University of Groningen M-theory and Gauged Supergravities Roest, D. IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 2004 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Roest, D. (2004). M-theory and Gauged Supergravities. s.n. Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Download date: 28-09-2021 hep-th/0408175 UG-04/05 M-theory and Gauged Supergravities D. Roest1 Centre for Theoretical Physics, University of Groningen Nijenborgh 4, 9747 AG Groningen, The Netherlands E-mail: [email protected] ABSTRACT We present a pedagogical discussion of the emergence of gauged supergravities from M- theory. First, a review of maximal supergravity and its global symmetries and supersymmetric solutions is given. Next, different procedures of dimensional reduction are explained: reductions over a torus, a group manifold and a coset manifold and reductions with a twist. Emphasis is placed on the consistency of the truncations, the resulting gaugings and the possibility to generate field equations without an action. Using these techniques, we construct a number of gauged maximal supergravities in diverse dimensions with a string or M-theory origin. One class consists of the CSO gaugings, which comprise the analytic continuations and group contractions of SO(n) gaugings. We construct the corresponding half-supersymmetric domain walls and discuss their uplift to D- and M-brane distributions. Furthermore, a number of gauged maximal supergravities are constructed that do not have an action. arXiv:hep-th/0408175 v2 31 Aug 2004 1Based on the author’s Ph.D. thesis, defended cum laude on June 25, 2004. Contents 1 Introduction 6 2 Supergravity 9 2.1 Supersymmetry ................................... 9 2.1.1 SuperalgebraandSupercharges . ... 9 2.1.2 PossibleSupergravityTheories. ..... 11 2.2 MaximalSupergravitiesin11Dand10D . ...... 12 2.2.1 Supergravityin11D ............................ 12 2.2.2 MinimalSupergravityin10D . 13 2.2.3 IIAandIIBSupergravity. 14 2.2.4 SupergravityRelationsandDualities. ....... 17 2.3 Scalar Cosets and Global Symmetries in D 9 .................. 18 2.3.1 ScalarCosets ................................≤ 19 2.3.2 Example: SL(2, R) SymmetryofIIB ................... 20 2.3.3 GlobalSymmetriesofMaximalSupergravities . ....... 21 2.4 SupersymmetricSolutions . .... 23 2.4.1 GenericBraneSolutions . 23 2.4.2 BraneswithLittleTransverseSpace . ..... 26 2.4.3 MaximallySupersymmetricSolutions . ..... 27 2.4.4 Half-supersymmetricSolutions . .... 28 2.4.5 Relations between Half-Supersymmetric Solutions . .......... 30 3 Dimensional Reduction 32 3.1 Introduction.................................... 32 3.1.1 ScalarFieldandKaluza-KleinStates . ..... 32 3.1.2 ConsistencyofTruncations. ... 33 3.2 ToroidalReduction ............................... .. 34 3.2.1 GravityonaCircle ............................. 34 3.2.2 GravityonaTorus ............................. 36 3.2.3 InclusionofGaugePotentials . ... 37 3.2.4 GlobalSymmetryEnhancement . 38 3.2.5 FermionicSector .............................. 39 3.3 ReductionwithaTwist ............................. .. 39 3.3.1 BoundaryConditionsandTwistedExpansions . ...... 39 Contents 3 3.3.2 GlobalSymmetriesandMonodromy. .. 40 3.3.3 GravityandGaugings ........................... 42 3.3.4 EnhancedGaugings ............................ 43 3.4 ReductionoveraGroupManifold . .... 44 3.4.1 GroupManifolds .............................. 44 3.4.2 GravityonaGroupManifold . 45 3.4.3 GaugingsfromGroupManifolds. .. 47 3.4.4 ConsistencyofReductionoverGroupManifolds . ....... 47 3.4.5 Twistedvs.GroupManifoldReductions . ..... 48 3.5 ReductionoveraCosetManifold. ..... 49 3.5.1 CosetManifolds .............................. 49 3.5.2 CosetReductions.............................. 49 3.6 Lagrangianvs.FieldEquations . ...... 52 3.6.1 ToyExample ................................ 52 3.6.2 GeneralRequirements . 55 4 Gauged Maximal Supergravities 56 4.1 Introduction.................................... 56 4.2 MassiveandGaugedIIASupergravity . ...... 57 4.2.1 IIASupergravity .............................. 58 4.2.2 MassiveIIASupergravity . 58 4.2.3 GaugedIIASupergravity. 60 4.2.4 Combinations of Mass Parameters and α′ Corrections. 61 4.3 Gauged Maximal Supergravities in D =9 ..................... 63 4.3.1 Maximal Supergravity in D =9 ...................... 63 4.3.2 Twisted Reduction of IIB using SL(2, R) ................. 