Smooth tensionful higher-codimensional brane worlds with bulk and brane form fields
PPreprint typeset in JHEP style - HYPER VERSION
Smooth tensionful higher-codimensional brane worldswith bulk and brane form fields
Olindo Corradini
Dipartimento di Fisica, Universit`a di Bolognaand INFN, Sezione di BolognaVia Irnerio 46, Bologna I-40126, Italy&Centro de Estudios en F´ısica y Matem´aticas Basicas y AplicadasUniversidad Aut´onoma de ChiapasTuxtla Guti´errez, Chiapas, MexicoE-mail: [email protected]
Abstract:
Completely regular tensionful codimension- n brane world solutions are dis-cussed, where the core of the brane is chosen to be a thin codimension-( n −
1) shell in aninfinite volume flat bulk, and an Einstein-Hilbert term localized on the brane is included(Dvali-Gabadadze-Porrati models). In order to support such localized sources we enrichthe vacuum structure of the brane by the inclusion of localized form fields. We find thatphenomenological constraints on the size of the internal core seem to impose an upperbound to the brane tension. Finite transverse-volume smooth solutions are also discussed. a r X i v : . [ h e p - t h ] D ec ontents
1. Introduction and Summary 12. Vanishing bulk cosmological constant 2 p -form field 52.3 Comments 7
3. Non-vanishing bulk cosmological constant 9
1. Introduction and Summary
Brane world models have drawn a lot of attention in the last years since they provide aninteresting scenario for the search of solutions to long standing particle physics puzzles asthe cosmological constant problem and the hierarchy problem. In cosmology they mightprovide alternatives to dark matter and/or dark energy (see e.g. [1, 2]).In the present manuscript we study brane world models with codimension larger thantwo, for a variety of situations. However, we are mostly interested in flat bulk modelswhere the extra-dimensional volume is infinite and 4d gravity is brane-induced on thebrane at short scales [3, 4, 6] (see [5] for an orientifold derivation). Thin tensionful higher-codimensional solutions in flat space are known to give rise to singular backgrounds [7]and need to be regulated. One possible way to regulate such singularity is to ”resolve” thebrane, by giving it a non-trivial core, in the extra-dimensions, e.g. a thin spherical shell; inthis latter case the brane results effectively codimension-one. This method has proven to bequite efficient in the codimension-two case, both for finite volume rugby-ball [8] solutionsand for infinite-volume induced gravity ones. For codimension larger than two it had been shown that a naive regularization of thehigher-codimension brane by blowing the thin brane to a thin spherical shell lead to a no-gotheorem [11], that we review later. A possible way-out to such no-go theorem was thenfound in [12] by employing bulk higher curvature terms to regulate the bulk singularity.Another recent proposal for smoothing out higher-codimensional singularities is to considera bulk Einstein-Skyrme model [13]. Here we present a different way-out: we keep bulkEinstein-Hilbert gravity but consider a richer brane vacuum structure by the inclusionof higher-rank (form) fields (this was suggested in [14] for a Z Tensionful codimension-two singularities are milder (conical) and do not, a priori, need to be regu-lated [9]. However, regularization via smoothing out the brane profile is often invoked in order to avoidsubtleties associated to purely conical radially symmetric extra-dimensional space [10]. – 1 –imilarly to the codimension-two models that involve an axion field [16, 17, 18, 19] whosevacuum expectation value cancels the tangential (to the brane profile) component of thepressure. We explicitly show here that the inclusion of higher-rank fields works as well forour higher-codimensional solutions.Another reason behind the present work is the study of new (higher-codimensional)brane cosmology models: as said above codimension-one regularization seems necessary forcosmological setups, at least in bulk Einstein gravity (for Lovelock gravity and/or brokenspherical symmetry the situation might improve [20, 21, 22]). Such regularization allowedto study some cosmological properties of codimension-two setups using the moving braneapproach [23] or weak field limit [17, 24]. We can thus also see the present work as apossible framework where study cosmology on a generic-codimension brane world.Finally we consider higher-codimensional induced gravity brane world models, in thelight of more recent results [25] where it was found that cascading higher-codimensionalinduced-gravity models are ghost-free, hence shedding new light on such induced gravitymodels, which have been sources of several controversies regarding their classical and quan-tum stability. In [26] it was also suggested that cancellation of ghost excitations might aswell take place for resolved brane setups with codimension larger that two, provided tan-gential pressures are cancelled. We show later that, opposed to the codimension-two case,in our setup tangential pressures do not have to vanish and no strong fine-tuning betweenflux field and tension is a priori needed. However, phenomenological constraints on thesize of the internal brane profile seem to impose, for this class of models, an upper bound(cfr. eq. (2.37)) to the brane tension, as opposed to the codimension-two case where theupper bound for the tension is due to a topological constraint (the conical deficit angle isbounded to be less than 2 π ).
