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The homogeneity theorem for ten- and eleven-dimensional supergravities Jos e Figueroa-OFarrill 13 February 2013 Jos e Figueroa-OFarrill Homogeneous supergravity backgrounds 1 / 26 Supergravity result of ongoing effort to marry GR

  1. Supersymmetry Eleven-dimensional supergravity has local supersymmetry manifests itself as a connection D on the spinor bundle S D is not induced from a connection on the spin bundle the field equations are encoded in the curvature of D : � e i · R D ( e i , −) = 0 i geometric analogies: ∇ ε = 0 = ⇒ Ric = 0 Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 5 / 26

  2. Supersymmetry Eleven-dimensional supergravity has local supersymmetry manifests itself as a connection D on the spinor bundle S D is not induced from a connection on the spin bundle the field equations are encoded in the curvature of D : � e i · R D ( e i , −) = 0 i geometric analogies: ∇ ε = 0 = ⇒ Ric = 0 ∇ X ε = 1 2 X · ε = ⇒ Einstein Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 5 / 26

  3. Supersymmetry Eleven-dimensional supergravity has local supersymmetry manifests itself as a connection D on the spinor bundle S D is not induced from a connection on the spin bundle the field equations are encoded in the curvature of D : � e i · R D ( e i , −) = 0 i geometric analogies: ∇ ε = 0 = ⇒ Ric = 0 ∇ X ε = 1 2 X · ε = ⇒ Einstein a background ( M , g , F ) is supersymmetric if there exists a nonzero spinor field ε satisfying Dε = 0 Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 5 / 26

  4. Supersymmetry Eleven-dimensional supergravity has local supersymmetry manifests itself as a connection D on the spinor bundle S D is not induced from a connection on the spin bundle the field equations are encoded in the curvature of D : � e i · R D ( e i , −) = 0 i geometric analogies: ∇ ε = 0 = ⇒ Ric = 0 ∇ X ε = 1 2 X · ε = ⇒ Einstein a background ( M , g , F ) is supersymmetric if there exists a nonzero spinor field ε satisfying Dε = 0 such spinor fields are called Killing spinors Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 5 / 26

  5. Killing spinors Not every manifold admits spinors: so an implicit condition on ( M , g , F ) is that M should be spin Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 6 / 26

  6. Killing spinors Not every manifold admits spinors: so an implicit condition on ( M , g , F ) is that M should be spin The spinor bundle of an eleven-dimensional lorentzian spin manifold is a real 32-dimensional symplectic vector bundle Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 6 / 26

  7. Killing spinors Not every manifold admits spinors: so an implicit condition on ( M , g , F ) is that M should be spin The spinor bundle of an eleven-dimensional lorentzian spin manifold is a real 32-dimensional symplectic vector bundle The Killing spinor equation is 12 ( X ♭ ∧ F ) · ε + 1 D X ε = ∇ X ε + 1 6 ι X F · ε = 0 which is a linear, first-order PDE: Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 6 / 26

  8. Killing spinors Not every manifold admits spinors: so an implicit condition on ( M , g , F ) is that M should be spin The spinor bundle of an eleven-dimensional lorentzian spin manifold is a real 32-dimensional symplectic vector bundle The Killing spinor equation is 12 ( X ♭ ∧ F ) · ε + 1 D X ε = ∇ X ε + 1 6 ι X F · ε = 0 which is a linear, first-order PDE: linearity: solutions form a vector space Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 6 / 26

  9. Killing spinors Not every manifold admits spinors: so an implicit condition on ( M , g , F ) is that M should be spin The spinor bundle of an eleven-dimensional lorentzian spin manifold is a real 32-dimensional symplectic vector bundle The Killing spinor equation is 12 ( X ♭ ∧ F ) · ε + 1 D X ε = ∇ X ε + 1 6 ι X F · ε = 0 which is a linear, first-order PDE: linearity: solutions form a vector space first-order: solutions determined by their values at any point Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 6 / 26

  10. Killing spinors Not every manifold admits spinors: so an implicit condition on ( M , g , F ) is that M should be spin The spinor bundle of an eleven-dimensional lorentzian spin manifold is a real 32-dimensional symplectic vector bundle The Killing spinor equation is 12 ( X ♭ ∧ F ) · ε + 1 D X ε = ∇ X ε + 1 6 ι X F · ε = 0 which is a linear, first-order PDE: linearity: solutions form a vector space first-order: solutions determined by their values at any point the dimension of the space of Killing spinors is 0 � n � 32 Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 6 / 26

