We investigate the effect of parallelizable property on supergravity equations of motion, therefore we will extract the exact solutions of Conformal Liouville Theory from D=10 Parallelizable IIB Keywords: Conformal Liouville Theory, Parallelizability, supergravity

Boukili, Abdellah EL

EL Boukili, Abderrahman

Sedra, Moulay Brahim

English

International Institute for Science, Technology and Education (IISTE): E-Journals

Advances in Physics Theories and Applications                                                              www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol.16, 2013          46 Conformal Liouville Theory from D=10 Parallelizable type IIB supergravity with a 5-form Ramond-Ramond.  Abderrahman EL Boukili1*, Abdellah EL Boukili§ and Moulay Brahim Sedra* *Université Ibn Tofail, Faculté des Sciences, Département de Physique, Laboratoire des Hautes énergies, Sciences de l'Ingénierie et Réacteurs (LHESIR), Kénitra, Morocco. §Université Mohamed V, Faculté des Sciences Agdal Département de Chimie (LECA), Rabat, Morocco. Abstract We investigate the effect of parallelizable property on supergravity equations of motion, therefore we will extract the exact solutions of Conformal Liouville Theory from D=10 Parallelizable IIB Keywords: Conformal Liouville Theory, Parallelizability, supergravity.  1. Introduction The conformal two-dimensional Liouville field theory is related string fields theories [1] and provide an effective theory of 2D quantum gravity[2]. However , the parallelizable manifold is one of different categories of manifolds[3], which has a lot of applications in physics namely, Parallelizable supergravity[4], Parallelizable supersymmetry[5] and Parallelizable plane waves[6]. In our studies we will introduce the parallelizability condition in the action of D=10 supergravity in presence of a 5 form Ramon-Ramon and we will see under which conditions we get a conformal Liouville equations.  The outline of our paper is as follows. In section 2, we review the parallelizability according to different definition. In section 3, we introduce the parallelizability in NS-NS part of the supergravity action. In section 4, we will extract the Conformal Liouville theory from Parallelizable IIB supergravity in the presence of a 5-form Ramond-Ramond. A brief conclusion is reported in Sec. 5.                                                           1Corresponding author: aelboukili@gmail.com Advances in Physics Theories and Applications                                                              www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol.16, 2013          47 2. Review on Parallelizability According to a definitions given in the references [7], a manifold M is said to be parallelizable if there exists a smooth section of the frame bundle, or equivalently, if there exist n smooth sections of the tangent bundle T(M), such that they are linearly independent at each point of M.  More intuitively, a manifold is parallelizable if one can cover the whole manifold with a single non-degenerate coordinate system. There is another definition of parallelizable manifolds due to Cartan-Schouten [8]: A manifold is called parallelizable if there exists a torsion which ``flattens'' the manifold, i.e. makes the Riemann curvature tensor vanish. Let us make explicit the decomposition of the connection into a Christoffel piece and a torsion contribution[8;5] :  ,ˆ λµνλµνλµν T+Γ=Γ  (1) where λµνΓ  is symmetric in µν  indices and λµνT  (torsion) is anti-symmetric. The curvature  µναβRˆ  may be decomposed in a similar way, into a piece which comes only from the Christoffel connection and the torsional contributions: .ˆ ρνβµαρρναµβρµναβµνβαµναβµναβ TTTTTTRR −+∇−∇+≡  (2) The parallelizability condition simply becomes 0ˆ =µναβR . If the modified Ricci tensor λρνµλραµναµνµν TTTRR −∇+≡ˆ  (3)  is zero, the manifold is said to be Ricci-parallelizable . Note that the generalized Ricci tensor  µνRˆ  is not symmetric in its µν  indices. For a manifold to be Ricci-parallelizable, the symmetric and anti-symmetric parts of µνRˆ  should both vanish, namely 0=∇αµναT  and 0=−λρνµλρµν TTR . With the above definitions, the parallelizing torsion for a group manifold is given by the structure constants of the group algebra [8]. 