Deformations of Cosmological Solutions of D=11 Supergravity
aa r X i v : . [ h e p - t h ] A p r Deformations of Cosmological Solutions of D=11 Supergravity
Nihat Sadik Deger and Ali Kaya Dept. of Mathematics, Bogazici University, Bebek, 34342, Istanbul-Turkey Dept. of Physics, Bogazici University, Bebek, 34342, Istanbul-Turkey
E-mails: [email protected] , [email protected]
ABSTRACT
We study Lunin-Maldacena deformations of cosmological backgrounds of D=11 supergravitywhich gives an easy way to generate solutions with nonzero 4-form flux starting from solutionsof pure Einstein equations which possess at least three U (1) isometries. We illustrate this onthe vacuum S-brane solution from which the usual SM2-brane solution is obtained. Applyingthe method again, one either gets the recently found S-brane system where contribution of theChern-Simons term to field equations is non-zero or the SM2 ⊥ SM2(0) intersection, dependingon which U (1) directions are used during the process. Repeated usage of the procedure gives riseto configurations with several overlapping S-branes some of which are new. We also employ thismethod to construct two more new solutions and make comments about accelerating cosmologiesthat follow from such deformed solutions after compactification to (1+3)-dimensions. ontents Using symmetries to find new solutions is quite an old and powerful idea. A string theorybackground which is independent of d -toroidal coordinates has an O ( d, d ) T-duality symmetry(for a review see [1]). Exploiting this makes it possible to obtain new solutions starting froman existing one. This approach, also played an important role in the AdS/CFT context inunderstanding the gravity duals of deformed conformal field theories [2]. If the ten dimensionalgravity dual has a U (1) × U (1) symmetry in its geometry then such a solution can be deformedusing the so called TsT transformation where a shift is sandwiched in between two T-dualities.For 11-dimensional solutions one just needs an extra U (1) for the dimensional reduction to tendimensions [2]. After that, the TsT transformation is applied using the remaining two directionsand the result is lifted back to 11-dimensions. If the initial D=11 background has more than3 U (1) isometries then this process can be repeated to obtain a multiparameter deformation.In [3] general formulas for these deformed solutions in D=11 were obtained. With the help ofthese, one can write down deformed solutions directly in D=11 without going through detailsof this rather lengthy calculation. To use these formulas, it does not matter where these U (1)directions lie in the geometry however the initial solution should satisfy certain conditions whichare not too restrictive as will be seen.Solution generating techniques were applied frequently to time-dependent backgrounds inthe past (see for example [4, 5]) and actually in one of the first papers on the construction ofS-brane solutions [6] to supergravity theories, such a method was employed [7]. In this paper wewill study Lunin-Maldacena deformations [2] of time-dependent solutions of D=11 supergravity,which was previously carried on for static M2 and M5 branes in [8]. Using a formula derivedin [3] it is apparent that if the initial solution has zero 4-form field strength, then after thedeformation, one gets a 3-form potential along the U (1) directions which are used during theprocess. This is an interesting result since, in particular it means that if we have a cosmologicalsolution of pure D=11 supergravity with the geometry R × M , we can easily generalize thisto a solution which has a nonzero 4-form flux along R using U (1) directions of R whichcan be obtained by periodic identifications. The new solution is electrically charged and canbe interpreted as a generalized SM2-brane [9, 10, 11, 12, 7] with an arbitrary transverse space1 . Such geometries are quite popular in attempts to realize inflation using String/M-theorycompactifications. The fact that, hyperbolic compactifications produce a transient acceleratingphase in 4-dimensional Einstein frame was first noted in [13]. Shortly after, it was noticed thatthe