν withΛ = −3|κWeκK2 |2.⋆We do not choose the commonly used K¨ahler gauge which introduces the potential func-tion[21] G(T, ¯T) = K(T, ¯T) + ln|W(T )|2, since it is not adequate for situations in which thesuperpotential is allowed to vanish.† We use the conventions: γµ = eµaγa where γa are the flat spacetime Dirac matrices satisfying{γa, γb} = 2ηab; eaµeµb = δab ; a = 0, ...3; µ = t, x, y, z; ψ = ψ†γt; (+, −, −, −) space-timesignature; and write κ = 8πGN.4

2.1. ADM Mass DensityIn the following we obtain a lower bound on the mass density of domain wallsliving in this theory.

In that regard, we employ the results of[22] and[23] who ad-dressed the positivity of the ADM mass in general relativity, as well as certaingeneralizations to anti-de Sitter backgrounds[16]. We note that the ADM mass forspatially infinite objects is not well-defined[24].

However, as a weaker requirement,we will assume that the ADM procedure is valid for the mass per unit area ratherthan the mass of the domain wall. Indeed, this is the energy which is of interestsince the total mass is, by definition, infinite.Consider the supersymmetry charge densityQ[ǫ′] =Z∂Σ¯ǫ′γµνρψρdΣµν(2.2)where ǫ′ is a commuting Majorana spinor, ψρ is the spin 3/2 gravitino field, and Σis a spacelike hypersurface.

Taking a supersymmetry variation of Q[ǫ′] with respectto another commuting Majorana spinor ǫ′ yieldsδǫQ[ǫ′] ≡{Q[ǫ′], ¯Q[ǫ]}=Z∂ΣNµνdΣµν = 2ZΣ∇νNµνdΣµ(2.3)where Nµν = ¯ǫ′γµνρ ˆ∇ρǫ is a generalized Nester’s form[23]. Here ˆ∇ρǫ ≡δǫψρ =[2∇ρ + ieK2 (WPR + ¯WPL)γρ −Im(KT∂ρT)γ5]ǫ and ∇µǫ = (∂µ + 12ωabµ σab)ǫ.

In(2.3) the last equality follows from Stoke’s law.We consider an Ansatz for the space-time metric ds2 = A(z, t)(dt2 −dz2) +B(z, t)(−dx2 −dy2) characteristic of space-times with a domain wall where z isthe coordinate transverse to the wall. However, we do not assume a priori thatthe metric is symmetric about the plane z = 0.

Nor do we assume a particularbehavior of A and B at |z| →∞except that the asymptotic metric satisfies thevacuum Einstein equations with a zero or negative cosmological constant.We are concerned with supercharge density and thus insist upon only SO(1, 1)covariance in the z and t directions. This in turn implies that the space-like hyper-surface Σ in eq.

(2.3) is the z−axis with measure dΣµ = (dΣt, 0, 0, 0) = |gttgzz|12dz.The boundary ∂Σ are then the two asymptotic points z →±∞. Technical de-tails in obtaining the explicit form of eq.

(2.3) are given in reference[18] and will beomitted here.5

After some algebra, the volume integral yields:2ZΣ∇νNµνdΣµ =∞Z−∞[−δǫ′ψ†i gijδǫψj + KT ¯T δǫ′χ†δǫχ]dz(2.4)where δǫψi and δǫχ are the supersymmetry variations of the fermionic fields in thebosonic backgrounds. Upon setting ǫ′ = ǫ the expression (2.4) is a positive definitequantity which in turn (through eq.

(2.3) ) yields the bound δǫQ[ǫ] ≥0.Analysis of the surface integral in (2.3) yields two terms: (1) The ADM massdensity of the configuration, denoted σ and (2) The topological charge density,denoted C. Positivity of the volume integral translates into the boundσ ≥|C|(2.5)which is saturated iffδǫQ[ǫ] = 0. In this case the bosonic backgrounds are super-symmetric; i.e.

they satisfy δψµ = 0 and δχ = 0 (see eq. (2.4) ).