3. Parallelizable supergravity  Since in most string theories, a torsion field naturally arises, one may look for implications of parallelizability for supergravities and their solutions [9]. Here we mainly focus on type II Advances in Physics Theories and Applications                                                              www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol.16, 2013          48 theories, however, most of our arguments can be used for heterotic theories as well. First we recall that the NS-NS part of the supergravity action is of the form[5] .121)(41 222108  −∇+−= −∫ HRegxdlS pφµφ  (4) Since we are interested in solutions involving only the metric and torsion, we set the dilaton field φ  to a constant. Then the supergravity equations of motion for the metric and µνB  field are .,0)(,041dBHHgHHR==−∇=−αµναρλνµρλµν (5) If we define αµνH21  as torsion αµνT , the supergravity equations for the metric and µνB  fields are nothing but the Ricci-parallelizability condition. Hence all parallelizable manifolds (which are obviously also Ricci-parallelizable) satisfying the constant dilaton constraint [10] 2121HR =  (6) and have a closed torsion (i.e. 0=dH ), are solutions of supergravity . For any supergravity solution one may wonder about sg  and α ′  exactness as well as (classical) stability. The parallelizable solutions which we are interested in are non-dilatonic and hence they are gs  independent. The α ′ -exactness of these supergravity solutions were studied long ago in reference [11]. Computing the second α ′  contributions to the string theory β -functions, it was shown that such contributions are zero for parallelizable manifolds. Therefore the parallelizable solutions of supergravity are exact up to order 2′α . It has been argued that this property is expected to remain to all orders in α ′ [12]. 4. Conformal Liouville theory from Parallelizable supergravity Let's consider the effective action of D=10 type IIB supergravity with a Ramond-Ramond 5-form field strength [13]: Advances in Physics Theories and Applications                                                              www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol.16, 2013          49 ,!5.4121)(422210−−Φ∇+−=ΦΦ−∫ µνρστµνρστµνρµνρµ FFSeHHRegxdeff  (7) where g is the metric of spacetime, Φ a dilaton, the field strength H=dB with B an anti-symmetric NS- NS field and F=∂C(4) where C(4) is the corresponding R-R 4-form.  The derived supergravity equations of motion in the σ-model frame ( )( )( )∇=∇=+Φ∇Φ∇−Φ∇∇=−++Φ∇∇−=Φ−Φ,0),(0,20,2)(2121101.!4.4.412µνκλρµµνρµµνρµνρµµµµκλρστκλρστµνκλρσνµκλρσρσνµρσνµµνFFFFFHeHHgHHRIe (8) introducing parallelizability conditions equ.(6) we obtain the following system of equations ( )( )( )∇=∇=+Φ∇Φ∇−Φ∇∇=−+Φ∇∇−=Φ−Φ,0,0,20,20)(2121101.!4.42µνκλρµµµνρµνρµνρµµµµκλρστκλρστµνκλρσνµκλρσνµFFFFFeHHHgIIe (9) As for the R-R 5-form, we assume that )( ixdu ϕ∧=F  (10) where )( ixϕ  is an anti-self-dual closed 4-form in the transverse eight-dimensional space, i.e., 0)(),()( =−= ∗ iii xdxx ϕϕϕ  (11) Such as FF ∗= . Taking the (uu)-component of the equation for the Ricci tensor(ie. νµµ == ,xu ), we obtain  ijklijkluueϕϕ!4.4202Φ+Φ∇∇−=  (12) where 2ϕϕϕ ≡ijklijkl . The covariant derivative u∇  is defined as follows [5;6;13]: uuababuuu ww ˆ4141+∂=Γ+∂=∇  (13) where ababuu ww Γ=ˆ  is the spin connection. By inserting the expression of covariant derivative in equation (12) we find the following equation  Advances in Physics Theories and Applications                                                              www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol.16, 2013          50 )2exp()().ˆ41ˆ(41).ˆ41(. 22 Φ=Φ+∂+Φ∂+Φ∂ ϕχuuuuuu www  (14) where8!.42)(ϕϕχ ≡ .  Case 1: For 0w =uˆ , equation (14) became )2exp()(.2 Φ=Φ∂ ϕχu  (15) this equation has been well studied in references [6;13] which corresponds to a Liouville equation whose solution is given by ,)(1)(.)(2)2exp(22 −∂=Φufufuϕχβ  (16) where f  is an arbitrary analytic function.  Case 2: If we set 0w ≠uˆ  with the following constraints: with the following constraints: 0).ˆˆ().ˆ( 2414141 =Φ+∂+Φ∂ uuuuu www  (17) equation (14) becomes Φ=Φ∂=Φ+∂+Φ∂)2exp()(.0).ˆˆ().ˆ()(22414141ϕχuuuuuu wwwIII  (18) by integrating the system (III), we find the following solution ( )=Φ∫=Φ−∂−2)(1)()(2ˆ241)2exp(ˆ.)