transverse space could also be flat [14] or spherical [15] when a nonzero flux is present. Anintuitive understanding of this fact was given in [15] using the 4-dimensional effective theorydescription, where the problem reduces to studying a potential function of scalar fields. Thecontribution of 4-form flux to this potential is always positive which enhances the amount ofacceleration. Hence, it is desirable to add flux to a vacuum solution.The plan of our paper is as follows. In the next section, we will recall the formula derived in[3] to obtain 1-parameter Lunin-Maldacena deformations [2] of D=11 backgrounds. In section3, we illustrate this method on some examples. Our first starting point is the vacuum S-branesolution [12, 13, 16] of D=11 supergravity whose 4-form field strength is zero. It contains several U (1) directions and by using three of them, we demonstrate that one obtains the SM2-branesolution [9, 10, 11, 12, 7] in a form which has a direct zero flux limit. We also explain thetransformation of this to the familiar SM2-brane metric. Applying the deformation furtheron the SM2-brane, one either gets the recently found SM2-SM2-SM5 Chern-Simons S-branesystem where contribution of the Chern-Simons term to the field equations is non-zero [17] orthe SM2 ⊥ SM2(0) intersection [18], depending on which U (1) directions are used during theprocess. Changing the initial solution to an SM5-brane [10, 6] one gets either the same Chern-Simons S-brane system [18] or the SM2 ⊥ SM5(1) intersection [18]. Successive application ofthe method produces solutions with more S-branes including standard intersections of S-branesand overlappings of Chern-Simons S-brane systems. The latter ones are new. There are alsoother intersections where there is no supersymmetric analog. In the subsection 3.2, we usethis method to construct two more new solutions one of which is a new Chern-Simons S-branesystem where there are two non-intersecting SM2-branes inside an SM5. The second one is anSM2-brane solution with a different transverse space than the usual one. In section 4, we studyconsequences of such deformations for accelerating cosmologies in (1+3)-dimensions. We showthat the cosmology of a deformed solution differs from the original one only when we compactifyon deformation directions. We also find that for the two new solutions the number of e-foldingsis of order unity. We conclude in section 5 with some comments and possible extensions of thiswork.
In this section we will explain how to generate Lunin-Maldacena deformations [2] of an 11-dimensional background using a formula derived in [3]. The bosonic action of the 11-dimensionalsupergravity is S = Z d x ( √− gR − . √− gF − F ∧ F ∧ A ) , (1)whose equations of motion are R AB = 12 . F ACDE F BCDE − . g AB F , (2) d ∗ F = 12 F ∧ F . (3)There is also the Bianchi identity dF = 0. This is quite a simple theory in terms of number offields and for a solution it is enough to specify its metric and the 4-form field strength only.2ssume that we have a solution of these equations with the following two properties:( i ) Its metric contains I ≥ ii ) Its 4-form field strength has at most one overlapping with these I directions.Then, one can obtain a new solution by reducing along one of these U (1)’s and then usingthe other two for the TsT (2 T-dualities and a shift in between) transformation [2]. Once the 3 U (1) directions for the deformation process are decided, the choice of the reduction direction andT-duality directions from these does not affect the final answer. If one uses only a single 3-torus,one gets a 1-parameter deformation. When I > U (1)’s and obtain a multiparameter deformation. In [3] general formulas for thesenew solutions were obtained subject to two conditions above which considerably simplify thenecessary calculations. We will now review the formula for the 1-parameter case below.Suppose that there are 3 U (1) directions { x , x , x } which possibly mix among themselvesbut not with any other coordinate in the metric and let T denote the 3 × T mn = g mn . Then, starting with a solution { F , g AB } where each term in F has at mostone