2. Vanishing bulk cosmological constant
The brane world model we study in this section is described by the following action: S = (cid:99) M D − (cid:90) d D x (cid:112) − (cid:98) g (cid:98) R + (cid:90) Σ d D − x √− g (cid:34) M D − (cid:18) R − Λ − · p ! F p ] (cid:19) + 2 (cid:99) M D − K ± (cid:35) (2.1)Σ = R D − n − , × S n − (cid:15) Here Σ is a fat codimension- n source brane, whose geometry is given by the product R D − n − , × S n − (cid:15) , where R D − n − , is the ( D − n )-dimensional Minkowski space, and S n − (cid:15) is a ( n − (cid:15) (in the following we will assume that n ≥ M D − Λ plays the role of the tension of the brane Σ and F [ p ] is the field strength of a( p − A [ p − F [ p ] = dA [ p − , F m ...m p = p∂ [ m A m ...m p ] = ∂ m A m ...m p + cyclic (2.2)Also, g mn ≡ δ mM δ nN (cid:98) g MN (cid:12)(cid:12)(cid:12) Σ , (2.3)– 2 –here x m are the ( D −
1) coordinates along the brane (the D -dimensional coordinates aregiven by x M = ( x m , r ), where r ≥ D -dimensional metric is ( − , + , . . . , +)); finally, the ( D − R is constructed from the ( D − g mn and K is the extrinsic curvature, with K ± ≡ K + + K − . In the following we will use the notation x i = ( x α , r ), where x α are the ( n −
1) angular coordinates on the sphere. Moreover, themetric for the coordinates x i will be (conformally) flat: δ ij dx i dx j = dr + r γ αβ dθ α dθ β , (2.4)where γ αβ is the metric on a unit ( n − D − n ) Minkowskicoordinates on R D − n − , via x µ (note that x m = ( x µ , θ α )).The bulk equations of motion are clearly given by (cid:98) G MN = 0 (2.5)and the boundary conditions for the fat brane can be obtained using Israel junction con-ditions (cid:68) K mn − δ mn K (cid:69) ± = − (cid:99) M D − T mn (2.6)where T mn = − M D − (cid:16) G mn + g mn Λ (cid:17) + T mn ( F ) (2.7)is the total energy-momentum tensor for the “matter” localized on the fat brane, with T mn ( F ) = M D − ( p − (cid:18) − p F g mn + F ml ...l p F n l ...l p (cid:19) . (2.8) In order to better clarify our results let us first re- R ε Σ * Figure 1:
Pictorial representationof the infinite-volume smooth braneworld. vise the no-go theorem associated to radially symmet-ric solutions in absence of the p -form term [11]. Letus consider the following ansatz for the backgroundmetric: ds = exp(2 A ) η µν dx µ dx ν + exp(2 B ) δ ij dx i dx j , (2.9)where A and B are functions of r but are indepen-dent of x µ and θ α (that is, we are looking for solutionsthat are radially symmetric in the extra dimensions).The bulk equations of motion then read (here primedenotes derivative w.r.t. r ):( D − n ) (cid:20) D − n −
12 ( A (cid:48) ) + n − r A (cid:48) + ( n − A (cid:48) B (cid:48) (cid:21) – 3 –( n − n − (cid:20)
12 ( B (cid:48) ) + 1 r B (cid:48) (cid:21) = 0 (2.10)( D − n ) (cid:20) A (cid:48)(cid:48) + D − n + 12 ( A (cid:48) ) + n − r A (cid:48) + ( n − A (cid:48) B (cid:48) (cid:21) +( n − (cid:20) B (cid:48)(cid:48) + n −
32 ( B (cid:48) ) + n − r B (cid:48) (cid:21) = 0 (2.11)Above, equation (2.11) is the ( αβ ) equation, while equation (2.10) is the ( rr ) equation.Note that the latter equation does not contain second derivatives of A and B . The solutionfor B (cid:48) is given by (we have chosen the plus root, which corresponds to solutions withinfinite-volume extra space): B (cid:48) = − r − D − nn − A (cid:48) + (cid:114) r + 1 κ ( A (cid:48) ) , (2.12)where we have introduced the notation1 κ ≡ ( D − n )( D − n − n − (2.13)to simplify the subsequent equations.Here we are interested in non-singular solutions such that A and B are constant for r < (cid:15) , and asymptote to some finite values as r → ∞ . For r > (cid:15) the solution for A and B is given by A ( r ) = − κn − (cid:18) f ( r )1 − f ( r ) (cid:19) , r > (cid:15) , (2.14) B ( r ) = − D − nn − A ( r ) + 1 n − (cid:0) − f ( r ) (cid:1) , r > (cid:15) , (2.15)where f ( r ) ≡ (cid:0) r ∗ r (cid:1) n − and r ∗ is the integration constant, and where we have set