Curved Dyonic Domain Walls in Four Dimensions

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1 ISSN: Indonesan Journal of Appled Physcs (201) Vol.0 No.1 Halaman 7 Aprl 201 Curved Dyonc Doman Walls n Four Dmensons Bobby Eka Gunara Indonesa Center for Theoretcal and Mathematcal Physcs (ICTMP) and Theoretcal Physcs Laboratory, Theoretcal Hgh Energy Physcs and Instrumentaton Research Group, Faculty of Mathematcs and Natural Scences, Insttut Teknolog Bandung Jl. Ganesha No. 10 Bandung bobby@f.tb.ac.d Receved , Revsed , Accepted , Publshed ABSTRACT In ths short paper we show the exstence of soltonc solutons of four dmensonal ungauged N=1 supergravty coupled to arbtrary vector and chral multplets whose Rcc scalar curvature s constant. The Rcc scalar of spacetmes ndeed depends on the σ-model, namely the complex scalars and ther frst dervatve. Then, we gve two explct models, namely statc doman walls and statc sphercal symmetrc black holes whch are related to our prevous works. Keywords: Ensten-Maxwell-Hggs theory, Rcc Scalar, Doman Walls. ABSTRAK Dalam paper n dtunjukkan eksstens solus soltonk dar teor N=1 supergravtas ungauged berdmens empat terkopel multplet vektor dan skalar secara sembarang dmana solus tersebut mempunya skalar Rcc konstan. Skalar Rcc tersebut bergantung pada geometr nternal medan skalar (σ-model) yang kompleks beserta turunan pertamanya. Kemudan, dberkan dua contoh eksplst, yatu doman walls statk dan lubang htam statk bersmetr bola dmana keduanya berkatan dengan pekerjaan kam sebelumnya.. Kata Kunc: Teor Ensten-Maxwell-Hggs, Rcc Scalar, Doman Walls. INTRODUCTION Topologcal defects such as doman wall solutons of supergravty have acqured a large nterest due to ther dualty wth renormalzaton group (RG) flows descrbed by a beta functon of feld theory n the context of AdS/CFT correspondence [1]. In partcular there has been a lot of study consderng these solutons whch preserve some fracton of supersymmetry n fve dmensonal supergravty theory. Inspred by the development of supergravty theory t s of nterest to extend the case to more general theory wthout supersymmetry. In ths short paper, we study soltonc solutons of Ensten-Maxwell-Hggs n four dmensons agan nspred by four dmensonal N=1 supergravty wth general gauge-scalar couplngs. In partcular, we consder a class of solutons called charged doman wall. If the electrc and magnetc charges are ncluded n the setup, then the doman walls nterpolate two ground states of constant scalar curvature whch are not Ensten. These generalze our prevous results n neutral doman walls dscussed [2-6] n whch they connect two Ensten spaces, partcularly ant-de Stter spaces.

2 Curved Dyonc Doman... Halaman 8 Ths paper provdes our prelmnary results on curved dyonc doman walls of four dmensonal Ensten-Maxwell-Hggs theory. The dyonc means that the soltonc object has both electrc and magnetc charges. Moreover, n the theory we should turn on the functon called scalar potental of the theory n order to have a doman wall soluton. The structure of the paper can be mentoned as follows. In secton 2 we state our general results on charged doman walls. In secton we dscuss shortly some aspects of four dmensonal Ensten-Maxwell-Hggs theory nspred by four dmensonal N=1 supergravty coupled to vector and chral multplets. Then we dscuss feld equatons of motons of the theory n secton. Secton 5 provdes dscusson on charged doman walls n the asymptotc regons. METHOD CURVED DOMAIN WALLS Now we turn our attenton to consder some geometrcal aspects of doman walls n (pseudo)- Remannan geometry. In general, the doman wall ansatz metrc can be wrtten down as 2 2 a b 2 ˆab ds a u g dx dx du, (1) where ab, 0,1, 2 and a u s the warp factor. The components of Rcc tensor have the form ' 2 2 ˆ ˆ ab ab ab R a g R a a, (2) R a ' 2 a, () whch mples that Rcc scalar s gven by R ' 2 Rˆ a a, () a da where. The quanttes R ˆab and ˆR are the components of Rcc tensor and Rcc scalar du n three dmensons, respectvely. Frstly, let us consder a case when the four-manfold s Ensten, R g, (5) where, 0,... and s real. Theorem 1 Suppose we have the condton (5) whch follows that the three dmensonal hypersurface s also Ensten wth constant. Then, there are only two cases:

3 Curved Dyonc Doman... Halaman 9 ku ku 1. For 0, then a u A0 e B0 e where A0, B 0 are real constants, k, and AB For 0, then a u C0 cos k ' u D0 sn k ' u where C0, D 0 are real constants, k ', and CD 0 0. Snce doman wall nterpolates between ground states of dfferent cosmologcal constant n the asymptotc lmt, both cases are possble are possble. In the context of feld theory the results n Theorem 1 descrbe an uncharged doman wall [2-6]. Next we turn to a case of constant Rcc scalar R. (6) Theorem 2 Suppose we have the condton (6) such that the three dmensonal hypersurface s of a constant scalar curvature wth constant. Then, we also two cases: 2 ku ku 1. For 0, then a u + A0e B0 e where A0, B 0 are real constants and k For 0, then a u C0 cos k ' u D0 sn k ' u where C0, D 0 are real constants and k '. Smlar as n Theorem 1, the results n Theorem 2 descrbe a charged doman walls n the asymptotc regons nterpolatng between the spaces of constant scalar curvature whch are not Ensten. EINSTEIN-MAXWELL-HIGGS THEORY In ths secton we gve a short descrpton of Ensten-Maxwell-Hggs theory n four dmensons. Ths theory s nspred by bosonc parts of N=1 supergravty coupled to arbtrary vector and chral multplets n four dmensons. For nterested reader, the complete N=1 supergravty can be found, for example, n [7]. Let us now dscuss the ngredents of Ensten-Maxwell-Hggs theory n four dmensons. Ths theory conssts of a gravty coupled to n v vector and n c real scalar felds. Furthermore, the Lagrangan of the theory has the form L = - R + g j m z m z j +Â LS F L mn F Smn L + Á LS F mn F Smn - V ( z) (7)