Such configurationssaturate the previous bound thus establishing their stability.2.2. Self-Dual EquationsWe now concentrate on solving for the space-time metric and matter fieldconfiguration in the supersymmetric case.

This calculation involves an analysisof the first order equations δǫψµ = 0 and δǫχ = 0 which is discussed in[18].⋆Theself-dual equation for the matter field T(z) follows from δǫχ = 0:∂zT(z) = ieiθ√AeK2 KT ¯T DTW(2.6)with a constraint on the ǫ-spinor:ǫ1 = eiθǫ∗2. (2.7)The undetermined phase eiθ is in general a space-time coordinate-dependent func-tion.Since we wish to define the ADM mass per unit area of the domain wall unam-biguously, we look for a time-independent metric solution.

For the walls studiedin[13], the resulting reflection symmetric metric is time-dependent even though the⋆We call these first order differential equations the Bogomol’nyi[25] or self-dual equations.Their square gives the classical equations of motion.6

energy-momentum tensor of the domain wall is time-independent (unless one takesa special value of mass to tension ratio that is not realized by generic field theoryexamples). With no assumed reflection symmetry of the space-time metric, a priorione cannot say if there exist nontrivial time-independent domain wall solutions.However, in order for our assumption of the time independence of the T-field tobe consistent with the Bogomol’nyi equation (2.6) , the metric component A mustbe time-independent.The self-dual equations for the metric components, following from δǫψt =δǫψx = 0, are∂zA−1/2 = ∂zB−1/2 = −κ(ie−iθ)WeκK2 .

(2.8)Since the metric functions A and B are real, the phase eiθ is required to meet alocal constraintW = −iζeiθ|W|(2.9)where ζ = ±. Assuming continuity, ζ = ± can change only at points where Wvanishes.

This connection between the metric and matter superpotential restrictsthe possible W admitting walls in the local theory. This result is in contrast tothe global case in which all W with degenerate vacua admit wall solutions.

Wecomment on this result later.δǫψz = 0 yields the differential equation for the z dependent phase θ:∂zθ = −Im(KT ∂zT). (2.10)Consistency of (2.6) , (2.8) and (2.10) with (2.9) leads to the following sufficientconditions for the existence of a static supersymmetric domain wall:Im(∂zT DTWW) = 0,(2.11)∂zT(z) = −ζ√A|W|eκK2 KT ¯T DTWW,(2.12)∂zA−1/2 = ∂zB−1/2 = κζ√A|W|eκK2 ,(2.13)as well as the explicit expression for the ADM mass density (energy per area orsurface tension) of the supersymmetric domain wall configurationσ = |C| ≡2|(ζ|WeκK2 |)z=+∞−(ζ|WeκK2 |)z=−∞| ≡2√3κ−1|∆(ζ|Λ|1/2)|(2.14)where Λ ≡−3|κWeκK2 |2 is the cosmological constant for the supersymmetric vac-uum.7

Figure 1: The path in superpotial space traversed as the scalar field interpolatesbetween degenerate vacua. The wall is realized in both the global and local theoriesfor path (A) and just for the global theory in path (B).We now comment on these equations.It follows from (2.14) that there are no static domain walls saturating theBogomol’nyi bound that interpolate between two supersymmetric vacua with zerocosmological constant.

In this case W(+∞) = W(−∞) = 0 and thus there is noenergy associated with such a domain wall since |C| ≡0. This result is in agreementwith the results of reference[13], where for infinitesimally thin domain walls withasymptotically Minkowski space-times only time-dependent metric solutions wereobtained.

The result from (2.14) implies that static supersymmetric domain wallsolutions exist only if at least one of the vacua is AdS.Eq. (2.11) is a consistency constraint which specifies the geodesic path betweentwo supersymmetric vacua in the supergravity potential space eκK2 W ∈C whenmapped from the z-axis (−∞, +∞).

