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PHYSICAL REVIEW D 67, 046006 ͑2003͒ 3-form induced potentials, dilaton stabilization, and running moduli Andrew R. Frey* Department of Physics, University of California, Santa Barbara, California 93106 Anupam Mazumdar† CHEP, McGill University, 3600 University Road, Montre´al, Quebec, Canada H3A 2T8 ͑Received 8 November 2002; published 27 February 2003͒ We study the potential induced by imaginary self-dual 3-forms in compactifications of string theory and the cosmological evolution associated with it. The potential contains exponentials of the volume moduli of the compactification, and we demonstrate that the exponential form of the potential leads to a power law for the scale factor of the universe. This power law does not support accelerated expansion. We explain this result in terms of supersymmetry and comment on corrections to the potential that could lead to inflation or quintes- sence. DOI: 10.1103/PhysRevD.67.046006 PACS number͑s͒: 11.25.Mj, 04.65.ϩe, 98.80.Cq I. INTRODUCTION so these models have the phenomenology of the Randall- Sundrum models ͓17–19͔. The warp factor depends on the If we believe that string theory ͑or M theory͒ is the fun- position of D3-branes ͑and orientifold planes͒ on the com- damental description of interactions in our Universe, then we pact space and also determines the 5-form field strength. The are obviously forced to place the basic processes of cosmol- condition ͑1͒ gives rise to a potential for many of the light ogy into a string theoretic framework. Important steps have scalars, including the dilaton generically, which vanishes at been made in this direction by examining four dimensional the classical minimum and furthermore has no preferred supergravity models for potentials that could support the compactification volume. We will be interested in the behav- early phase of the accelerated expansion of the Universe, ior of these systems above the minimum, and the 4D metric known as inflation, which solves some of the outstanding will generalize !g . problems of the hot big bang cosmology ͓1͔. See, for recent For simplicity, we will mainly consider the case where the ͓ ͔ 6 ͓ examples, 2,3 . Other work has identified string theory mod- internal manifold is a T /Z2 orientifold, as described in 20– els in which D-brane physics leads to inflation ͓3–6͔.1 At the 22͔͑or in dual forms in ͓16,23͔͒. We take the torus coordi- mӍ mϩ same time, however, it has proven challenging to incorporate nates to have square periodicities, x x 2 ls , so that cosmological acceleration into string theory backgrounds be- the geometric structure is encoded in the metric. On this cause they tend to relax to supersymmetric vacua ͓7,8͔.In torus, the 3-form components must satisfy the Dirac quanti- this paper, we ask whether a stringy potential generated by zation conditions higher dimensional magnetic fields can give rise to acceler- ated expansion. We restrict our analysis to the classical po- 1 1 ϭ ϭ tential of supergravity. Hmnp hmnp , Fmnp f mnp , hmnp , f mnp Z. 2 ls 2 ls We study a class of exact solutions to type IIB supergrav- ͑3͒ ity that have a vacuum state ͓denoted by superscript (0)] with 3-form magnetic fluxes that satisfy a self-duality rela- Boundary conditions at the orientifold planes give large tion Kaluza-Klein masses to many fields ͑including the metric ͒ components gm , for example , and the remaining theory is Ϫ⌽(0) Nϭ Ã(0)͑FϪC(0)H͒ϭe H ͑1͒ described by an effective 4D gauged 4 supergravity with 6 completely or partially broken supersymmetry via the super- Higgs effect ͓24–29͔. ͑ ͒ on the compact space, which should be Calabi-Yau CY In the following section, we discuss the dimensional re- ͓ ͔ space 9,10 . These vacua were described in some detail in duction of the type IIB superstring in toroidal compactifica- ͓ ͔ ͓ ͔ 11 and in dual versions in 12–16 . The metric is of tions with self-dual 3-form flux, ignoring the warp factor, ‘‘warped product’’ form, paying particular attention to the potential for a subset of the light scalars. Next, in Sec. III, we find the cosmological evo- (0)2ϭ A ϩ ϪA m n ͑ ͒ ds e dx dx e gmndy dy , 2 lution driven by our potential based on known inflationary models; we find that our potentials do not lead to an accel- erating universe. Finally, in Sec. IV, we comment on the *Electronic address: [email protected] generalization of our results to more complicated models, †Electronic address: [email protected] compare our results to other models that do lead to inflation, 1As did ͓47͔, which found inflation in a small region of moduli and discuss corrections to our potential that might or might space. not lead to inflation. 