JHEP09(2007)038 hep092007038.pdf June 21, 2007 of non-linear August 23, 2007 September 14, 2007 Received: Accepted: IIB string theory to find ects is determined by the U.S.A. t, and of any topology. We Published: urations of expanding or con- ysics, constraint on the total tension. http://jhep.sissa.it/archive/papers/jhep092007038 /j Published by Institute of Physics Publishing for SISSA [email protected] , F-Theory, p-branes, Sigma Models. We construct a general class of new time dependent solutions [email protected] Center for Cosmology and Particle Physics, Department of Ph New York University, 4E-mail: Washington Place, New York, NY 10003, SISSA 2007 -models coupled to gravity. These solutions describe config c Matthew Kleban and Michele Redi Expanding F-theory Abstract: ° tracting codimension two solitons which areThe not subject two to dimensional a metricLiouville on equation. the This space spaceshow transverse can to that be the this compact def orbackgrounds construction non-compac describing can a be number of applied 7-branes naturally larger than in 24. Keywords: type σ JHEP09(2007)038 3 4 6 8 1 2 5 9 10 xample) . π = 1) [1]. If the xactly 4 N atic ansatz. This is lowing the metric to tions for codimension πG construction allows us lications in string theory gravity — or relativistic , but is naturally foliated l mass of the defects is less , and allows it to have any nerate a conical space with e defects has zero curvature cting towards) a big bang at negative curvature of this two numbers of defects and general hese solutions describe 2+1 di- reless point masses, allows them – 1 – is the mass of the particle (we set 8 m -model solitons (unlike abelian Higgs model vortices, for e σ , where m = α 1 CP -models are ubiquitous. In particular we will show that this , or compact and of spherical topology when the total mass is e σ π -model solitons with tension and topology not allowed by a st σ In [2] it was shown that these constraints can be relaxed by al In this note we will make use of this fact to find expanding solu Our interest in these types of configurations arises from app 2.1 Sigma model2.2 solitons 3.1 Sen’s limit 3.2 Other topologies = 0. 1. Introduction It is well known that point particles in pure 2+1 dimensional 2. Construction codimension two objects indeficit higher angle dimensional theories — ge Contents 1. Introduction be time dependent.by The two space dimensional remains locally flatdimensional hyperbolic outside slices. space the accommodates The defects defects constant topology with (subject any to total somemensional mass mild cosmologies restrictions). uniformly expanding Physicallyt from t (or contra two possible because the where have a scale invariance which,to like expand distributions uniformly of (see pressu also [3]). to generalize the stringy cosmic strings of [4] to arbitrary 3. F-theory revisited 4. Corrections 5. Discussion metric is static, theaway from two the dimensional singularities space andthan transverse 2 is to non-compact when th the tota JHEP09(2007)038 (2.2) (2.1) (2.3) (2.4) (2.5) ) is related to ¯ z z, es can be avoided if the ich is boost invariant along avitational solutions corre- ence of sources means that ¯ 0] for a review and additional z. c cosmic string — generates tive curvature, but in certain ¯ z heory [5] backgrounds of type is the Euler character. Using on of the defect. Choosing a wo objects coupled to gravity. lly flat. It would be interesting tained by orbifolding flat space χ . dzd . uum Einstein’s equations reduce dzd i ) ) ) ¯ z i ¯ z m z z, z, ( πχ, ( φ − φ N =1 . Non-compact asymptotically conical e i X e z 2 π = 4 t ( 2 2 + 1 = 0 away from the defects. We can apply δ i + − = 4 2 i K i i dx R Σ m i πχ dx ∂ m i Z – 2 – dx N =1 i X P dx = 4 + , if the space is compact and without boundary the 1 + 2 = g + -function sources: 2 2 δ 2 K 2 φ dt R − ¯ Σ z dt − ∂ ∂ Σ − z Z Z . = ∂ = 2 2 , with = π 2 χ 2 φ 2 ds ≤ , eq. (2.2) implies ds i has φ ¯ z g m of the two dimensional space Σ parametrized by ( ∂ z 2 P i∂ R 2 − )= γ ( is the extrinsic curvature of the boundary and 2 1 γR K √ Our solutions are reminiscent of those of refs. [6 – 9] (see [1 In [2] it was shown that these restrictions on the deficit angl by Since the curvature appropriate for a distributionto of the parallel Poisson’s defects, equation the for vac IIB string theory with a number of 7-branes largerreferences), than which 24. considered timefoliated dependent solutions with ob negativelythe curved transverse slices. metric does not Inlimits in the our general sources case have are constantto the point-like nega investigate and pres the our connection solutions with are these loca works. 2. Construction Our results follow insponding part to from an arbitrary [2], numberTo which of see constructed point-like how codimension this new t works, gr its recall that world a volume codimension directionsa two defect — locally wh such flat as metricmetric a with ansatz a straight conical relativisti singularity at the locati transverse topology. This in turn can be exploited to find F-t the Gauss-Bonnet theorem to Σ: where φ Since a surface of genus only solution has spherical topology and eqs. (2.2) and (2.3) one finds solutions exist when two dimensional metric is uniformly expanding: JHEP09(2007)038 2 t (2.8) (2.6) (2.9) , is a , and (2.10) ´ π ¯ z 4 = 4 this i are com- − 2 d z > i ). The two τ ³ i ¯ z In m . z, ) ( i j P ¯ τ τ i = 2ln τ = the entire manifold φ K i ture term on the left- τ . admit soliton solutions det( Positivity of the volume given by = 0 ¶ j q i . Integrating the Liouville ¯ ates: τ n τ anti-holomorphic) functions , which arises due to the uation, e energy can be expressed in µ ¯ d to a holomorphic mapping τ z . n the presence of defects the φ n for the conformal factor: n from higher dimension, and 2 ) ology only if ∂ ∂ e choose the metric ansatz (2.5), i form: i z 2 1 12]). 0 is the only constraint on the z τ he global solution is non-trivial. ∂ µ k − ¯ ...d ∂ τ i 1) (2.7) j 1 z τ ¯ τ ( τ i V > 2 2 − K τ δ d g K i ( Z π m + ) + 2 l 2 1 τ R N =1 + 4 z i − X 1 2 i ∂ i – 3 – µ − τ m ¯ z x φ )) = ∂ d e N =1 z i X (¯ 1 2 + ) is time dependent, but nonetheless it cancels in the g d j with constant negative curvature. The full metric is ¯ l ¯ τ z = τ − = , 2 -model coupled to gravity. Here the fields ¯ z ) z, √ V σ φ ∂ z ( H i ( ¯ z φ τ i Z ∂ 2 z τ t z ( ∂ − ∂ ( K l ¯ τ z = k ∂ ¯ τ z i S τ z ∂ K 2 d determines the metric of the target space complex manifold. Z i 2 K j − ¯ τ ∂ = i τ ∂ E ≡ Under some topological conditions actions of the form (2.8) The constraint on the total mass is modified because the curva j ¯ ), but one should make sure that the solution is well defined on τ i z τ ( i describing a complex non-linear The vacuum Einstein’s equations become a Liouville equatio metric for the hyperbolic plane This differs from the Poisson equation (2.2) by the term spanned by the scalars.of The the solutions spacetime surface so Σ obtaineda into Bogomol’nyi-like correspon the form target as space the manifold. integral Th of the (1,1) K¨ahler plex scalars which could ariseK as moduli of a compactificatio describing codimension two objects [11].and To see assume this the one scalars can depend only on the transverse coordin is the bosonic part of a supersymmetric action. flat space written inspace a away two from dimensional the singularities hyperbolic is slicing. still locally I flat,hand but side t of (2.3)equation now (2.6) and makes using a (2.3) negative one definite finds contribution that the volume of Σ is existence of solutions for all topologies (see2.1 theorem Sigma A of model [ solitons Consider the following action (at least for closed and compact spaces) in general (from here on weimplies will that drop there the are compact boundary solutions term with for spherical simplicity). top dimensional conformal factor dependence. In the absence of sources the solution of this eq These equations are triviallyτ solved by any holomorphic (or equations of motion for the scalars: JHEP09(2007)038 - δ int (2.13) (2.11) (2.14) counting N -model with σ . The expanding space is positive. π = 4 E that Einstein’s equations 0, and the energy density he solutions of eq. (2.11) olitons of this model cor- ization corresponds to the pace sphere. The simplest he right-hand side of (2.12) 2.5) they reduce to a single q. (2.12) would be zero, and ed above to a , ergy χ< ng the Liouville equation and )) oting that the mapping (2.14) xistence for arbitrary topology pology the total energy should proven when the sources are , without common factors. The z (¯ pace times an integer j q ¯ τ , ) , ] z 1) (2.12) and ( ¯ -vortex solution is given in terms of a τ i τ τ − ) ) p N ( z z is positive if the metric K¨ahler is positive g ( ( ( K ¯ z P Q π E ∂ z log[1 + ∂ + 4 – 4 – )= 2 0. z − a E ( ≥ . From direct integration of (2.10) we see that the φ τ = z = e χ 1 2 0). = . The general V K 2 -function sources replaced by the smooth energy density = τ δ πa φ ). The energy E > ¯ z z ∂ ( i z τ ∂ , and for higher genus surfaces there is evidently no constra we obtain π E ) are polynomials of degree z ( Q ) and 0. This last is guaranteed if the metric K¨ahler on the target z ( 1 P . target space [14]. The potential K¨ahler is K > K CP eq. (2.7)). In the static ansatz (2.1) the left-hand side of e Having solved the scalar equations of motion, we need to show As in the case of point particles, some mild restrictions on t Physically we expect solutions of (2.11) to exist so long as t ¯ ¯ z z 1 ∂ ∂ z z cf. rational function, energy of this configuration is 2 so that the metricrespond is to that mappings of ofexample a the of round spacetime such sphere. sphere a into map The the is topological target simply s s 2.2 As an explicit example, we can apply the construction outlin CP functions [12]. In the smoothof the case transverse the space results Σ, of at [13] least when prove e it is closed, compact ∂ definite (as required bytotal supersymmetry), deficit and angle. with our normal can be solved consistently.equation: With the time dependent ansatz ( The situation is more complex when arise depending on theusing total the energy definition of of the source. Integrati ∂ the degree of the mappings energy (2.10) of these solutions can be easily evaluated by n The right-hand side is essentially the volume of the target s which is similar to (2.6) with the the only allowed compact manifold would be spherical with en ( is positive. As we mentioned previously, existence has been where ansatz allows much more generalagain solutions. be For spherical greater to thanfrom 4 Gauss-Bonnet (at least if JHEP09(2007)038 (3.2) (3.3) (3.1) (2.17) (2.16) (2.15) -model is σ = 1. Since this is N 2, which is nothing but hen the symmetry is the modular > enient to parametrize the , 2 ¶ a get space manifold were con- (2.12). 2 ¯ τ ) dent solutions is in the context ) monodromies as it varies on , where the relevant ing point for F-theory [5]. µ needs to exploit the symmetries tion of the theory, the solutions ¯ τ Z , tained by erecting a torus at each τ∂ − U(1) manifold. Static codimension µ / τ , ∂ ) ( )]. Topological solitons of this theory . 3 g R ¯ τ 2 f 3 , ) − . determine the complex structure of the ) + ¯ z 4 − f N. z g R + 4 τ 2 1 − 1 2 fx ( 2 i 2 g µ πa + 4(24 a − and (1 + x ) times, so that 3 27 – 5 – ) invariant, but quantum effects break this to the discrete κ x f = 2 10 R = 2 log[ can be interpreted as the modular parameter of a , p,q = = E κ )= g d τ − 2 φ τ − e ( y on the torus. = j √ τ K =max( Since Z N 1 − to undergo non trivial SL(2 = τ S . The complex numbers 2 ). This is manifest in the twelve dimensional picture, where C Z , To construct these configurations more explicitly it is conv An explicit solution of the Liouville equation can be found w At the supergravity level the relevant part of the action is At the classical level the action (3.1) is SL(2 1 the compactification manifold. so constructed describe an ellipticallypoint fibered on manifold ob the base manifold [5]. torus as the complex surface “hidden torus” associated with a 12 dimensional interpreta torus: embedded in corresponding to mappings of the Riemann sphere into the tar sidered in [4]. To constructof solutions with the finite theory energy one by allowing From the fact that the metric must be positive it follows that which has the form of (2.8) with covers the target space sphere a trivial mapping from sphere to sphere, the natural guess is By plugging into eq. (2.11) one finds that this is a solution if the constraint implied by the positivity of the volume3. in eq. F-theory revisited Arguably the most interesting applicationof of our string time theory. depen The simplest case is type IIB string theory subgroup SL(2 provided by the axion-dilaton, which spans an SL(2 two configurations with varying axion-dilaton are the start transformation of the complex structure JHEP09(2007)038 ) t (3.5) (3.4) old so . This ransfor- n ) to the Z , ) [4, 15]. For L(2 z ( τ we conclude that laced on a sphere to be polynomials and and 3 2 g n F-theory solutions is /g 3 6 in our conventions, it f the associated Liouville and his type is obtained by π/ f io lle equation with point-like larities in es. For the case where the e internal space (at given ollows that it takes precisely rly acute worry for our time model [4]. 2 complex parameters. Note n. The solution for the scalars ing pendent ansatz (2.5), however, ) is on is that the classical config- space is nonetheless regular [4]. symmetry is broken. Since the − =25 defects to find a compact e base of the elliptic fibration is ) 7-brane charge [5, 16]. For the Z − σ , N n have degree 2 p,q 3 ) g z ( f , and 6 π , as in these cases the solution is regular. + 4 f n 2 N ) z = ( – 6 – = 6 g E , and the metric on the base manifold can be found N 3 K ∆ = 27 the polynomials singularities depends on 5 n n ) to be polynomials of degree 8 and 12 respectively. The manif of the base manifold. The zeros of the polynomial z + 2 complex free parameters. Taking into account conformal t ( z g n ) and ) is the modular function mapping the fundamental domain of S is the degree of ∆. z τ ( ( f j N In the type IIB context these configurations carry ( When the number of 7-branes is greater than 24 they cannot be p The Gauss-Bonnet theorem would require at least The number of parameters necessary to specify our expanding constructed is an elliptic fibration of solution. However in this case there will be branch cut singu where case relevant to the simplestthe F-theory Riemann compactifications sphere, th therefore24 from eqs. 7-branes (3.5) to andtaking close (2.12) the it f space. The most general solution of t this reason we will focus on the case explicitly by solving the Poisson equation sourced by the Riemann sphere. Multi-vortex solutions are obtained by tak where without some source of negative curvature.the Using problem the is time de of the formis we the discussed in same the as previousequation sectio eq. (2.11). (3.3), while the metric is now determined by As we show below, insingularities. a certain limit this reduces to the Liouvi similar to the dimensionnumber of of the singularities moduli is space 6 of static 7-bran corresponds to 5 mations and the fact that the solution only depends on the rat follows immediately from eq. (2.10) that the total energy is in the coordinate determine points where the torus degenerates,Noting but where that the the area of the fundamental domain of SL(2 the general solution with 6 however that, contrary to the F-theory case, the volume of th is now fixed by eq. (2.12). 