otherintegrations constants such that A ∞ = B ∞ = 0. A pictorial representation of such setupis given in Fig. 1, where the gray disk describes the extra-dimensional shape of the insidebulk ( r < (cid:15) ), the bell-shaped part is the asymptotically-flat outside bulk ( r > (cid:15) ), the circleΣ is the fat brane and the ”star” represents the would-be naked singularity r = r ∗ . Israeljunction conditions (2.6) provide the equations at the location of the fat brane, r = (cid:15) ;including the contribution of the induced EH term G µν = − ( n − n − R (cid:15) δ µν (2.16) G αβ = − ( n − n − R (cid:15) δ aβ (2.17)where R (cid:15) ≡ (cid:15) e B ( (cid:15) ) is the physical radius of the ( n − − sphere, we obtain( n −
2) 2 f ( (cid:15) )1 − f ( (cid:15) ) + (cid:15)L e B ( (cid:15) ) (cid:20) Λ − ( n − n − R (cid:15) (cid:21) = 0 (2.18)( n −
1) 2 f ( (cid:15) )1 − f ( (cid:15) ) − D − n − κf ( (cid:15) )1 − f ( (cid:15) ) + (cid:15)L e B ( (cid:15) ) (cid:20) Λ − ( n − n − R (cid:15) (cid:21) = 0 (2.19)– 4 –or the ( αβ ) and ( µν ) components respectively, with L ≡ M D − c M D − . We can rewrite theprevious matching conditions in a more useful way as follows:2 f ( (cid:15) )1 − f ( (cid:15) ) + L R (cid:15) ( λ − n + 3) = 0 , (2.20) D − n − κf ( (cid:15) )1 − f ( (cid:15) ) + L R (cid:15) ( λ + n −
1) = 0 . (2.21)where we have defined Λ ≡ λ n − R (cid:15) . Let us study possible solutions to these matchingconditions with r ∗ < (cid:15) for which 0 < f ( (cid:15) ) <
1: they would be non-singular solutions asthe would-be naked singularity r = r ∗ is cut away. The second matching conditions canonly be satisfied if Λ <
0. Hence λ must be a negative parameter; in other words thereare no non-singular solutions of this type with positive tension. Moreover, from the firstcondition we have: f ( (cid:15) ) = − λ + n − − λ − n + 1 (cid:114) ( n − D − D − n . (2.22)For n ≥
3, the condition 0 < f ( (cid:15) ) < λ . Hence, theabove matching conditions cannot be simultaneously satisfied within this class of solutions.For a different class of solutions that is curved both on the inside bulk and on the outsidebulk it is possible to overcome the previous no-go theorem [15]. In [14] an upgradedversion of the model [15], that suggested the use of brane form fields, was considered. Inthe next section we will see that changing the structure of the vacuum brane stress tensor,with the inclusion of higher-rank tensors is crucial also for type of geometry describedabove, as it allows smooths solutions. This type of geometry is the higher-codimensionalversion of that considered in [19]. Such type of regularization was studied in [27], in thecontext of compact codimension-two brane worlds, in order to obtain codimension-twoeffective actions. For the sake of generality we will thus consider in Section 3 some higher-codimensional generalizations of the backgrounds considered in [27], that will require bulkhigher-rank tensors as well as non-vanishing bulk cosmological constant or a bulk σ − modelmatter action [28]. p -form field In order to enrich the vacuum structure of our brane world we include a p -form field in theworldvolume of the blown-up brane Σ. We consider the case of a ( n − D − n )-form as in the string landscape [30]. Werequire its energy-momentum tensor to have the block-diagonal form T mn ( F ) = (cid:32) T δ µν T (cid:48) δ αβ (cid:33) (2.23)with T, T (cid:48) constant. In order to achieve that let us use spherical coordinates γ αβ dθ α dθ β = dθ + sin θ dθ + sin θ sin θ dθ + · · · + sin θ · · · sin θ n − dθ n − (2.24)– 5 –o parameterize the ( n − A [ n − ± = √ R n − (cid:15) (cid:16) ± c + h ( θ ) (cid:17) E [ n − (2.25)with Φ constant, E [ n − being the volume form of the equatorial ( n − h (cid:48) ( θ ) = sin n − θ , and h ( π ) = − h (0); the field strength F [ n − = √ R n − (cid:15) S [ n − (2.26)is thus proportional to the volume form of the unit ( n − S n − and the integrationconstant c is fixed by regularity conditions at the poles [29]. From (2.26) one immediatelyobtains T mn ( F ) = M D − Φ (cid:32) − δ µν
