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D=6 supergravity with R 2 terms Antoine Van Proeyen K.U. Leuven Dubna, 16 December 2011 collaboration with F. Coomans, E. Bergshoeff and E. Sezgin

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The map: dimensions and # of supersymmetries Dsusy MM 10MWIIAIIBI 9MN=2N=1 8MN=2N=1 7SN=4N=2 6SW(2,2)(2,1)(1,1)(2,0)(1,0) 5SN=8N=6N=4N=2 4MN=8N=6N=5N=4N=3N=2N=1

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Plan 1. D=6 supersymmetry: what is special ? 2. Higher derivative actions 3. Construction of actions (main part) 4. Solutions 5. Conclusions and outlook

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1. D=6 supersymmetry n Spinors are symplectic – Weyl, therefore sometimes called ‘chiral supergravity’ n Minimal algebra has 8 generators: like N=2 n R-symmetry is USp(2)=SU(2) n 2-forms have dual formulation, 4-forms are like scalars n There is an off-shell and superconformal formulation

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Symplectic-Weyl spinors Conventions: metric mostly +, hermitian ° ¹ for ¹ spacelike directions, anti-hermitian ° 0 n 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 n Therefore symplectic Majorana: n consistently combined with Weyl condition: symplectic-Weyl spinor: doublet of 4 –component spinors with reality condition → 8 real components spinors have 8 components for D=6

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Symmetry properties n Also different symmetry properties: n therefore we define such that

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R-symmetry n Transformations between supersymmetry parameters: n preserving symplectic structure n is group USp(2) = SU(2)

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p-form gauge fields n they are all gauge fields: n 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 Reducible symmetry

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On- and off- shell degrees of freedom n off-shell degrees of freedom : # of field components − # gauge transformations (are SO(D-1) representations) n On-shell= # of helicity states or count # initial conditions and divide by 2. E.g. -scalar: field equation ¹ ¹ Á =0. Initial conditions Á (t=0,x i ) and 0 Á (t=0,x i ) - fermions: eom linear in derivative: ½ of components (are SO(D-2) representations)

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12.3 Multiplets n There is an argument that # bosonic d.o.f. = # fermionic d.o.f., based on {Q,Q}=P (invertible) Q n Should be valid for on-shell multiplets if eqs. of motion are satisfied n for off-shell multiplets counting all components:

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2. Higher derivative actions n Why interested in higher-derivative terms n methods: perturbative or not

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Interest in higher-derivative terms appear as ® 0 terms in effective action of string theory n corrections to black hole entropy n higher order to AdS/CFT correspondence n compactification to D=3 : make graviton a (massive) propagating mode (graviton not prop. without higher-derivative terms) Bergshoeff, Hohm, Rosseel, Sezgin, Townsend,

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Perturbative or ‘toy model’ (1) perturbative as in string theory: supersymmetric only order by order in ® 0 n (2) off-shell exactly supersymmetric invariants have been constructed then ‘auxiliary fields’ are propagating. n In method (2): ghosts. n If we put small parameter before invariants, then auxiliary fields can be eliminated perturbatively n Open question: is this on-shell Lagrangian related to compactified string Lagrangian (which has no auxiliary fields) n We will consider (2) with arbitrary (not necessarily small) parameter, ‘toy model’.

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3. Construction of actions n The off-shell super-Poincaré action using superconformal methods n Coupling to vector multiplets and gauging the R-symmetry n Alternative off-shell formulation n R 2 invariant n total action

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Constructions of actions n order by order Noether transformations: the only possibility for the maximal theories (Q>16) n superspace: -very useful for rigid N=1: shows structure of multiplets. -very difficult for supergravity. Needs many fields and many gauge transformations n (super)group manifold: -Optimal use of the symmetries using constraints on the curvatures n 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) Possible constructions:

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For minimal supergravity D=6 n Component, Noether procedure: Nishino, Sezgin, 1984 n Superspace: Awada, Townsend, Sierra, 1985 n Group manifold: R. D'Auria, P. Fré, T. Regge, 1983: ‘Consistent supergravity in six dimensions without action invariance’ n Superconformal: Bergshoeff, Sezgin and AVP, 1985; F. Coomans and AVP,

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Conformal gauge fields n Constraints determine two gauge fields ‘Weyl multiplet’: K-gauge choice: remains dilatation as extra gauge symmetry

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Gravity as a conformal gauge theory The strategy scalar field (compensator) n First action is conformal invariant, n gauge-fixed one is Poincaré invariant. Scalar field had scale transformation (x)= ¸ D (x) (x) conformal gravity: dilatational gauge fixing

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Schematic: Conformal construction of gravity conformal scalar action (contains Weyl fields) Gauge fix dilatations and special conformal transformations Poincaré gravity action

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Superconformal algebra n In general

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D=6, N=2 superconformal gauge multiplet determined by constraints ‘Weyl multiplet’: remaining ‘extra’ symmetries: D, K a, SU(2), S i PS: there is also another choice of extra fields, i.e. another Weyl multiplet, but this one is chosen to obtain an invariant action

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Compensating multiplet: linear multiplet gauge fix: fixes D and SU(2) →SO(2)fixes S i fixes K a we also split the gauge field SU(2) = traceless + SO(2)

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The strategy superconformal action of linear multiplet (contains Weyl multiplet) Gauge fix extra symmetries Poincaré supergravity action

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To obtain R- symmetry gauging we add a vector multiplet superconformal invariant action uses fields of Weyl multiplet coupling with linear multiplet

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Gauged supergravity e.g. E ¹º½¾ field equation: 4-form becomes scalar U(1) R £ U(1) gauge symmetry remainder from SU(2) gauged by V ¹ ; param. ¸ from gauge mult gauged by W ¹ ; param. ´ fix Á = Á 0 : W ¹ gauges R-symmetry

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PS: to Salam-Sezgin model n dualize

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Alternative formulation n other gauge fixing then: ¾ and Ã i replaced by L and i

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The R ¹ºab R ¹ºab invariant n Trick from E.Bergshoeff, M. Rakowski, 1987: n Transformation laws equal with vector multiplet Define

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The R ¹ºab R ¹ºab invariant off-shell: every term is separately invariant. No auxiliary fields eliminated yet: U(1) R £ U(1) gauge symmetry

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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

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4. Solutions Salam-Sezgin, 1984: no 6D Minkowki or (A)dS solution, but a Mink 4 £ S 2 preserving N=1 in D=4. (1/2 susy) n 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 AdS 3 × S 3 solutions

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Solutions and supersymmetry Consider ) (boson) = fermion ) (fermion) = boson n We consider solutions with fermions=0 ! (such that at least some Lorentz symmetry is preserved) To check susy of a solution: just check ) (fermion) = boson = 0 This restricts and the boson configuration ) (fermion) = 0 includes differential equation for (x) ! In general not any more (x), but spinors dependent on constants n Algebra reduces to a Lie algebra with global supersymmetry

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Solutions without flux n scalar L is a constant L 0 > 0, arbitrary n We find in general R=g 2 L 0 : hence Mink 6 can only be a solution for g=0. For g 0: neither a solution for any constant curvature, i.e. no (A)dS 6 Solutions M 1 £ M 2, with e.g. have fixed M 2 = mg 2 all non-susy in g 2 L 0

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Solutions with 2-form flux Supersymmetric solution: Mink 4 £ S 2 preserving N=1 in D=4. (1/2 susy) with fluxes of gauge fields W and of V on the sphere. M 2 not fixed ! n other non-susy solutions with -Mink 4 £ S 2 and M 2 =-4g 2 -AdS 4 £ S 2 -dS 4 £ S 2 -dS 4 £ H 2

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Solution with 3-form flux M 1 £ M 2, both 3-dimensional n flux of B on both factors g=0: no effect of higher-derivative terms; AdS 3 £ S 3 susy solution g 0: found one solution with M 2 fixed: AdS 3 £ S 3 non-susy solution

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5. Conclusions and outlook n using off-shell formulation constructed R- symmetry gauged minimal D=6 supergravity with higher derivative terms n auxiliary fields can be eliminated perturbatively n potential is not modified by Riem 2 terms the supersymmetric Mink 4 £ S 2 solution is still valid n other solutions exist

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Outlook n D=6 is highest that allows an off-shell formulation: worthwile to investigate further n adding Yang-Mills multiplets and hypermultiplets n anomalies (grav. CS term is part of the Riem 2 invariant) n ghost problem n Weyl invariant exists ? n black hole solutions ? -higher derivatives important for connection microscopic – macroscopic entropy -so far only Gibbons – Maeda for ungauged and without higher derivative action.

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