Flux formulation of Double Field Theory

1. Flux formulation of DoubleFieldTheory Quantum Gravity in theSouthernCone VI Maresias, September 2013 Carmen Núñez IAFE-CONICET-UBA

2. Outline • IntroductiontoDoubleFieldTheory • Applications • Flux formulation • Doublegeometry • Open questions and problems • Workwith G. Aldazabal, W. Baron, D. Geissbhuler, D. Marqués, V. Penas

3. T-duality • Closedstringtheoryon a torusTdexhibitsO(d,d)symmetry • Stringsexperiencegeometry in a ratherdifferentwaytopointparticles. • T-dualityestablishesequivalenceof theoriesformulatedonverydifferentbackgrounds • Isthere a more appropriategeometricallanguagewith whichtounderstandstringtheory ?

4. DOUBLE FIELD THEORY • DFT isconstructedfromthe idea toincorporatetheproperties of T-dualityinto a fieldtheory • Conservedmomentum and winding quantum numbershaveassociatedcoordinates in Td • Doubleallcoordinates • Everyobject in a dualityinvarianttheorymustbelongtosomerepresentation of thedualitygroup. In particular, xihavetobesupplementedwith XM  fundamental rep. O(D,D) • Raise and lowerindiceswiththe O(D,D) metric • Introduce doubledfields and write withmanifest global O(D,D) symmetry

5. Fieldcontent • Focusonbosonic universal gravity sectorGij, Bij,  • Fields are encoded in a 2D × 2D GENERALIZED METRIC , O(D,D) INVARIANT GENERALIZED DILATON

6. ThegeneralizedmetricspacetimeactionHull and Zwiebach (2009)O. Hohm, C. Hull and B. Zwiebach (2010) O(D,D) symmetryismanifest • DFT also has a gauge invariancegeneratedby a pair of parameters • Gauge invariance and closure of the gauge algebra lead to a set of • differentialconstraintsthatrestrictthetheory. In particular, theconstraints • can besolvedenforcing a strongerconditionnamedstrongconstraint

7. Strongconstraint • Allfields, gauge parametersandproducts of themsatisfy • Itimpliesthereissome dual framewherefields are notdoubled • Stronglyconstrainted DFT displaysthe O(D,D) symmetrybutitis notphysicallydoubled • Gauge invariance and closure of gauge transformationsweakercondition • Certainbackgroundsallowrelaxations of thestrongconstraint, producing a trulydoubledtheory: • Massivetype IIA O. Hohm, S. Kwak (2011) • SuggestedbyScheck-Schwarzcompactifications of DFT G. Aldazabal, W. Baron, D. Marqués, C.N. (2011) D. Geissbhuler (2011) • Sufficientbutnotnecessaryfor gauge invariance and closure of gauge algebra M. Graña, D. Marqués (2012) • Explicitdoublesolutionsfound in D. Geissbhuler, D. Marqués, C.N., V. Penas (2013)

8. Applications of DFT • DFT has been a powerfultoolto explore stringtheoreticalfeaturesbeyondsupergravity and Riemaniangeometry • Somerecentdevelopmentsinclude: • Geometricinterpretation of non-geometricgaugings in flux compactifications of stringtheoryG. Aldazabal, W. Baron, D. Marqués, C.N. (2011) D. Geissbhuler (2011) • Identification of new geometricstructures D. Andriot, R. Blumenhagen, O. Hohm, M. Larfors, D. Lust, P. Patalong (2011, 2012) • Description of exoticbraneorbitsF. Hassler, D., Lust (2013) J.de Boer, M. Shigemori (2010, 2012), T. Kikuchi, T. Okada, Y. Sakatani (2012) • Non-commutative/non-associativestructures in closedstringtheory R. Blumenhagen, E. Plauschinn, D. Andriot, C. Condeescu, C. Floriakis, M. Larfors, D. Lust , P. Patalong (2010-2012) • New perspectiveson‘ corrections, O. Hohm, W. Siegel, B. Zwiebach (2012,2013) • New possibilitiesforupliftings, modulifixing and dS vacua, Roest et al. (2012)

9. Application I: Missinggaugings in geometriccompactifications(seeAldazabal’stalk) 10D stringsugra SS reduction ontwisted T6 4D gaugedsugraT-duality4D gaugedsugra geometricfluxesin d-dimall dual (geometric & abc & Habc non- geometric) gaugings Modulifixing & dS vacua T-duality in D-dim Double Field Theory SS reduction on twisted T6,6 ???