This geodesic equation has qualitatively newfeatures in comparison with the geodesic equation in the global supersymmetriccase[9 ,10 ,18]. While in the global case geodesics are arbitrary straight lines in theW−plane, the local geodesic equation in the limit κ →0 (global limit of the localsupersymmetric theory) leads to the geodesic equation Im(∂zWW ) ≡∂zϑ = 0 whereW has been written as W(z) = |W|eiϑ.

This in turn implies that as κ →0 thelocal geodesic equation reduces to the constraint that W has to be a straight linepassing through the origin; i.e. the phase of W has to be constant mod π. Figure1 illustrates these points.

This observation in turn implies that the introduction ofgravity imposes a strong constraint on the type of domain wall solutions realized.In particular, domain wall solutions in the global case interpolating between vacuain the eκK2 W plane that do not lie along a straight line passing through the origindo not have an analogous solution in the local case. This result is a manifestationof the singular nature of a perturbation in Newton’s constant as seen in (2.13).

Another way to understand the inability of all global walls to be realized inthe local theory is that the space-time metric introduces an extra field degree offreedom to the local theory which allows for an extra direction to connect previouslydisconnected vacua.Eq. (2.12) for the T field (the “square root” of the equation of motion for T)and eq.

(2.13) for the metric (the “square root” of the Einstein’s equation) areinvariant under z translation as well as under rescalings of (A, B) →λ2(A, B) andz →λ−1z. Additionally, eq.

(2.12) implies that ∂zT(z) →0 as one approachesthe supersymmetric minima which are points where DTW = 0, thus indicating8

Figure 2: The projection of a typical scalar potential on a particular complexT direction indicating the supersymmetric minima (DWT = 0) between which thedomain wall interpolates. These minima are in general not degenerate.

However,since they are supersymmetric, when gravitational energy is included they becomeenergetically the same thus allowing for domain wall solutions.a solution smoothly interpolating between supersymmtric vacua. In general, thefield T reaches the supersymmetric minimum exponentially fast as a function of z.3.

Classification of the WallsWe now concentrate the equation (2.13) for the metric. Our aim is to classifyall the qualitatively different metric configurations.

First, we set A(z) = B(z)without loss of generality which implies that the metric is conformally flat. Also,we emphasize in (2.13) the singular limit when gravity is turned off(κ →0).

Asnoted earlier, the same singular limit (κ →0) is also responsible for the restrictivegeodesics in the W-plane compared to a global theory which contains no gravita-tional information (κ = 0). For κ = 0, the conformal factor factor A is constantin the whole space; i.e.

we have flat space-time everywhere. However, the momentκ > 0, A varies with z.

Thus, our aim is to study the nature of the conformal factorA(z). We classify three types of static domain wall configurations which depend onthe nature of the potential of the matter field.

For illustrative purposes to indicatethe nature of the minima between which a wall interpolates, we sketch a typicalscalar potential in Figure 2. The non-degeneracy, as emphasized throughout thepaper, is deceptive since degeneracy is based on both gravity and matter energy;the scalar potential only involves the matter part.

(I) A wall interpolating between a supersymmetric AdS vacuum (|W+∞| ̸= 0)and a Minkowski supersymmetric vacuum (|W−∞| = 0). From (2.13) one sees thaton the Minkowski side the conformal factor approaches a constant which can benormalized to unity; i.e.A(z) →1,z →−∞.

(3.1)On the AdS side A(z) falls offas z−2 with the strength of the fall-offdeterminedby the strength of the cosmological constant; i.e.A(z) →3|Λ+∞|z2,z →+∞. (3.2)The surface energy of this configuration as determined from (2.14) isσI = 2√3κ−1|Λ+∞|1/2.

(3.3)9

Here, the cosmological constant of the supersymmetric AdS vacuum is Λ+∞=−3|κWeκK2 |2+∞. (II) A wall interpolating between two supersymmetric AdS vacua and wherethe superpotential passes through zero in between.

