A stable vacuum of the tachyonic E8 string
aa r X i v : . [ h e p - t h ] A p r arXiv:0710.1628 [hep-th] A stable vacuum of the tachyonic E string Simeon Hellerman and Ian Swanson
School of Natural Sciences, Institute for Advanced StudyPrinceton, NJ 08540, USA
Abstract
We consider tachyon condensation in unstable ten-dimensional heterotic string theory withgauge group E . In the background of a lightlike linear dilaton rolling to weak coupling,we find an exact solution in which the theory decays to a stable ground state. The finalstate represents a new, modular-invariant perturbative string theory, tachyon-free in ninespacetime dimensions with a spacelike dilaton gradient, E gauge group and no spacetimesupersymmetry.October 9, 2007 ontents E string in 10 dimensions 4 T . . . . . . . . . . . . . . . . . . . . 124.2 Effective worldsheet theory at large X + . . . . . . . . . . . . . . . . . . . . . 144.3 Dynamical readjustment of the metric and dilaton gradient . . . . . . . . . . 144.4 Stability properties of the tachyon solution . . . . . . . . . . . . . . . . . . . 16 Introduction
Weakly coupled heterotic string backgrounds with ten-dimensional Poincar´e symmetry wereclassified some time ago [1–8]. Among them is the unstable heterotic string with a single E gauge group realized on the worldsheet as a level-two current algebra (hereafter, theUHE string). This background has been the subject of much interest. In contrast to thesupersymmetric 10D heterotic string with E × E gauge group, the UHE string has a singlereal tachyon T and broken supersymmetry.One prominent role for the UHE string emerges from the study of spacetime-destroyingdecay modes of nonsupersymmetric M-theory backgrounds [9]. The supersymmetric het-erotic background has been interpreted as the R → R , on which the 11-dimensional gravitino has su-persymmetric boundary conditions. Changing the boundary condition so that the gravitinohas half-integral rather than integral moding on the interval produces a nonsupersymmetrictheory in 10 dimensions, with a nonzero Casimir energy causing the two endpoints of theinterval to attract one another.In [9], Fabinger and Hoˇrava proposed that the UHE background could be a limit ofsuch a nonsupersymmetric configuration, following a phase transition in which the E × ¯ E gauge group is broken spontaneously to a diagonal subgroup. It was also conjectured thatthe remaining singlet tachyon can condense, generating a “decay to nothing,” in the senseof [10–17]. Utilizing and extending the methods of [18], an exact solution was studied recentlyby Hoˇrava and Keeler [19, 20] describing such a decay in the background of a lightlike lineardilaton rolling to weak coupling in the future. In this paper we follow the logic of [18, 21–24] to analyze the condensation of the singlettachyon of the UHE theory. We show that, in addition to the solution studied in [19, 20],there exists another exact solution with the same initial state, leading to a notably differentoutcome. Our solution is dimension-changing in the sense described in [21], with a final statedescribed by a stable string theory in nine dimensions with no spacetime supersymmetry.We analyze the UHE theory in the background of a lightlike linear dilaton rolling toweak coupling in the direction X − ≡ √ ( X − X ). This theory admits deformations with atachyon T growing exponentially in the complementary lightcone direction X + ≡ √ ( X + X ). These deformations are exact solutions to the beta-function equations, and the motion The first exact heterotic bubble of nothing solution was written in [21].
2f a string in this theory is integrable. It has been suggested [18] that such null propagationof the tachyon arises automatically from the evolution of localized field inhomogeneities inthe background of a linear dilaton.The organization of the paper is as follows. After reviewing the UHE theory and thegeneral properties of tachyon condensation in Section 2, we consider two cases: first, wereview a solution in which the tachyon T depends only on the lightcone direction X + ; second,we go on to consider a more generic case, in which the tachyon profile is not assumed topreserve any special symmetry. We review the more symmetric case (studied in [19, 20]) inSection 3, where 8D Poincar´e invariance is imposed on the directions X i transverse to thelightcone. Under this restriction, the most generic solution to the equation of motion is ofthe form Φ = − q √ X − , T = µ exp ( βX + ), with qβ = √ /α ′ , possibly including fields thatdecay to zero at late times.In Section 4 we break Poincar´e invariance in the eight directions X i to obtain late-timeendpoints that differ qualitatively from the bubble of nothing. If the real tachyon T isallowed to depend in an arbitrary way on X i , T generically develops zeroes along loci of realcodimension one in the eight directions spanned by X i . To simplify the analysis, we take thelimit in which the field T varies on long distance scales in the X i directions, as comparedto the string scale. In this limit, each component of the zero locus is well approximated bya flat, isolated component of codimension one, with the tachyon T varying linearly in thedirection transverse to the component. We show that the physics of our solution is that of dimension quenching [21], where the number of spacetime dimensions reduces dynamicallyfrom ten to nine. In the limit X + → ∞ , strings propagate in only 8+1 dimensions, confinedto the locus T = 0 by a potential barrier that becomes infinitely steep and high.In Section 5, we show that the stable, late-time limit is described by a novel perturbativeheterotic string theory with E gauge symmetry, a linear dilaton gradient in a spacelikedirection, and no spacetime supersymmetry in nine dimensions. To simplify the exposition,we refer to this nine-dimensional heterotic theory as the HE9 theory. We calculate theone-loop partition function in the HE9 theory and verify that it is indeed modular invariant.Unlike its 10D parent, the 9D heterotic theory supported inside the bubble is stable . It followsthat the dynamical dimension-reducing solution is indeed generic in the space of solutions:any normalizable on-shell perturbation decays or disperses at late times. Section 6 containsconclusions and general comments. 3 Review of the unstable E string in 10 dimensions In this section we review several properties of the UHE string theory that are salient to thepresent discussion. We establish conventions, present the worldsheet supersymmetry algebraand outline the free-fermion construction of the level-two E current algebra. The UHE string [1, 9], like all perturbative heterotic string theories, is described in (su-per)conformal gauge as a (0,1) superconformal field theory (SCFT). There are ten embed-ding coordinates X µ , transforming in the standard way under 10D Poincar´e invariance. Eachembedding coordinate is paired with a right-moving worldsheet superpartner ψ µ , and thetotal central charge of the right-moving SCFT is equal to c R = 15.The left-moving side has an E current algebra at level two, and a single Majorana-Weylfermion ˜ λ . The current algebra has central charge c alg = 31 /