64 4.3.3 ToroidalReductionofMassiveIIA. .... 66 4.3.4 OverviewofMassiveDeformationsin9D . .... 66 4.3.5 Combining Massive Deformations in 9D and α′ Corrections . 68 4.3.6 Quantisation Conditions on SL(2, R) MassParameters . 69 4.4 Gauged Maximal Supergravities in D =8 ..................... 71 4.4.1 TheBianchiClassificationof3DGroups . .... 71 4.4.2 Reductionovera3DGroupManifold . .. 74 4.4.3 Supersymmetry Transformations and Global Symmetries......... 77 4.4.4 LagrangianforClassATheories. ... 78 4.4.5 LagrangiansforTruncationsofClassBTheories . ........ 79 4.4.6 Nine-dimensionalOrigin . .. 80 4.5 CSO GaugingsofMaximalSupergravities . 81 4.5.1 CSO AlgebrasandGroups. 81 4.5.2 GaugedMaximalSupergravity. .. 83 4.5.3 ScalarPotential............................... 86 4.5.4 GroupContractionandDimensionalReduction . ....... 87 4.5.5 OtherGaugedMaximalSupergravities. ..... 88 4 Contents 5 Domain Walls 90 5.1 D8-braneinMassiveIIA . .. .. .. .. .. .. .. .. 90 5.1.1 D8-braneSolution ............................. 90 5.1.2 Source Terms and Piecewise Constant Parameters . ........ 91 5.1.3 TypeI′ StringTheoryandSupergravity . 92 5.2 DomainWalls in CSO GaugingsandtheirUplift . 94 5.2.1 TheDW/QFTCorrespondence. 94 5.2.2 DomainWalls................................ 95 5.2.3 Higher-dimensionalOriginandHarmonics . ...... 97 5.2.4 Brane Distributions for SO(n) Harmonics................. 98 5.3 DomainWallsin9DandUplifttoIIB . .100 5.3.1 Domain Walls in SL(2, R) GaugedSupergravities . 100 5.3.2 Seven-branesandOrientifoldPlanes . .. ..101 5.3.3 UplifttoIIBandD7-branes . 102 5.3.4 DomainWallsforotherGaugedSupergravities . .......105 5.4 DomainWallsin8DandUplifttoIIAand11D . .105 5.4.1 DomainWallsofClassATheories. 105 5.4.2 UplifttoIIAandD6-branes . 107 5.4.3 Upliftto11DandKK-monopoles . 108 5.4.4 Isometriesofthe3DGroupManifold . 110 5.4.5 DomainWallsforClassBTheories . 112 5.5 DomainWallswithStringsAttached . .. ..113 5.5.1 D8-F1-D0Solution.............................113 5.5.2 Domain Walls with SL(2, R) Strings in D =9 ..............115 Acknowledgements 121 A Conventions 122 A.1 Generalities .................................... 122 A.2 Spinors and Γ-matricesinVariousDimensions. 123 B Supergravity and Reductions 126 B.1 11DSupergravity ................................. 126 B.1.1 11D Supersymmetry Transformations and Field Equations ........126 B.1.2 11DReductionAnsätzetoIIA . 126 B.2 IIASupergravity ................................. 127 B.2.1 IIA Supersymmetry Transformations and Field Equations ........127 B.2.2 IIAReductionAnsätzeto9D. 128 B.3 IIBSupergravity ................................. 129 B.3.1 IIB Supersymmetry Transformations and Field Equations.........129 B.3.2 IIBReductionAnsätzeto9D. 130 B.4 9DMaximalSupergravity . 131 B.4.1 9D Supersymmetry Transformations and Field Equations.........131 B.4.2 Twisted Reduction of IIA using β .....................132 B.4.3 Twisted Reduction of IIA using α .....................133 Contents 5 B.4.4 Twisted Reduction of IIB using δ .....................134 B.4.5 ToroidalReductionofGaugedIIA . 134 Bibliography 136 Chapter 1 Introduction This review article1 deals with the construction of different gauged supergravities that arise in the framework of string and M-theory. The latter are thought to be consistent theories of quan- tum gravity, unifying the four different forces. One of their particular features is their critical dimension: these theories necessarily live in ten or eleven dimensions. For this reason one needs a procedure to obtain effective four-dimensional descriptions, which goes under the name of Kaluza-Klein theory or dimensional reduction. In this article we will discuss a number of pos- sible dimensional reductions and the resulting lower-dimensional descriptions. It will be useful to be acquinted with perturbative string theory (see e.g. [1–3]) and the basic concepts of string dualities (see e.g. [4–6]); in this article, emphasis will be placed on supergravity aspects. Proposed in 1974 [7], the idea of string theory as a theory of quantum gravity was not really picked up until the "first superstring revolution" in the mid 1980s. After this period, there were five different perturbative superstring theories: four of closed strings (type IIA, IIB and heterotic with gauge group E8 E8 or SO(32)) and one of open and closed strings (type I). This