3.1 Sen’s limit A general concern abouturation solutions might with be varying strongly axion-dilat dependent corrected. configurations because This (as seems we a will discuss) particula super JHEP09(2007)038 (3.8) (3.6) (3.7) in (3.3) termined tacks of six f, g there are re- esponding to possible. Close to 2 ) transformation ) or its double cover, global Z y is associated with the Z , , e metric, (2 n a constant axion-dilaton becomes constant and the solutions where rface with constant negative ese solutions are fully under nsequence only globally it is τ lows an arbitrary number of n particular this might allow SL usual F-theory. As in [17] by ng theory. We now show that described pertubatively by an ation theory cannot be applied ral to interpret our solutions as solutions. ions strongly hints that they will e total number of singularities is 2 e full SL(2 vial of the solution in the limit is still / . 1 ) of the torus (3.2) moving around 3 − E controls the string coupling, and can . | f i 2 z ) x,y α ¯ z + 4 − ! z 2 z 1 g 1 | − (log =1 4 1 0 ¯ z Y 0 – 7 – m z − 2 Ã ∼ R φ e = . We thank Simeon Hellerman for pointing this out to us. 2 2 n sources (2.6). Around the singularities the metric is just ds − δ = 24 . The free parameter N α = 2 -model solitons collapse to point-like singularities corr /g σ 3 controls the string coupling, it follows from eq. (3.3) that each [17]. For the static F-theory solutions there are four s f τ 5 we can pack them into stacks of six each. The metric is then de π ≥ is invariant. conical defects (known as parabolic singularities) become n τ π with n In [17] Sen considered a special limit of F-theory where In this limit the In the perturbative string theory framework such a monodrom Another interesting feature of this solutions is that on a su ) 7-branes each. The manifold is topologically a sphere whos We should note that since some of the fields transform under th = 6 2 p,q obstructions may in fact require be chosen so that the coupling is small. The energy solution can be understoodthe as same an limit orientifold can of beone type used must IIB in go the stri to time a dependent particular case. limit of To obtai the parameter space of our determined by the degree of the polynomial 27 imaginary part of are chosen so that gions of space where theeven string in coupling principle. is large While and thesurvive perturb topological in origin the of full thecontrol. solut theory, it is useful to find a limit where th is locally flat butN with four singularities. Similarly, if th a deficit angle of a reparametrization of flatpossible space with to conical distinguish deficit. theselooking solutions As at a from the co the transformation onesthe of in singularities the the one discovers coordinates that ( there exists a non-tri by the Liouville equation with ( presence of an orientifoldO7-plane plane and so four that 4 eachdefects 7-branes and singularity [17]. locally is the Since spacean is our the orientifold construction same of as al string in theoryone [17], it to with is give five natu a or perturbative more world-sheet O7-planes. description I of these under which curvature 2 such a singularity the metric is given by JHEP09(2007)038 n 0. sing (3.9) E > rus, i.e. function − ℘ Z) six times (this , is non-compact. A ere the base of the interesting to study the pology. In this case the s” of the 12 dimensional toroidal topology, we can diverges, but the volume is anding torus. We can find rom eq. (2.12) we see that . It would be interesting to n of SL(2 z 3 he simplest solution is given l, but as we have seen these erm in eq. (2.4) compensates ite number of defects. In the ℘ )) describes a “stringy cosmic s to perturbative string theory. ζ ee whether tachyon instabilities = ( ng array of defects. More general s of the Weierstrass nt. h g . ( 3 1 3 ) − z )) and ( j z ( ℘ 2 ℘ ) = α℘ ζ = 0 to any finite ( = 4(24 27 + 4 τ z f – 8 – 1, while the toroidal topology simply needs by considering mappings between the fundamental )) = g z ( g > τ does not have orbifold singularities and is well defined = 0 and two single zeros in the fundamental region of ( ) is a double cover of the Riemann sphere), so evidently 3 j z τ ) is a meromorphic map from a Riemann surface Σ with z ( ζ singularities together draws that point out into a cusp. The ( ℘ h π = z -function with a map from the Riemann surface into the Rieman j in (3.3) must be mereomorphic functions defined on the base to ) and the Riemann surface. We can construct such maps by compo g Z , into the complex plane, then (2 and ζ f SL has one double pole at ) transformation around these points is trivial. It would be ℘ Z , We can find generalizations of the solutions of [4] to cases wh This can be done rather explicitly for the case of toroidal to We can also find solutions in cases where the transverse space We would like to thank Allan Adams for discussions on this poi 3 region of fibration is a Riemann surface of genus The proper distance from the singularity at the inverse of the the spacetime torus (not to be confused with the “hidden toru description) one can check that on the torus. The mapping (3.9) covers the fundamental regio behavior of string theoryappear close along to the these lines singularities in and ref. s [19]. 3.2 Other topologies With the static ansatzconstraints the are only avoided allowed when topologythere the is is ansatz no spherica topological is constraint time for dependent. F sphere. In particular, if Since finite. Evidently moving two SL(2 coordinate an analog of the Sen limitThis that could should be connect these done solution for example by taking this configuration can be understood as six 7-branes on an exp follows from the fact that the elliptic functions. These are in general rational function string” configuration with transverse space Σ. functions simple example follows fromalways re-interpret the the configuration above: as an givenconfigurations infinite a expandi should solution also with exist,case either where with the a total finitefor mass or the is bulk infin finite, curvature contribution. the extrinsic curvature t study these configurations more in detail. (with periods determined byby the torus) and its derivative. T JHEP09(2007)038 , c (4.1) (4.3) = 6). N ak super- strings where N k that this condi- e locally unstable ome negligible. At lars the variation of ǫ ¢ ppens. For simplicity we i Sen’s limit. For the latter energy in the supergravity ¯ ent stacks of 7-branes. At τ -Schwarz compactifications nce implies that supersym- itions. µ dentical to the supersymmet- ∂ at late times. c, the string coupling can be ity dilutes with the expansion fficient to guarantee the exis- t case). Since supersymmetry l. j the non-compact case s become very massive, so we ¯ τ iations of the gravitino, due to of Calabi-Yau manifolds are su- or equations. There are however onfigurations of K cally satisfied by the background. − = 0. This proves that the expanding j τ corrections. As we argued above the ˆ ǫ µ φ ′ ∂ e α j 2 = 0 (4.2) τ ǫ /t K ] ˆ 1 ¡ and ν 1 4 ∝ s – 9 – , D g i − µ R τ µ µ D [ ¯ ǫ∂ ǫω µ = 0 this will presumably turn in a tachyon instability − σ t 2 ǫ . The gravitino equation implies an integrability equation µ √ ǫ ∂ i ¯ z with four supercharges following [18]. For a supersymmetri γ singularities. = = i = π µ D χ ǫ ψ ǫ ǫ δ δ singularities have no flat space analog, we expect them to bre π is zero for ˆ i χ all the correction due to higher powers of the curvatures bec is defined by the variation of the gravitino above. On can chec is the spin connection. For holomorphic solutions of the sca are non-supersymmetric even in flat space, and are likely to b t µ µ n ω D In Sen’s limit, the background around the singularities is i Since the 2 Finally our solutions will have both model coupled to gravity the variations of the fermions are 6= 6 − σ the spinors persymmetric [5, 20]. Inmetry the is expanding broken. case theconsider It time the depende is problem in interesting 4 to see in detail how this ha 4. Corrections F-theory backgrounds constructed from elliptic fibrations where where tion is just the LiouvilleHowever equation the (2.11), so integrability it condition istence is automati of necessary Killing but spinors. notthe su time In dependence fact of the writing ansatz, explicitly one the finds var that solutions are not supersymmetric.and However, the therefore energy the dens supersymmetry is approximately restored ric case, so locally oneno can find globally solutions well of defined the Killing Killing spin spinors (except possibly in The breaking of supersymmetrywhere here supersymmetry is is broken reminiscent non-locally of by Scherk boundary cond symmetry locally and they are likely unstable. Similarly, c N least in the Sen’staken limit, arbitrarily the small, space andexpect is the these locally solutions non-supersymmetric to supersymmetri state be under good perturbative contro first ones become underit control is everywhere sufficient in to the note space that in the the curvatures scale as, (but see [15] foris an interesting broken exception at in looplate the level times non-compac there this will effectapproximation. be could just a As be force we computed between approach from the the differ Casimir similar to the one of the 2 At large JHEP09(2007)038 = 0. Here t ]. D gauged chiral 6 68. mple in F-theory, solu- In particular elliptically oeff, Jose Blanco-Pillado, tring theory, generalizing singularity at e of the metric allows us densation to supercritical cription of these solutions ]. Here we have considered hep-th/9602022 hese solutions under pertur- supergravity approximation ls au aa´n Augusto Rabad´an, olas, Ra´ul d the Liouville equation plays ro relative velocities. Perhaps re likely to appear. One pos- efects and differing transverse ion at the initial stages of this onfigurations. In particular we ons to higher dimensional base ]. n for very useful conversations. gularity. A related conjecture is ]. re work. Scaling solutions to Stringy cosmic strings and noncompact (1996) 403 [ Gravitational backreaction of matter B 469 (1990) 1. hep-th/0608083 – 10 – astro-ph/0609553 Three-dimensional Einstein gravity: dynamics of flat -model solutions corresponding to supermassive codi- B 337 σ Nucl. Phys. , (2006) 324 [ (1984) 220. (2006) 001 [ 8 152 12 Nucl. Phys. , JCAP , New J. Phys. , Evidence for F-theory Ann. Phys. (NY) , Calabi-Yau manifolds supergravity space inhomogeneities On the gravitational side, one could study the stability of t We have left many interesting questions unanswered. For exa [4] B.R. Greene, A.D. Shapere, C. Vafa and S.-T. Yau, [3] A.J. Tolley, C.P. Burgess, C. de Rham and D. Hoover, [5] C. Vafa, [1] S. Deser, R. Jackiw and G. ’t Hooft, [2] A. Gruzinov, M. Kleban, M. Porrati and M. Redi, higher curvature corrections willbreaks become down. important Since and supersymmetrysibility the is is that broken tachyon instabilities condensationthat a could these resolve backgrounds the are sin string dual theories [23]. or We connected leave via these investigations tachyon to con futu Acknowledgments We would like to thankAlbion Allan Lawrence, Adams, John Ofer McGreevy, Massimo Aharony, Porrati, Eric OriolSagnotti, Bergsh Puj Eva Silverstein,and especiallyM.R. Simeon would like Hellerma to thankwork. J.J. The Blanco-Pillado work for of collaborat M. R. is supported byReferences the NSF grant PHY-02450 tions can be viewedfibered as Calabi-Yau manifolds a of compactification anythe from dimension simplest can case 12 where be dimensions. the useda base [20 space key is role. two dimensional, Itmanifolds. an would be Another interesting direction tousing consider is brane generalizati probes to [17, study 21, the 22]. field theorybations, des for example where thethe defects most are pressing given some question non-ze is what happens as we approach the mension two solitons coupledto construct to solutions gravity. withtopologies, an The arbitrarily evading time the large dependenc constraints numberhave of which shown d apply that to this staticF-theory c construction backgrounds can to be an arbitrary applied number to of type 7-branes. IIB s 5. Discussion In this note we have presented new JHEP09(2007)038 , , ]. ]; ns , Phys. JHEP , , , 99 B 518 B 387 ]. (2006) B 299 , Mod. Phys. , Fortschr. Phys. ]. , D 73 hep-th/9608005 hep-th/0612121 Ann. Math. Nucl. Phys. Dimensional duality gularities orbifold Phys. Lett. , , ) ]. , Nucl. Phys. (2003) 061303 Z , , 91 hep-th/9605150 Phys. Rev. erstein, , Hyperbolic space cosmologies SL(2 -manifolds (1997) 2501 [ hep-th/9603161 2 (2007) 044025 [ (1991) 793. D 55 (1996) 562 [ hep-th/9502107 Seven-branes and supersymmetry 324 ]. D 75 (1978) 341. Phys. Rev. Lett. Compactifications of F-theory on (1996) 437 [ , ]; B 475 ]. B 74 Don’t panic! Closed string tachyons in ALE Phys. Rev. Existence of global strings coupled to gravity Three-dimensional supergravity and the New dimensions for wound strings: the modular , Probing F-theory with branes B 476 – 11 – ]. (1995) 6603 [ Phys. Rev. Accelerating cosmologies from compactification ]. , hep-th/0108075 Exceptional groups from open strings Nucl. Phys. D 51 , Phys. Lett. ]. ]. Bogomolny bounds for cosmic strings , hep-th/9602114 Time-dependent orbifolds and string cosmology hep-th/0303097 ]. ]. Nucl. Phys. Compactifications of F-theory on Calabi-Yau threefolds — I The supersymmetric nonlinear sigma model in four-dimensio , Curvature functions for compact ]; hep-th/0305032 (2001) 029 [ Phys. Rev. hep-th/0306291 , (1996) 74 [ 10 . (1989) 1546. ]; hep-th/0612072 hep-th/0310099 Dimensional mutation and spacelike singularities Prescribing curvature on compact surfaces with conical sin (2003) 061302 [ JHEP B 473 Cosmological string models from Milne spaces and hep-th/9605199 hep-th/9709013 (2004) 421 [ D 39 , 91 (2003) 058 [ Accelerating cosmologies from S-branes hep-th/0510044 BPS states on a three brane probe F-theory and orientifolds 10 A 19 (2007) 003 [ (2004) 145 [ hep-th/0303238 cosmological constant Calabi–Yau threefolds — II (1996) 278 [ (1998) 151 [ Nucl. Phys. 02 space-times and its coupling to supergravity (1974) 14. JHEP Phys. Rev. 086004 [ Transactions of the American Mathematical Society transformation of geometry to topology J. McGreevy, E. Silverstein and D. Starr, D. Green, A. Lawrence, J. McGreevy, D.R. Morrison and E. Silv Lett. [ C.-M. Chen, P.-M. Ho, I.P. Neupane, N. Ohta and J.E. Wang, (1988) 719; G.W. Gibbons, M.E. Ortiz and F.R. Ruiz, Rev. Lett. arXiv:0705.0550 52 [8] J.G. Russo, [7] N. Ohta, [6] P.K. Townsend and M.N.R. Wohlfarth, [9] E. Silverstein, [22] A. Sen, [17] A. Sen, [18] K. Becker, M. Becker and A. Strominger, [21] T. Banks, M.R. Douglas and N. Seiberg, [16] M.R. Gaberdiel and B. Zwiebach, [20] D.R. Morrison and C. Vafa, [15] E.A. Bergshoeff, J. Hartong, T. Ortin and D. Roest, [19] A. Adams, J. Polchinski and E. Silverstein, [13] J. Kazdan and F. Warner, [14] E. Cremmer and J. Scherk, [12] M. Troyanov, [11] A. Comtet and G.W. Gibbons, [10] L. Cornalba and M.S. Costa, JHEP09(2007)038 ; , hep-th/0612051 , Charting the landscape of ; . hep-th/0612116 , – 12 – Cosmological solutions of supercritical string theory arXiv:0705.0980 , Dimension-changing exact solutions of string theory ; Cosmological unification of string theories supercritical string theory hep-th/0611317 [23] S. Hellerman and I. Swanson,