2, and the fermion ˜ λ contributescentral charge 1 /
2, so the total central charge of the left-moving side, including the tenbosonic coordinates X µ , is c L = 26. The level-two E current algebra has a free-fermionrepresentation based on 31 left-moving Majorana-Weyl fermions ˜ λ A .The worldsheet degrees of freedom transform as follows under the (0 ,
1) worldsheet su-persymmetry: [
Q, X µ ] = i r α ′ ψ µ , { Q, ψ µ } = r α ′ ∂ + X µ , { Q, ˜ λ } = F , [ Q, F ] = i ∂ + ˜ λ , { Q, ˜ λ A } = F A , [ Q, F A ] = i ∂ + ˜ λ A . (2.1) To better understand the discrete gauge symmetry of the UHE string, we review the free-fermion construction of the E current algebra at level two. It was observed in [1] that thereexists a discrete symmetry ( ZZ ) L (where the subscript L indicates the group acting onlyon left-moving excitations) acting on a set of 31 free fermions ˜ λ A , such that gauging ( ZZ ) L E . With ( ZZ ) L taken to be generated by g , · · · , g , the action of ( ZZ ) L on left-moving fermions is specified by: g = σ ⊗ ⊗ ⊗ ⊗ , g = 1 ⊗ σ ⊗ ⊗ ⊗ ,g = 1 ⊗ ⊗ σ ⊗ ⊗ , g = 1 ⊗ ⊗ ⊗ σ ⊗ ,g = 1 ⊗ ⊗ ⊗ ⊗ σ , (2.2)where these matrices act in the basis (˜ λ, ˜ λ , · · · , ˜ λ ). The single fermion ˜ λ does not partic-ipate in the current algebra, and is neutral under all elements of ( ZZ ) L .The generators g i act with a minus sign on subsets of (˜ λ, ˜ λ A ) in blocks of 16 at a time.As a result, every sector twisted by an element of ( ZZ ) L is level-matched. As usual, we alsogauge the operation ( − F w , which acts on all left- and right-moving fermions simultaneouslywith a −
1. The action of ( − F w defines an R-parity, meaning that it acts with a − G ( σ + ).There are no left-moving currents in the untwisted sectors, since any fermion bilinear(without derivatives) is odd under at least some element in ( ZZ ) L . All E currents comefrom the twisted sectors. Each of the 31 non-identity elements of ( ZZ ) L defines a twistedNS sector in which 16 of the 32 current algebra fermions are periodic. Prior to imposingprojections, the number of fermion ground states is 2 = 256. Imposing five independent g i projections cuts the number of ground states by a factor of 2 , leaving eight ground statesin each twisted sector, for a total of 8 ·
31 = 248. The ground-state weight in each of thetwisted sectors is from 16 periodic fermions, so there are exactly 248 weight-one currents,generating the 248-dimensional Lie algebra E . The remaining fermion ˜ λ is invariant under( ZZ ) L and transforms as an E singlet, but it plays a role as the left-moving matter part ofthe tachyon vertex operator. We can summarize the spectrum of the UHE string by computing its one-loop partitionfunction. We will take the standard notation for the path integral on a torus of complexstructure τ , with two real fermions transforming with signs of ( − a +1 and ( − b +1 aroundthe spacelike and (Euclidean) timelike cycles, respectively. The partition functions are: Z ab ≡ η ( τ ) θ ab (0 , τ ) , ˜ Z ab ≡ η (¯ τ ) θ ab (0 , ¯ τ ) , (2.3)5or right- or left-movers, respectively.In the untwisted sector – that is, with no action of ( ZZ ) L on fermions when transportedaround the spacelike cycle – the partition functions for the left-moving fermions (˜ λ, ˜ λ A ) are L . ) (¯ τ ) = 132 (cid:16) ˜ Z (cid:17) (cid:20)(cid:16) ˜ Z (cid:17) + 31 (cid:16) ˜ Z (cid:17) (cid:21) ,L . ) (¯ τ ) = 132 (cid:16) ˜ Z (cid:17) (cid:20)(cid:16) ˜ Z (cid:17) + 31 (cid:16) ˜ Z (cid:17) (cid:21) ,L . ) (¯ τ ) = 132 (cid:16) ˜ Z (cid:17) ,L . ) (¯ τ ) = 0 . (2.4)Here we have averaged over insertions of elements of ( ZZ ) L to implement the projection ontoinvariant states. In the sectors twisted by a nontrivial element g ∈ ( ZZ ) L , we have L (¯ τ ) = 132 (cid:16) ˜ Z (cid:17) (cid:16) ˜ Z (cid:17) ,L (¯ τ ) = 132 (cid:16) ˜ Z (cid:17) (cid:16) ˜ Z (cid:17) ,L (¯ τ ) = 132 (cid:16) ˜ Z (cid:17) (cid:20)(cid:16) ˜ Z (cid:17) + (cid:16) ˜ Z (cid:17) (cid:21) ,L (¯ τ ) = 0 . (2.5)Summing over twisted sectors, we obtain L (¯ τ ) = 132 (cid:16) ˜ Z (cid:17) (cid:20)(cid:16) ˜ Z (cid:17) + 31 (cid:16) ˜ Z (cid:17) + 31 (cid:16) ˜ Z (cid:17) (cid:21) ,L (¯ τ ) = 132 (cid:16) ˜ Z (cid:17) (cid:20)(cid:16) ˜ Z (cid:17) + 31 (cid:16) ˜ Z (cid:17) + 31 (cid:16) ˜ Z (cid:17) (cid:21) ,L (¯ τ ) = 132 (cid:16) ˜ Z (cid:17) (cid:20)(cid:16) ˜ Z (cid:17) + 31 (cid:16) ˜ Z (cid:17) + 31 (cid:16) ˜ Z (cid:17) (cid:21) ,L (¯ τ ) = 0 . (2.6)6s usual, the partition function for the right-moving fermions and superghosts is ( Z ab ) .The full path integrals for the fermions and superghosts are then F ab ( τ, ¯ τ ) ≡ ( Z ab ) L ab , (2.7)where we have summed over twisted sectors and projected onto ( ZZ ) L -invariant states. Under τ → τ + 1, the nonvanishing F ab transform as F → − F , F → − F , F → + F , (2.8)while under τ → − /τ , the F ab transform classically: F ab → F ba . (2.9)The path integrals in the NS ± and R ± sectors are therefore I NS ± ≡ (cid:0) F ∓ F (cid:1) , (2.10)where the sign flip comes from the odd fermion number of the superghost ground state inthe NS sector. The partition function in the Ramond sector is I R+ = I R − = 12 F . (2.11)The path integral over bosons and reparametrization ghosts is independent of the sector,and equal to iV (4 π α ′ τ ) − | η ( τ ) | − , (2.12)for ten free embedding coordinates, where V is the (infinite) volume of the flat ten-dimensional spacetime. Multiplying the two partition functions against the factor of dτ d ¯ τ / τ from the gauge-fixed path integral measure [25] and integrating over the modular fundamen-tal region F , we find Z F dτ d ¯ τ π α ′ τ (4 π α ′ τ ) − | η ( τ ) | − ( I NS+ − I R ± ) , (2.13)with the minus sign in front of I R ± implementing the fermionic spacetime statistics of theRamond states. 