00 + δ αβ (cid:33) (2.27)and T µν = − M D − n − R (cid:15) ( λ + ϕ ) δ µν (2.28) T αβ = − M D − n − R (cid:15) ( λ − ϕ ) δ αβ (2.29)where we have defined Φ ≡ ( n − ϕ /R (cid:15) . Hence the boundary conditions (2.20) and (2.21)still hold with the replacements λ → λ − ϕ and λ → λ − (2 n − ϕ respectively. We thushave 2 f ( (cid:15) )1 − f ( (cid:15) ) = L R (cid:15) (cid:18) − λ + ϕ + n − (cid:19) , (2.30) D − n − κf ( (cid:15) )1 − f ( (cid:15) ) = L R (cid:15) (cid:18) − λ + (2 n − ϕ − n + 1 (cid:19) (2.31)so that λ can be either positive or negative, provided ϕ is large enough. The secondcondition gives R (cid:15) = L (cid:115) D − n ( n − D −
2) 1 − f ( (cid:15) )4 f ( (cid:15) ) (cid:18) − λ + (2 n − ϕ − n + 1 (cid:19) (2.32)that replaced into the first condition yields f ( (cid:15) ) = (cid:114) ( n − D − D − n − λ + ϕ + n − − λ + (2 n − ϕ − n + 1 (2.33)that is the equivalent of (2.22). Note however that now there are smooth solutions with f ( (cid:15) ) < λ ). Forexample let us consider n = 3 , D = 7: in such a case we have f ( (cid:15) ) = (cid:113) − λ + ϕ − λ +3 ϕ − thatcan be smaller than one, provided ϕ is large enough.– 6 –he flux ϕ increases the value of the physi- R ε −2R(r) r Figure 2:
Pictorial representation of theextra-dimensional space. cal four-dimensional vacuum energy density thatcan be obtained by integrating (2.28) over thebrane profile E d = ( n − S ( n − ( λ + ϕ ) M n R n − (cid:15) (2.34)where S ( n − is the volume of the unit-radius( n − − sphere. Coupling of the form potentialto a localized extended object leads to a quanti-zation condition for the flux [29, 30, 31, 32]; wecome back to this point in the next section.Let us point out a crucial difference betweenour setup ( n >
2) and previously consideredcodimension-two smooth solution. In the codi-mension two case [19] the smooth solution con-sidered has A = constant and B ∼ ln r so that the junction condition coming from the( αβ ) equation of motion (2.11) is trivial (there are no second derivatives in B in such acase) and this can only be satisfied if we “tune” Λ = Φ . In our case the only requirementfor the “flux” is a lower bound. Note however that, once λ and ϕ are chosen, the value ofthe physical radius is fixed in terms of (2.32).In the present 4d-Poincar´e-invariant Ricci-flat setup the inclusion of bulk fluxes isproblematic because of the no-go theorem [34]. In other words the present solutions avoidsuch no-go theorem in a trivial way: no bulk fluxes, and presence of discontinuities in thederivatives of the warp factors that are absorbed by localized fluxes and brane tension. We comment here on the possible physical scales involved in the model; we focus on the case d = 4 , n = 3 for it displays all the details of these models. There is a variety of scenariosthat might appear according to the different values of tension and flux and it is beyond thescopes of the present manuscript to give a detailed study of such issues. Let us howeverpoint out a few interesting features. The present model has codimension larger than two andthere is, a priori, no critical value for the tension. However, phenomenological constraintsimpose that the internal radius of the brane satisfies R (cid:15) < ( T eV ) − . Since the scale L willbe related to the crossover scale after which brane gravity turns higher-dimensional and itmust thus be taken to be enormously large, it is natural to assume that the internal radiusof the brane be extremely smaller than L . Hence, noting that a tiny value in the roundparenthesis of (2.32) would yield to an inconsistent value for f ( (cid:15) ), equation (2.32) implies f ( (cid:15) ) ∼ < ϕ ∼ > − (cid:16)(cid:112) / − (cid:17) λ − (cid:112) / λ ≤ λ M ≡ / ( (cid:112) / −