10. Application II: New geometricstructuresNon geometry, GeneralizedGeometry • Diffeomorphisms of GR and gauge transformations of 2-form are combined in generalizeddiffeomorphisms and Lie derivatives ; New termneeded so that • Gauge transformations Theaction of thegeneralizedmetricformulationis gauge invariantbecauseR(H,d)is a generalizedscalarunderthestrongconstraint

11. DFT vs GeneralizedGeometry • Thedoublegeometryunderlying DFT differsfromordinarygeometry. • DFT is a smalldeparturefromGeneralizedGeometry(Hitchin, 2003; Gualtieri, 2004) • Given a manifold M, GG putstogethervectors Vi and one-formsi as V +   TM T*M . Structuresonthislargerspace TheCourantbracketgeneralizesthe Lie bracket V and  are nottreatedsymmetrically • DFT puts TM and T*M on similar footingbydoublingtheunderlyingmanifold. Gauge parameters and thenC-bracket For non-doubled Mthe C-bracket reduces totheCourantbracket

12. Geometry, connections and curvature • Theactionwastendentiouslywritten as It can beshownthattheaction and EOM of DFT can beobtainedfrom traces and projections of a generalizedRiemann tensor RMNPQ • TheconstructiongoesbeyondRiemanniangeometrybecauseitisbasedongeneralizedratherthanstandard Lie derivatives • Thenotions of connections, torsion and curvaturehavetobegeneralized • E.g.thevanishingtorsion and compatibilityconditions do notcompletely determine theconnections and curvatures, butonlyfixsome of theirprojectionsI. Jeon, K. Lee, J. Park (2011), O. Hohm, B. Zwiebach (2012) • Strongconstraintwasassumed in theseconstructions. Can itberelaxed?

13. Flux formulation of DFTW. Siegel (1993) D. Geissbhuler, D. Marqués, C.N., V. Penas (2013) • Basic fields are generalizedvielbeins EAM and dilaton • EAM can beparametrized in terms of vielbein of D-dimensional metric D-dimensional Minkowskimetric • Arrangethefields in dynamicalfluxes: • Fielddependent and non-constantfluxes, thatgiverisetogaugingsorconstantfluxesuponcompactification (e.g.Fabc=Habc)

14. Theaction Theactiontakestheform of theelectric sector of thescalarpotential of N=4 D=4 gaugedsupergravity Vanishesunderstrongconstraint Generalizedmetric DFT actionmodulo onestrongconstraintviolatingterm Thisactiongeneralizesthegeneralizedmetricformulation, includingallterms thatvanishunderthestrongconstraint

15. Generalizeddiffeomorphisms • Theclosureconstraints (generalized Lie derivativesgenerateclosedtransformations) taketheform: • and they asure that FABC, FAtransform as scalars and S is gauge invariant • Imposingtheseconditionsonlyrequiresa relaxedversionof strong • constraintthetheoryadmitstrulydoublefields • Constraints can beinterpreted as Bianchi identitiesforgeneralized • Riemann tensor

16. Geometricformulation of DFT • Define covariantderivativeontensors • Determine theconnectionsimposing set of conditions: • Compatibilitywithgeneralizedframe: • Compatibilitywith O(D,D) invariantmetric • Compatibilitywithgeneralizedmetric • Covarianceundergeneralizeddiffeomorphisms: • CovarianceunderdoubleLorentztransformations: Lorentzscalar • Vanishinggeneralizedtorsion: Standard torsion non covariant • Compatibilitywithgeneralizeddilaton Only determine someprojections of theconnections

17. Generalizedcurvature • ThestandardRiemann tensor in planarindicesisnot a scalarundergeneralizeddiffeomorphisms • It can bemodifiedadding new terms, leadingto • Projectionswithgive and similarly EOM • Bianchi identities

18. Scherk-Schwarzsolutions • Alltheconstraints can besolvedrestrictingthefields and gauge parameters as where and quadraticconstraints of N=4 gaugedsugra • Fortheseconfigurationsalltheconsistencyconstraints are satisfied. • Thedynamicalfluxesbecome: • Thisansatzcontainsthe usual decompactifiedstrongcontrained case (U=1, =0, xi, i=1,…, D). Itis a particular limit in whichallthe compact dimensions are decompactified.

19. Conclusions • Presentedformulation of DFT in terms of dynamical and fielddependentfluxes. • The gauge consistencyconstraintstaketheform of quadraticconstraintsforthefluxes, thatadmitsolutionsthatviolatethe SC  allowstogobeyondsupergravity • Computedconnections and curvaturesonthedoublespaceunderassumptionthatcovarianceisachievedupongeneralizedquadraticconstraints, ratherthan SC, which can beinterpreted as BI. • Interestingly, thisproceduregivesrisetoallthe SC-violatingterms in theaction, which are gauge invariant and appearsystematically • This completes the original formulation of DFT, incorporatingthemissingtermsthatallowtomakecontactwithhalf-maximalgaugedsugra, containingalldualityorbits of non-geometricfluxes (FABCFABC).

20. Open questions • Someelements of the O(D,D) geometryhavebeenunderstood, butitisimportanttobetterunderstandthegeometryunderlying DFT • Can thisconstructionbe extended beyondtori? Calabi-Yau? • ’ corrections. Innerproduct and C-bracket are corrected deformation of Courantbracket and otherstructures in GG • Beyond T-duality? U-duality? • Relationbetween DFT and stringtheory. Isthis a consistenttruncation of stringtheory? No massivestates, butfullyconsistent • Worldsheettheory?

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