7he measure dτ d ¯ τ /τ is itself modular invariant, as is the combination τ | η ( τ ) | . Thefermionic partition functions transform under τ → τ + 1 as I NS ± → ± I NS ± , I R ± → + I R ± . (2.14)Since the F ab transform classically under τ → − /τ , (2.9) the combination I NS+ − I R ± = 12 (cid:0) F − F − F (cid:1) (2.15)remains unchanged. It follows that the integral (2.13) is modular invariant, for either choiceof GSO projection in the Ramond sectors.The factor iV (4 π α ′ τ ) − in the modular integrand comes from the integral over bosoniczero modes: i.e., the momentum integral for physical states. Removing this and the measurefactor, and performing the τ integral that implements level matching, we obtain the partitionfunction for the masses of physical states: Z mass ( τ, ¯ τ ) ≡ X physicalstates ( − F S exp (cid:0) − πα ′ m τ (cid:1) = Z dτ | η ( τ ) | − ( I NS+ − I R ± ) ≡ Z NSmass ( τ, ¯ τ ) − Z Rmass ( τ, ¯ τ ) . (2.16)The expansion of these functions yields Z NSmass ( τ ) = ( q ¯ q ) − + 2 ,
048 + 148 ,
752 ( q ¯ q ) + O (cid:0) ( q ¯ q ) (cid:1) ,Z Rmass ( τ ) = 3 ,
968 + 9 , ,
416 ( q ¯ q ) + O (cid:0) ( q ¯ q ) (cid:1) , (2.17)where we have defined q ≡ exp (2 πiτ ).The first term in the NS partition function corresponds to the tachyon T in the spectrum,with a single state at mass m = − α ′ . The second term corresponds to several differentmassless fields, including the graviton G µν , with 35 polarizations, the B-field B µν with 28polarizations, the dilaton with 1 state, and the E gauge field A µ with 8 ·
248 = 1,984 masslessstates, for a total of 2,048 massless states. The next term represents the first massive level,with 148,752 states at m = α ′ .The leading term in the Ramond partition function represents a pair of Majorana-Weylfermions (one of each chirality), each in the adjoint of E , for a total of 2 · ·
248 = 3,968physical states at m = 0. The next term appears at m = α ′ . In particular, there is noroom for a gravitino, which, if present, would represent another 56 physical states at m = 0.8 The Hoˇrava-Keeler bubble of nothing
In this section we review the Hoˇrava-Keeler exact bubble of nothing, with the intent ofcontrasting it with the dimension-reducing solution described in the next section.
We consider the heterotic string in a lightlike linear dilaton background, with dilaton gradientgiven by V + = V i = 0 and V − = − q/ √
2. We take q to be positive, but its magnitude isframe dependent: assuming it is positive, we can boost it to any other positive value undera Lorentz transformation.The worldsheet theory is conformally invariant, with a free, massless Lagrangian givenby L kin = 12 π G µν (cid:20) α ′ ( ∂ + X µ )( ∂ − X ν ) − iψ µ ( ∂ − ψ ν ) (cid:21) − i π ˜ λ A ( ∂ + ˜ λ A ) − i π ˜ λ ( ∂ + ˜ λ ) . (3.1)The dilaton coupling takes the form L dilaton = 14 π (cid:18) Φ − q √ X − (cid:19) R (2) , (3.2)where R (2) is the worldsheet Ricci scalar, and Φ is the value of the dilaton at X − = 0.To allow the closure of worldsheet supersymmetry off shell, we supplement the action withkinetic terms for the nondynamical auxiliary fields F and F A : L aux = 12 π (cid:0) F + ( F A ) (cid:1) . (3.3) To obtain the Hoˇrava-Keeler solution, we must deform the lightlike linear dilaton backgroundby letting the tachyon acquire a nonzero value obeying the equations of motion. The singlereal tachyon T is a singlet of E , and couples to the worldsheet as a superpotential W ≡ ˜ λ : T ( X ) : , (3.4)9here the component action comes from integrating the superpotential over a single Grass-mann direction θ + (to form a (0 ,
1) superspace integral):∆
L ≡ − π Z dθ + W = − π F : T ( X ) : − i r α ′ ∂ µ T ( X ) : ˜ λψ µ ! . (3.5)Integrating out F yields a potential of the form∆ L = − π : T ( X ) : , (3.6)and a modified supersymmetry transformation { Q, ˜ λ } = F = : T :.The linearized equation of motion for the tachyon is ∂ µ ∂ µ T − V µ ∂ µ T + 2 α ′ T = 0 . (3.7)The Hoˇrava-Keeler solution takes the tachyon gradient to lie in a lightlike direction: T = µ exp (cid:0) βX + (cid:1) , (3.8)with qβ = √ /α ′ . For diagrammatic reasons (discussed in [18, 21, 22]), the worldsheettheory defined by this superpotential has exactly vanishing beta function, so the tachyonprofile represents a solution beyond linearized order.The bosonic potential on the string worldsheet is positive-definite and increasing expo-nentially toward the future:∆ L = − π µ exp (cid:0) βX + (cid:1) + i π r α ′ βµ exp (cid:0) βX + (cid:1) ˜ λψ + . (3.9)The classical worldsheet potential of this model is identical to that described in [18, 26, 27].Consequently, classical solutions for strings moving in the UHE bubble of nothing are thesame as those in the bubble of nothing in Ref. [18]. That is, the solution describes a Liouvillewall moving at the speed of light in the negative X direction. Every string state eventuallymeets the wall and is accelerated outward to the left.The worldsheet theory of the UHE bubble of nothing differs from that of the bosonic bub-ble of nothing due to the presence of worldsheet fermions. However, it is not straightforwardto understand the physical meaning of quantum effects contributed by the fermionic degrees10f freedom. Fermionic interactions are only supported in the region of positive X + , wherestring states are energetically forbidden from penetrating. In [19, 20], worldsheet techniqueswere developed for studying the UHE bubble of nothing deep inside the tachyon condensate.In particular, the authors of [19, 20] made a non-standard gauge choice for the local world-sheet supersymmetry in which the region of nonzero tachyon condensate could be studiedwith greater ease. The aim of [20] was to understand the “nothing phase” of the UHE string,possibly in terms of a conjectured topological phase of M-theory [28] describing the regionbehind the Hoˇrava-Witten wall [29].Our