1) (2.37)that yields a critical value for the brane tension.Casting the bulk metric into the form ds = V ( ρ ) dx µ dx µ + dρ + W ( ρ ) d Ω one cancheck that there exists an allowed configuration ¯ ρ at which W ( ρ ) is critical. For such avalue, corresponding to ¯ r ∼ r ∗ the physical radius W ( ρ ) = r e B ( r ) assumes its minimumvalue, after which the bulk radius asymptotically approaches the flat limit R ( r ) ∼ r : theshape of the extra-dimensional space thus looks like a ”throat” ending on the brane, likedepicted in Figure 2 and this scenario is somewhat similar to the “near-critical” limit of [19]where the bulk looks like a thin cylindrical sliver that ends up on the brane and opens upnon trivially at very large scales. Then gravity on the brane should behave 4 d at distancesshorter than L C = M M = M n R n − c M n R n − = L , then an intermediate 5 d behavior should takeover at distances ∼ > L , till the bulk finally opens up at a scale L (cid:48) related to ¯ r , and branegravity behaves seven-dimensionally, provided the scale at which sources on the brane feelthe whole seven-dimensional bulk is larger than L . In other words the bulk scale ¯ r mustbe seen as a very large scale from the point of view of an observer on the brane. Howeverregardless of the specific details of the crossover physics we see that brane tension mustbe bounded from above at least for this class of smooth solutions. Also, as mentionedabove, the form potential may be coupled to a charged particle eM (cid:82) W A , where W is theparticle worldline (for n = 4 it would be a string worldsheet and so on). When W wrapsthe horizon of the two-sphere, single-valuedness of the amplitude leads to a quantizationcondition for the flux [29] eM (cid:82) S F = 2 πk , that yields ϕR (cid:15) = k √ eM , k ∈ Z . (2.38)Hence, the above fine-tuning relation (2.36) can be attained only by the portion of tensionthat is quantized accordingly, and the excess of tension δλ ∼ eM seems to either gravitateor blow the internal radius to an unacceptably large value. Notice also that flux conser-vation due to Bianchi identity sets the conservation of ϕR (cid:15) , similarly to what discussedin [33] for the finite-volume rugby-ball model. In the present setup, unlike what happensin [33], the flux is not fixed in terms of bulk parameters and this would, a priori , allowan eventual phase transition that locally changes the value of the brane tension. However,since the internal radius of the brane must locally change in order to “absorb” the differenttension and keep the four-dimensional part of the brane flat, this would lead to a scenariowhere different four-dimensional domains (characterized by different values of tension) havedifferent Planck masses. One may worry about the fact that parallel directions are necessarily warped and it may happen that aRS-like localization [35] takes place at those scales. More precisely one might expect an interplay of effectsbetween induced gravity and RS localization, such as the one described in [36]. However it is easy to seethat sign( V (cid:48) (¯ ρ )) = sign( Q ( (cid:15) )) > d “bulk” behaves as a brane-to-boundary chunk of AdS and the five-dimensional length is thus infinite. It is thus natural to expect thatno RS localization takes place. – 8 –nother issue concerns the stability of such solutions. Although such important pointwould require a detailed investigation let us here mention a few related results obtainedin the past in similar models. In the absence of localized fluxes, instabilities were indeedfound in the models discussed in [5, 14]. However, in [14] it was also shown that localizedinduced stress tensors of the form (2.23) do indeed lead to a stabilization and such effectis quite likely to take place in the present solutions as well.To conclude this section, let us briefly mention that, for generic values of parameters,it seems plausible that the higher codimension resolved brane solutions discussed herebehave more like the “subcritical” codimension-two cases [19], and the crossover scale from4 d gravity and (4 + n )-gravity is expected to be given by r nC ∼ M c M n .