The cosmological constant neednot be the same in both vacua. The point where W = 0 can be chosen at z = 0without loss of generality due to the translational symmetry of the system.

At thispoint ζ changes sign and thus ζ+∞= −ζ−∞= 1. The conformal factor has thesame asympotic behaviour on both sides of the domain wall:A(z) →3|Λ±∞|z2,z →±∞(3.4)while at z = 0, i.e.

when W = 0, the conformal factor levels out, i.e. ∂zA(z)z=0 = 0.In other words A(z) has a characteristic (in general asymmetric) bell-like shape.The surface energy of this configuration isσII = 2√3κ−1(|Λ|1/2−∞+ |Λ|1/2+∞)(3.5).

(III) A wall interpolating between two AdS vacua, while the superpotentialdoes not pass through zero. Again, the cosmological constant need not be thesame in both vacua.

In this case, since |W| is never zero, ζ has the same signin the whole region, say, +1. Eq.

(2.13) in turn implies that the conformal factornecessarily blows up at some coordinate z∗. In general, the matter field T has longsince interpolated between the two vacua by the time the metric reaches z∗.

Thus,the domain wall, defined as the region over which T moves from one vacuum toanother, lies entirely within the coordinate region z∗< z < +∞. The conformalfactor has the asymptotic behaviour:A(z) →3|Λ+∞|z2,z →+∞A(z) →3|Λz∗|(z −z∗)2 z →z∗.

(3.6)The surface energy of this configuration isσIII = 2√3κ−1||Λ|1/2z∗−|Λ|1/2+∞|. (3.7)Note that the point z∗is an infinite proper spatial distance away from any otherpoint z > z∗sinceRdzA1/2 →ln|z −z∗|.10

In order to understand this singularity as well as the distinctive z−2 behaviourof the conformal factor on the AdS side of a wall, it is appropriate at this pointto study AdS space-time in a coordinate system which singles out the z direction.For this purpose, we consider the metricds2 = (αz)−2(dt2 −dx2 −dy2 −dz2)(3.8)with z > 0. As noted above, this is the form of the metric on the AdS side ofthe domain wall when the T field has reached its supersymmetric vacuum.

In thiscontext, α is related to the cosmological constant by Λ = −3α2.Eq. (3.8) is the form for the metric describing AdS space-time where thetranslational invariance is broken in the z direction.The curvature tensor, bydefinition of a maximally symmetric space-time, satisfies Rµνσρ = α2(gµσgνρ −gµρgνσ).

One can represent four dimensional AdS space-time as the hyperboloidηABY AY B = α−2 embedded in the five dimensional space with flat metric ηAB =diag(+ −−−+). We found that the following choice of coordinatesY 0 = teα˜z,Y 1 = xeα˜z,Y 2 = yeα˜zY 3 = (α)−1 sinh(α˜z) −12αeα˜z(x2 + y2 −t2)Y 4 = (α)−1 cosh(α˜z) + 12αeα˜z(x2 + y2 −t2)(3.9)yield the metric intrinsic to the surfaceds2 = e2α˜z(dt2 −dx2 −dy2) −d˜z2.

(3.10)This choice of intrinsic coordinates is motivated from the cosmological form forthe metric in de Sitter space (see, for example[26]). By choosing z = α−1e−α˜z werecover the form of the metric in (3.8) .These coordinates cover one-half of the AdS manifold since Y 3 + Y 4 > 0.

Bychoosing (Y 3, Y 4, z) →(−Y 3, −Y 4, −z), we cover the Y 3+Y 4 < 0 region and havethe metric (3.8) for z < 0. This choice should be contrasted with the standardset of coordinates respecting spherical symmetry about an origin which completelycovers AdS space-time[27].