focus is more pedestrian. The easiest observables to understand are associated withregions whose future infinity has vanishing potential on the string worldsheet. In the nextsection, we will focus on solutions to the UHE string that contain such regions. We studya different class of exact bubble solutions to the UHE string theory, where the interior ofthe bubble supports a stable, nine-dimensional string theory rather than a topological or“nothing” phase. The bubble of nothing solution described above imposes 8D Poincar´e symmetry in the direc-tions transverse to the lightcone defined by the dilaton gradient V µ . In this section we relaxthis assumption and examine solutions that break the 8D Poincar´e symmetry. Within thismore general ansatz there are exact classical solutions of the UHE model where the totalnumber of spacetime dimensions decreases from 9+1 to 8+1. These solutions are qualita-tively similar to the decays of the type HO + background [30] (i.e., the unstable supercriticalheterotic model with nondiagonal GSO projection and gauge group SO (32) × SO ( D − +(1) string described in [30], with the exception that in the present case thetachyon and dilaton gradients are lightlike, rather than timelike.Our initial conditions are such that the (9 + 1)-dimensional theory at X = −∞ hasFRW spatial slices in the form of the maximally Poincar´e-invariant nine-dimensional IR .The final theory will be a stable heterotic theory with gauge group E (at level two) in 8 + 111imensions, with spacelike dilaton gradient and flat string-frame metric. The solutions westudy are exact to all orders in α ′ . Furthermore, there is a readjustment of the dilaton andthe string-frame metric that comes from a one-loop effect on the worldsheet, just as in thedimension-changing transitions discussed in [21]. T At this point we allow the tachyon T to vary in the eight dimensions X , . . . , X transverseto the lightcone directions X ± . We assume that the tachyon has a smooth vanishing locus T = 0, and we take the limit in which the typical scale of variation | k − | − of the tachyonis large compared to the string length √ α ′ . We can then approximate the tachyon as alinear function of the direction normal to the locus T = 0. Having done so, we can alwaysperform an 8D Poincar´e transformation to set the zero locus precisely at X = 0. We loseno generality, then, by taking T ( X ) = r α ′ exp (cid:0) βX + (cid:1) (cid:2) µX + O ( kX ) (cid:3) , (4.1)with qβ = √ α ′ . (4.2)Taking the long-wavelength limit and assuming a smooth vanishing locus for the tachyonamounts to dropping the O ( kX ) terms. The superpotential is then of the form W = µ r α ′ exp (cid:0) βX + (cid:1) ˜ λ X , (4.3)which means that the interaction Lagrangian equals L int = − µ πα ′ exp (cid:0) βX + (cid:1) : X : + iµ π exp (cid:0) βX + (cid:1) ˜ λ (cid:0) ψ + βX ψ + (cid:1) . (4.4)This superpotential is the same as that discussed in the heterotic section (Section 3) of [21],restricted to the case in which a single real tachyon acquires a vev.12he equations of motion take the form (with i = 2 , · · · , ∂ + ∂ − X + = ∂ − ψ + = ∂ + ∂ − X i = ∂ − ψ i = ∂ + ˜ λ A = 0 ,∂ + ∂ − X = − µ (cid:0) βX + (cid:1) X + iµβα ′ (cid:0) βX + (cid:1) ˜ λ ψ + ,∂ + ∂ − X − = µ β (cid:0) βX + (cid:1) X − iµβα ′ (cid:0) βX + (cid:1) ˜ λ (cid:0) ψ + β X ψ + (cid:1) ,∂ + ˜ λ = µ (cid:0) βX + (cid:1) (cid:0) ψ + βX ψ + (cid:1) ,∂ − ψ − = βµ (cid:0) βX + (cid:1) X ˜ λ . (4.5)This 2D worldsheet theory is integrable at both the classical and quantum levels. As aresult, the properties of string trajectories in this theory are particularly simple, and can besummarized as follows: • In infinite worldsheet volume, the general classical solution can be written in closedform. • In finite worldsheet volume, trajectories with X ± , ψ ± independent of the spatial world-sheet coordinate σ can be written in a simple closed form involving Bessel functions(see, e.g., Eqn. (2.17) of Ref. [21]). The Virasoro constraints and null-state gaugeequivalences are sufficient to put a general physical solution into such a form. • The classical and quantum behavior of these solutions can be understood from generalprinciples, using the adiabatic and virial theorems. The excitation numbers in themassive modes X , ψ , ˜ λ are frozen into constant values at late times. • String behavior relative to the bubble wall at X + ∼ X + → ∞ phase if theoccupation numbers in the massive modes are all equal to zero. Otherwise, the stringis pushed outward along the bubble wall at X ∼ − X , accelerated to the speed oflight. 13 .2 Effective worldsheet theory at large X + As with the examples studied in [18, 21–24], the properties that render the theory exactlysolvable at the classical level also make the quantum theory particularly simple. Namely, allconnected correlators of free fields have quantum perturbation expansions that terminate atone-loop order. The structure of the quantum corrections can be summarized as follows: • The two-point function for the light-cone fields X ± and their superpartners ψ ± areproportional to G µν , so the propagators only connect “+” fields to “ − ” fields. Prop-agators for the massless X ± multiplets are therefore oriented, and we represent themas dashed lines with arrows pointing from + to − . The massive multiplet X , ψ , ˜ λ is correlated with itself, so its propagators are represented by solid, unoriented lines. • Fundamental vertices representing the classical potential and Yukawa couplings havearbitrary numbers of outgoing dashed lines emanating from them, and exactly twosolid lines. • Every connected tree diagram with multiple vertices therefore has the structure of anordered sequence of vertices with a single solid line passing through, and arbitrarynumbers of dashed lines emanating from each vertex. • Interaction vertices have only outgoing (dashed) lines, and no two vertices can beconnected with a dashed line. A connected Feynman diagram can have either zero orone loop, but not two loops or more. • Connected loop diagrams consist strictly of a closed solid line with dashed lines emanat-ing from an arbitrary number of points. This exhausts the set of connected Feynmandiagrams in the theory, and every connected correlator is exact at one-loop order.