  11. Killing spinors Not every manifold admits spinors: so an implicit condition on ( M , g , F ) is that M should be spin The spinor bundle of an eleven-dimensional lorentzian spin manifold is a real 32-dimensional symplectic vector bundle The Killing spinor equation is 12 ( X ♭ ∧ F ) · ε + 1 D X ε = ∇ X ε + 1 6 ι X F · ε = 0 which is a linear, first-order PDE: linearity: solutions form a vector space first-order: solutions determined by their values at any point the dimension of the space of Killing spinors is 0 � n � 32 a background is said to be ν -BPS , where ν = n 32 Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 6 / 26

  12. Which values of ν are known to appear? ν = 1 backgrounds are classified JMF+Papadopoulos (2002) Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 7 / 26

  13. Which values of ν are known to appear? ν = 1 backgrounds are classified JMF+Papadopoulos (2002) ν = 31 32 has been ruled out Gran+Gutowski+Papadopolous+Roest (2006) JMF+Gadhia (2007) Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 7 / 26

  14. Which values of ν are known to appear? ν = 1 backgrounds are classified JMF+Papadopoulos (2002) ν = 31 32 has been ruled out Gran+Gutowski+Papadopolous+Roest (2006) JMF+Gadhia (2007) ν = 15 16 has been ruled out Gran+Gutowski+Papadopoulos (2010) Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 7 / 26

  15. Which values of ν are known to appear? ν = 1 backgrounds are classified JMF+Papadopoulos (2002) ν = 31 32 has been ruled out Gran+Gutowski+Papadopolous+Roest (2006) JMF+Gadhia (2007) ν = 15 16 has been ruled out Gran+Gutowski+Papadopoulos (2010) No other values of ν have been ruled out Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 7 / 26

  16. Which values of ν are known to appear? ν = 1 backgrounds are classified JMF+Papadopoulos (2002) ν = 31 32 has been ruled out Gran+Gutowski+Papadopolous+Roest (2006) JMF+Gadhia (2007) ν = 15 16 has been ruled out Gran+Gutowski+Papadopoulos (2010) No other values of ν have been ruled out The following values are known to appear: 0, 1 32 , 1 16 , 3 32 , 1 8 , 5 32 , 3 16 , . . . , 1 4 , . . . , 3 8 , . . . , 1 2 , . . . , 9 16 , . . . , 5 8 , . . . , 11 16 , . . . , 3 4 , . . . , 1 Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 7 / 26

  17. Which values of ν are known to appear? ν = 1 backgrounds are classified JMF+Papadopoulos (2002) ν = 31 32 has been ruled out Gran+Gutowski+Papadopolous+Roest (2006) JMF+Gadhia (2007) ν = 15 16 has been ruled out Gran+Gutowski+Papadopoulos (2010) No other values of ν have been ruled out The following values are known to appear: 0, 1 32 , 1 16 , 3 32 , 1 8 , 5 32 , 3 16 , . . . , 1 4 , . . . , 3 8 , . . . , 1 2 , . . . , 9 16 , . . . , 5 8 , . . . , 11 16 , . . . , 3 4 , . . . , 1 where the second row are now known to be homogeneous! Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 7 / 26

  18. Supersymmetries generate isometries The Dirac current V ε of a Killing spinor ε is defined by g ( V ε , X ) = ( ε , X · ε ) Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 8 / 26

  19. Supersymmetries generate isometries The Dirac current V ε of a Killing spinor ε is defined by g ( V ε , X ) = ( ε , X · ε ) More generally, if ε 1 , ε 2 are Killing spinors, g ( V ε 1 , ε 2 , X ) = ( ε 1 , X · ε 2 ) Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 8 / 26

  20. Supersymmetries generate isometries The Dirac current V ε of a Killing spinor ε is defined by g ( V ε , X ) = ( ε , X · ε ) More generally, if ε 1 , ε 2 are Killing spinors, g ( V ε 1 , ε 2 , X ) = ( ε 1 , X · ε 2 ) V := V ε is causal : g ( V , V ) � 0 Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 8 / 26