In this case the metric has the formds2 = (α cos ρ)−2(dt2c −dρ2 −sin2 ρ(dθ2 + sin2 θdφ2))(3.11)with 0 ≤ρ < π/2,0 ≤θ ≤π,0 ≤φ < 2π, and −π ≤tc ≤π.11

The time-like coordinate tc in (3.11) is a periodic coordinate. However, thecoordinates (3.9), in which time ranges over −∞< t < ∞, exhibit no periodicstructure.

What we have effectively done in choosing the planar coordinates (3.9)is to sacrifice a complete covering of AdS for a non-periodic time-like variable. Thecoordinates (3.8) are extendible whereas those of (3.11) are not.The previous discussion of the metric (3.8) now allows for a straightforwardinterpretation of the singular wall (type III) configuration.What we have isa domain wall separating two distinct regions of a generalized AdS space-timepossessing a z dependent cosmological parameter which never passes through zero.The singular point z∗corresponds to the origin z = 0 in the metric (3.8) .

On the“other side” of z∗lives an AdS space-time symmetric to the z > z∗side. Togetherthese two sides completely cover the whole of the generalzed AdS space-time justas the regions z > 0 and z < 0 in the planar coordinates leading to (3.8) cover allof AdS.4.

ExamplesThe above discussion of the three types of domain wallsœfoot In Ref. œref-markœ the stringy examples based on the SL(2, ⋄bfZ) duality symmetry of thestring theory is also discussed.

is illustrated by a simple polynomial form for thesuperpotential, a flat K¨ahler manifold: K = T ¯T, and a real T. We choose thesuperpotentialW = γT[15T 4 −13T 2(a2 + b2) + a2b2]. (4.1)where γ is a mass dimension −2 parameter which we set to unity and a2 and b2 arepositive dimension 2 parameters.

Depending on the value of the parameters a andb, the superpotential (4.1) provides us with a set of theories which accommodatethe above three classes of the domain walls.Note that the geodesic constraint Im(∂zT DT WW ) = 0 is always satisfied forT = ¯T. The supersymmetric vacuum satisfies DTW ≡WT + κKT W = 0, whereWT = (T 2 −a2)(T 2 −b2).

Thus, for a, b << 1/√κ, the supersymmetric vacua takeplace for real values of T near ±a, ±b. Figures 3, 4 and 5 display the conformalfactor A for the these three classes of the domain walls.

Each example correspondsto a different choice of the parameters a and b, which we took for simplicity to bein the range << 1/√κ.12

Figure 3: Type (I) conformal factor A(z) for a space-time with Λ−∞= 0(Minkowski: z < 0) separated by a domain wall from a space-time with Λ+∞< 0(AdS: z > 0). The wall, i.e.

the region over which the matter field T changes iscentered at z = 0 and has thickness ≈200 in √κ units. The superpotential (4.1)has parameters a2 = 0, b2 = 0.1 and T interpolates between T−∞= 0 = a andT∞= .318 ≈b.Figure 4: Type (II) Conformal factor A(z) for a space-time with negativecosmological constant separated by a domain wall from its mirror image (i.e.

a Z2configuration). The wall is centered at z = 0 and has thickness ≈200 in √κ units.The superpotential (4.1) has parameters a2 = .025, b2 = 0.1. and T interpolatesbetween T∓∞= ±.1598 ≈±a.Figure 5: Type (III) conformal factor A(z) for a space with negative cosmo-logical constant separated by a domain wall from a space with a different negativecosmological constant.

The superpotential W never passes through a zero as Tinterpolates from one vacuum to another. The domain wall is centered at z = 0and has thickness ≈200 where z is measured in √κ units.

The singularity is atz∗≈−5600. The superpotential (4.1) has parameters a2 = .025, b2 = 0.1 and Tinterpolates between T−∞= .315 ≈b and T∞= .160 ≈a.5.

Geodesic StructureWe now turn to the study of the geodesic structure for the induced space-time. To do so, we analyze the motion of test particles in the background of asupersymmetric domain wall.The motion of massless particles is trivial since the metric is conformally flat;they simply define the usual 45◦null rays in a space-time diagram.