3. Non-vanishing bulk cosmological constant
In this section we consider some finite-volume counterparts of the solutions found in theprevious section. What follows is to be understood as higher-codimension generaliza-tions of the smooth codimension-two solution described in [17] of which we also use theconventions. The bulk part of the action (2.1) now gets generalized to S bulk = (cid:99) M D − (cid:90) d D x (cid:112) − (cid:98) g (cid:34) (cid:98) R − (cid:98) Λ − p + 1)! (cid:98) H p +1] (cid:35) (3.1)whereas the brane part remains the same as before. We seek for a solution with spheri-cally symmetry in the extra n -dimensional space and Poincar´e invariance in the d paralleldirections ds = η µν dx µ dx ν + R (cid:16) dθ + cos θd Ω n − (cid:17) , − π < θ < π d Ω n − is the line elements of the ( n − θ . In fact in generalwe might have more branes localized at different angles. For simplicity we will assume Z symmetry along θ and a pair of identical branes located at ± ¯ θ : the symmetry allows usto concentrate only on the northern hemisphere θ >
0. Again, we start considering thecase where p = n −
1. Similarly to [17] we assume to have an “inside bulk” ¯ θ < θ < π/ < θ < ¯ θ with different radii, R i = Rβ and R o = R , and differentcosmological constants, Λ i and Λ o respectively and we take the magnetic monopole ansatzfor the n -form field strength, with (cid:98) H [ n ] = (cid:40) (cid:98) Q i ( βR ) n S [ n ] , inner bulk (cid:98) Q o R n W [ n ] , outer bulk (3.3)with S [ n ] and W [ n ] respectively being the volume forms of the unit-radius n -sphere and ofthe unit-radius wedged n -sphere whose line element is given by dθ + β cos θd Ω n − . The Higher-codimensional brane solutions with bulk higher-rank tensors were considered in [31] where theregularization consisted in blowing up n − n to 2. Moreover, in [31], the “dual” form ˜ H [ d ] was considered, instead of H [ n ] . – 9 –ulk equations of motion fix the value of the cosmological constants and magnetic fields interms of the radii R − a = (cid:98) Λ a ( n − = (cid:98) Q a n − , a = i, o . (3.4)It is easy to see, that using (3.4), and redefining coordinates as θ ( l ) = ¯ θ − (cid:18) ¯ θ − lR (cid:19) (cid:104) Θ(¯ θR − l ) + β − Θ( l − ¯ θR ) (cid:105) (3.5) z α = βRθ α , α = 1 , . . . , n − βR ) n S [ n ] and R n W [ n ] are bothgiven by V [ n ] = (cos θ ( l )) n − dl ∧ Ω [ n − ( z ) , ds = dl + cos θ ( l ) d Ω n − ( z ) (3.7)and (cid:98) H [ n ] = (cid:112) n − θ (cid:48) ( l ) V [ n ] (3.8)and θ (cid:48) ( l ) is discontinuous at the location of the brane ¯ l = ¯ θR . The integration by partsassociated to the equation of motion for the potential form field (cid:98) ω [ n − = (cid:104) ± c + f ( θ ( l )) (cid:105)(cid:112) n −
1) Ω [ n − ( z ) , f (cid:48) ( θ ) = cos n − θ (3.9)whose field strength is (cid:98) H , will thus give rise to a jump condition at the location of thebrane δ ω S ( H ) ⊇ − (cid:99) M D − ( n − (cid:90) M d D x (cid:112) − (cid:98) g ∇ M δ (cid:98) ω M ··· M n − (cid:98) H M ··· M n − = (cid:99) M D − ( n − (cid:90) ∂M d D − x √− g δω m ··· m n − (cid:68) (cid:98) H l m ··· m n − (cid:69) ± (3.10)where ω m ··· m n − = (cid:98) ω m ··· m n − (¯ θ ), and comes from the ”-” branch of (3.9) as the brane sitsinside the northern hemisphere. We thus need to ameliorate the ( n − ω [ n − , namely˜ F [ n − = F [ n − + eM ω [ n − (3.11)and δ ω S ( ˜ F ) = − M D − ( n − (cid:90) Σ d D − x √− g δω m ··· m n − e ˜ F m ··· m n − , (3.12)so that (cid:68) (cid:98) H l m ··· m n − (cid:69) ± = L M e ˜ F m ··· m n − (3.13)– 10 –s the jump condition for the form field, with (cid:68) (cid:98) H l α ··· α n − (cid:69) ± = (cid:112) n −