common direction with { x , x , x } , after the deformation we find [3]:˜ F = F − γi i i ⋆ F + γd (cid:0) KdetT dx ∧ dx ∧ dx (cid:1) ,d ˜ s = K − / g µν dx µ dx ν + K / g mn dx m dx n , (4) K = [1 + γ detT ] − , where m, n = { , , } and µ, ν denote the remaining directions. The new solution is given by { ˜ F , ˜ g AB } . The Hodge dual ⋆ is taken in the 11-dimensions, with respect to the undeformedmetric and i m is the contraction with respect to the isometry direction ∂/∂x m , i.e. i m ≡ i ∂/∂x m .Here γ is a real deformation parameter and when γ = 0 we go back to the original solution.From the last term in ˜ F note that such a deformation always generates a 3-form potential alongthe deformation directions even when the original F = 0, provided that detT is not constant.For time-dependent solutions this term corresponds to the flux of a generalized SM2-brane lyingalong the { x , x , x } directions. Its charge is proportional to the deformation parameter γ . Now, we will study Lunin-Maldacena [2] deformations of some cosmological solutions of D=11supergravity using the above formula (4). We will first begin with the vacuum S-brane solutionto clarify our method and to establish its connection to SM2-brane and Chern-Simons S-branesystem [17] through this deformation. Repeated usage of this method gives rise to a largenumber of configurations with several S-branes that can be divided into 3 classes. The firstset contains standard intersections of S-branes [18] which have supersymmetric analogs. In thesecond one we have configurations where each S-brane pair makes a standard intersection butoverall intersection has no supersymmetric analog, which are different than those studied in [19]since brane charges are independent. The last category consists of overlappings between Chern-Simons S-brane systems and S-branes. Solutions in the first two groups can be constructed usingintersection rules found in [18], whereas those in the third one are new. In part 3.2, we will usethis deformation to construct two additional new solutions. The first one is a new Chern-SimonsS-brane system and the second one is an SM2 brane with a different transverse space.3 .1 S-branes
The vacuum S-brane solution [12, 13, 16] of D=11 supergravity is given as: ds = e λ ( t − t ) / ( dx + dx + dx ) + e − λ ( t − t ) / k X i =1 e b i t − c i ) dθ i + e − λ ( t − t ) / e b t − c ) G − nn − n,σ ( − dt + G n,σ d Σ n,σ ) ,F = 0 , (5)where d Σ n,σ is the metric on the n -dimensional unit sphere, unit hyperbola or flat space and G n,σ = m − sinh [( n − m ( t − t )] , σ = − ,m − cosh [( n − m ( t − t )] , σ = 1 (sphere) , exp[2( n − m ( t − t )] , σ = 0 (flat) , (6)with k + n = 7 and n ≥
2. Constants satisfy b t − c = − n − k X i =1 b i t + 1 n − k X i =1 c i , n ( n − m = 2 n − k X i =1 b i ! + 2 k X i =1 b i + λ . (7)Here, we took the exponentials multiplying { x , x , x } directions the same since after the de-formation we want to have a homogeneous 3-dimensional space which will correspond to theworldvolume of an SM2. We can set one of the constants { λ, b , ..., b k } to 1 by a rescaling andone of the integration constants { t , t , c , ..., c k } to zero by a shift in the time coordinate. Sincethere is no mixing in the metric and the 4-form is zero, we can use any 3 from x or θ coordinatesfor the deformation by assuming that they are periodic. Choosing deformation directions as { x , x , x } , we see that the 3 × T is diagonal with detT = e λ ( t − t ) and applying(4) to the vacuum S-brane solution (5) we find: d ˜ s = e λ ( t − t ) / (1 + γ e λ ( t − t ) ) − / ( dx + dx + dx )+ (1 + γ e λ ( t − t ) ) / e − λ ( t − t ) / " k X i =1 e b i t − c i ) dθ i + e b t − c ) G − nn − n,σ ( − dt + G n,σ d Σ n,σ ) ˜ F = γd [ e λ ( t − t ) (1 + γ e λ ( t − t ) ) − dx ∧ dx ∧ dx ] , (8)where γ is the deformation parameter and when γ = 0 we go back to the vacuum S-branesolution (5). Note also that even though we started with a solution with no 4-form field, afterthe deformation we have a solution with ˜ F = 0. This is an SM2-brane solution located at { x , x , x } , however its metric is not in the familiar form which contains