  21. Supersymmetries generate isometries The Dirac current V ε of a Killing spinor ε is defined by g ( V ε , X ) = ( ε , X · ε ) More generally, if ε 1 , ε 2 are Killing spinors, g ( V ε 1 , ε 2 , X ) = ( ε 1 , X · ε 2 ) V := V ε is causal : g ( V , V ) � 0 V is Killing: L V g = 0 Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 8 / 26

  22. Supersymmetries generate isometries The Dirac current V ε of a Killing spinor ε is defined by g ( V ε , X ) = ( ε , X · ε ) More generally, if ε 1 , ε 2 are Killing spinors, g ( V ε 1 , ε 2 , X ) = ( ε 1 , X · ε 2 ) V := V ε is causal : g ( V , V ) � 0 V is Killing: L V g = 0 L V F = 0 Gauntlett+Pakis (2002) Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 8 / 26

  23. Supersymmetries generate isometries The Dirac current V ε of a Killing spinor ε is defined by g ( V ε , X ) = ( ε , X · ε ) More generally, if ε 1 , ε 2 are Killing spinors, g ( V ε 1 , ε 2 , X ) = ( ε 1 , X · ε 2 ) V := V ε is causal : g ( V , V ) � 0 V is Killing: L V g = 0 L V F = 0 Gauntlett+Pakis (2002) L V D = 0 Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 8 / 26

  24. Supersymmetries generate isometries The Dirac current V ε of a Killing spinor ε is defined by g ( V ε , X ) = ( ε , X · ε ) More generally, if ε 1 , ε 2 are Killing spinors, g ( V ε 1 , ε 2 , X ) = ( ε 1 , X · ε 2 ) V := V ε is causal : g ( V , V ) � 0 V is Killing: L V g = 0 L V F = 0 Gauntlett+Pakis (2002) L V D = 0 ε ′ Killing spinor = ⇒ so is L V ε ′ = ∇ V ε ′ − ρ ( ∇ V ) ε ′ Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 8 / 26

  25. Supersymmetries generate isometries The Dirac current V ε of a Killing spinor ε is defined by g ( V ε , X ) = ( ε , X · ε ) More generally, if ε 1 , ε 2 are Killing spinors, g ( V ε 1 , ε 2 , X ) = ( ε 1 , X · ε 2 ) V := V ε is causal : g ( V , V ) � 0 V is Killing: L V g = 0 L V F = 0 Gauntlett+Pakis (2002) L V D = 0 ε ′ Killing spinor = ⇒ so is L V ε ′ = ∇ V ε ′ − ρ ( ∇ V ) ε ′ L V ε = 0 JMF+Meessen+Philip (2004) Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 8 / 26

  26. The Killing superalgebra This turns the vector space g = g 0 ⊕ g 1 , where Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 9 / 26

  27. The Killing superalgebra This turns the vector space g = g 0 ⊕ g 1 , where g 0 is the space of F -preserving Killing vector fields, and Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 9 / 26

  28. The Killing superalgebra This turns the vector space g = g 0 ⊕ g 1 , where g 0 is the space of F -preserving Killing vector fields, and g 1 is the space of Killing spinors Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 9 / 26

  29. The Killing superalgebra This turns the vector space g = g 0 ⊕ g 1 , where g 0 is the space of F -preserving Killing vector fields, and g 1 is the space of Killing spinors into a Lie superalgebra JMF+Meessen+Philip (2004) Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 9 / 26

  30. The Killing superalgebra This turns the vector space g = g 0 ⊕ g 1 , where g 0 is the space of F -preserving Killing vector fields, and g 1 is the space of Killing spinors into a Lie superalgebra JMF+Meessen+Philip (2004) It is called the symmetry superalgebra of the supersymmetric background ( M , g , F ) Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 9 / 26

  31. The Killing superalgebra This turns the vector space g = g 0 ⊕ g 1 , where g 0 is the space of F -preserving Killing vector fields, and g 1 is the space of Killing spinors into a Lie superalgebra JMF+Meessen+Philip (2004) It is called the symmetry superalgebra of the supersymmetric background ( M , g , F ) The ideal k = [g 1 , g 1 ] ⊕ g 1 generated by g 1 is called the Killing superalgebra Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 9 / 26