Now we would like to study the dynamics of states that penetrate into the far interior of thebubble, at X + → ∞ . These strings necessarily have all massive modes in their ground states,frozen out with exponentially increasing mass. It is possible to integrate out X , ψ and ˜ λ exactly to obtain an effective action for the remaining worldsheet degrees of freedom. All buta finite number of terms have canonical dimension greater than 2, coming from derivatives14nd fermions. By scale invariance, such terms always appear dressed with real exponentialsof X + , with negative exponent. We are ultimately interested in the limit X + → ∞ , so allterms in the action of canonical dimension greater than two can be ignored. Furthermore, noterms of canonical dimension less than two are generated, since there are no such operatorsthat are invariant under the ( ZZ ) L symmetry and (0 ,
1) supersymmetry.The only terms generated that survive the X + → ∞ limit are thus a renormalization ofthe Ricci term √ g R (2) , and a renormalization of the kinetic term for X + , ψ + . In the languageof spacetime physics, these represent dynamical readjustments of the string-frame metric andthe dilaton due to the backreaction of the condensing tachyon. As described in [21], thesereadjustments arise entirely from one-loop renormalizations on the string worldsheet, so theyare exactly calculable. The relevant diagrams are depicted in Fig. 1, where solid lines aredrawn to indicate massive fields, and dashed oriented lines indicate the massless X ± fields.We will refer to quantities in the final-state theory with a hatted notation. The readjustedmetric and dilaton are thus denoted by ˆ G and ˆΦ, respectively, with the gradient of thereadjusted dilaton denoted by ˆ V µ . ∆( ∂ + Φ) = (cid:1) ∆ G ++ = (cid:2) Figure 1: Diagrams contributing to the nonvanishing renormalizations of the dilaton andmetric, ∆( ∂ + Φ) and ∆ G ++ . Solid lines indicate massive fields, while dashed oriented linesrepresent propagators of the massless lightcone fields X ± .The calculation of the renormalized worldsheet couplings is straightforward (see Refs. [21,31] for details). Defining M ≡ µ exp ( βX + ), the dilaton readjustment is∆Φ = 14 ln (cid:18) M ˜ µ (cid:19) , (4.6)where ˜ µ is an arbitrary mass scale entering the definition of the path integral measure. The15enormalization can be expressed as∆Φ = (const . ) + β X + , ∆ V + = β . (4.7)As noted, there is also a nonzero renormalization of the string-frame metric. For a generalizedmass term πα ′ M ( X ) X , where M ( X ) depends arbitrarily on all coordinates other than X , the metric G µν is renormalized by an amount∆ G µν = α ′ ∂ µ M ∂ ν MM . (4.8)For M = µ exp ( βX + ), this gives the renormalized metricˆ G ++ = − ˆ G −− = α ′ β , (4.9)with all other components unrenormalized.The linear dilaton central charge at X + → ∞ is therefore given by (including the renor-malized dilaton gradient ˆ V µ ): c dilaton = 6 α ′ ˆ G µν ˆ V µ ˆ V ν = 3 qβα ′ √ − β q α ′ . (4.10)With qβ = √ /α ′ , the final dilaton contribution to the central charge is c dilaton = 32 . (4.11)We therefore find that exactly 3 / X , ψ , ˜ λ into the strength of the dilaton gradient at X + = ∞ . The total centralcharge, including free-field and dilaton contributions, is again the same at X + = ∞ as at X + = −∞ , and in particular equal to (26 , We will demonstrate explicitly in Section 5 that the nine-dimensional string theory describingthe final state of the transition is stable, meaning it has no spatially normalizable perturba-tions that grow with time. For the purposes of the discussion in this subsection, however, wewill assume the stability of the final state. No normalizable perturbation of the transitionas a whole should be able to affect the qualitative nature of the endpoint: a perturbation of16he full, time-dependent background will always evolve into a perturbation of the late-timebackground. The latter is stable, so the late-time limit of such a perturbation cannot lead toa further instability changing the phase of the final state. In Ref. [23], transitions having thisproperty were referred to as stable transitions , and many of the transitions studied in [21] arein fact stable. In particular, the transitions whose final states are spacetime-supersymmetricor two-dimensional are such that a generic perturbation (satisfying the appropriate normaliz-ability condition) of the full, time-dependent solution will not alter the solution qualitativelyat late times. In this subsection we wish to refine this classification by drawing a furtherimportant distinction between different types of stable transitions. For purposes of our dis-cussion, we will consider bubble of nothing solutions to be stable transitions, in that nonormalizable perturbation changes their qualitative behavior.
Absolute vs. local stability
We first define absolutely stable transitions as those that exhibit a stable final endpoint whosequalitative nature is completely determined in advance by the nature of the initial state. Forinstance, the transitions from supercritical type HO + / (which has diagonal GSO projection)on certain orbifolds to supersymmetric type HO string theory in ten dimensions have theproperty of absolute stability. Even a large, non-infinitesimal change in the initial configu-ration of the tachyon will not result in any final state other than a single supersymmetrictype HO theory in ten dimensions. This supercritical orbifold does have other possible end-states. For instance, it can fragment into disconnected baby universes, of which preciselyone supports supersymmetric type HO theory, with the others containing unstable type HO / in various dimensions. However, such alternate endpoints are always fine-tuned. Untunedtransitions lead to a unique and universal final state: a single component, supporting su-persymmetric type HO in ten dimensions. A phase space of absolute-stable transitions isdepicted in Fig. 2.We define a second type of stable transition as possessing the weaker property of being locally stable in the space of solutions. That is, a linearized perturbation around a locallystable solution will always preserve the qualitative nature of the final state. However, asufficiently large deformation of a locally stable solution can lead to a qualitatively differentbut stable outcome. A model phase space of locally stable transitions is presented in Fig. 3.The string theories of type HO + discussed in [30] were shown to have the property of local17 on−genericunstableendpointsUniversal stable endpoint Figure 2: Schematic phase diagram of endpoints in absolute-stable transitions. Contoursrepresent loci of non-generic unstable endpoints of the transition, which are always fine-tuned. Untuned transitions land in a universal stable theory, away from the loci of unstableendpoints.stability, but not absolute stability. Starting with type HO + in supercritical dimensionsleads generically to a number of disconnected universes supporting stable, supersymmetrictype HO string theory. However, the number of such universes, and the chirality of thegravitini and gaugini in the baby universes, depends on the number of zeroes of the tachyonconfiguration, as well as on tachyon derivatives at the vanishing points. In some open setsof tachyon configuration space there are no zeroes at all, and the universe is destroyed fromwithin by a bubble of nothing. Local stability of dimension-changing type UHE transitions
We now wish to demonstrate that our dimension-reducing solutions of UHE string theoryare locally stable, rather than absolutely stable transitions. Consider a tachyon profile ofthe form T ( X ) ≡ µ exp ( βX + ) f ( X i , X + ), with i = 2 , · · · ,
9, and f ( X i ,
0) taken to be acompletely arbitrary real function. We assume that : T ( X ) : is an operator of definiteweight ( , ), which means that f ( X i , X + ) satisfies the diffusion equation ∂ + f = βα ′ ∂ i f . (4.12)For µ exp ( βX + ) ≫
1, the background is described by a string theory in nine dimensions,18 on−generic endpointsUnstable, fine−tuned,Inequivalent stable endpoints
Figure 3: Phase diagram of endpoints in locally stable transitions. Contours depict loci ofnon-generic, fine-tuned unstable endpoints of the transition. The regions bounded by theseloci indicate stable but inequivalent endpoints. For these purposes, the “nothing” state willbe considered a stable endpoint, in that nearby solutions converge to it at large X + .supported on the vanishing locus of f . For √ α ′ ≫ | ∂ i f | , higher orders of conformal pertur-bation theory can be neglected, and the leading order, given by equation (4.12), is a goodapproximation to the dynamics. Using equation (4.12), we demonstrate that two initialtachyon profiles can lead to different outcomes for the final behavior of the system.Consider one initial profile, given by f ( X i ,
0) = X + X − R , where we assume R ≫ α ′ . Given this initial condition, the subsequent evolution of the tachyon profile is f ( X i , X + ) = 1 α ′ (cid:0) X + X − R + 2 βα ′ X + (cid:1) . (4.13)At a particular lightcone time X +crit = R / (2 βα ′ ), the circle (defined by T = 0) shrinks tozero size, and the minimum of the classical worldsheet potential rises above zero: V ws = T = µ exp (cid:0) βX + (cid:1) f = µ α ′ exp (cid:0) βX + (cid:1) (cid:2) X + X + 2 βα ′ ( X + − X +crit ) (cid:3) , (4.14)which is greater than 4 β µ ( X + − X +crit ) when X + > X +crit . A positive-definite worldsheetpotential increasing exponentially with lightcone time has qualitative behavior akin to the19ubble of nothing described previously. It does not describe the static (8 + 1)-dimensionalheterotic string theory at large X + , or anything in the universality class thereof.For a second initial profile, one may change the sign of the X term from +1 to − a , with a >
1. The tachyon profile then evolves as: f ( X i , X + ) = f ( X i ,
0) = 1 α ′ (cid:2) X − aX − R + (1 − a ) βα ′ X + (cid:3) . (4.15)The effective value of R increases as X + → ∞ , so the qualitative behavior of the back-ground at late times is that of the stable nine-dimensional heterotic theory propagating intwo disconnected baby universes, described by the two branches of the hyperbola in the X , plane where f ( X i , X + ) vanishes.This establishes that two simple choices of initial conditions in the same linear dilatonbackground of the same theory can lead to qualitatively different behaviors at late times,each of which is stable under small perturbations. In one case, we have a universe-destroyingbubble of nothing. In the second case we obtain a bubble of new vacuum in which the universebifurcates into two stable nine-dimensional components. This indeterminate behavior, withmultiple possible stable endpoints following from the same initial starting point, is suggestiveof a randomly populated landscape of vacua. We have established that a final state of our dimension-reducing solution describes a nine-dimensional theory, which we refer to as the HE9 theory. In the transition, the fields X , ψ and the neutral fermion ˜ λ have decoupled from the worldsheet, and the dilaton