cosh functions. Tounderstand the relation, we scale { x , x , x } coordinates with γ − / and all other coordinatesand constants { m, λ, b , ..., b k } with γ / in (5) before performing the deformation, which makes γ disappear in K . Then, deforming this rescaled metric using { x , x , x } directions we get: d ˜ s ′ = H − / ( dx + dx + dx ) + H / [ k X i =1 e b i t − c i ) dθ i + e b t − c ) G − nn − n,σ ( G n,σ d Σ n,σ − dt )]˜ F ′ = qλH − dt ∧ dx ∧ dx ∧ dx ,H ≡ q cosh λ ( t − t ) , q ≡ γ , (9)4hich is the standard SM2-brane solution [9, 10, 11, 12, 7] with k -smearings whose transversespace is of the form R × ... × R k × Σ n,σ . Note that γ → q = 0. This analysis clarifies the passage from the SM2-branesolution (8) to the vacuum S-brane solution (5).From this point it is possible to continue with more deformations since there are ( k + 3)appropriate coordinates in the initial solution (5). We have two options which are consistentwith our rules: either we choose one U (1) direction from the worldvolume of SM2 and two fromoutside or we choose all of them transverse to the SM2. For the first choice, if we take { x , θ , θ } directions for deforming solution given in (9), then this adds another SM2 along these directionsand we get the standard SM2 ⊥ SM2(0) intersection [18] where the worldvolume of the secondSM2 is inhomogeneous with some exponentials of time which is due to the choice that we madefor homogeneous directions in our initial vacuum (5). For the latter, without loss of generalitylet us use { θ , θ , θ } coordinates to deform the solution (9). Since these do not overlap withany of the directions of the 4-form field strength of (9) and they do not mix with any coordinatein the metric, we are allowed to use the deformation formula (4). Again the 3 × T is diagonal and after the deformation of the SM2 solution (9) with the parameter γ we find: d ˆ s = K / H / X i =1 e b i t − c i ) dθ i + K − / H / k X i =4 e b i t − c i ) dθ i (10)+ K − / H − / ( dx + dx + dx ) + K − / H / G − nn − n,σ e b t − c ) (cid:2) − dt + G n,σ d Σ n,σ (cid:3) ,K = [1 + γ q cosh λ ( t − t ) e bt − c ) ] − , b ≡ ( b + b + b ) , c = ( c + c + c ) , ˆ F = ˜ F ′ − γ qλ Vol( θ k ) ∧ Vol(Σ n,σ )+ γ q cosh λ ( t − t )[ λ tanh λ ( t − t ) + 2 b ] e bt − c ) [1 + γ q cosh λ ( t − t ) e bt − c ) ] dt ∧ dθ ∧ dθ ∧ dθ . This is nothing but a slight generalization of the solution given in [17] which was previouslyobtained by directly solving the field equations (2) and (3). It is more general because exponen-tials in front of the { θ , θ , θ } coordinates in the metric are not all equal and it is possible tohave two smearings instead of one. Moreover, in [17] the constant γ does not appear explicitly,and hence it does not reduce to the single SM2-brane (9) by setting a constant to zero unlike thesolution above. If we take constants as b = b = b ≡ d/ c = c = c ≡ t / n = 4, then after some further redefinitionsof constants and rescaling of coordinates two solutions agree completely. Looking at ˆ F we seethat this new solution describes two SM2-branes located at { x , x , x } and { θ , θ , θ } and anSM5-brane at { x , x , x , θ , θ , θ } . Note that ˆ F ∧ ˆ F = 0 and therefore the contribution of theChern-Simons term to the field equations (3) is non-zero [17].From the deformation formula (4) we see that there is no way to obtain the SM5-branesolution [10, 6] from any vacuum solution. However, we can start directly with the SM5-branesolution. Since the 4-form field of an SM5-brane lies along the transverse space there are twooptions for deformations that are compatible with our rules: Either all 3 are chosen from theworldvolume of SM5 or two are chosen from the worldvolume and one from the outside. Inthe first case, after applying (4) one gets again (10) after some obvious choices of constants.Whereas, from the latter starting from an SM5 with inhomogeneous worldvolume and 1-smearing To go from the above solution (10) to the one found in [17] first set λ = 1 and define γ qe − t → e − t . Thenperform the changes θ i → θ i γ / , q → − qγ − , x i → x i γ − / , i = 1 , ,