  32. The Killing superalgebra This turns the vector space g = g 0 ⊕ g 1 , where g 0 is the space of F -preserving Killing vector fields, and g 1 is the space of Killing spinors into a Lie superalgebra JMF+Meessen+Philip (2004) It is called the symmetry superalgebra of the supersymmetric background ( M , g , F ) The ideal k = [g 1 , g 1 ] ⊕ g 1 generated by g 1 is called the Killing superalgebra It behaves as expected: it deforms under geometric limits (e.g., Penrose) and it embeds under asymptotic limits. Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 9 / 26

  33. The Killing superalgebra This turns the vector space g = g 0 ⊕ g 1 , where g 0 is the space of F -preserving Killing vector fields, and g 1 is the space of Killing spinors into a Lie superalgebra JMF+Meessen+Philip (2004) It is called the symmetry superalgebra of the supersymmetric background ( M , g , F ) The ideal k = [g 1 , g 1 ] ⊕ g 1 generated by g 1 is called the Killing superalgebra It behaves as expected: it deforms under geometric limits (e.g., Penrose) and it embeds under asymptotic limits. It is a very useful invariant of a supersymmetric supergravity background Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 9 / 26

  34. A crash course on homogeneous geometry “manifold”: smooth, connected, finite-dimensional Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 10 / 26

  35. A crash course on homogeneous geometry “manifold”: smooth, connected, finite-dimensional “Lie group”: finite-dimensional with identity 1 Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 10 / 26

  36. A crash course on homogeneous geometry “manifold”: smooth, connected, finite-dimensional “Lie group”: finite-dimensional with identity 1 G acts on M (on the left) via G × M → M , sending ( γ , p ) �→ γ · p Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 10 / 26

  37. A crash course on homogeneous geometry “manifold”: smooth, connected, finite-dimensional “Lie group”: finite-dimensional with identity 1 G acts on M (on the left) via G × M → M , sending ( γ , p ) �→ γ · p actions are effective: γ · p = p , ∀ p = ⇒ γ = 1 Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 10 / 26

  38. A crash course on homogeneous geometry “manifold”: smooth, connected, finite-dimensional “Lie group”: finite-dimensional with identity 1 G acts on M (on the left) via G × M → M , sending ( γ , p ) �→ γ · p actions are effective: γ · p = p , ∀ p = ⇒ γ = 1 M is homogeneous (under G ) if either Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 10 / 26

  39. A crash course on homogeneous geometry “manifold”: smooth, connected, finite-dimensional “Lie group”: finite-dimensional with identity 1 G acts on M (on the left) via G × M → M , sending ( γ , p ) �→ γ · p actions are effective: γ · p = p , ∀ p = ⇒ γ = 1 M is homogeneous (under G ) if either G acts transitively: i.e., there is only one orbit; or 1 Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 10 / 26

  40. A crash course on homogeneous geometry “manifold”: smooth, connected, finite-dimensional “Lie group”: finite-dimensional with identity 1 G acts on M (on the left) via G × M → M , sending ( γ , p ) �→ γ · p actions are effective: γ · p = p , ∀ p = ⇒ γ = 1 M is homogeneous (under G ) if either G acts transitively: i.e., there is only one orbit; or 1 for every p ∈ M , G → M sending γ �→ γ · p is surjective 2 Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 10 / 26

  41. A crash course on homogeneous geometry “manifold”: smooth, connected, finite-dimensional “Lie group”: finite-dimensional with identity 1 G acts on M (on the left) via G × M → M , sending ( γ , p ) �→ γ · p actions are effective: γ · p = p , ∀ p = ⇒ γ = 1 M is homogeneous (under G ) if either G acts transitively: i.e., there is only one orbit; or 1 for every p ∈ M , G → M sending γ �→ γ · p is surjective 2 given p , p ′ ∈ M , ∃ γ ∈ G with γ · p = p ′ 3 Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 10 / 26

