Affine symmetries in supergravity work with Hermann Nicolai, Martin - PowerPoint PPT Presentation

Affine symmetries in supergravity work with Hermann Nicolai, Martin Weidner, Thomas Ortiz IHES 05/2013 
 Henning Samtleben 


1998 motivation : 2D supergravity symmetries classically integrable field theory affine symmetry group E 9 — solution generating (transitive) infinite-dimensional symmetries : E 9 E 10 E 11 deformations affine symmetry also organizes the deformations of the theory infinite-dim. HW representations of non-propagating fields supersymmetry SO(9) supergravity : first example of such a 2d deformation : IIA on S 8 matrix model holography Henning Samtleben ENS Lyon

motivation : SO(9) supergravity Domain wall / QFT correspondence [H.J. Boonstra, K. Skenderis, P. Townsend, 1999] holography for Dp-branes : AdS p+2 x S 8-p warped dual to SYM p+1 theory gaugings of maximal supergravity [Salam, Sezgin, 1984] D6 IIA AdS 8 x S 2 d=8, SO(3) [Samtleben, Weidner, 2005] D5 IIB AdS 7 x S 3 d=7, SO(4) [Pernici, Pilch, van Nieuwenhuizen, 1984] D4 IIA AdS 6 x S 4 d=6, SO(5) D3 IIB AdS 5 x S 5 d=5, SO(6) [Günaydin, Romans, Warner, 1985] D2 IIA AdS 4 x S 6 d=4, SO(7) [Hull, 1984] D1 IIB AdS 3 x S 7 d=3, SO(8) [de Wit, Nicolai, 1982] D0 IIA AdS 2 x S 8 d=2, SO(9) ?? Henning Samtleben ENS Lyon

plan Affine symmetries in supergravity motivation D=4 supergravity : symmetries and deformations D=2 supergravity : symmetries and deformations example : SO(9) supergravity conclusions Henning Samtleben ENS Lyon

1998 D=4 supergravity symmetries and deformations Henning Samtleben ENS Lyon

1998 D=4 supergravity: some generic features L = R + G ij ( φ ) ∂ µ φ i ∂ µ φ j + I ΛΣ ( φ ) F Λ µ ν F µ ν Σ + R ΛΣ ( φ ) F Λ ∗ F µ ν Σ + · · · µ ν bosonic sector of maximal (N=8) D=4 supergravity Henning Samtleben ENS Lyon

1998 D=4 supergravity: symmetries L = R + G ij ( φ ) ∂ µ φ i ∂ µ φ j + I ΛΣ ( φ ) F Λ µ ν F µ ν Σ + R ΛΣ ( φ ) F Λ ∗ F µ ν Σ + · · · µ ν � scalar sector: G/H coset space sigma model E 7 SU(8) V ∈ E 7 V ≈ V · H H ∈ SU(8) triangular gauge exp { φ m N m } exp φ λ h λ � ⇥ V = Cartan nilpotent grading E 7 action → G V H G, V V − φ m → φ m + λ m shift symmetries G = exp { λ m N m } : G = exp { λ m N † ‘hidden’ symmetries φ i m } non-linear! (on ) (linear on ) V Henning Samtleben ENS Lyon

1998 D=4 supergravity: self-duality L = R + G ij ( φ ) ∂ µ φ i ∂ µ φ j + I ΛΣ ( φ ) F Λ µ ν F µ ν Σ + R ΛΣ ( φ ) F Λ ∗ F µ ν Σ + · · · µ ν self-duality (D=4: electric-magnetic duality for vectors) ∂ L Λ = 2 ∂ [ µ A ν ] Λ field strength: dual: F µ ν G µ ν Λ = − ε µ νρσ ∂ F ρσ Λ Λ = 0 Bianchi: ∂ [ µ F νρ ] dual vectors: G µ ν Λ = 2 ∂ [ µ A ν ] Λ eom: ∂ [ µ G νρ ] Λ = 0 symplectic rotation non-local (on ) ! A Λ F Λ U ΛΣ Z ΛΣ F Σ � � � � � � µ − → V ΛΣ ( A Λ G Λ G Σ (local on ) W ΛΣ µ , A µ Λ ) choice of an electric frame, analogous pattern for (n—1)-forms in D=2n E 7 is realized (on-shell) on the combined set of 28 electric +28 magnetic vectors Henning Samtleben ENS Lyon

1998 D=4 supergravity: gauging L = R + G ij ( φ ) ∂ µ φ i ∂ µ φ j + I ΛΣ ( φ ) F Λ µ ν F µ ν Σ + R ΛΣ ( φ ) F Λ ∗ F µ ν Σ + · · · µ ν self-duality (D=4: electric-magnetic duality for vectors) electric gauging (“standard”) gauging (embedding tensor) M Θ M α t α = ∂ µ − A µ Λ Θ Λ α t α − A µ Λ Θ Λ α t α D µ = ∂ µ − A µ magnetic gauging (“non-standard”) consistency encoded in a set of algebraic constraints on the embedding tensor α Θ M P Ω K ) P = 0 α t α ,N linear: (susy / consistent tensor hierarchy) Θ ( M 56 x 133 = 56 + 912 + 6480 quadratic: (generalized Jacobi / locality) β + ( t α ) N γ = 0 γ Θ M α Θ N P Θ M α Θ P f αβ Ω MN Θ M α Θ N β = 0 ⇐ ⇒ Henning Samtleben ENS Lyon

D=4 supergravity: gauging 1998 L = R + G ij ( φ ) ∂ µ φ i ∂ µ φ j + I ΛΣ ( φ ) F Λ µ ν F µ ν Σ + R ΛΣ ( φ ) F Λ ∗ F µ ν Σ + · · · µ ν self-duality (D=4: electric-magnetic duality for vectors) electric gauging (“standard”) gauging M Θ M α t α = ∂ µ − A µ Λ Θ Λ α t α − A µ Λ Θ Λ α t α D µ = ∂ µ − A µ magnetic gauging (“non-standard”) off-shell formulation � � 8 Θ Λ α B α ∧ 2 ∂ A Λ + X MN Λ A M ∧ A N − 1 β B β − 1 L top = 4 Θ Λ + · · · upon introduction of additional two-forms (dual to scalars) and BF couplings gauging of on-shell symmetries [de Wit, HS, Trigiante ] Henning Samtleben ENS Lyon

1998 D=2 supergravity affine symmetries Henning Samtleben ENS Lyon

1998 D=2 supergravity ungauged Lagrangian � ⇥ ˙ √− g ρ − R + tr[ P µ P µ ] + L ferm ( ψ I , ψ I A ) − 1 L = 2 , χ 4 V − 1 ∂ µ V = Q µ + P µ coset space sigma model coupled to dilaton gravity off-shell symmetry (target space isometries): E 8 field equations dilaton scalars J µ ≡ ρ V P µ V − 1 ∂ µ J µ conserved E 8 Noether current M = 0 � ρ = 0 has a remarkable structure : (infinite tower of) dual scalar potentials classical integrability, affine Lie-Poisson symmetry E 9 duality ⇤ µ ˜ ⇥ = � µ ν ⇤ ν ⇥ dual dilaton dual (D–2) forms ∂ µ Y M ≡ ε µ ν J ν dual scalars M Henning Samtleben ENS Lyon

1998 D=2 supergravity ungauged duality ⇤ µ ˜ ⇥ = � µ ν ⇤ ν ⇥ dual dilaton dual (D–2) forms ∂ µ Y M ≡ ε µ ν J ν dual scalars M classical integrability, affine Lie-Poisson symmetry E 9 (1) δ 1 ˜ ρ = λ shift symmetries (248) Λ α δ α , 1 Y 1 = Λ Λ α δ α , 1 V = 0 ρ V [ V − 1 Λ V ] p Λ α δ α , − 1 V = [ Λ , Y 1 ] V − ˜ (248) ‘hidden’ symmetries � extends to an infinite tower: � � ρ + 1 V P ± V − 1 + 1 dual scalars 2 ρ 2 ∂ ± Y 2 = ± ρ ˜ 2[ Y 1 , ∂ ± Y 1 ] , � � ∓ 1 V P ± V − 1 + [ Y 1 , ∂ ± Y 2 ] − 1 2 ρ 3 ∓ ρ ˜ ρ 2 − ρ 2 ˜ ∂ ± Y 3 = 6[ Y 1 , [ Y 1 , ∂ ± Y 1 ]]] . ρ V V − V V V � � � 1 � [ Λ , Y 2 ] + 1 2 ρ 2 + ˜ ‘hidden’ symmetries (248) Λ α δ α , − 2 V = ρ 2 V [ V − 1 Λ V ] p 2[[ Λ , Y 1 ] , Y 1 ] − ˜ ρ [ Λ , Y 1 ] V + etc... close into (half of) the affine algebra ! Henning Samtleben ENS Lyon

D=2 supergravity ungauged 1998 linear system the equations of motion can be encoded as integrability conditions of a linear system [Belinskii, Zakharov / Maison / Julia / Nicolai, Warner] V = Q ± + 1 ⇥ γ V − 1 ∂ ± ˆ ˆ ( light-cone-coord. ) 1 ± γ P ± x ± ˆ for a group-valued function and the spectral parameter V ( γ ) γ = 1 � ⇥ ⇤ w + ˜ ( w + ˜ ρ ) 2 − ρ 2 ρ − ρ expansion in w gives rise to the infinite series of dual scalars V = . . . e Y 3 w − 3 e Y 2 w − 2 e Y 1 w − 1 V ˆ ± ρ V P ± V − 1 = ∂ ± Y 1 2 ρ 2 ) V P ± V − 1 + 1 ρ + 1 = − ( ± ρ ˜ 2 [ Y 1 , ∂ ± Y 1 ] ∂ ± Y 2 = ∂ ± Y 3 . . . Henning Samtleben ENS Lyon

1998 D=2 supergravity ungauged affine symmetry group E 9 action parametrized by a meromorphic function Λ ( w ) � � V − 1 Λ ( w )ˆ ˆ V = ˜ Λ h + ˜ Λ ( w ) ∂ w ˆ V ( w )ˆ V − 1 ( w ) Λ k δσ = κ − tr w dw � � 1 2 γ ( w ) ⇥ f ( w ) ⇤ w � 2 π i f ( w ) V − 1 δ V = � � ˜ Λ k ( w ) ρ (1 − γ ( w ) 2 ) w extends to the set of dual scalars � � V V − δ ˜ ρ = λ Λ h ( v ) + γ ( v ) (1 − γ 2 ( w )) � 1 � �� V − 1 δ ˆ ˆ V ( w ) = λ ˆ V − 1 ∂ w ˆ V ( w ) + ˜ ˜ ˜ Λ ( w ) − Λ k ( v ) γ ( w ) (1 − γ 2 ( v )) v − w v (1 coset action E 9 / K(E 9 ) } m < 0 hidden symmetries { t α m , L 1 , k } — off-shell Virasoro L 1 ˜ ρ = 1 } shift central extension [Julia ] k σ = 1 symmetries m > 0 deformations : gauge part of this nonlinear, nonlocal, on-shell symmetry Henning Samtleben ENS Lyon

1998 D=2 supergravity deformations [HS, Martin Weidner ] Henning Samtleben ENS Lyon