3. After these, the constant γ disappears. ⊥ SM5(1) intersection [18]. Thus, we have all the standard doubleintersections between SM2 and SM5 branes [18] except SM5 ⊥ SM5(3).Now, a large number of S-brane configurations can be constructed by applying more defor-mations that are compatible with our conditions. To increase this number one can use SM2,SM5 and SM5 ⊥ SM5(3) as a basis and systematically perform deformations. In finding the listof resulting configurations it is enough to remember the following set of rules about positions ofavailable deformation directions:SM2 −−−−−−−→
SM2 ⊥ SM2(0) , SM2 −−−−−−−→
CSSSM5 −−−−−−−−−→
SM2 ⊥ SM5(1) , SM5 −−−−−−−−−→
CSSwhere CSS stands for the Chern-Simons S-brane system in which there are 2 non-intersectingSM2-branes inside an SM5. Of course, when there are more than one brane in the initial sys-tem these two rules should be used simultaneously. In this way, we can get all the standardS-brane intersections listed in [18] which have static supersymmetric analogs. There are alsointersections where each S-brane pair makes a standard intersection but overall intersection hasno supersymmetric analog, however their construction still follows intersection rules found in[18]. Besides these, overlappings between CSS systems and CSS systems with extra S-branesare allowed which are new in the S-brane literature. For example, consider the SM2 ⊥ SM2(0)intersection that we mentioned above where SM2’s are located at { x , x , x } and { x , θ , θ } .If we deform this using { θ , θ , θ } we get an overlapping of two CSS systems (10) where thereis an additional SM2 at { θ , θ , θ } and two SM5’s are located at { x , x , x , θ , θ , θ } and { x , θ , ..., θ } . Instead of this, if we use { θ , θ , θ } then we find a CSS system with an addi-tional SM2 at { θ , θ , θ } where the SM5-brane is located at { x , x , x , θ , θ , θ } . Similarly,using { x , θ , θ } we get SM2 ⊥ SM2 ⊥ SM2(0) intersection [18], whereas { x , θ , θ } gives anotherSM2 ⊥ SM2 ⊥ SM2(-1) solution where each pair has one common direction but there is no overallcommon intersection where -1 indicates this fact. The last one is different than the solutionfound in [19] since SM2-brane charges are independent.
Of course, it is straightforward to generate additional new solutions using this mechanism. Forexample, deforming the power-law solution given in [20] (see also [11, 21]) we arrive at: d ˜ s = (1 + γ α t − / ) − / α t − / ( dx + dx + dx ) (11)+ (1 + γ α t − / ) / h α t − / ( dx + ... + dx ) + α t − / d Σ , − − α t − / dt i , ˜ F = λt − dt ∧ Vol(Σ , − ) + γd " α t − / γ α t − / dx ∧ dx ∧ dx − γλα − dx ∧ · · · ∧ dx , where constants are fixed as ( α ) = 27 λ / α ) = 2 / (7 λ ) and α = ( α ) ( α ) . This isanother SM2-SM2-SM5 Chern-Simons system for which ˜ F ∧ ˜ F = 0. The initial SM2-brane hashyperbolic worldvolume Σ , − and can be called a flux SM6-brane too [21]. The other SM2 islocated at { x , x , x } and SM5 worldvolume contains both of them.6s a second example we deform the vacuum solution found in [22] and find: d ˜ s = e λt/ (1 + γ e λt ) − / ( dx + dx + dx ) (12)+ (1 + γ e λt ) / e − λt/ G − / ,σ ( − e β/ dt + G ,σ n X i =1 e β i / d Σ m i ,σ ) , ˜ F = 2 γλe λt (1 + γ e λt ) dt ∧ dx ∧ dx ∧ dx , where the Σ m i ,σ ’s are m i ≥ σ and P ni =1 m i = 7. This represents an SM2-brane located at { x , x , x } witha transverse space of the form M m ,σ × · · · × M m n ,σ . The function G ,σ is given by (6) with m = λ/ √
84 and warping constants β i and β are determined as β i = 12 ln m i − n Y j =1 (cid:18) m i − m j − (cid:19) m j , β = n X i =1 m i β i = 72 ln " n Y i =1 ( m i − − m i / . (13)When the transverse space is only one piece (that is, n = 1), all β i ’s and β become zero andthe above solution reduces to the usual SM2-brane (8) with no smearings. If desired, this againcan be put into a form where cosh functions appear in the metric as we did above in (9), whichreplaces the transverse part of (9) with the above product space structure. Hyperbolic compactifications to 4-dimensions may lead to a short period of accelerating cos-mologies [13]. Unfortunately, in all the examples studied so far, such as [13, 14, 15, 22, 17], theamount