  42. A crash course on homogeneous geometry “manifold”: smooth, connected, finite-dimensional “Lie group”: finite-dimensional with identity 1 G acts on M (on the left) via G × M → M , sending ( γ , p ) �→ γ · p actions are effective: γ · p = p , ∀ p = ⇒ γ = 1 M is homogeneous (under G ) if either G acts transitively: i.e., there is only one orbit; or 1 for every p ∈ M , G → M sending γ �→ γ · p is surjective 2 given p , p ′ ∈ M , ∃ γ ∈ G with γ · p = p ′ 3 γ defined up to right multiplication by the stabiliser of p : H = { γ ∈ G | γ · p = p } , a closed subgroup of G Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 10 / 26

  43. A crash course on homogeneous geometry “manifold”: smooth, connected, finite-dimensional “Lie group”: finite-dimensional with identity 1 G acts on M (on the left) via G × M → M , sending ( γ , p ) �→ γ · p actions are effective: γ · p = p , ∀ p = ⇒ γ = 1 M is homogeneous (under G ) if either G acts transitively: i.e., there is only one orbit; or 1 for every p ∈ M , G → M sending γ �→ γ · p is surjective 2 given p , p ′ ∈ M , ∃ γ ∈ G with γ · p = p ′ 3 γ defined up to right multiplication by the stabiliser of p : H = { γ ∈ G | γ · p = p } , a closed subgroup of G M ∼ = G/H , hence M is a coset manifold Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 10 / 26

  44. A crash course on homogeneous geometry “manifold”: smooth, connected, finite-dimensional “Lie group”: finite-dimensional with identity 1 G acts on M (on the left) via G × M → M , sending ( γ , p ) �→ γ · p actions are effective: γ · p = p , ∀ p = ⇒ γ = 1 M is homogeneous (under G ) if either G acts transitively: i.e., there is only one orbit; or 1 for every p ∈ M , G → M sending γ �→ γ · p is surjective 2 given p , p ′ ∈ M , ∃ γ ∈ G with γ · p = p ′ 3 γ defined up to right multiplication by the stabiliser of p : H = { γ ∈ G | γ · p = p } , a closed subgroup of G M ∼ = G/H , hence M is a coset manifold H → G is a principal H -bundle ↓ M Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 10 / 26

  45. Homogeneous supergravity backgrounds A diffeomorphism ϕ : M → M is an automorphism of a supergravity background ( M , g , F ) if ϕ ∗ g = g and ϕ ∗ F = F Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 11 / 26

  46. Homogeneous supergravity backgrounds A diffeomorphism ϕ : M → M is an automorphism of a supergravity background ( M , g , F ) if ϕ ∗ g = g and ϕ ∗ F = F Automorphisms form a Lie group G = Aut ( M , g , F ) Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 11 / 26

  47. Homogeneous supergravity backgrounds A diffeomorphism ϕ : M → M is an automorphism of a supergravity background ( M , g , F ) if ϕ ∗ g = g and ϕ ∗ F = F Automorphisms form a Lie group G = Aut ( M , g , F ) A background ( M , g , F ) is said to be homogeneous if G acts transitively on M Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 11 / 26

  48. Homogeneous supergravity backgrounds A diffeomorphism ϕ : M → M is an automorphism of a supergravity background ( M , g , F ) if ϕ ∗ g = g and ϕ ∗ F = F Automorphisms form a Lie group G = Aut ( M , g , F ) A background ( M , g , F ) is said to be homogeneous if G acts transitively on M Let g denote the Lie algebra of G : it consists of vector fields X ∈ X ( M ) such that L X g = 0 and L X F = 0 Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 11 / 26

  49. Homogeneous supergravity backgrounds A diffeomorphism ϕ : M → M is an automorphism of a supergravity background ( M , g , F ) if ϕ ∗ g = g and ϕ ∗ F = F Automorphisms form a Lie group G = Aut ( M , g , F ) A background ( M , g , F ) is said to be homogeneous if G acts transitively on M Let g denote the Lie algebra of G : it consists of vector fields X ∈ X ( M ) such that L X g = 0 and L X F = 0 ( M , g , F ) homogeneous = ⇒ the evaluation map ev p : g → T p M are surjective Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 11 / 26