gradientand string-frame metric have acquired renormalizations associated with one-loop quantumcorrections to the free worldsheet theory. As in the cases studied in [21–24], the effect is totransfer the central charge contributions of the decoupled worldsheet degrees of freedom intothe strength of the dilaton gradient, keeping the total central charge constant.In this section we analyze directly the HE9 final state of our solution. The final vacuumhas a flat string-frame metric and linear dilaton with spacelike gradient. Furthermore, thefinal vacuum has no tachyon degree of freedom. The lightest excitations are the metric ˆ G ,an NS two-form ˆ B , the dilaton ˆΦ, an E gauge field ˆA and a fermion ˆΛ transforming asa 16-real-dimensional Majorana spinor of SO (8 , of E .20ll of these fields descend in obvious ways from the degrees of freedom present in the initial10D UHE theory.We have also seen that the final (8 + 1)-dimensional state has a dilaton gradient that isspacelike, with norm-squared ˆ V = α ′ . We can choose a new set of coordinates Y m , m =0 , · · · , X ± variables as Y = −√ α ′ qX − + β √ α ′ X + ,Y = 2 β √ α ′ X − ,Y m = X m , m = 2 , · · · , . (5.1)In the Y m system, the m , n , · · · indices are raised and lowered with the hatted metric ˆ G mn ,and the dilaton is given byˆΦ = ˆΦ + ˆ V m Y m , ˆ V , , , ··· , = 0 , ˆ V = ˆ q , (5.2)with ˆ q = 12 √ α ′ , ˆ G mn = η mn . (5.3) The computation of the partition function for the HE9 theory proceeds as for the UHE theory.The major difference is that the contributions from the boson X and the fermions ψ , ˜ λ are absent. The linear dilaton does not couple to the torus, and has no effect on the pathintegral. The fermions ψ , ˜ λ are neutral under ( ZZ ) L , and have the same transformationsunder ( − F w . The path integral on the torus with spin structure a, b is therefore the sameas that in the UHE theory, with one less factor of (4 π α ′ τ ) − | η ( τ ) | − in the boson pathintegral, one less factor of (cid:16) ˜ Z ab (cid:17) in the left-moving fermion path integral, and one lessfactor of ( Z ab ) in the right-moving fermion path integral.21he left-moving fermion path integrals in the HE9 theory are thus L (¯ τ ) = 132 (cid:16) ˜ Z (cid:17) (cid:20)(cid:16) ˜ Z (cid:17) + 31 (cid:16) ˜ Z (cid:17) + 31 (cid:16) ˜ Z (cid:17) (cid:21) ,L (¯ τ ) = 132 (cid:16) ˜ Z (cid:17) (cid:20)(cid:16) ˜ Z (cid:17) + 31 (cid:16) ˜ Z (cid:17) + 31 (cid:16) ˜ Z (cid:17) (cid:21) ,L (¯ τ ) = 132 (cid:16) ˜ Z (cid:17) (cid:20)(cid:16) ˜ Z (cid:17) + 31 (cid:16) ˜ Z (cid:17) + 31 (cid:16) ˜ Z (cid:17) (cid:21) ,L (¯ τ ) = 0 . (5.4)The full path integral for fermions and superghosts, with spin structure a, b is F ab ( τ, ¯ τ ) ≡ ( Z ab ) L ab . (5.5)These are then combined into projected partition functions in NS and R sectors: I NS ± ≡ (cid:0) F ∓ F (cid:1) ,I R+ ≡ I R − = 12 F . (5.6)The path integral for bosons and reparametrization ghosts is iV (4 π α ′ τ ) − | η ( τ ) | − , (5.7)and the measure for moduli takes the usual form. The total partition function is then: Z F dτ d ¯ τ π α ′ τ (4 π α ′ τ ) − | η ( τ ) | − ( I NS+ − I R ± ) . (5.8)The functions F ab , I NS ± and I R ± exhibit the same modular transformation properties astheir ten-dimensional counterparts (see, e.g., Eqns. (2.14, 2.15)), so the modular invarianceof Eqn. (5.8) follows. Defining mass partition functions for the HE9 theory in parallel withthose of the UHE theory, we find Z NSmass ( τ ) = ( q ¯ q ) + h ,
785 + 108 ,
500 ( q ¯ q ) + O ( q ¯ q ) i ,Z Rmass ( τ ) = 1 ,
984 + 4 , ,
880 ( q ¯ q ) + O (cid:0) ( q ¯ q ) (cid:1) . (5.9)22e now pause to emphasize several points regarding the spectrum. As expected, theNS sector is tachyon-free. The lowest NS states gain an effective mass-squared of ˆ V = α ′ from their coupling to the background dilaton gradient [32], as is usual in subcritical stringtheory [33, 34]. The first NS mass level consists of · − · B-field polarizations, and one dilaton, together with 7 ·
248 = 1,736 polarizations of the E gauge field, for a total of 1,785 physical states. These states would be massless in abackground with constant dilaton.The lowest Ramond mass level is a massless Majorana spinor in the adjoint of E . AMajorana spinor in 9D has 16 degrees of freedom off shell, which reduces to eight uponimposing the on-shell conditions. Multiplying by the dimension of the adjoint representation,we obtain 8 ·
248 = 1,984 physical states. Interestingly, the spin- fermion does not obtaina nonzero mass from its coupling to the dilaton gradient.The masslessness of the lowest state is an inevitable consequence of the effective fieldtheory. There is only a single Majorana adjoint fermion at the lowest level. A single Majoranafermion can have no mass term with itself in 8 k spatial and one time dimension: if C isthe charge-conjugation matrix acting on spinors, then C and C Γ m are both symmetric.A Majorana spinor ˆΛ obeys C αβ ¯ˆΛ β = ˆΛ α , so terms such as M ¯ˆΛ ˆΛ and ¯ˆΛ ( ∂/ ˆΦ) ˆΛ vanishidentically by Fermi statistics. Therefore, the rescaling from string frame to a canonicallynormalized fermion cannot introduce couplings that give ˆΛ a nonzero physical mass. Wesummarize the field content and spectral properties in the low-lying energy levels of boththe ten-dimensional parent UHE theory and the HE9 final state in Table 5.1. The HE9 theory has no unbroken spacetime supersymmetry. One way to see this is todemonstrate the complete absence of Bose-Fermi mass degeneracy. The m = 0 adjointfermions ˆΛ are split from the gauge field ˆA in the spectrum by an amount ∆ m = α ′ ;the canonically normalized modes of ˆA have an effective mass equal to √ α ′ . Furthermore,the multiplicities of the gauge field and adjoint fermions do not agree: for each of the 248gauge generators, there are seven physical polarizations of the gauge field, but eight physicalpolarizations of ˆΛ , which transforms as a Majorana spinor of SO(8,1). There are also nospin-1 / / B or graviton ˆ G . Indeed, the lightest spin-3 / / m = − /α ′ T m = 0 Φ(1) + G (35) + B (28) + A (1984) 2048R m = 0 Λ + (1984) + Λ − (1984) 3968HE9 NS m = +1 / (4 α ′ ) ˆΦ(1) + ˆ G (27) + ˆ B (21) + ˆA (1736) 1785R m = 0 ˆΛ (1984) 1984Table 1: Summary of field content and multiplicities in the lowest-lying mass levels of the ten-dimensional UHE string