of e-foldings is only of order 1 and hence these are not useful for explaining early timeinflation. Yet, such solutions might be relevant for the presently observed acceleration of ouruniverse [23].After compactification from D=11 to (1+3)-dimensions, the 4-dimensional part of all theabove S-brane solutions in the Einstein frame has the form: ds E = − S dt + S ds M , (14)where S is some function of time that depends on the solution and M is a three dimensionalhomogeneous space. This universal structure is due to a particular property of these solutions.Namely, the function in front of the time coordinate in the metric is given as multiplication ofpowers of other functions appearing in the metric where powers are the dimensions of spacesthat these functions multiply. Now, the proper time is given by dτ = S dt and the expansionand acceleration parameters can be found respectively as H = S − dSdτ = S − dSdt , a = d Sdτ = − S − d dt S − . (15)An accelerating phase requires H > a > K . If after the defor-mation we compactify on a (1+3)-dimensional space whose spatial part M was not used forthe deformation, then it is immediately seen that in the Einstein frame (14) the factor K does7ot appear in the function S . Hence, the cosmology of compactication on ( t, M ) will be thesame before and after the deformation. On the other hand, if we use three coordinates of M for the deformation, then we get a factor of K − / in the S function and the cosmology willnow be different. This is actually not a surprise, since the main effect of such a deformation isto produce a 3-form potential along the deformation directions (possibly with some additionalfluxes) which will change the 4-dimensional cosmology only if we compactify on these coordi-nates. This argument, together with the fact that we want M to be a homogeneous spaceimply that increasing the number of standard intersections [18] will not improve the amount ofe-foldings as was explicitly checked for double intersections in [22].The above discussion shows that for the SM2-brane solution (8) only compactification on { t, x , x , x } may lead to a result different than the vacuum (5). In this case the S function isgiven as S = e − λt/ (1 + γ e λt ) / e − ( b ... + bk ) t n − e ( c ... + ck )2( n − G − n n − n,σ , (16)where k + n = 7 and k ≤
5. The enhancement of the 4-form flux on acceleration [15] can beclearly seen in this example. For the vacuum case ( γ = 0) an accelerating phase happens onlywhen σ = − γ = 0 the expansion factor gets slightly bigger for σ = − σ = 0 and σ = 1 [15, 14, 22]. For γ = 0 previously only k = 0[14] and k = 1 [22] cases were analyzed explicitly. We found that increasing the number of flatproduct spaces does not lead to a significant modification and still the acceleration is of order 1.Similarly, for the Chern-Simons S-brane system (10) compactification on { t, x , x , x } willgive the same answer like SM2-brane (9) as was explicitly observed in [17]. However, com-pactification on { t, θ , θ , θ } in (9) and (10) will give different S functions and hence differentcosmologies. For this case, choosing b = b = b ≡ d/ c = c = c ≡ t / S = [1 + γ q cosh λ ( t − t ) e dt − t ] / e − ( n +2)( dt − t n − e − ( b b t − ( c c n − G − n n − n,σ , (17)where 2 ≤ n ≤
4. In [17] this compactification with no smearings ( n = 4) were studied and anaccelerating interval was found for each σ . However, the amount of e-folding was again of orderunity. When n = 3 and n = 2 there is a short period of acceleration too, however there is nomajor change in the expansion factor.For the new Chern-Simons S-brane system that we constructed above (11), after the compact-ification we get a different cosmology from the original one only if we compactify on { t, x , x , x } .In this case we have S = (1 + γ t − / ) / t − / . (18)However, from (15) we find that acceleration is always negative with or without deformation.Even though, we have flux along { t, x , x , x } and some part of the transverse space is hyperbolicwe do not get an accelerating phase.Finally, for the new SM2 solution with product transverse space (12), if we compactify on { t, x , x , x } the S function is given as S = e β/ e − λt/ (1 + γ α e λt ) / G − / ,σ , (19)whose form is almost identical with the SM2-brane case (16). Hence, it immediately follows thatwhen γ = 0 there is acceleration only for σ = − σ = 1 and σ = 0 after the 4-form becomes nonvanishing, alas only of order 1.8 Conclusions