  50. Homogeneous supergravity backgrounds A diffeomorphism ϕ : M → M is an automorphism of a supergravity background ( M , g , F ) if ϕ ∗ g = g and ϕ ∗ F = F Automorphisms form a Lie group G = Aut ( M , g , F ) A background ( M , g , F ) is said to be homogeneous if G acts transitively on M Let g denote the Lie algebra of G : it consists of vector fields X ∈ X ( M ) such that L X g = 0 and L X F = 0 ( M , g , F ) homogeneous = ⇒ the evaluation map ev p : g → T p M are surjective The converse is not true in general: if ev p are surjective, then ( M , g , F ) is locally homogeneous Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 11 / 26

  51. Homogeneous supergravity backgrounds A diffeomorphism ϕ : M → M is an automorphism of a supergravity background ( M , g , F ) if ϕ ∗ g = g and ϕ ∗ F = F Automorphisms form a Lie group G = Aut ( M , g , F ) A background ( M , g , F ) is said to be homogeneous if G acts transitively on M Let g denote the Lie algebra of G : it consists of vector fields X ∈ X ( M ) such that L X g = 0 and L X F = 0 ( M , g , F ) homogeneous = ⇒ the evaluation map ev p : g → T p M are surjective The converse is not true in general: if ev p are surjective, then ( M , g , F ) is locally homogeneous This is the “right” working notion in supergravity Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 11 / 26

  52. The homogeneity theorem Empirical Fact Every known ν -BPS background with ν > 1 2 is homogeneous. Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 12 / 26

  53. The homogeneity theorem Homogeneity conjecture known ν -BPS background with ν > 1 Every //////// 2 is homogeneous. Meessen (2004) Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 12 / 26

  54. The homogeneity theorem Homogeneity conjecture known ν -BPS background with ν > 1 Every //////// 2 is homogeneous. Meessen (2004) Theorem Every ν -BPS background of eleven-dimensional supergravity with ν > 1 2 is locally homogeneous. JMF+Meessen+Philip (2004), JMF+Hustler (2012) Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 12 / 26

  55. The homogeneity theorem Homogeneity conjecture known ν -BPS background with ν > 1 Every //////// 2 is homogeneous. Meessen (2004) Theorem Every ν -BPS background of eleven-dimensional supergravity with ν > 1 2 is locally homogeneous. JMF+Meessen+Philip (2004), JMF+Hustler (2012) In fact, vector fields in the Killing superalgebra already span the tangent spaces to every point of M Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 12 / 26

  56. Proof We fix p ∈ M and show ev p : k 0 → T p M is surjective Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 13 / 26

  57. Proof We fix p ∈ M and show ev p : k 0 → T p M is surjective Assume, for a contradiction, ∃ 0 � = X ∈ T p M such that X ⊥ V ε 1 , ε 2 for all ε 1,2 ∈ g 1 Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 13 / 26

  58. Proof We fix p ∈ M and show ev p : k 0 → T p M is surjective Assume, for a contradiction, ∃ 0 � = X ∈ T p M such that X ⊥ V ε 1 , ε 2 for all ε 1,2 ∈ g 1 0 = g ( V ε 1 , ε 2 , X ) = ( X · ε 1 , ε 2 ) Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 13 / 26

  59. Proof We fix p ∈ M and show ev p : k 0 → T p M is surjective Assume, for a contradiction, ∃ 0 � = X ∈ T p M such that X ⊥ V ε 1 , ε 2 for all ε 1,2 ∈ g 1 0 = g ( V ε 1 , ε 2 , X ) = ( X · ε 1 , ε 2 ) X · : g 1 → g ⊥ 1 Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 13 / 26

  60. Proof We fix p ∈ M and show ev p : k 0 → T p M is surjective Assume, for a contradiction, ∃ 0 � = X ∈ T p M such that X ⊥ V ε 1 , ε 2 for all ε 1,2 ∈ g 1 0 = g ( V ε 1 , ε 2 , X ) = ( X · ε 1 , ε 2 ) X · : g 1 → g ⊥ 1 ⇒ dim g ⊥ dim g 1 > 16 = 1 < 16, so ker X · � = 0 Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 13 / 26

  61. Proof We fix p ∈ M and show ev p : k 0 → T p M is surjective Assume, for a contradiction, ∃ 0 � = X ∈ T p M such that X ⊥ V ε 1 , ε 2 for all ε 1,2 ∈ g 1 0 = g ( V ε 1 , ε 2 , X ) = ( X · ε 1 , ε 2 ) X · : g 1 → g ⊥ 1 ⇒ dim g ⊥ dim g 1 > 16 = 1 < 16, so ker X · � = 0 ( X · ) 2 = − g ( X , X ) = ⇒ X is null Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 13 / 26