theory and its nine-dimensional endpoint, following the transition.The tachyon is absent in the HE9 final state. Fields in the massless NS sector acquirean effective mass m = 1 / (4 α ′ ) after the transition, due to the coupling to the dilatonbackground. Furthermore, the massless R sector loses half of its particle content in thetransition. Interestingly, the lightest state in the HE9 theory is a fermion rather than aboson. (Hatted quantities are reserved for the nine-dimensional theory.) string state m ≡ − k m k m field name transversality / Dirac equation gauge invariance e mn ˜ α m − ψ n − / | k ; 0 i α ′ ˆ G mn , ˆ B mn , ˆΦ ( k + i ˆ V ) m e mn =( k + i ˆ V ) n e mn = 0 ∆ e mn = ˜ ξ m ( k − i ˆ V ) n +( k − i ˆ V ) m ξ n e m ψ m − / | k ; 0 i g L α ′ ˆA m ( k + i ˆ V ) m e m = 0 ∆ e m = ( k − i ˆ V ) m ξ | k ; ˆ α i g L · ( − Fw ˆΛ ˆ α ∂/ ˆΛ = 0 - Table 2: The lowest-lying normalizable string modes in the HE9 theory. The canonicallynormalized modes of the graviton, dilaton and B-field acquire an effective mass ∆ m = √ α from their coupling to the background dilaton gradient, as does the E gauge field. The E adjoint fermion, on the other hand, does not acquire a positive mass: the canonicallynormalized field ˆΛ obeys the massless Dirac equation.24elds enter at m = α ′ , which is four times as heavy as the normalizable excitations of ˆΦ , ˆ B and ˆ G .Despite the lack of spacetime supersymmetry, our theory is tachyon-free. The tachyonvertex operator would have to be built from a matter primary of weights (˜ h, ˜ h − / h < h < λ A , which have ˜ h = 1 /
2. (All twisted operators in the currentalgebra have weight at least one.) Each of the 31 fermions ˜ λ A transforms nontrivially undersome element of the left-moving gauge group ( ZZ ) L , and none can therefore enter a physicalvertex operator unaccompanied by other current algebra fermions. Hence, there are notachyons in the HE9 final state.Finally, the background has no moduli. The lightest field is the dilaton, whose normal-izable excitations have m = √ α ′ . Even the constant mode of the dilaton δ ˆΦ = const . doesnot represent a modulus in the spacelike linear dilaton background. Shifting ˆΦ can be com-pensated by a redefinition of the spatial coordinate Y , so even this degree of freedom is nottruly a modulus; rather, it is pure gauge. In addition to the bubble of nothing solution studied by Hoˇrava and Keeler, the UHE theoryin ten initial dimensions admits exact solutions that transit to a stable, nine-dimensionaltheory with no moduli and no spacetime supersymmetry. This transition is depicted schemat-ically in Fig. 4, where our solution focuses on the upper left-hand region of the spacetimediagram.Generally, the absence of supersymmetry with no tachyons and no moduli is interesting.Few completely stable nonsupersymmetric string theories are known in dimensions above D = 2. The O (16) × O (16) theory in ten dimensions is nonsupersymmetric and tachyon-free at tree level, but its massless fields (such as the dilaton and scale factor of the metric)acquire potentials from higher-genus string diagrams. The HE9 theory, by contrast, hasno moduli at tree level, and the effective mass shift due to the dilaton gradient rendersthe background stable against quantum corrections. Background shifts due to higher-genusstring diagrams are finite away from the strong coupling region, and can be incorporated viathe Fischler-Susskind mechanism [35, 36]. 25 (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(c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Time X Figure 4: The dynamical spacetime transition from ten dimensions outside a bubble wall tonine dimensions in the interior. Our solution focuses on the upper left-hand corner of thediagram, where the bubble is a domain wall moving to the left at the speed of light. Thefinal phase is a stable theory with a single E gauge group and no spacetime supersymmetry.The qualitative nature of the final state of the UHE string depends on details of theinitial tachyon profile. This situation is reminiscent of the behavior of an eternally in-flating universe with a complicated scalar potential, in which different, inequivalent vacuaare populated by the random evolution of scalar fields across a jagged landscape of localminima [37, 38]. Tachyonic starting points that can make locally stable (but not absolutelystable) transitions suggest a possible arena for studying the string landscape concept in aweakly-coupled limit.Cosmological evolution in quantum gravity can produce qualitatively different outcomesfrom the same initial conditions. The stable HE9 theory is separated in phase space from thetheory studied in [19,20], though both theories descend from the same unstable parent theory(the UHE background). Two such basins of attraction must be separated by a transitionwith a characteristic critical behavior. It would be interesting to find the set of unstable,fine-tuned endpoint theories that separate these two stable attractors in phase space. These As noted above, similar non-deterministic behavior also occurs in the supercritical type HO + string [30]. SO (32), O (16) × E , O (8) × O (24), ( E × SU (2)) , U (16) and E , the last belonging to the UHE theory studiedin this paper. It would be interesting to analyze the remaining unstable backgrounds in thisclassification, with the hope of finding new string vacua of the type studied here.Of the six, the UHE theory is distinguished by having a sensible 11-dimensional inter-pretation [9]. In this interpretation, the string-theoretic bubble of nothing is lifted to 11dimensions as a cosmological spacetime with a particular geometry and topology. Giventhat the same UHE state can also transition to the HE9 background, the 11-dimensionaldescription of type UHE and its instabilities could provide insight into noncritical and non-supersymmetric string vacua through the lens of M-theory. Acknowledgments
S.H. is the D. E. Shaw & Co., L. P. Member at the Institute for Advanced Study. I.S. is theMarvin L. Goldberger Member at the Institute for Advanced Study. The authors gratefullyacknowledge additional support from U.S. Department of Energy grant DE-FG02-90ER40542(S.H.) and U.S. National Science Foundation grant PHY-0503584 (I.S.).
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