In this paper we looked at applications of Lunin-Maldacena deformations [2] to cosmologicalsolutions of D=11 supergravity. The method becomes especially useful if we have a solutionof pure Einstein equations which has an R part in its geometry. Then, using the deformationthis can be generalized to a solution with a 3-form potential along these directions. To realizeinflation is a big challenge for String/M-theory and compactifications with different transversespace geometries and fluxes is a promising way to attack this puzzle. Furthermore, this alsoshows that to construct flat SM2-brane solutions with general transverse spaces, it is enough toconcentrate only on Einstein equations (2) with F = 0. We hope that with this simplificationit will be easier to construct such solutions which may have better cosmological features.As we saw, using the deformation repeatedly it is possible to obtain configurations withseveral S-branes some of which are new solutions. If we extend our basis of initial solutionsto include SM2 ⊥ SM2(-1) and SM2 ⊥ SM5(0) intersections found in [19] which have no super-symmetric analogs, then we will obtain intersections between these, standard S-branes andChern-Simons S-brane systems too. Cosmological aspects of such solutions need further ex-amination. It may also be interesting to apply this method to intersections of S-branes withp-branes [24, 25].We carried on our investigation using the formula (4) derived in [3] which makes the cal-culations very simple. In fact, once U (1) directions are decided it only remains to calculatea determinant. In this formula it is assumed that 3 U (1) directions have no mixing with anyother coordinate in the metric. However, this condition can be relaxed. In [3] formulas wheremixing of these with 1 or 2 more directions which are not necessarily U (1) are also provided.In employing (4) we can in principle use any 3 U (1)’s in the geometry which are consistentwith our rules. For example, if there is an n -dimensional maximally symmetric piece in thegeometry, than its SO ( n ) Cartan generators can be deformed, which will give an SM2-branewith an unconventional worldvolume. Another consequence that follows from (4) is that it isnot possible to obtain the SM5-brane solution from a vacuum or to add a single SM5 to anexisting solution. Moreover, there seems no way to construct nonstandard SM2 ⊥ SM2(-1) andSM2 ⊥ SM5(0) intersections [19]. For such cases, it may be necessary to use a more generalU-duality transformation which is worth exploring.In the last decade an elegant way to construct cosmological solutions of D=11 supergravity,which includes SM2-branes [26] and their intersections [27] with flat transverse spaces, usingKac-Moody algebras have been developed (for a review and more refences see [28, 29]). Itwill be interesting to work out generalization of this approach to cover also SM5-branes and tounderstand the action of Lunin-Maldacena deformations in this framework.We can of course apply our method to static solutions as well. However, deformation of astatic vacuum does not give an M2-brane but a static SM2 [21] whose worldvolume is Euclidean.The connection between one of the static versions of our Chern-Simons S-brane system [17] andcomposite M-brane solution [30] was already noted in [17]. The composite M-brane solution [30]and its intersections [31] were obtained using U-duality and their anisotropic black generaliza-tion were also studied [32]. We expect our approach to be useful in construction of Poincar´esymmetric versions of these black brane solutions. We aim to examine these problems in thenear future. 9 cknowledgements
NSD is grateful to the Abdus Salam ICTP and especially to its associates scheme, where somepart of this paper was written.
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