(ufufwuuuewKuϕχβ (19) where K and β  are two arbitraries constants.  if we combine the two equations of (19) we obtain the following expression for the function f    1ˆ1]ˆ.2exp[2)(2exp21)(2141−−+ ∫−= ∫uwuewKTuf βϕχ (20) Case 3: If we set 0w ≠uˆ  with the following constraints: Advances in Physics Theories and Applications                                                              www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol.16, 2013          51 Φ∂=Φ+∂+Φ∂ 22 ).().ˆ41ˆ(41).ˆ41( uuuuuu ucwww  (21) with this constraint, equation (14) is transformed to the following system =Φ∂=Φ+Φ∂+Φ∂ϕϕζ 2exp).(0)()()(22uuu ubuaIV  (22) with )(.4ˆ)(ucwuua −= , )()( )(ˆ41)(ˆ412ucwucw uuuub +−= ∂  and )(1)()(uc+=ϕχϕζ  . We 1Φ  consider a particular solution of the system (22), for example the solution given in the reference [13] )).cos(log()(1 uau −=Φ  (23) where ).(2 ϕζ=a  Taking z.1Φ=Φ  and zu∂=θ  then equation (22) becomes [ ] 02)(. 111 =Φ∂+Φ+∂Φ θθ uu ua  (24) which has as solution ( ) ( ) )exp(21 uuAu αθ −Φ=  (25) where )(ua∫=α  and A is a real constant. Consequently, the solution Φ  is given by [ ] +∫−=Φ ∫ BA).)((cos(log)(.).)(cos(log)(2 uuauuϕζϕζ  (26) Case 4: If we set 0w ≠uˆ  and )2exp().().ˆ41ˆ(41).ˆ41( 2 Φ=Φ+∂+Φ∂ ϕγuuuuu www  In this case we have the system of equations ( )Φ−=Φ∂Φ=Φ+∂+Φ∂)2exp()()(.)2exp()().ˆ41ˆ(41).ˆ41(22ϕγϕχϕγuuuuuu www (27) The second equation of this system corresponds to a Liouville equation which has the following solution Advances in Physics Theories and Applications                                                              www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol.16, 2013          52 ,)(1)(.)()(2)2exp(22 −∂−=Φufufuϕγϕχβ  (28) Where 1ˆ1]ˆ.2exp[2)()(2exp21)(2141−−+ ∫−−= ∫uwuewKTuf βϕγϕχ (29) and )2exp().ˆ41ˆ(.ˆ41)( 2 Φ− Φ+∂+Φ∂= uuuuu wwwϕγ  (30) 5. Conclusion In our studies we have study some applications of parallelizability in physics. Such property is introduced as an ansatz in the action of D=10 supergravity in presence of a 5 form Ramon-Ramon. We have proposed and discussed the different cases of analytic expression for the solutions of conformal Liouville equation obtained from the background of parallelizable supergravity.  References [1] A. Polyakov, Quantum Geometry of Bosonic Strings, Phys. Lett., B103 (1981) 207-210. [2] E. D’ Hoker and R. Jackiw, Classical and quantal Liouville field theory, Phys. Rev., D26 (1982) 3517 [3] T. Lance, Differentiable structures on manifolds, Surveys on Surgery Theory, vol. 1, Ann. of Math. Stud. 145, Princeton Univ. Press, NJ, 2000, pp. 73-104. 38 [4] M.J. Duff, B.E.W. Nilsson and C.N. Pope , Kaluza-Klein Supergravity, Phys.Rept. 130 (1986) 1-142 [5] Darius Sadri and M. M. Sheikh-Jabbari, String Theory on Parallelizable PP-Waves, JHEP 0306:005,2003 [6] M. 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Roiban, Supergravity pp-wave solutions with 28 and 24 supercharges, hep-th/0206195. [13] I. Bakas and J. Sonnenschein, On integrable models from pp-wave string backgrounds, JHEP 0212 (2002) 049   This academic article was published by The International Institute for Science, Technology and Education (IISTE).  The IISTE is a pioneer in the Open Access Publishing service based in the U.S. and Europe.  The aim of the institute is Accelerating Global Knowledge Sharing.  More information about the publisher can be found in the IISTE’s homepage:  http://www.iiste.org  CALL FOR PAPERS The IISTE is currently hosting more than 30 peer-reviewed academic journals and collaborating with academic institutions around the world.  There’s no deadline for submission.  Prospective authors of IISTE journals can find the submission instruction on the following page: http://www.iiste.org/Journals/ The IISTE editorial team promises to the review and publish all the qualified submissions in a fast manner. 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Conformal Liouville Theory from D=10 Parallelizable type IIB supergravity with a 5-form Ramond-Ramond.

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Conformal Liouville Theory from D=10 Parallelizable type IIB supergravity with a 5-form Ramond-Ramond.

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