  62. Proof We fix p ∈ M and show ev p : k 0 → T p M is surjective Assume, for a contradiction, ∃ 0 � = X ∈ T p M such that X ⊥ V ε 1 , ε 2 for all ε 1,2 ∈ g 1 0 = g ( V ε 1 , ε 2 , X ) = ( X · ε 1 , ε 2 ) X · : g 1 → g ⊥ 1 ⇒ dim g ⊥ dim g 1 > 16 = 1 < 16, so ker X · � = 0 ( X · ) 2 = − g ( X , X ) = ⇒ X is null dim ( ev p (k 0 )) ⊥ = 1 Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 13 / 26

  63. Proof We fix p ∈ M and show ev p : k 0 → T p M is surjective Assume, for a contradiction, ∃ 0 � = X ∈ T p M such that X ⊥ V ε 1 , ε 2 for all ε 1,2 ∈ g 1 0 = g ( V ε 1 , ε 2 , X ) = ( X · ε 1 , ε 2 ) X · : g 1 → g ⊥ 1 ⇒ dim g ⊥ dim g 1 > 16 = 1 < 16, so ker X · � = 0 ( X · ) 2 = − g ( X , X ) = ⇒ X is null dim ( ev p (k 0 )) ⊥ = 1 V ε ⊥ X = ⇒ V ε = λ ( ε ) X for some λ : g 1 → R Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 13 / 26

  64. Proof We fix p ∈ M and show ev p : k 0 → T p M is surjective Assume, for a contradiction, ∃ 0 � = X ∈ T p M such that X ⊥ V ε 1 , ε 2 for all ε 1,2 ∈ g 1 0 = g ( V ε 1 , ε 2 , X ) = ( X · ε 1 , ε 2 ) X · : g 1 → g ⊥ 1 ⇒ dim g ⊥ dim g 1 > 16 = 1 < 16, so ker X · � = 0 ( X · ) 2 = − g ( X , X ) = ⇒ X is null dim ( ev p (k 0 )) ⊥ = 1 V ε ⊥ X = ⇒ V ε = λ ( ε ) X for some λ : g 1 → R V ε 1 , ε 2 = 1 2 ( V ε 1 + ε 2 − V ε 1 − V ε 2 ) = 1 2 ( λ ( ε 1 + ε 2 )− λ ( ε 1 )− λ ( ε 2 )) X Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 13 / 26

  65. Proof We fix p ∈ M and show ev p : k 0 → T p M is surjective Assume, for a contradiction, ∃ 0 � = X ∈ T p M such that X ⊥ V ε 1 , ε 2 for all ε 1,2 ∈ g 1 0 = g ( V ε 1 , ε 2 , X ) = ( X · ε 1 , ε 2 ) X · : g 1 → g ⊥ 1 ⇒ dim g ⊥ dim g 1 > 16 = 1 < 16, so ker X · � = 0 ( X · ) 2 = − g ( X , X ) = ⇒ X is null dim ( ev p (k 0 )) ⊥ = 1 V ε ⊥ X = ⇒ V ε = λ ( ε ) X for some λ : g 1 → R V ε 1 , ε 2 = 1 2 ( V ε 1 + ε 2 − V ε 1 − V ε 2 ) = 1 2 ( λ ( ε 1 + ε 2 )− λ ( ε 1 )− λ ( ε 2 )) X dim ev p (k 0 ) = 1 ⇒⇐ Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 13 / 26

  66. Generalisations Theorem Every ν -BPS background of type IIB supergravity with ν > 1 2 is homogeneous. Every ν -BPS background of type I and heterotic supergravities with ν > 1 2 is homogeneous. JMF+Hackett-Jones+Moutsopoulos (2007) JMF+Hustler (2012) Every ν -BPS background of six-dimensional ( 1, 0 ) and ( 2, 0 ) supergravities with ν > 1 2 is homogeneous. JMF + Hustler (2013) Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 14 / 26

  67. Generalisations Theorem Every ν -BPS background of type IIB supergravity with ν > 1 2 is homogeneous. