Wrapped brane gas as a candidate for Dark Matter
aa r X i v : . [ h e p - t h ] J a n EPHOU 09-004
Wrapped brane gas as a candidate for Dark Matter
Masakazu Sano and Hisao Suzuki
Department of Physics, Hokkaido University, Sapporo, Hokkaido 060-0810 Japan
We consider brane gas models based on type II string theories and analyze the mass, the Ramond-Ramond charge and the charge on moduli fluctuations of branes wrapping over cycles of a compacti-fied space in the four-dimensional Einstein frame. A six-dimensional torus and Calabi-Yau threefoldsare considered for the Kaluza-Klein reduction. A large volume of the compactified space and a weakstring coupling gives rise to point particles of the wrapped branes which have a light mass and asmall charge of the Ramond-Ramond flux and of the moduli fluctuations, while the particles becomevery heavy in the string frame. We find that the masses and the charges satisfy the sea-saw likedual relations which become time-independent in the four-dimensional Einstein frame.
PACS numbers: 04.50. -h, 11.25.Mj, 11.25. -w
I. INTRODUCTION
The string gas model [1] is one of interesting scenariosin string cosmologies (see [2–6] for recent reviews). Thesimplest of such model treats the space as T . If manystrings wrap over cycles of a nine-dimensional torus, T at a very early universe, the cosmological singularity canbe resolved by T-duality [7]. The string gas models alsoprovide an intriguing idea on a realization of the largethree-dimensional space by annihilation of the strings,while the scale of the six-dimensional torus is stabilizedby the tension of remaining strings. The brane gas mod-els [8] are extension of the idea and have been considereda cosmological role of windings of D-branes.Although these gas models are very interesting, one ofthe serious problems is the moduli stability. This prob-lem is a serious obstacle to build String/Brane gas modelsconsistent with a realistic phenomenology and cosmol-ogy. This is resulting from massless scalar fields appearon the three dimensions without the moduli stabiliza-tion. There are many interesting works on the modulistabilization [9–24]. Among the stability of many mod-uli fields, the stabilization of the dilaton field has beendifficult in brane gas models. Recently, we have shownthat electric fields on D-brane, windings of NS5-braneand Kaluza-Klein monopole (KK5-monopole) can stabi-lize radial moduli fields of a six-dimensional torus T and the dilaton field simultaneously in brane gas modelsbased on type II string theories [25].According to the result of [25], gases of branes wrap-ping over only cycles of T make the energy density ofa pressureless matter, namely the energy density is in-versely proportional to the third power of the scale fac-tor. This implies that we observe those wrapped branesas point particles on the three-dimensional space. Thisfact leads to the appearance of the dark matter of thewrapped branes, because wrapped branes are not ordi-nary matters. The possibility of the dark matter of thewrapped branes has been investigated in various models.The appearance of light particles of the wrapped branesin the string frame, if the string scale, √ α ′ is larger thanthe Planck length in Ref.[26] . The large scale structure and the accelerating phase were discussed in [12, 27–30]without an influence of the Ramond-Ramond (RR) flux,although D-branes have the RR charges. However, themass and the charge of RR flux are not estimated simul-taneously in the four-dimensional Einstein frame whichis required for obtaining realistic models.We take the four-dimensional Einstein frame, whenwe consider the time-independent four-dimensional grav-itational constant after the Kaluza-Klein reduction. Inthis frame, the realization of light wrapped branes is nottrivial, since it is known that a large volume of the six-dimensional torus and a weak string coupling give riseto heavy wrapped branes in the string frame for a smallstring length [31].The purpose of this paper is to study wrapped branesin the brane gas cosmology based on type II string the-ories as dark matter, using the effective field theory ofbrane gas models. We estimate the mass and the chargeon the RR flux and on moduli fluctuations. These fluc-tuations also provide forces between the branes. We findsome models where branes can be realized as dust par-ticles whose masses can be around 10TeV scales. Wetake the string scale as 1 / √ α ′ ∼ m Planck and consider acompactification with T and the Calabi-Yau threefold(CY ), using the four-dimensional Einstein frame whichis the key idea to obtain a light mass and a weak inter-action.The description of the effective field theory requires alarge volume of a compactified space and a weak stringcoupling, because quantum corrections should be sup-pressed and the perturbative expansions should be valid.In the string frame, those conditions lead to the appear-ance of heavy wrapped branes, as the world volume ac-tion of the brane is proportional to its volume and theinverse of the string coupling. It seems apparent thatthe dark matter candidate of the wrapped branes may bedifficult in this frame. However, in the four-dimensionalEinstein frame, we will show that there are cases wherethe wrapped branes obtain the light mass and the smallcharge on the RR flux and on the moduli fluctuations un-der the large volume and the weak string coupling. Wewill show that the effective masses and charges satisfy thesea-saw like dual relations which are time-independent,while each quantity depends on the time variable throughmoduli fields. The dual relations are one of reasons forthe existence of the various light wrapped branes withthe small charges.From the phenomenological point of view, the estima-tion of the mass and charge is mainly done by controllingscales of the compactified space by hand. However, inthis paper, the string scale is of the order of the Planckscale and then the existence of light particles of wrappedbranes is quite non-trivial under the large volume. In onemodel, we find that the mass of the D0-brane is of theorder of O (10) TeV, if we take the six-dimensional com-pactified volume as ( O (10 ) × π √ α ′ ) where √ α ′ is thestring length and holds m Planck ∼ / √ α ′ in this paper.The square of the effective charges of the RR flux and ofthe moduli fluctuations is of order O (10 − ) by the samescale of the compactification. Then, the D0-branes is apossible candidate for the dark matter at a late time.We also consider the masses of wrapped branes by aD1-KK5 brane gas systems [25], because the system canbe stabilized by the scale of the T and the dilaton field.It is not trivial that both the weak string coupling andthe large scale of the compactification are realized simul-taneously in the specific model after the moduli stabiliza-tion. We find such realization of those conditions existsif the number density of D1 and KK5 satisfy a specificcondition which is given later. In the case of the Calabi-Yau compactification, we cannot prove the moduli stabi-lization in this model, however the dual relation is alsosatisfied and then we can show that light particles aregenerated for the large volume of the CY .This paper is organized as follows. In section II, wewill give the definition of the four-dimensional Einsteinframe. In section III, we will consider the effective worldvolume action of wrapped branes in the four-dimensionalEinstein frame. In section IV, we will show the electricmagnetic dual relation between masses of the wrappedbranes. The dual relation explains the existence of thelight wrapped branes. In section V, we will see a casein which the electric-magnetic dual relation of wrappedbranes is satisfied in the CY compactification. In sec-tion VI, we will investigate the effective RR charge inthe four-dimensional Einstein frame. The effective cou-pling of D p - and D(6 − p )-brane also satisfies the electric-magnetic dual relation. In section VII, we will commenton the charge of fluctuations of the moduli fields, becausethose fluctuations also give interactions between variousbranes. In section VIII, we would like to analyze an ex-plicit model constructed by a D1-KK5 brane gas system.Sec. IX will be devoted to the summary and some dis-cussions. II. FOUR-DIMENSIONAL EINSTEIN FRAME
In this section, we shall define the four-dimensionalEinstein frame which gives the time-independent four-dimensional gravitational constant after a compactifica- tion. In general, the Kaluza-Klein reduction gives rise toa coupling between the four-dimensional Einstein-Hilbertterm and various moduli fields. If those moduli fields arefunctions of coordinates of the compactified space, wecan integral out moduli fields and renormalize the factorinto the higher dimensional gravitational constant. How-ever, in cosmologies, moduli fields depend on the timeand, therefore, we cannot integral out the moduli fieldscompletely. The time-independent Newton constant canbe easily realized in the four-dimensional Einstein framewhich is useful to analyze a behavior of scale factors andvarious fields.We consider a homogeneous ten-dimensional metricand a six-dimensional torus, T as a compactified space.The scale factors are functions of the time coordinate.The T has six scale factors corresponding to six cycles.The string-frame metric is given by the following equa-tion: ds = − e λ ( t ) dt + e λ ( t ) d x + X m =4 e λ m ( t ) ( dy m ) (1)where d x ≡ P i =1 ( dx i ) is the line element of the flatthree-dimensional space, R or T and the cycle of T isdefined as 0 ≤ y m ≤ π √ α ′ . (2) α ′ is related with a string length as l s ∼ √ α ′ . The scalefactor e λ m ( t ) describes the scaling of the cycle defined by(2). The four-dimensional Einstein frame is defined bythe following transformation: λ ( t ) = n ( t ) + β ( t ) , β ( t ) ≡ φ ( t ) − λ ( t ) ,λ ( t ) = A ( t ) + β ( t ) , λ ( t ) ≡ X m =4 λ m ( t ) (3)where φ ( t ) is dilaton field. e n ( t ) and e A ( t ) are the lapsefunction and the scale factor of the three-dimensionalspace, respectively. For the Einstein frame the propertime of the four-dimensional space-time is defined by e n ( t ) = 1 which is not equivalent to the proper time of theten-dimensional space-time in the string frame because of e λ ( t ) = e β ( t ) .By (3) the dilaton gravity sector of the string frameaction becomes [7]: S = 116 πG Z M d X √− Ge − φ ( t ) ( R + 4( ∇ φ ) )= 116 πG Z M d xe n ( t )+3 A ( t ) × e − n ( t ) (cid:16) − A ( t ) + 2 ˙ β ( t ) + X m =4 ˙ λ m ( t ) (cid:17) (4)where the four-dimensional Newton constant and the ten-dimensional gravitational constant are given by [31] G = G (2 π √ α ′ ) = α ′ , (5) κ = 8 πG = 12 (2 π ) α ′ . (6)Eq. (4) shows that the four-dimensional Einstein-Hilbert term which is given by the first term of the secondline does not involve moduli fields and the dilaton. By(5) we find that the string length is of the order of thePlanck length. An advantage of the new variable β ( t ) isthat the kinetic term of the above action becomes diag-onal. III. POINT PARTICLES FROM WRAPPEDBRANES
In the previous section, we have defined the four-dimensional Einstein frame which gives the four-dimensional Einstein-Hilbert term with the time-independent Newton constant and diagonal kinetic termsof moduli fields. We found that the dynamics of themoduli fields is governed by potential terms derived byflux terms and brane sources after the Kaluza-Klein re-duction. In this section, we will consider the effectiveaction of the wrapped D p -brane, NS5-brane and KK5-monopole, using the four-dimensional Einstein frame andderive the energy density of those ingredients which con-tribute as potential terms for the moduli fields.We consider branes wrapping over only cycles of T .After the compactification, we see those objects as pointparticles on the three-dimensional space. This expecta-tion is motivated by ref. [25] which shows that wrappedbrane gases give the energy density of a pressureless mat-ter. In [25], we gave the effective action without a depen-dence of the velocity along the three-dimensional space.We would like to derive the effective action with the ve-locity on the three-dimensional space. The effective ac-tion will be used to read off the mass and the charge ofa wrapped brane in later sections. To derive the effec-tive action we assume a cancellation of total RR chargesof the D-branes, because of homogeneous distributionsof many D-branes on the nine-dimensional space by thebrane gas approximation. Thus, we will assume wrappedbranes as gases of free particles.First, we will consider the world volume action ofD p -brane wrapping over a p -dimensional cycle of T .If the D p -brane wraps over a ( m , . . . , m p )-cycle (0 ≤ p ≤ p -brane exist along the( m , . . . , m p )-directions. The D p -brane can move alongtransverse directions. We assume that the distributionsof the branes are homogeneous and that the gauge fieldis Abelian. Then the gauge potential and transverse co-ordinates depend only on the time variable, i.e. A m a ( t ), X i ( t ) ( i = 1 , ,
3) and X m a ( t ) . We will adopt the co-ordinate system as ξ = t , ξ m a = y m a ( a = 1 , , . . . , p ). Then, the Dirac-Born-Infeld action of D p -brane wrappingover the ( m , . . . , m p )-cycle, Σ p is given by S ( m ··· m p )D p = − T p Z R × Σ p d p +1 ξe − φ ( t ) p − det( γ ab + 2 πα ′ F ab )= − (2 π √ α ′ ) p T p Z dte − φ ( t )+ λ ( t )+ P pa =1 λ ma ( t ) × n − e − λ ( t )+2 λ ( t ) 3 X i =1 ( ˙ X i ( t )) − (2 πα ′ ) p X a =1 e − λ ( t ) − λ ma ( t ) ( ˙ A m a ( t )) − X b = p +1 e − λ ( t )+2 λ mb ( t ) ( ˙ X m b ( t )) o (7)where the tension of the D p -brane is given by [31] T p = (2 π ) − p ( α ′ ) − ( p +1) / . (8) P i =1 ( ˙ X i ( t )) denotes the velocity along the three-dimensional space. The world volume action is propor-tional to exp( − φ ) √ det γ ab and then the mass becomesheavy for e φ ≪ e λ m ≫ S ( m ··· m p )D p = − (2 π √ α ′ ) p T p Z dte n ( t ) − λ ( t )+ P pa =1 λ ma ( t ) × n − e − n ( t )+2 A ( t ) 3 X i =1 ( ˙ X i ( t )) − A ( t ) o (9)where A ( t ) is defined by the following equation: A ( t ) ≡ (2 πα ′ ) p X a =1 e − β ( t ) − n ( t ) − λ ma ( t ) ( ˙ A m a ( t )) + X b = p +1 e − β ( t ) − n ( t )+2 λ mb ( t ) ( ˙ X m b ( t )) . (10)Compared with (9) to (7), we find the basic difference be-tween the string frame and Einstein frame. In the stringframe, the winding mode of the world volume dependson the dilaton, whereas the winding mode of D p -branedoes not have the coupling of the dilaton field in thefour-dimensional Einstein frame. This fact implies thatthe winding mode cannot stabilize the dilaton field in thefour-dimensional Einstein frame. Namely, if we take A ( t )defined by (10), we also cannot stabilize the dilaton, since A ( t ) includes the dilaton with e − β ( t ) only. Therefore,the D p -brane wrapping over cycles of T cannot stabilizethe dilaton field in the four-dimensional Einstein frame,using the cosmological background defined by (1).The wrapped branes can be regarded as the ideal gasbased on the dilute gas approximation. We can solve theequations of motion of A m a ( t ) and X m b ( t ) derived from(9) as follows: e n ( t ) − λ ( t )+ P pa ′ =1 λ ma ′ ( t ) × (2 πα ′ ) e − β ( t ) − n ( t ) − λ ma ( t ) ˙ A m a ( t )= f m a πα ′ n − e − n ( t )+2 A ( t ) 3 X i =1 ( ˙ X i ( t )) − A ( t ) o , (11) e n ( t ) − λ ( t )+ P pa ′ =1 λ ma ′ ( t ) × e − β ( t ) − n ( t )+2 λ mb ( t ) ˙ X m b ( t )= v m b n − e − n ( t )+2 A ( t ) 3 X i =1 ( ˙ X i ( t )) − A ( t ) o (12)where we have the equations of motions and f m a and v m b are constants of integration. f m a and v m b are inte-gers and not continuouse numbers, because of a cyclicityof the T . For exmaple, v m b is related a conjugate mo-mentum P m ··· m p D p, m b ≡ ∂ L m ··· m mp D p /∂ ˙ X m b as P m ··· m p D p, m b = v m b / √ α ′ by (12). The conjugate momentum must bean integer on a cycle of the T therefore v m b is an in-teger. By the T-dulaity f m a and v m b is mapped eachother [25] and then f m a must be also an integer. Thesimilar property on a quantized momentum is satisfiedfor a fundamental string as shown in Appendix C.We can solve the equations (11) and (12) on A ( t ): A ( t ) = e λ − P pa ′ =1 λ ma ′ ( t ) e A ( t )1 + e λ − P pa ′ =1 λ ma ′ ( t ) e A ( t ) × n − e − n ( t )+2 A ( t ) 3 X i =1 ( ˙ X i ( t )) o , e A ( t ) ≡ p X a =1 e β ( t )+2 λ ma ( t ) f m a πα ′ ! + X b = p +1 e β ( t ) − λ mb ( t ) ( v m b ) . (13)In the four-dimensional Einstein frame, the potentialterm is derived by u ( m ··· m p )D p ≡ − T = 2 g p − g ( x ) δ L ( m ··· m p )D p δg = − e − n ( t ) − A ( t ) δ L ( m ··· m p )D p δn ( t ) . (14)Using (9), (13) and (14), we obtain the potential of the D p -brane wrapping over the ( m · · · m p )-cycle: u ( m ··· m p )D p = e − A ( t ) (2 π √ α ′ ) p T p × n e − λ ( t )+2 P pa =1 λ ma ( t ) + e A ( t ) o × n − e − n ( t )+2 A ( t ) 3 X i =1 ( ˙ X i ( t )) o − . (15)This corresponds to the energy density of a relativisticparticle with the mass given by m ( m ··· m p ) p =(2 π √ α ′ ) p T p × n e − λ ( t )+2 P pa =1 λ ma ( t ) + e A ( t ) o . (16)Thus, we see D p -branes behave as point particles onthe three-dimensional space as Fig.1. The world volumedecays with e − A ( t ) as the pressureless matter.The mass given by (16) has the following role as apotential of the moduli fields. The winding part of (16)is proportional toexp 12 + p X a =1 λ m a ( t ) − X b = p +1 λ m b ( t ) ! . This indicates that the wrapped D p -brane binds the( m · · · m p )-cycle where the D p -brane expands and the( m p +1 · · · m )-direction is stretched like a rubber bandas in Fig.2. Similarly, by (13) it is found that the elec-tric fields in e A ( t ) also bind the ( m · · · m p )-cycle. Thetransverse velocities in e A ( t ) make a pressure bringingthe expansion to the transverse directions.We will derive the action of the point particle forNS5-brane and KK5-monopole in type II string theorywhich has five spatial dimensions. In this paper, weconsider only winding modes of NS5-brane and KK5-monopole moving along the three-dimensional space.The world volume action of KK5-monopole wrapping FIG. 1: Wrapped brane gases can be seen as point particleson three dimensions. Those point particles decay as the pres-sureless dust with e − A ( t ) . FIG. 2: A cycle where branes wrap is bound by the tensionof branes. Cycles where branes do not wrap are stretchedlike a rubber tube in the four-dimensional Einstein frame.The cycles are homogeneously bound under the brane gasapproximation, although this figure shows one of branes asan example. over the ( m · · · m )-cycle is given by [32, 33] S ( m ··· m )KK5 = − T KK5 Z R × Σ d ξe − φ ( t ) k p − det e γ ab = − (2 π √ α ′ ) T KK5 Z dte − φ ( t )+2 λ ( t )+ λ ( t )+ P a =1 λ ma ( t ) × n − e − λ ( t )+2 λ ( t ) 3 X i =1 ( ˙ X i ( t )) o , e γ ab ≡ ∂X m ∂ξ a ∂X m ∂ξ a ( G mn − k − k m k n ) . (17)where k m ≡ δ mm is the killing vector of the S isometry along the ( m )-cycle and k ≡ G mn k m k n =exp (2 λ m ( t )). We also consider the NS5-brane. Theworld volume action of the NS5-brane [34] wrapping over( m · · · m )-cycles is given by S ( m ··· m )NS5 = − T NS5 Z R × Σ d ξe − φ ( t ) p − det γ ab = − (2 π √ α ′ ) T NS5 Z dte − φ ( t )+ λ ( t )+ P a =1 λ ma ( t ) × n − e − λ ( t )+2 λ ( t ) 3 X i =1 ( ˙ X i ( t )) o . (18)We consider the transformation of fields as in (3) toobtain the potential term of the NS5-brane and the KK5-monopole in the four-dimensional Einstein frame. Sub-stituting (3) for (17) and (18) and replacing L ( m ··· m p )D p with L ( m ··· m )NS5 / KK5 in (14), the energy density of the NS5- brane and the KK5-monopole are given by u ( m ··· m )NS5 = e − A ( t ) (2 π √ α ′ ) T NS5 × e − β ( t ) − λ ( t )+ P a =1 λ ma ( t ) × n − e − n ( t )+2 A ( t ) 3 X i =1 ( ˙ X i ( t )) o − , (19) u ( m ··· m )KK5 = e − A ( t ) (2 π √ α ′ ) T KK5 × e − β ( t ) − λ ( t )+2 λ m + P a =1 λ ma ( t ) × n − e − n ( t )+2 A ( t ) 3 X i =1 ( ˙ X i ( t )) o − , (20)where T D5 = T NS5 = T KK5 [33]. The mass of the particlesis given by m ( m ··· m )NS5 = (2 π √ α ′ ) T NS5 × e − β ( t ) − λ m ( t ) , (21) m ( m ··· m )KK5 = (2 π √ α ′ ) T KK5 × e − β ( t )+ λ m ( t ) . (22)It is expected that the world volume of NS5 and KK5contributes to the stabilization of the dilaton, since thoseobjects have the opposite dependence on the dilaton,compared with (16). In fact, [25] have suggested thata D1-KK5 brane gas model stabilizes the dilaton field aswell as the radial moduli fields, simultaneously. IV. MASS OF WRAPPED BRANES IN THEFOUR-DIMENSIONAL EINSTEIN FRAME
In section III, the effective mass of the wrapped braneis derived in the four-dimensional Einstein frame. In thissection, we would like to consider a behavior of the massderived in the previous section, taking a large scale of the T and a weak string coupling related with the dilatonfield as g s = exp( φ ). The condition is required to consideran effective field theory. It is not necessarily correct thatD p -brane is very heavy with a large volume and a weakstring coupling, since the world volume action has thenontrivial coupling to the dilaton and the scale of T inthe four-dimensional Einstein frame. We take the stringscale as 1 / √ α ′ ∼ m Planck ∼ O (10 ) GeV to analyze themass.First, we will consider winding modes of branes and e λ m ( t ) = e λ ′ ( t ) . By (16), (21) and (22) the mass of branesis as follows: m ( m ··· m p ) p, winding = 1 √ α ′ × e ( p − λ ′ ( t ) ,m ( m ··· m )NS5 , winding = 1 √ α ′ × e − φ ( t )+2 λ ′ ( t ) ,m ( m ··· m )KK5 , winding = 1 √ α ′ × e − φ ( t )+4 λ ′ ( t ) . (23)We find that the mass of NS5 and KK5 becomes veryheavy for e λ ′ ≫ e φ ≪
1. In the case of p ≥
4, themass of D p -brane also becomes heavy. The D3-brane hasthe Planck mass. On the other hand, for p ≤
2, the D p -brane has the light mass under the condition of e λ ′ ≫ e φ ≪
1. For example, we will consider exp( λ ′ ) ∼O (10 ) which implies that the compactification is aroundGUT scale, V T = (2 π √ α ′ exp( λ ′ )) ∼ ( O (10 − )GeV − ) . The mass of the D p -brane is of the order of m ( m ··· m p ) p, winding ∼ O (10 p − ) GeV . (24)The mass ranges from O (10) TeV for p = 0 to O (10 )GeV for p = 2. This is the non-trivial result, since thestring frame gives rise to heavy branes for the large vol-ume. We should include those light states in string cos-mologies. Taking the four-dimensional Einstein framegives the interesting result for the mass of wrapped D-branes.Secondly, we will consider the anisotropic case, λ m = λ n satisfying exp( λ m ) ≫ p - and D(6 − p )-brane can be represented as m ( m ··· m p ) p, winding = 1 √ α ′ exp p X a =1 λ m a ( t )2 − X b = p +1 λ m b ( t )2 ! ,m ( m p +1 ··· m )6 − p, winding = 1 √ α ′ exp X a = p +1 λ m a ( t )2 − p X b =1 λ m b ( t )2 ! . (25)Those masses explicitly satisfy the following electric-magnetic dual relation: m ( m ··· m p ) p, winding × m ( m p +1 ··· m )6 − p, winding = (cid:16) √ α ′ (cid:17) . (26)The right hand side of the above equation is independenton scale factors, while masses depend on the time. Thus,masses of D p - and D(6 − p )-brane are not independenteach other and satisfy a sea-saw like condition. A D p -brane becomes light when a D(6 − p )-brane has a heavymass and vice versa. The electric-magnetic dual relationis one of the reason for this scaling property. The be-havior of the mass can be classified into three parts by adynamics of the moduli fields as follows.(1) P pa =1 λ m a ( t ) − P b = p +1 λ m b ( t ) > p -brane is heavy and D(6 − p ) becomeslight.(2) P pa =1 λ m a ( t ) − P b = p +1 λ m b ( t ) = 0:the masses of D p - and D(6 − p )-brane are of order O (1 / √ α ′ ) ∼ O ( m Planck ). (3) P pa =1 λ m a ( t ) − P b = p +1 λ m b ( t ) < − p )-brane is heavy and D p -branebecomes light.A comment is in order about the dimensionality ofspace-times. The time-independent electric-magneticdual relation (26) is realized only in the four-dimensionalEinstein frame. For example, the mass of wrapped D p -brane in the string frame is given by m (string | m ··· m p ) p, winding = 1 √ α ′ e − φ ( t )+ λ m ( t )+ ··· + λ mp ( t ) and the dual relation is [35] m (string | m ··· m p ) p, winding × m (string | m p +1 ··· m )6 − p, winding = (cid:16) √ α ′ (cid:17) exp (cid:16) − φ ( t ) + λ ( t ) (cid:17) . (27)This relation explicitly depends on time. If the righthand side of (27) becomes large, each mass of wrappedbranes can take a large value. Similarly, using the ( d +1)-dimensional Einstein frame given by (A2), the mass ofthe wrapped D p -brane is given by m ( d +1 | m ··· m p ) p, winding = 1 √ α ′ × exp − dd − φ ( t ) + p X a =1 λ m a ( t ) − d − X m = d +1 λ m ( t ) ! (28)and the dual relation is m ( d +1 | m ··· m p ) p, winding × m ( d +1 | m p +1 ··· m − ( d +1) )10 − ( d +1) − p, winding = (cid:16) √ α ′ (cid:17) exp d − d − − φ ( t ) + X m = d +1 λ m ( t ) ! . (29)The right hand side also involves time-dependent scalefactors. Therefore, the four-dimensional Einstein frameis a special case where the electric-magnetic dual relationbecomes scale-free.By the S- and T-duality, it is expected that the massof the wrapped NS5-brane and KK5-monopole has adual relation as (26). Using S-dual rule in the four-dimensional Einstein frame (B3) and (B4), the windingmodes of D1- and D5-brane are mapped to the windingmodes of fundamental string given by the first term of(C5) with w m a = 1 and NS5-brane, respectively. Thenwrapped fundamental string and NS5-brane satisfy thefollowing dual relation: m ( m )F1 , winding × m ( m ··· m )NS5 = (cid:16) √ α ′ (cid:17) (30)where we have used (21) and (C5). According to T-dualalong ( m )-cycle ( λ m → − λ m , φ → φ − λ m ), the massof wrapped NS5-brane is mapped to the mass of KK5-monopole as m ( m ··· m )NS5 → m ( m ··· m )KK5 [25]. According toT-dual along ( m )-cycle ( λ m → − λ m , φ → φ − λ m ),the mass of wrapped NS5-brane is mapped to the massof KK5-monopole as m ( m ··· m )NS5 → m ( m ··· m )KK5 [25]. Thewinding mode of the fundamental string is mapped tothe momentum mode as m ( m )F1 , winding → m ( m )F1 , momentum = √ α ′ e β ( t ) − λ m ( t ) , ( n m = 1). Therefore we obtain m ( m )F1 , momentum × m ( m ··· m )KK5 = (cid:16) √ α ′ (cid:17) . (31)The right hand side of (30) and (31) is scale freein the four-dimensional Einstein frame. For example,if m ( m )F1 , winding and m ( m )F1 , momentum are lighter than thePlanck mass, the mass of the the wrapped NS5-braneand the wrapped KK5-monopole becomes very heavy.The dual relations (30) and (31) are related with thestabilization of the dilaton field. The winding and themomentum mode of the fundamental string is propor-tional to e + β . By dual relations, the mass of NS5 andKK5 must have the dependence of e − β on the dilatonfield. NS5 and KK5 are essential for the stabilizationof the dilaton [25]. Thus the dual relation of wrappedbranes plays an important role for the moduli stabiliza-tion. V. MASS OF WRAPPED BRANES WITHCALABI-YAU THREEFOLD
In the previous section, we have found that there arecases where the mass of the brane wrapping over cyclesof the T becomes light in the four-dimensional Einsteinframe. One may expect a light wrapped branes for theCY compactification. In this section, we will investigatemasses of D p -branes wrapping over cycles of the CY ,taking the four-dimensional Einstein frame. To analyzethe mass we control scales of the Calabi-Yau by handbecause we cannot show the moduli stabilization of theCalabi-Yau in this paper. It is also interesting to knowwhether masses satisfy the electric-magnetic dual relationfound in the previous section.The CY has real six dimensions, 2-, 3- and 4-cycles.D0-, D2-, D3-, D4- and D6-brane can live in the CY ,while D1-, D5-, NS5-brane and KK5-monopole cannotwrap a cycle of the CY . In this section, we assume alarge volume (complex structure) limit and ignore anyquantum correction. We do not consider instabilities ofwrapped branes and assume the supersymmetric cyclecondition well behaves. To obtain the four-dimensionalEinstein frame with a CY compactification we redefine β ( t ) as β ( t ) = φ ( t ) −
12 ln V CY ( t )(2 π √ α ′ ) ! (32)where we have defined V CY ( t ) ≡ R d y p g CY ( t ). Thevolume of the CY has a time dependence through mod-uli fields. If D p -branes ( p = 0 , , ,
6) are included minimally inthe CY or its cycle, we can represent winding modes ofthe D p -branes as m = (2 π √ α ′ ) T V − / ,m ( A )2 = (2 π √ α ′ ) T V − / × Z Σ A J,m , ( A ) = (2 π √ α ′ ) T V − / × Z Σ , A J ∧ J,m = (2 π √ α ′ ) T V / , (33)where J is the K¨ahler form, Σ A and Σ , A are dual toharmonic form ω A and ˜ ω A . The properties of the K¨ahlerform are explained in Appendix D. The Poincar´e dualgives rise to Z Σ A J = Z CY J ∧ ˜ ω A = v A , Z Σ , A J ∧ J = Z CY J ∧ J ∧ ω A ≡ K ABC v B v C . Using above equations and (D2), we obtain the followingrelations between masses of wrapped branes: m × m = √ α ′ ! , X A ∈ h , m ( A )2 × m , ( A ) = 3 √ α ′ ! . (34)Eq.(34) implies that m ( m ( A )2 ) is not independent of m ( m , ( A ) ). In fact, by (D4) and (D5), there are relationsbetween those masses as follows: m = − (2 π √ α ′ ) − Im N m ,m , ( A ) = − (2 π √ α ′ ) − Im N AB m ( B )2 . (35)Note that the dual relation for D0- and D6-branes issatisfied quite naturally, since the dual relation does notrequire details of the CY . For instance, a large CY witha condition as V CY ∼ ( O (10 ) × π √ α ′ ) gives m ∼O (10) TeV. We consider a large volume by a constantscaling v A → exp(2 s ) v A (36)for all cycles. The m ( A )2 and m , ( A ) have the followingscalings: m ( A )2 → (2 π √ α ′ ) T V − / × e − s v A ,m , ( A ) → (2 π √ α ′ ) T V − / × e + s K ABC v B v C (37)where V CY → exp(+6 s ) V CY . Then light D2-branes andheavy D4-branes are realized for a positive constant s . Ifexp(2 s ) v A is chosen as O (10 ) × (2 π √ α ′ ) , The m ( A )2 isof order O (10 ) GeV. Those results have same scalingsgiven by (23) and (24).The behavior of the D2- and D4-branes is more com-plicated than the D0-D6 case, because the dual rela-tion for D2- and D4-branes cannot restrict all degreesof freedom of moduli fields. If we choose a scaling of v A → exp(2 s A ) v A , the masses of D2- and D4-branes varycorrespondingly. Using (34) and (35), tuning the scale ofcycles of the CY , we can find models having light andheavy D2- and D4-branes.Quite similarly, we can consider a D3-brane wrappingover a three-cycle. To derive the volume of the 3-cyclefrom CY data we will use a property of the specialLagrangian submanifold (supersymmetric cycle) [36–38].Here, we assume that the condition of the special La-grangian submanifold well behaves.In general we call N a special Lagrangian with a con-stant phase exp( iθ ), if [36, 38] J | N = 0 , [sin θ ReΩ − cos θ ImΩ] | N = 0 , [cos θ ReΩ + sin θ ImΩ] | N = d Vol N (38)where the holomorphic three-form is given byΩ = Z ˆ K α ˆ K − F ˆ K β ˆ K , J ∧ J ∧ J = 3 i ∧ Ω . (39)( α ˆ K , β ˆ K ) is the dual cohomology basis of ( A ˆ K , B ˆ K ) andthe real basis of H (CY ) in that they satisfy R CY α ˆ K ∧ β ˆ L = δ ˆ L ˆ K with all other intersections vanishing.We will consider D3-brane wrapping over a basis of the3-cycle A ˆ K or B ˆ K ( ˆ K = 0 , , · · · , h , ) as the specialLagrangian submanifold. Taking into account (38), weassume the existence of special Lagrangian cycles definedby the following conditions:ReΩ | A ˆ K = d Vol A ˆ K , ImΩ | A ˆ K = 0 , ( θ A ˆ K = 0) (40) − ImΩ | B ˆ K = d Vol B ˆ K , ReΩ | B ˆ K = 0 . ( θ B ˆ K = − π Z ˆ K α ˆ K − i Im F ˆ L β ˆ L (42)and Im Z ˆ K = 0 , Re F ˆ K = 0 . (43)Eq.(43) and Eq.(E3) give rise to Re M ˆ K ˆ L = 0 and thenthe Hodge dual of ( α ˆ K , β ˆ K ) defined by (E1) and (E2) is ∗ α ˆ K = − (Im M ) ˆ K ˆ L β ˆ L , ∗ β ˆ K = (Im M ) − K ˆ L α ˆ L . (44)In general, a Hodge dual of a basis includes a linearcombination of α ˆ K and β ˆ K as (E1). On the other hand, Eq.(44) indicates that ( α ˆ K , β ˆ K ) is decomposedwith respect to the Hodge dual, if D3-branes can wraparound A ˆ K and B ˆ K as special Lagrangian cycles withthe conditions (40) and (41). By (42), (44) and (E4) ∗ ReΩ | A ˆ K = ImΩ | B ˆ K is realized and then (40) and (41)give rise to ∗ d Vol A ˆ K = − d Vol B ˆ K .The world volume action of the wrapped D3-brane isdescribed by S D3 = − T Z R × A ˆ K d ξe − φ √− γ = − Z dte n × (2 π √ α ′ ) T V − / Re Z ˆ K (45)whereVol A ˆ K = Z A ˆ K ReΩ = Z CY ReΩ ∧ β ˆ K = Re Z ˆ K . Then the mass is given by m ( ˆ K )3 = (2 π √ α ′ ) T V − / Re Z ˆ K . (46)The dual volume is also defined byVol B ˆ K = Z B ˆ K ( − ImΩ) = − Z CY ImΩ ∧ α ˆ K = − Im F ˆ K and the dual mass is m , ( ˆ K ) = − (2 π √ α ′ ) T V − / Im F ˆ K . (47)Using (42) and the following relation4 V CY = Z CY ReΩ ∧ ImΩ= − Re Z ˆ K Im F ˆ K , (48) m ( ˆ K )3 and m , ( ˆ K ) satisfy X ˆ K ∈ h , +1 m ( ˆ K )3 × m , ( ˆ K ) = 4 √ α ′ ! . (49)Those masses also have the following relation: m , ( ˆ K ) = − Im M ˆ K ˆ L m (ˆ L )3 . (50)For h , + 1 > Z → Z ˆ K has a scaling as exp(3 s )Re Z ˆ K for a constant s , F ˆ K also has a scaling as exp(3 s ) F ˆ K ,because F is the holomorphic function with the degreetwo. Then, by (46), (47) and (48) it is found that themass is of order of the Planck mass for a large volumegiven by exp(3 s )Re Z ˆ K ∼ ( O (10 ) × π √ α ′ ) . This resultcorresponds to (24).In Sec. IV and Sec. V, we have considered the scalingbehavior of the mass of wrapped branes. For the stringframe, a large volume of the compactified space and aweak string coupling give rise to heavy branes, while, inthe four-dimensional Einstein frame, various light parti-cles of the wrapped branes arise after the T and the CY compactification. Those light particles are given by thestring scale which is of order the Planck length. For ex-ample, it has been shown that the mass of a D0-brane isof order O (10) TeV for V T or CY ∼ ( O (10 ) × π √ α ′ ) .Therefore, the four-dimensional Einstein frame is quitenontrivial. If the light particles have a weak RR inter-action, the wrapped branes may become a component ofthe dark matter. We will discuss the RR charge in thenext section. VI. RR CHARGE IN THEFOUR-DIMENSIONAL EINSTEIN FRAME
In the section IV and V, we have shown that variousD p -branes become light in the four-dimensional Einsteinframe. The light D p -brane locally interact through theRR flux. We have to estimate th RR charge, becausethe dark matter should have a very weak interaction. Toestimate the coupling of the RR flux, we will analyzethe RR flux and the Wess-Zumino (WZ) term by thefour-dimensional Einstein frame. If D p -brane has a weakcoupling, the light D p -brane is a dark matter candidate.We do not consider a contribution of the D-instanton( p = −
1) potential, for simplicity. In this section, theRR potential is a function of the four-dimensional coor-dinates, while the fields on the D p -brane and the scalefactor depend only on the time coordinate.The action of the RR flux and the WZ term of a D p -brane wrapping over a ( m · · · m p )-cycle is given by [31] S ( m ··· m p )RR = − κ Z M d X √− G | F p +2 | , (51) S ( m ··· m p )WZ = µ p Z R × Σ p exp(2 πα ′ F (2) ) ∧ X q C q (52)where | F p +2 | ≡ p + 2)! F µ ··· µ p +2 F µ ··· µ p +2 , (53) F p +2 = dC p +1 is the RR flux and F (2) is the gauge fieldon the D-brane. In the string frame the RR charge hasa relation given by [31] µ p = T p . (54)We will consider the dimensional reduction of the WZterm. The gauge field on the D p -brane is given by F (2) = ˙ A a ( t ) dt ∧ dξ a . (55)This equation leads to a relation F (2) ∧ · · · ∧ F (2) = 0.The RR potential of a D p -brane wrapping the specific ( m · · · m p )-cycle can be expanded as C p +1 = C ( m ··· m p ) µ ( x ) dX µ dt dt ∧ dξ m ∧ · · · ∧ dξ m p + X b = p +1 C ( m ··· m p ) m b ( x ) dX m b dt dt ∧ dξ m ∧ · · · ∧ dξ m p (56)where X i and X m b are transverse coordinates of thewrapped D p -brane. After the Kaluza-Klein reduction, C ( m ··· m p ) m b ( x ) is a scalar field on the four-dimensionalspace-time and the WZ term is S ( m ··· m p )WZ = Z M d x (2 π √ α ′ ) p µ p δ ( x − X ( t )) × C ( m ··· m p ) µ ( x ) dX µ dt + X b = p +1 C ( m ··· m p ) m b ( x ) dX m b dt ! . (57)First, we will consider the T compactification. Weperform the Kaluza-Klein reduction of the kinetic termof the RR flux and expand the RR potential on theten-dimensional space-time as C p +1 = ( C ( m ··· m p ) µ dx µ + P b = p +1 C ( m ··· m p ) m b dy m b ) ∧ dy m ∧ · · · ∧ dy m p . Substitut-ing this equation for (51) and using (3), (5), (6), (8) and(54), we obtain the following effective action of the RRflux in the four-dimensional Einstein frame: S ( m ··· m p )RR = Z M d x √− g × − g ( m ··· m p ) p ) ˜ F ( m ··· m p ) µν ˜ F ( m ··· m p ) µν − X b = p +1 g ( m ··· m p , m b ) p ) ∂ µ ˜ C ( m ··· m p ) m b ∂ µ ˜ C ( m ··· m p ) m b ! , (58)( g ( m ··· m p ) p ) ≡ π exp( − λ + 2 p X a =1 λ m a ) , (59)(˜ g ( m ··· m p , m b ) p ) ≡ π exp( − φ + 2 p X a =1 λ m a + 2 λ m b )(60)where˜ F ( m ··· m p ) µν = ∂ µ ˜ C ( m ··· m p ) ν − ∂ ν ˜ C ( m ··· m p ) µ , ˜ C ( m ··· m p ) µ ≡ (2 π √ α ′ ) p µ p C ( m ··· m p ) µ , ˜ C ( m ··· m p ) m b ≡ (2 π √ α ′ ) p µ p C ( m ··· m p ) m b . (61)The effective action of ˜ F ( m ··· m p ) µν has no dependence ofthe dilaton. This is the fact on N = 2 supergravity [40].The dilaton field lives in a hypermultiplet and does notcouple with a vector multiplet.0We find that g ( m ··· m p ) p satisfies the electric-magneticdual relation: g ( m ··· m p ) p × g ( m p +1 ··· m )6 − p = 2 π. (62)By (62) the coupling g ( m ··· m p ) p can take a small value,while the dual coupling g ( m p +1 ··· m )6 − p becomes large. Thedual relation (62) is also understood as follows. If we adda trivial quantity which is proportional to R F p +2 ∧ G − p , G − p = dD − p to the action (51) and eliminate F p +2 , weobtain R G − p ∧ ∗ G − p by the dual field G − p = ∗ F p +2 .Performing the Kaluza-Klein reduction, we can take thecoupling of the dual field, g − p in term of the definition(59). We consider ˜ g ( m ··· m p , m b ) p later.Secondly, we would like to discuss the CY compact-ification. The property considered above is general andit is expected that the charge relations can be satisfiedin the cases of the CY compactification. However, inthe CY compactification, the square of the coupling be-comes a matrix defined by the K¨ahler or the complexstructure moduli after the Kaluza-Klein reduction [40–44]. For example, we would like to consider the type IIAsupergravity compactified on a CY . After rescaling as C p +1 = µ − p ˜ C p +1 , the RR flux is given by F = µ − ˜ F , F = µ − ˜ F A ω A . Then the gauge kinetic term of the vector multiplets isgiven by − κ Z M
12 ( F ∧ ∗ F + F ∧ ∗ F )= Z M − "( − π Im N (2 π √ α ′ ) ) ˜ F ∧ ∗ ˜ F + ( − π Im N AB (2 π √ α ′ ) ) ˜ F A ∧ ∗ ˜ F B (63)where ( A = 1 , · · · , h , ). Im N and Im N AB are definedby (D5). Introducing a dual field ˜ G = ˜ F dual A ˜ ω A and˜ G = ˜ F dual d Vol CY which are related to the flux of D4-and D6-brane , we will consider the following term:12 π Z M ˜ F ∧ ˜ G = 12 π Z M V CY ˜ F ∧ ˜ F dual , π Z M ˜ F ∧ ˜ G = 12 π Z M ˜ F A ∧ ˜ F dual A (64)where we have used (D7). Adding (64) to (63) and in-tegrating out ˜ F and ˜ F A ( A = 1 , , · · · , h , ), the gaugekinetic term of (63) is mapped to Z M − "( − π (2 π √ α ′ ) (Im N − ) ) ˜ F dual ∧ ∗ ˜ F dual + ( − π (2 π √ α ′ ) (Im N − ) AB ) ˜ F dual A ∧ ∗ ˜ F dual B (65) where the inverse matrix on the couplings is defined by(D6) and ∗ ˜ F dualˆ A = − ˜ F dualˆ A has been used for a four-dimensional Lorentz manifold. We can read off the gaugecouplings is as follows:( g − ) = − π Im N (2 π √ α ′ ) , ( g − ) AB = − π Im N AB (2 π √ α ′ ) , ( g − ) = − π (2 π √ α ′ ) (Im N − ) , ( g − ) AB = − π (2 π √ α ′ ) (Im N − ) AB . (66)Those couplings satisfy( g − ) ˆ A ˆ B ( g − ) ˆ B ˆ C = (2 π ) − δ ˆ C ˆ A (67)where ( ˆ A = 0 , , , · · · , h , ). The above relation hasthe same structure to (62). In type IIB theory, the cou-pling of the flux of a D3-brane wrapping over a three-cycle is given by replacing N with M defined by (E3)[44] and taking a suitable normalization constant.We will consider the scaling behavior of g ( m ··· m p ) p and˜ g ( m ··· m p , m b ) p , assuming a constant dilaton and constantmoduli fields. We should notice that ˜ g ( m ··· m p , m b ) p is notthe true coupling of C ( m ··· m p ) m b , since the kinetic termof X m b ( t ) has a coupling with moduli fields such as e − β +2 λ ′ ( dX m b /dt ) in (10). To normalized the kineticterm, we define the following equations: dX m b dt ≡ e β − λ mb d ˜ X m b dt , ˜ C ( m ··· m p ) m b ≡ e − β + λ mb Ψ ( m ··· m p ) m b . (68)Then, the kinetic term is proportional to e − n ( d ˜ X m b /dt ) in (9). Using the above relations,the kinetic term of Ψ ( m ··· m p ) m b is given by X b = p +1 − g ( m ··· m p , m b ) p ) ∂ µ Ψ ( m ··· m p ) m b ∂ µ Ψ ( m ··· m p ) m b (69)where the coupling is correspond with (59):( g ( m ··· m p ) p ) = ( g ( m ··· m p , m b ) p ) . (70)We will consider the isotropic case such as λ m = λ n . g ( m ··· m p ) p and g ( m ··· m p , m b ) p can take a small value for p ≤ e λ ′ ≫
1. We will take e λ ′ ∼ O (10 ) forinstance. The coupling is given by( g ( m ··· m p ) p ) = ( g ( m ··· m p , m b ) p ) ∼ O (10 p − ) . (71)According to the relation, D0-brane has the very weakinteraction g ∼ O (10 − ). For wrapped D1- and D2-brane the coupling is given by ( g ( m )1 ) ∼ O (10 − ) and1( g ( m m )2 ) ∼ O (10 − ), respectively. The coupling ofD3-brane has g ( m m m )3 ∼ O (1). In the anisotropic case, λ m = λ n , Eq.(59) with (62) has the same classificationon the mass of D-branes in section IV .We also investigate scaling of the charges, controllingscales of the Calabi-Yau by hand. In the case with aCY compactification small couplings of wrapped D0-and D2-branes are realized by a large − Im N ˆ A ˆ B , becauseof − Im N ˆ A ˆ B ∝ ( g − ) ˆ A ˆ B . For example, using (D4) and(D5), we consider a large volume of the CY by a scal-ing, v A → exp(2 s ) v A for a positive constant s . Thiscondition gives rise toIm N → exp(6 s )Im N , Im N AB → exp(2 s )Im N AB . (72)Using (66) and (D5), the coupling of the D0-brane can be g ∼ O (10 − ) for V CY ∼ ( O (10 ) × π √ α ′ ) . Then theweak coupling is realized for wrapped D0- and D2-branes.The coupling of D3-branes which is related with the ma-trix M ˆ K ˆ L in (E3) has different behavior from the flux oftype IIA. The coupling involves the holomorphic functionwith the degree two, F (exp(3 s ′ ) Z ˆ K ) = exp(6 s ′ ) F ( Z ˆ K )by Z ˆ K → exp(3 s ′ ) Z ˆ K . Then, (E3) implies that the cou-pling matrix M does not change for the scaling of Z ˆ K .We may require details of the topological data of the CY to analyze the scaling of M , however we do not considerthe details in this paper. VII. FLUCTUATIONS OF MODULI FIELDS
In previous sections, we have investigated the mass andthe RR charge of the wrapped branes to consider a pos-sibility of a dark matter candidate. It has been shownthat there are cases where the wrapped D p -brane has alight mass and a weak RR charge in the four-dimensionalEinstein frame, if the scale of the compactified space sat-isfies a condition discussed in section IV, V and VI. How-ever, we have to discuss an interaction in term of fluctu-ations of moduli fields, because the fluctuations give riseto propagations of scalar interactions between branes. Infact, from the ten-dimensional point of view, the radialmoduli is one of components of the gravitational fieldsand then the fluctuations of the moduli also give the in-teraction between the D-branes [31]. In this section, wewould like to comment on a charge of the interactionwithout the analysis of the cosmological perturbation.We consider the T compactification only.The kinetic term of the moduli fields is diagonal in(4). To canonically normalize the kinetic term of thefluctuations, we define λ m = λ (0) m + p πG δ ˜ λ m (73)where λ (0) m is a fixed constant. Then, using (9), the source term of the interaction is given by δS ( m ··· m p )D p ∝ − (2 π √ α ′ ) p T p p πG exp (cid:16) p X a =1 λ (0) m a − λ (0) (cid:17) × Z dte n ( t ) − δ ˜ λ + p X a =1 δ ˜ λ m a ! . (74)The coefficient of the above equation gives the charge interm of the fluctuation. The magnitude of the interactionis given by the square of the coefficient: π exp (cid:16) p X a =1 λ (0) m a − λ (0) (cid:17) . (75)It is found that the scaling of (75) is same to (59) andthe similar dual relation such as (62) is satisfied. If2 P pa =1 λ (0) m a − λ (0) < λ (0) m = λ (0) n also realizes a weak inter-action on the fluctuations at a large scale of T for D0-,D1- and D2-branes.Taking into account results discussed in previous sec-tions, we find that there are cases in which wrapped D p -branes have the light mass, the weak RR charge andthe weak interaction on the fluctuations of the modulifields in the four-dimensional Einstein frame, consideringa large volume of the compactified space. For instance,the mass of a D0-brane is of the order of O (10) TeVfor V T or CY ∼ ( O (10 ) × π √ α ′ ) . The square of thecharge on the RR flux and the fluctuation of the modulifields is also of order O (10 − ). Therefore, a possibilityof the dark matter arises from the wrapped D p -branes inthe four-dimensional Einstein frame. VIII. D -KK BRANE GAS SYSTEM
We have considered the behavior of masses and chargesof wrapped branes in the four-dimensional Einsteinframe, adjusting the compactification scale by hand.However, it is not necessarily possible to tune the scaleof the compactification freely, because this tuning shouldbe consistent with a weak string coupling to consider theeffective field theory at a low energy after the moduli sta-bilization. In this section, we will investigate a modulistabilization in the T compactification, using the branegas system of D1-branes and KK5-monopoles in type IIBstring theory [25]. A condition for a large volume and aweak string coupling will be explained. We will estimatethe mass of the wrapped branes.At a matter-dominated era, the velocity of wrappedbranes may vanish, on average, in a very large scale.Then, we will consider the case of ˙ X i ( t ) ≃ f m a = 2 πα ′ f, v m a = 0 (76)to obtain the analytic value of λ a ( t ) and β ( t ) at a mini-mum. The choice of this initial condition represents thefact where the initial gauge fields on the D1-branes aresame for each cycle. This condition may be natural forthe isotropic expansion of the internal space. The abovecondition provides us the following energy density: U IIB = e − A ( t ) × ( N D1 m + N KK5 m KK5 ) (77)where N D1 and N KK5 are the number density of the pointparticles. This number density is a constant because thenumber density is decided at present as e A ( t =0) = 1. Thescaling of the number density is controlled by the scalefactor e A ( t ) . m D1 and m KK5 are given by m ≡ (2 π √ α ′ ) T X a =4 e − λ ( t )+ λ a ( t ) (1 + f e β ( t )+ λ ) , (78) m KK5 ≡ (2 π √ α ′ ) T KK5 9 X a =4 e − β ( t )+ λ a ( t ) . (79)If all moduli fields are fixed, U IIB is equal to the energydensity of a pressureless matter as ρ ∼ e − A .By (78) and (79) U IIB takes a positive value. Thusthe minimum is given by ∂U IIB /∂β = ∂U IIB /∂λ a = 0.( ∂∂λ a − ∂∂λ b ) U IIB = 0 gives e λ a ( t ) = e λ b ( t ) ≡ e λ ′ ( t ) (80)for ( a, b = 4 , , . . . , e β min. = r f N KK5 N D1 , (81) e λ ′ min. = f ! N D1 N KK5 ! . (82)As shown in Sec. III, f is an integer therefore stabilizedvalue of moduli fields cannot continuously connect to an-other stabilized values under the T compactification.These equations have a relation as e φ min. = e β min. +6 λ ′ min. = f − . The dilaton field at the minimumis given by e φ min. = 1 f (83) where we have used (3), (81) and (82). By (82) and (83)there is a relation between N D1 and N KK5 as follows: N D1 N KK5 = √ e − φ min. +6 λ ′ min. . (84)If a large volume of the T , exp( λ ′ mim. ) ≫ g s = exp( φ mim. ) = r f ≪ N D1 ≫ N KK5 . (86)(86) indicates that the wrapped D1-brane gas domi-nates the components. The number density of KK5has very low number density by the relation (84) andthen KK5-monopoles may survive the annihilation by theheavy mass and the low number density on the three-dimensional space.Eq.(82) and Eq.(83) show that the string coupling g s = e φ min. is related to the scale of the compactifica-tion through the initial condition of the electric fields onthe D1-brane. We require a weak string coupling g s ≪ e λ ′ min ≫
1. Ifwe take the specific initial condition as 1 ≪ f in (83),the weak string coupling is realized. However, DBI ac-tion has upper bound for the electric fields. This boundis given by 1 − A ( t ) > X i ( t ) ≃
0. We have to check that theinitial condition satisfies (87) at the minimum. Substi-tuting (81) and (82) for (13), we obtain the followingcondition for a D1-brane:1 − A min. = 1 − f e β min. +6 λ ′ min. f e β min. +6 λ ′ min. = 12 . (88)Therefore we can take the initial condition satisfying1 ≪ f because (88) does not depend on any initial con-dition in this model. By (84) and (87) we can take theweak string coupling and the large volume of T simul-taneously.Finally, we would like to estimate the mass scale of thebranes. At the minimum given by (82) and (83), the scaleof T is isotropic, i.e. e λ a min. = e λ ′ min. for ( a = 4 , · · · , m ( m )1 | min. = e − λ ′ min. × m Planck ,m ( m ··· m )KK5 | min. = e − φ min. +4 λ ′ min. √ × m Planck (89)where m Planck ≡ G − / = 2 √ / √ α ′ ∼ O (10 ) GeV isthe four-dimensional Planck mass. If we take a weak3string coupling e φ min . ≪ e λ ′ min. ≫
1, the mass of KK5-monopole becomes heavier than thePlanck mass. On the other hand, D1-branes become lightparticles under the weak string coupling and the largescale of T . If we consider e λ ′ min. ∼ O (10 ), the order ofthe mass is given by m | min. ∼ O (10 ) GeV.In the present section, we have considered the branegas model constructed by D1-KK5 as a simple model ofthe moduli stabilization. It is found that (85) and (86)are required to realize a large volume and a weak stringcoupling at a minimum of the potential and the D1-branegas becomes light. IX. SUMMARY AND DISCUSSIONS
We have considered a possibility of the dark mattercandidate for wrapped branes in brane gas cosmologiesbased on the type II string theories. Using the four-dimensional Einstein frame, we have investigated themass, the RR charge and the interaction on fluctuationsof the moduli fields. This analysis has been done by thedescription of the effective field theory, taking the stringscale as 1 / √ α ′ ∼ m Planck . A large volume of the com-pactified space and a weak string coupling are requiredto suppress string excitations and to obtain the pertur-bative description.We have found models where D-branes wrapping overcycles of a compactified space has a light mass by thelarge volume and the weak string coupling, after the T and the CY compactification, while the mass be-comes very heavy in the string frame. The masses sat-isfy the electric-magnetic dual relation between a D p -and a D(6 − p )-brane. The four-dimensional Einsteinframe gives rise to the dual relation which is time-independent although each mass depends on the timecoordinate through moduli fields. The string and the( d + 1)-dimensional ( d = 3) Einstein frame, on the otherhand, derive the dual relation depending on the timevariable. Thus, the four-dimensional Einstein frame isthe special case and the electric-magnetic dual relationgives the mass hierarchy. For example, the mass of a D0-brane is of order O (10) TeV, if we take a large volumeas V T or CY ∼ ( O (10 ) × π √ α ′ ) . Similar dual rela-tion which is time-independent is realized for (F1, NS5)and for ( the momentum of F1, KK5) for the T com-pactification. The effective charges of the RR flux andof the moduli fluctuation have been investigated. Thecharges between the D p - and D(6 − p )-brane also sat-isfy the electric-magnetic dual relations and then thereare cases in which the wrapped D p -branes obtain a smallcharge for the large volume and the weak string coupling.For instance, the square of the charge of a D0-brane is oforder O (10 − ) for V T or CY ∼ ( O (10 ) × π √ α ′ ) .We have considered the behavior of the mass ofwrapped branes, using a toy model constructed by a D1-KK5 brane gas [25] where radial moduli fields of the T and the dilaton are simultaneously stabilized. The ef- fective field description requires a large volume of thecompactified space and a weak string coupling, howevera point where all moduli fields are stabilized depends onmodels and the realization of the condition is not trivial.In this model, the condition of the large volume and theweak string coupling imposes (85) and (86) at a mini-mum of the potential of the moduli fields and then lightD1-branes appear for a large volume of the T .Taking the four-dimensional Einstein frame is verysimple, however quite non-trivial results among massesand the charges are realized. We should consider thelight wrapped branes in string cosmologies because ofthe very small charges. Those ingredients may be a can-didate for the dark matter. We have considered no NSNStwo form and no quantum correction. Those quantitiesmay give the wrapped branes interesting results, usingexplicit CY compactifications.To check the possibility on the dark matter more rigor-ously we have to investigate the density perturbation ofthe dust of wrapped branes. If domains with dense gasesappear on the three-dimensional space, interactions onthe RR flux and on the moduli fluctuations may have aninteresting role for the evolution of the density, because of O (( g ( m ··· m p ) p ) ) ∼ O ( G ( m ( m ··· m p ) p, winding ) ) by (5), (16) and(59). The density perturbation has been considered invarious brane gas models [12, 27–29, 45, 46].In the brane world model, the wrapped branes dis-tribute in a bulk space as well as on the brane expand-ing the three dimensions. Then, various bound statesmay arise between many branes and the dark matter ofwrapped branes may have interactions with the dark en-ergy through the couplings of moduli fields. It may beinteresting to investigate the dynamics of the universewith the light branes in the four-dimensional Einsteinframe for various models [47–65]. Acknowledgments
The authors thank the Yukawa Institute for Theo-retical Physics at Kyoto University. Discussions duringthe YITP workshop YITP-W-08-04 on “Development ofQuantum Field Theory and String Theory” were usefulto complete this work. We also thank Particle TheoryGroup of the Yukawa Institute for Theoretical Physicsfor fruitful discussions. We would like to thank TohruEguchi, Kenji Hotta, Tetsuji Kimura, Hideo Kodama,Shinji Mukohyama, Misao Sasaki, Naoki Sasakura, SeijiTerashima for useful and helpful comments. M.S. is sup-ported by Sasagawa Scientific Research Grant from TheJapan Science Society and by Hokkaido University ClarkMemorial Foundation.