0556-2821/2003/67͑4͒/046006͑7͒/$20.0067 046006-1 ©2003 The American Physical Society ANDREW R. FREY AND ANUPAM MAZUMDAR PHYSICAL REVIEW D 67, 046006 ͑2003͒ II. DIMENSIONAL REDUCTION AND POTENTIAL M 2 1 ␥mnץ ␥ ץ ϭ P ͵ 4 ͱϪ ͫ ϩ S d x g R kin 16 E E 4 mn Here we will review the dimensional reduction of 10D type IIB supergravity in compactifications with imaginary 1 1 mn qp m n ␣ ץ ␣ץ ␥ ϩ ␥ ␥ D D  Ϫ self-dual 3-form flux on the internal manifold. For simplicity 4 mp nq 2 mn I I and specificity, we will concentrate on the toroidal compac- ͓ ͔ 1 1 ͬ ץ ץ⌽Ϫ 2⌽ץ⌽ ץ tifications of 21,22 , extending our analysis to more general Ϫ e C C , cases in Sec. IV. We will ignore the warp factor, which as- 2 2 sumes that the compactification radius is large compared to the string scale.2 1 M 2 ϭ . ͑6͒ P 2 2 8 ls A. Kinetic terms Here, M is the Planck mass, and we are using a coset space We will start with the kinetic terms, mostly following the P covariant derivative analysis of ͓22͔, using the Nϭ4 SO(6,22) ϫSU(1,1)/SO(6)ϫSO(22)ϫU(1) language because we 1 are studying configurations away from the moduli space at mn mn m n n m ␣ ͒ ͑7͒ץ ␣␣ Ϫץ ␣͑  ϩץD ϵ the bottom of the potential. Our main purpose is to identify 2 I I I I the physical interpretation of the canonically normalized sca- lars, so we will skip the algebraic details. which arises from the magnetic coupling of the D3-branes to As was shown in ͓22͔, the moduli must be tensor densities ; this is the dimensionally reduced action for the heterotic in order to avoid double trace terms in the action, theory of ͓30,32͔, as one might expect. In deriving the ac- tion, one needs the identity ⌬ ⌬ ␥mnϭ Ϫ⌽ mn mnϭ ⑀mnpqrs ͒ ͑ ⌬ ץ ⌽Ϫ ץ mnϭ␥ ץ ␥ϭϪ ␥ ץe g , C pqrs , ␥mn 2 2ϫ4! mn mn 6 4 ln . 8 It is easiest to study the cosmology of canonically normal- ⌬ϵͱ ͑ ͒ ized scalars; so we will break down the geometric moduli. det gmn, 4 For simplicity we will consider only the factorized case T6 ϭ(T2)3. We can then parametrize the metric on an indi- 3 ␣mϭ m vidual 2-torus ͓say, the (4Ϫ7) torus͔ as along with the D-brane positions I XI /2 ls and the 10D dilaton-axion. For the purpose of cosmology, we want to Ϫ 2 work in the 4D Einstein frame ͑note that this is different than e ϩe d Ϫe d ␥mnϭe2ͫ ͬ. ͑9͒ in ͓22͔ because we are allowing the dilaton to vary͒ Ϫede Here, gives the overall size of the T2, gives the relative ⌬ E Ϫ2⌽ length of the two sides, and d controls the angle between the gϭ e g . ͑5͒ 2 two directions of periodicity. Then the ␥ kinetic term be- comes From stringy dualities, it can be seen that the moduli defini- M 2 3 1 ץ ץ ϩץ ץ ͫ ϭϪ P ͵ 4 ͱϪ ͒ ͑ tions 4 correspond to the geometric moduli gmn ,Bmn in a Skin d x gE ͚ 2 i i i i toroidal heterotic compactification, and the metric ͑5͒ is the 16 iϭ1 2 ͓ ͔ 4D ‘‘canonical metric’’ 30,31 in the heterotic description 1 ͒ ͑ ͬ ץ ץ ϩ 22͔.4͓ di di . 10 The kinetic action obtained from dimensional reduction of 2 type IIB supergravity ͑SUGRA͒ and the D3-brane action is then For canonical normalization, the coefficient of the kinetic terms should simply be Ϫ1/2, so a further rescaling is nec- essary. 2Actually, because the warp factor A scales like RϪ4 ͓14,22͔, the radius need only be a few times the string scale for our approxima- B. Potential tion to be reasonable. 3If the D-branes are coincident, the index I labels the adjoint rep- The scalar potential comes from dimensional reduction of resentation of U(N); the kinetic terms remain the same ͓29͔. the background 3-form terms in the type IIB action. After 4Strictly speaking, these are only the heterotic dual variables with converting to our variables, the potential for the bulk modes vanishing fluxes; see ͓16͔. is, in generality, 046006-2 3-FORM INDUCED POTENTIALS, DILATON . PHYSICAL REVIEW D 67, 046006 ͑2003͒ 2 4 M P M ϭ ͑ ␥ ͒␥mq␥nr␥ ps͓ ⌽͑ Ϫ ͒ ͑ P 2 Ϫ2͚ V det mn e F CH mnp F V ϭ h e i i ͭ coshͩ ⌽Ϫ͚ ͪ ϫ 0 i 4! 32 4͑8͒3 i Ϫ ͒ ϩ Ϫ⌽ ͑ ͒ CH qrs e HmnpHqrs͔ 11 1 ⌽ϩ͚ Ϫ ϩ e i i͓C2Ϫ2Cd d d ϩd2d2d2ϩe 2 3d2d2 2 1 2 3 1 2 3 1 2 along with an additional term that subtracts off the vacuum 5 Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ ϩ 2 2 2 2ϩ 2 1 2 2ϩ 2 2 2 3 2ϩ 2 1 2 3 2 energy.