1) 1 − βRβ cos n − ¯ θ √ Ω (cid:15) α ··· α n − . (3.14)For the metric we have (2.6) instead, that using (3.7), simply yields the following non-vanishing components for the extrinsic curvature K ± αβ = ± ∂ l g αβ = ∓ θ (cid:48) (¯ l ± ) tan ¯ θ g αβ .Then, choosing the brane magnetic field to be F [ n − = Φ cos n − ¯ θ Ω [ n − ⇒ ˜ F [ n − = ˜Φ cos n − ¯ θ Ω [ n − (3.15)with ˜Φ = Φ + eM (cid:112) n − f (¯ θ ) − c )(cos ¯ θ ) − n , we have the following junction conditionsΛ −
12 ˜Φ = 2( n − L − βRβ tan ¯ θ (3.16)Λ + 12 ˜Φ = 2( n − L − βRβ tan ¯ θ (3.17) e ˜Φ = (cid:112) n − LM − βRβ . (3.18)where (3.16,3.17) are the ( αβ ) and ( µν ) components of the junction condition for the metricand (3.18) is the junction condition for the form field. They can be solved to giveΛ = 2 n − L − βRβ tan ¯ θ = 2 n −
32 ˜Φ (3.19)˜Φ = 2 eM (cid:112) n −
1) tan ¯ θ (3.20)so that a brane of arbitrary tension can be accommodated on such a setup while maintaining4 d -Poincar´e invariance. A few observations are in order. First, it is easy to see that,contrarily to what happens in [17], for fixed 4 d vacuum energy density one cannot take thethin limit ¯ θ → π/
2. The vacuum energy density can be simply obtained from the integralof the l.h.s. of (3.17) over the internal profile of the brane. Up to irrelevant numericalconstants it reads T ∼ (cid:99) M D − R n − ( β cos ¯ θ ) n − sin ¯ θ (1 − β ) . (3.21)For n = 2 one recovers the result of [8, 17]. For n >
2, holding T fixed, the aforementionedlimit is impossible as β is bounded from above. In other words T → θ → π ; it is thusdifficult to imagine how to extend the approach of [27] to codimension higher that two, atleast within this class of spherically symmetric regularizations. Also, coupling of the formfields to extended objects leads to quantization conditions [29, 31, 32] for the fluxes thatin turn yield a quantization condition for the brane tension.Let us conclude by mentioning possible extensions of the latter solutions to the caseof negative bulk cosmological constant. It is obvious that an unwarped solution like (3.2)with an internal AdS is prohibited by Maldacena-Nunez no-go theorem [34]. However,– 11 –t least partial way-outs seem possible if, for instance, one allows the extra space to benon-compact. In fact, let us start from ds = R (cid:16) dξ ξ + ξ η µν dx µ dx ν (cid:17) + δ ab dz a dz b , z a ∼ = z a + 2 πl a (3.22)where the ( n − − torus parameterized by z a is the internal profile of the brane localized at ξ = 1. Bulk equation of motion in presence of negative cosmological constant and fluxesyields a similar fine-tuning condition like the one given in (3.4). Taking for simplicity a Z -symmetry and setting ξ = 1 + (cid:15) | u | it is easy to see that (with the exception of codimensionone, where there is no bulk flux and reduces to the RS2 model [35]) finite transverse volume( (cid:15) = −
1) implies positive brane tension, Λ > <
0, whereasinfinite volume ( (cid:15) = +1) implies negative brane tension, Λ < > Acknowledgments
This work was partly supported by the Italian MIUR-PRIN contract 20075ATT78. Theauthor would like to thank C. Bogdanos, C. Charmousis, C. Germani and A. Iglesias fordiscussions and G. Tasinato for help and critical reading of the manuscript. The author isgrateful to the LPT Orsay for hospitality while parts of this work were completed.
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