Particles movingin constant z planes will feel no force since the conformal factor is only a functionof the transverse coordinate z. In other words, the metric is invariant under x, yboosts and thus without loss we can move to an inertial frame in which there isno motion in these directions.

Therefore, the only interesting geodesics will comefrom the 1 + 1 metric ds2 = A(z)(dt2 −dz2). For massive particles, which live ontime-like geodesics, we can parametrize the motion with the proper-time elementds2 ≡dτ2 > 0.

Rearranging the metric and introducing the conserved energy permass ǫ ≡A dtdτ of the particle yields the equation for the world-line(dzdt )2 + Aǫ2 = 1. (5.1)On a time-like geodesic, 0 ≤(dz/dt)2 < 1, and so the turning point, i.e.

v ≡dz/dt = 0, of the motion is where A/ǫ2 = 1.13

A convenient way to understand massive particle motion is to consider a par-ticle with a given initial coordinate velocity vo at some coordinate zo; from (5.1) ǫfor such a particle is ǫ2 = A(zo)(1−v2o)−1. Equation (5.1) can be thought of as theconservation of energy with an effective potential V (z) ≡(1 −v2) = A(z)A(zo)(1 −v2o).Again, points where V (z) = 1 are turning points.For particles incident upon the type I wall from the Minkowski side, passagethrough to the AdS side is always allowed.

However, the reverse motion requiresthe initial velocity to satisfy v2 > 1 −A(zo); otherwise there is a turning point andthe particle returns to the AdS side. Motion in the other domain wall backgroundsis analogous: a sketch of the effective potential V (z) = A(z)(1 −v2o)/A(zo) makesthe motion clear.One can understand the repulsive nature of these space-times on the AdS sideby calculating the force on a test particle which has a fixed position z (also knownas a fiducial observer).

This force can be obtained through the geodesic equationpαpβ;α = mfβ with pα = mdxαdτ .The gravitational force acting on the fiducialobserver isfβ = (0, 0, 0, −m2 A−2∂zA). (5.2)For a metric which falls offas on the AdS side of a wall, this force is directedtowards the AdS vacuum (e.g.

z = +∞in the type I wall depicted in figure 3).The magnitude of the acceleration is given by|a|2 ≡|fαfα|/m2 = (12∂zlnAA1/2 )2 = (κ|W|eκK2 )2. (5.3)For fiducial observers in the region where T is essentially at its vacuum value;i.e.

far away from the wall, the proper acceleration has the constant magnitude|a|2 = |Λ±∞/3|. In this region, integration of (5.1) yields the hyperbolic worldline for freely falling test particles z2 −t2 = |Λ±∞/3|−1ǫ−2.

Therefore, a fiducialobserver situated far away from a type I or II wall in a Λ±∞̸= 0 region will feel aconstant acceleration |Λ±∞/3|1/2 directed away from the wall as well as see freelyfalling test particles moving away from the wall with the a hyperbolic world line.Such a world line is also exhibited by a particle with a constant proper accelerationmoving in Minkowski space-time. These particles, known as Rindler particles[17],are not freely falling in the Minkowski background; their acceleration is provided byan external non-gravitational force.

However, we see that AdS provides preciselythis force due to the non-trivial curvature of the space-time. We add that on theMinkowski side of the walls, free test particles experience no gravitational forceeven though there is an infinite object nearby.14

One can understand the no-force result for the particles living on the Minkowskiside of the walls through the formalism of singular hypersurfaces[28]. A straight-forward calculation⋆yields a negative effective gravitational mass/area due to thewall whereas AdS has exactly the opposite positive gravitational mass.

Thus theobserver on the Minkowski side of the wall does not feel any gravitational force.This above result should be contrasted with the observation in Ref. [13], where in-finitesimally thin reflection symmetric domain walls with asymptotically Minkowskispace-times always repell the fiducial observer with a constant acceleration κσ/4.Here, σ is the energy per unit area of the domain wall.†Recall that these do-main walls always produce a time dependent metric.

In our case everything isstatic. In particular, for the type (I) domain walls interpolating between AdS andMinkowski space-times, the asymptotic acceleration on the AdS side can be writtenas a = κσI/2, where σI is the energy per unit area of the domain wall (I) definedin (3.3) .

On the Minkowski side a = 0. For the type II domain wall when thepotential has Z2 symmetry, the energy per unit area (3.5) is σII = 4κ−1|Λ±∞/3|1/2and the fiducial observer is repelled on both sides of the domain wall with the sameacceleration a±∞→κσII/4 which resembles remarkably the form for the acceler-ation for the domain walls discussed in Ref.[13].

In our case the domain wall alsorespects the Z2 symmetry, however, it is completely static and its repulsive natureis due to the AdS nature of the asymptotic space-time.6. Anti-de Sitter–Minkowski wallsIn many ways, the walls separating flat Minkowski space-time from AdS are themost interesting.

For example, it is known that AdS has the topology S1(time) ×ℜ3(space) and thus has closed time-like curves.The common remedy for suchloss of causality is to unwrap the time direction and work on the covering spaceCAdS. Nevertheless, this does not allow AdS to be globally hyperbolic; i.e.

it hasno Cauchy hypersurface and thus boundary conditions must be imposed at spatialinfinity in order to properly specify the Cauchy problem[31 ,32]. In the juxtapositionof AdS and Minkowski space-times, one must consider a proper formulation of theCauchy problem in order to quantize a field living on this manifold.

The problemsof AdS in some ways are softened by the presence of the Minkowski side, yet oneunfortunately cannot erase the problems associated with a lack of a Cauchy surface.⋆See[29] for a nice example of this formalism applied to a planar geometry. In addition,[13]employed these ideas in solving for the space-time around their domain walls.†Domain walls which separate two Minkowski vacua yet satisfy the nonstandard relationσ = 2τ, where τ is the surface tension of the wall, produce no gravitational force on testparticles.

Walls of isotropically and uniformly distributed cosmic strings produce such anequation of state[30].15

Figure 6: Penrose diagram for the covering space of the extended Minkowski-AdS domain wall system. The regions M and A are the Minkowski and AdS sidesof the wall.

The vertical line is the time-like line of the domain wall. The nullsseparating AdS patches are sights of instabilities.

The point Ωis on one such null.At this point, the observer experiences the complete history of his/her preceedingMinkowski region.As an indication of the interesting causal structure obtained through the jux-taposition of AdS and Minkowski space-times, consider the 1 + 1 Minkowski-AdS2wall and place an observer on the Minkowski side. Now allow the observer to senda moving mirror through the wall on a geodesic.

The mirror will travel on a hy-perbolic trajectory as it falls into the AdS space-time. If the Minkowski observersends massless radiation at the mirror, s/he will receive more reflected radiationout than sent in due to the coupling of the mirror to the curved space-time and theresulting particle creation[26].

In this way the Minkowski observer could deduce thestructure of the space-time on the other side of the wall. In addition, it is knownthat the energy radiated from the mirror on a hyperbolic trajectory is zero[33] andthus the stability of the wall is not compromised.Note that the moving mirror will reach the end of the coordinates z, t within afinite proper time (i.e.

these coordinates must be extended in order to cover AdS).However, the Minkowski observer considers t his/her proper time. Such behaviouris true in the full 3 + 1 case as well.

Therefore, to construct the causal structure ofthe domain wall system, we must extend the coordinates on the AdS side, whichmeans an extension onto a new half of AdS is necessary. However, by allowing formore AdS, we have added more effective gravitational mass to the system whichmust be cancelled by another identical wall.

On the other side of the wall there isanother Minkowski space. Moving to the covering space to avoid the closed time-like curves inherited from AdS gives us an infinite tower of 2-wall systems.

ThePenrose diagram for this configuration is shown in figure 6. Of particular interestis the null separating the AdS patches.

It turns out that this surface correspondsto the Cauchy horizon[20]with the metric closely related to the one at the extremalReissner Nordst¨om (RN) black-hole horizon. Note the similarity of the Penrosediagram for our domain wall and the one of the extremal (RN) black hole; theonly major difference is that in our case the singularity of the RN black-hole insidethe Cauchy horizon is replaced by an identical wall.

Space-time induced by suchdomain walls thus serves as an example of asymptotical Minkowski space-timewith Cauchy horizon but without any singularity. Further study of the phenomenaassociated with the Minkowski-AdS wall is underway[20].16

7. SummaryWe studied the field theoretic realization of a new type of domain wall.

Thesewalls separate two maximally symmetric space-times of non-positive cosmologicalconstant where one or both is AdS and the other can be Minkowski. These walls arefound in N = 1, d = 4 supergravity and saturate a positive energy/area theoremthus providing stability to the classical configuration.

We classified three canonicalsystems differing by the path in superpotential space traced out as the scalar fieldinterpolated from one vacuum to the other. Equivalently, the form of the conformalfactor on the conformally flat metric characterizes the three walls.

Examples weregiven illustrating the three walls as realized by a particular superpotential. Andthe motion of test particles living in the background space-time induced by thewalls was discussed.

Finally, we pointed out some interesting behaviour in regardto the space-times induced from the Minkowski-AdS walls.We wish to thank R. Davis, S.-J. Rey, and H. H. Soleng for enjoyable collab-orations and many enlightening discussions.

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Figure CaptionsFigure 1: The path in superpotial space traversed as the scalar field interpolatesbetween degenerate vacua. The wall is realized in both the global and local theoriesfor path (A) and just for the global theory in path (B).Figure 2: The projection of a typical scalar potential on a particular complexT direction indicating the supersymmetric minima (DWT = 0) between which thedomain wall interpolates.

These minima are in general not degenerate. However,since they are supersymmetric, when gravitational energy is included they becomeenergetically the same thus allowing for domain wall solutions.Figure 3: Type (I) conformal factor A(z) for a space-time with Λ−∞= 0(Minkowski: z < 0) separated by a domain wall from a space-time with Λ+∞< 0(AdS: z > 0).

The wall, i.e. the region over which the matter field T changes iscentered at z = 0 and has thickness ≈200 in √κ units.

The superpotential (4.1)has parameters a2 = 0, b2 = 0.1 and T interpolates between T−∞= 0 = a andT∞= .318 ≈b.Figure 4: Type (II) Conformal factor A(z) for a space-time with negativecosmological constant separated by a domain wall from its mirror image (i.e. a Z2configuration).

The wall is centered at z = 0 and has thickness ≈200 in √κ units.The superpotential (4.1) has parameters a2 = .025, b2 = 0.1. and T interpolatesbetween T∓∞= ±.1598 ≈±a.Figure 5: Type (III) conformal factor A(z) for a space with negative cosmo-logical constant separated by a domain wall from a space with a different negativecosmological constant. The superpotential W never passes through a zero as Tinterpolates from one vacuum to another.

The domain wall is centered at z = 0and has thickness ≈200 where z is measured in √κ units. The singularity is atz∗≈−5600.

The superpotential (4.1) has parameters a2 = .025, b2 = 0.1 and Tinterpolates between T−∞= .315 ≈b and T∞= .160 ≈a.Figure 6: Penrose diagram for the covering space of the extended Minkowski-AdS domain wall system. The regions M and A are the Minkowski and AdS sidesof the wall.

The vertical line is the time-like line of the domain wall. The nullsseparating AdS patches are sights of instabilities.

The point Ωis on one such null.At this point, the observer experiences the complete history of his/her preceedingMinkowski region.20

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