4 Curved Dyonc Doman... Halaman 0 where, j 1,..., n c and, 1,..., n v. The real scalars z span a Remannan manfold wth metrc g j. The quantty F s an Abelan feld strength of A, and F L mn s a Hodge dual of F. The functons and are real gauge couplngs whch depend on the scalars z. The real functon V() z s referred to as the scalar potental. EQUATIONS OF MOTIONS Let us frst dscuss the equatons of motons of the felds whch can be obtaned by varyng the acton related to the Lagrangan (7) wth respect to g, A, and z. Then, we have three equatons. Frst, the Ensten feld equaton 1 R g R T, (8) 2 where the energy-momentum tensor T s gven by j j j T 2 g z z g g z z j F F g F F g V (9) Second, the gauge feld equaton of motons s ( ) e mnrs n Á LS F L rs - Â S LSFrs together wth the Banch denttes = 0, (10) F 0. (11) Thrd, the scalar feld equaton of motons g j -g n ( -g g mn m z j ) + G jk m z j m z k = Â LS F rs L F Srs + Á LS F rs L F Srs - V (12) Takng the trace of (12) we fnally get j R 2 g z z - V z, z (1) j whch shows that the Rcc scalar of the spacetmes depends only on the dynamcs of the real scalars z. The Rcc scalar (11) becomes a constant f the scalars z are frozen or n other 1, 1, words z 0 n partcular regons N M where M s a soluton of (6)-(10) descrbng a four dmensonal curved spacetme. Ths wll be dscussed n detal n the followng sectons.

5 Curved Dyonc Doman... Halaman 1 DYONIC DOMAIN WALLS In ths secton we partcularly dscuss a class of doman walls that has both electrc and magnetc charges, called dyonc doman wall, n Ensten-Maxwell-Hggs theory. As dscussed n the precedng secton ths case s related to the case n Theorem 2. To obtan a doman wall soluton one has to solve the set of equatons of motons (6), (8), and (10) on the metrc (1). So, we have three coupled nonlnear dfferental equatons. It s hard to us to fnd an exact soluton n general, but one can only consder the asymptotc soluton, namely around u. In ths lmt the scalars z becomes fxed whch are the soluton of (6), (8), and (10). In the context of (10) these frozen scalars z 0 extremze both the scalar potental V z and the gauge couplngs and. Therefore, z 0 can be vewed as crtcal ponts of the scalar potental V z and the gauge couplngs and. Pluggng ths z 0 nto (6) and (11) one concludes that we have spaces of constant scalar curvature whch are not Ensten n the asymptotc regon. At the level of Rcc scalar (11) the gauge felds A does not play an mportant role snce t does not depend A. Thus, we have a dyonc curve doman walls that nterpolate two spaces of dfferent constant scalar curvatures. CONCLUSIONS So far we have consdered some aspects of four dmensonal curved charged doman walls whch nterpolates two ground states whch are spacetmes of constant Rcc scalar curvatures. These scalar curvatures have dfferent values. In four dmensonal Ensten- Maxwell-Hggs theory, these vacua (ground states) le n the asymptotc regons n whch the complex scalars are frozen. In other words, Rcc scalar curvatures are determned by these frozen scalars that belong to modul spaces. In the future we wll extend the analyss to a case of hgher dmensonal theores ncludng ten dmensonal strng theores and the mysterous eleven dmensonal M theory. ACKNOWLEDGMENTS Ths work was supported by Rset KK ITB 2011 No. 222/I.1.C01/PL/2011 and s extended by Rset KK ITB 201 No /AL-J/DIPA/PN/SPK/201. REFERENCES 1 For a revew see, for example: Aharony, O., Gubser, S.S., Maldacena, J., Oogur, H. and Oz, Y Large N Feld Theores, Strng Theory and Gravty. Physcs Reports Vol. 2, pp Gunara, B.E., Zen, F.P. and Aranto BPS Doman Walls and Vacuum Structure of N=1 Supergravty Coupled to a Chral Multplet. Journal of Mathematcal Physcs Vol. 8, Gunara, B.E. and Zen, F. P Kähler-Rcc Flow, Morse Theory, and Vacuum Structure Deformaton of N=1 Supersymmetry n Four Dmensons. Advances n Theoretcal and Mathematcal Physcs Vol. 1, pp Gunara, B.E. and Zen, F. P Deformaton of Curved BPS Doman Walls and Supersymmetrc Flows on 2d Kähler-Rcc Solton. Communcatons n Mathematcal Physcs Vol. 287, pp

6 Curved Dyonc Doman... Halaman 2 5 Gunara B.E. and Zen F. P Flat BPS Doman Walls on 2d Kähler-Rcc Solton. Journal of Mathematcal Physcs Vol. 50, Gunara B.E., Zen F. P. and Aranto Sphercal Symmetrc Dyonc Black Holes and Vacuum Geometres n d N=1 Supergravty on Kähler-Rcc Solton, Reports on Mathematcal Physcs Vol. 67, pp D'Aura, R. and Ferrara, S On Fermon Masses, Gradent Flows and Potental n Supersymmetrc Theores. Journal of Hgh Energy Physcs, Vol. 0105, pp.0.