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D=6 supergravity with R2 terms Antoine Van Proeyen K.U. Leuven Dubna, 16 December 2011 collaboration with F. Coomans, E. Bergshoeff and E. Sezgin The map: dimensions and # of supersymmetries D susy 11 M 32 24 20 16 12 8 4 M 10 MW IIA IIB I 9 M N=2 N=1 8 M N=2 N=1 7 S N=4 N=2 6 SW (2,2) (2,1) 5 S N=8 N=6 4 M N=8 N=6 (1,1) (2,0) (1,0) N=4 N=5 N=4 N=2 N=3 N=2 N=1 Plan 1. 2. 3. 4. 5. D=6 supersymmetry: what is special ? Higher derivative actions Construction of actions (main part) Solutions Conclusions and outlook 1. D=6 supersymmetry Spinors are symplectic – Weyl, therefore sometimes called ‘chiral supergravity’ Minimal algebra has 8 generators: like N=2 R-symmetry is USp(2)=SU(2) 2-forms have dual formulation, 4-forms are like scalars There is an off-shell and superconformal formulation Symplectic-Weyl spinors spinors have 8 components for D=6 Conventions: metric mostly +, hermitian ° ¹ for ¹ spacelike directions, anti-hermitian °0 Projections Weyl spinor is projected spinor or ‘Reality’ for spinors is defined by a ‘charge conjugation’: ¸C ´ i °0 C-1 ¸ * Majorana spinors are ‘real’: ¸C=¸ but for D=6: (¸C)C= - ¸ unavoidable for consistency with Lorentz symmetry Therefore symplectic Majorana: consistently combined with Weyl condition: symplectic-Weyl spinor: doublet of 4 –component spinors with reality condition → 8 real components Symmetry properties Also different symmetry properties: therefore we define such that R-symmetry Transformations between supersymmetry parameters: preserving symplectic structure is group USp(2) = SU(2) p-form gauge fields they are all gauge fields: Reducible symmetry Degrees of freedom: off-shell: as antisymmetric tensor in SO(D-1), i.e. SO(5) (massive representation) on-shell: as antisymmetric tensor in SO(D-2), i.e. SO(4) (massless representation) • Duality: p-form ↔ (D-2-p)=(4-p) form: because field strength are related by Hodge duality On- and off- shell degrees of freedom off-shell degrees of freedom : # of field components − # gauge transformations (are SO(D-1) representations) On-shell= # of helicity states or count # initial conditions and divide by 2. E.g. -scalar: field equation ¹¹Á =0. Initial conditions Á(t=0,xi) and 0Á(t=0,xi) - fermions: eom linear in derivative: ½ of components (are SO(D-2) representations) 12.3 Multiplets There is an argument that # bosonic d.o.f. = # fermionic d.o.f., based on {Q,Q}=P (invertible) Q Should be valid for on-shell multiplets if eqs. of motion are satisfied for off-shell multiplets counting all components: 2. Higher derivative actions Why interested in higher-derivative terms methods: perturbative or not Interest in higher-derivative terms appear as ®0 terms in effective action of string theory corrections to black hole entropy higher order to AdS/CFT correspondence compactification to D=3 : make graviton a (massive) propagating mode (graviton not prop. without higher-derivative terms) Bergshoeff, Hohm, Rosseel, Sezgin, Townsend, 1005.3952 Perturbative or ‘toy model’ (1) perturbative as in string theory: supersymmetric only order by order in ®0 (2) off-shell exactly supersymmetric invariants have been constructed then ‘auxiliary fields’ are propagating. In method (2): ghosts. If we put small parameter before invariants, then auxiliary fields can be eliminated perturbatively Open question: is this on-shell Lagrangian related to compactified string Lagrangian (which has no auxiliary fields) We will consider (2) with arbitrary (not necessarily small) parameter , ‘toy model’. 3. Construction of actions The off-shell super-Poincaré action using superconformal methods Coupling to vector multiplets and gauging the R-symmetry Alternative off-shell formulation R2 invariant total action Constructions of actions Possible constructions: order by order Noether transformations: the only possibility for the maximal theories (Q>16) superspace: - very useful for rigid N=1: shows structure of multiplets. - very difficult for supergravity. Needs many fields and many gauge transformations (super)group manifold: - Optimal use of the symmetries using constraints on the curvatures superconformal tensor calculus: - keeps the structure of multiplets as in superspace but avoids its immense number of unphysical degrees of freedom - extra symmetry gives insight in the structure - ! only for #Q · 16 (i.e.when there are matter multiplets) For minimal supergravity D=6 Component, Noether procedure: Nishino, Sezgin, 1984 Superspace: Awada, Townsend, Sierra, 1985 Group manifold: R. D'Auria, P. Fré, T. Regge, 1983: ‘Consistent supergravity in six dimensions without action invariance’ Superconformal: Bergshoeff, Sezgin and AVP, 1985; F. Coomans and AVP, 1101.2403 Conformal gauge fields Constraints determine two gauge fields ‘Weyl multiplet’: K-gauge choice: remains dilatation as extra gauge symmetry Gravity as a conformal gauge theory The strategy • scalar field (compensator) conformal gravity: dilatational gauge fixing First action is conformal invariant, gauge-fixed one is Poincaré invariant. Scalar field had scale transformation df (x)= ¸ D(x)f(x) f Schematic: Conformal construction of gravity conformal scalar action (contains Weyl fields) Gauge fix dilatations and special conformal transformations Poincaré gravity action Superconformal algebra In general D=6, N=2 superconformal gauge multiplet determined by constraints ‘Weyl multiplet’: remaining ‘extra’ symmetries: D, Ka, SU(2), Si PS: there is also another choice of extra fields, i.e. another Weyl multiplet, but this one is chosen to obtain an invariant action Compensating multiplet: linear multiplet gauge fix: fixes D and SU(2) →SO(2) fixes Si fixes Ka we also split the gauge field SU(2) = traceless + SO(2) The strategy superconformal action of linear multiplet (contains Weyl multiplet) Gauge fix extra symmetries Poincaré supergravity action To obtain R- symmetry gauging we add a vector multiplet superconformal invariant action uses fields of Weyl multiplet coupling with linear multiplet Gauged supergravity e.g. U(1)R £ U(1) gauge symmetry remainder from SU(2) gauged by V¹ ; param. ¸ from gauge mult gauged by W¹ ; param. ´ E¹º½¾ field equation: 4-form becomes scalar fix Á = Á0 : W¹ gauges R-symmetry Alternative formulation other gauge fixing then: ¾ and Ãi replaced by L and i The R¹ºab R¹ºab invariant Trick from E.Bergshoeff, M. Rakowski, 1987: Transformation laws equal with vector multiplet Define The R¹ºab ¹ºab R invariant off-shell: every term is separately invariant. No auxiliary fields eliminated yet: U(1)R £ U(1) gauge symmetry Final action PS: not in Einstein frame. L=1 gauge would have been in Einstein gauge. For much of the analysis, the gauge ¾ =1 is easier 4. Solutions Salam-Sezgin, 1984: no 6D Minkowki or (A)dS solution, but a Mink4 £ S2 preserving N=1 in D=4. (1/2 susy) with higher derivatives: without flux - a 3 or 4-dimensional Minkowski or dS; non-susy with flux - 2-form flux with 4-dimensional Minkowski, dS or AdS SS solution survives ! - 3-form flux for AdS3 × S3 solutions Solutions and supersymmetry Consider d(e) (boson) = e fermion d(e) (fermion) = e boson We consider solutions with fermions=0 ! (such that at least some Lorentz symmetry is preserved) To check susy of a solution: just check d(e) (fermion) = e boson = 0 This restricts e and the boson configuration. d(e) (fermion) = 0 includes differential equation for e(x) ! In general not any more e(x), but spinors dependent on constants Algebra reduces to a Lie algebra with global supersymmetry Solutions without flux scalar L is a constant L0> 0, arbitrary We find in general R=g2 L0 : hence Mink6 can only be a solution for g=0. For g 0: neither a solution for any constant curvature, i.e. no (A)dS6 in g2 L0 Solutions M1£ M2, with e.g. have fixed M2= mg2 all non-susy Solutions with 2-form flux Supersymmetric solution: Mink4 £ S2 preserving N=1 in D=4. (1/2 susy) with fluxes of gauge fields W and of V on the sphere. M2 not fixed ! other non-susy solutions with - Mink4 £ S2 and M2=-4g2 AdS4 £ S2 dS4 £ S2 dS4 £ H2 Solution with 3-form flux M1£ M2 , both 3-dimensional flux of B on both factors g=0: no effect of higher-derivative terms; AdS3 £ S3 susy solution g 0: found one solution with M2 fixed: AdS3 £ S3 non-susy solution 5. Conclusions and outlook using off-shell formulation constructed Rsymmetry gauged minimal D=6 supergravity with higher derivative terms auxiliary fields can be eliminated perturbatively potential is not modified by Riem2 terms the supersymmetric Mink4 £ S2 solution is still valid other solutions exist Outlook D=6 is highest that allows an off-shell formulation: worthwile to investigate further adding Yang-Mills multiplets and hypermultiplets anomalies (grav. CS term is part of the Riem2 invariant) ghost problem Weyl invariant exists ? black hole solutions ? - higher derivatives important for connection microscopic – macroscopic entropy - so far only Gibbons – Maeda for ungauged and without higher derivative action.