Appendix A: ( d + 1) -dimensional Einstein frame We will decompose the ten-dimensional space-time as10 = ( d + 1) + (10 − ( d + 1)) and take the ( d + 1)-4dimensional Einstein frame. The (d+1)-dimensional Ein-stein frame is obtained by a toroidal compactification ofthe circumference of 2 π √ α ′ e λ m as116 πG Z d X √− Ge − φ R (A1)= 116 πG d +1 Z d d +1 x p − g d +1 R d +1 + · · · where the ( d + 1)-dimensional gravitational constant isgiven by G d +1 = G / (2 π √ α ′ ) − ( d +1) . To obtain thiseffective action we will consider the following variables: G µν = e β d +1 g d +1 , µν ,G mn = e λ m δ mn ,β d +1 = 2 d − φ − d − X m = d +1 λ m , (A2)where ( µ, ν = 0 , , · · · , d ) and ( m, n = d + 1 , d +2 , · · · , d + 1)-dimensional Ein-stein frame. If we consider d = 3 and β ≡ β , thefour-dimensional Einstein frame defined by (3) is recov-ered. Appendix B: S-duality in the four-dimensionalEinstein frame
The S-duality rule is defined by the ten-dimensionalEinstein frame. Using (A2) in the case of d = 9, weobtain G AB = e β g , AB ,β = 14 φ = 14 β + λ ! (B1)where ( A, B = 0 , , · · · , β is related with variables( β, λ ) which are defined by (3) as (B1). Then the S-duality is given by φ → − φ which means that the string-frame metric transforms as G AB −→ e − φ G AB . (B2)to find the transformation rule of the S-duality in thefour-dimensional Einstein frame we will assume λ m → λ m + f S-dual ,β → β + f S-dual . (B3)If this transformation represents the S-duality satisfying(B2), the action (4) must be invariant under this trans-formation. Substituting (B3) for (4), the the S-dual in-variance requires f S-dual = − φ. (B4)It is found that this solution satisfies (B2), substituting(B4) for (B1). Appendix C: fundamental string gas
If the fundamental string wraps over a cycle of T andwe choose σ = t , 0 ≤ σ ≤ π √ α ′ , we can assume X = t,X m a = X m a ( t ) + w m a σ , ( a = 1 , , , · · · , , (C1)where we have considered the only zero mode and w m a indicates the winding number of the string. The aboveassumption is motivated by X m a ( t, σ + 2 π √ α ′ ) = X m a ( t, σ ) + 2 π √ α ′ w m a . We do not consider a depen-dence of X i ( i = 1 , , γ = − e λ + X a =1 e λ ma ( ˙ X m a ) ,γ = X a =1 e λ ma ˙ X m a w m a ,γ = X a =1 e λ ma ( w m a ) . (C2)We will impose γ m = 0, the meaning of which is ex-plained later. Then the world volume action of a funda-mental string wrapping over a cycle of T is given by S F1 = − T F1 Z R × Σ dσ dσ √− γ = − (2 π √ α ′ ) T F1 Z dte λ × { X a =1 e λ ma ( w m a ) (1 − X a =1 e λ ma − λ ( ˙ X m a ) ) } = − (2 π √ α ′ ) T F1 Z dte β + n × { X a =1 e λ ma ( w m a ) (1 − X a =1 e λ ma − β − n ( ˙ X m a ) ) } (C3)where T F1 = T D1 . The equation of motion is( X a =1 e λ ma ( w m a ) ) / e n + β e λ ma − β − n ˙ X m a = n m a { − X a =1 e λ ma − β − n ( ˙ X m a ) } (C4)where n m a is a constant of integration. By the definitionof the energy-momentum tensor (14) and (C4), we obtain u F1 = e − A m F1 = e − A (2 π √ α ′ ) T F1 × ( X a =1 e β +2 λ ma ( w m a ) + X a =1 e β − λ ma ( n m a ) ) . (C5)5Substituting (C4) for γ m = 0, we obtain X a =1 n m a w m a = 0 . (C6)If n m a and w m a are integers, (C6) indicates the levelmatching condition, L − ˜ L = 0, for the only zero mode.The mass m F1 is invariant under the T-duality, ( λ m a →− λ m a , φ → φ − λ m a ) with ( n m a ←→ w m a ). Appendix D: K ¨ Ahler form
The K¨ahler form J is expanded by a harmonic form ω A which is a basis of H , (CY ) as J = v A ω A . (D1)where ( A = 1 , · · · , h , ). It is useful to define the fol-lowing quantities: K ABC = Z CY ω A ∧ ω B ∧ ω C , K AB = Z CY ω A ∧ ω B ∧ J = K ABC v C , K A = Z CY ω A ∧ J ∧ J = K ABC v B v C , K = Z CY J ∧ J ∧ J = K ABC v A v B v C = 6 V CY . (D2)Using the following relation [66]: ∗ ω A = − J ∧ ω A + 32 K A K J ∧ J, ∗ J = 12 J ∧ J, (D3)we obtain Z CY ω A ∧ ∗ ω B = −K AB + 32 K A K B K . (D4)Eq.(D4) is related to gauge couplings of the vector mul-tiplets in the low energy effective action of type IIA su-pergravity compactified on a Calabi-Yau threefold. If theNSNS 2-form B vanishes, the matrix of the gauge cou-plings is given by [41–44]Im N = − K − V CY , Im N AB = − Z CY ω A ∧ ∗ ω B . (D5)Introducing K AB by K AB K AC = δ AC , one derives theinverse matrix:(Im N − ) = − K , (Im N − ) AB = − Z CY ˜ ω A ∧ ∗ ˜ ω B (D6) where we defined the dual basis ˜ ω A ∈ H , (CY ) by Z CY ω A ∧ ˜ ω B = δ BA (D7)and Z CY ˜ ω A ∧ ∗ ˜ ω B = −K AB + 3 v A v B K , ∗ ˜ ω A = −K AB + 3 v A v B K ! ω B . (D8) Appendix E: Hodge dual of ( α ˆ K , β ˆ K ) { α ˆ K , β ˆ K } are both three-forms and those Hodge dualsare also three-forms which can be expanded in term of α ˆ K and β ˆ K according to ∗ α ˆ K = A ˆ L ˆ K α ˆ L + B ˆ K ˆ L β ˆ L , ∗ β ˆ K = C ˆ K ˆ L α ˆ L + D ˆ K ˆ L β ˆ L . (E1)Using Z CY α ˆ K ∧ β ˆ L = δ ˆ L ˆ K , one derive B ˆ K ˆ L = Z CY α ˆ K ∧ ∗ α ˆ L = Z CY α ˆ L ∧ ∗ α ˆ K = B ˆ L ˆ K , − C ˆ K ˆ L = Z CY β ˆ K ∧ ∗ β ˆ L = Z CY β ˆ L ∧ ∗ β ˆ K = − C ˆ L ˆ K , − A ˆ L ˆ K = Z CY β ˆ L ∧ ∗ α ˆ K = Z CY α ˆ K ∧ ∗ β ˆ L = D ˆ L ˆ K . The matrices A , B , C , D are determined in term of amatrix M [67, 68] A = (Re M )(Im M ) − ,B = − (Im M ) − (Re M )(Im M ) − (Re M ) ,C = (Im M ) − , (E2)where M ˆ K ˆ L = F ˆ K ˆ L + 2 i (Im F ) ˆ K ˆ M Z ˆ M (Im F ) ˆ L ˆ N Z ˆ N Z ˆ M (Im F ) ˆ M ˆ N Z ˆ N . (E3)The matrix M satisfies F ˆ K = M ˆ K ˆ L Z ˆ L (E4)where we have used the following relations on a holomor-phic prepotential F with respect to Z ˆ K : F = 12 Z ˆ K F ˆ K , F ˆ K = ∂ F ∂Z ˆ K = Z ˆ K F ˆ K ˆ L , F ˆ K ˆ L = ∂ F ˆ L ∂Z ˆ K . (E5)6 [1] R. H. Brandenberger and C. Vafa: Superstrings in theEarly Universe: Nucl. Phys. B316
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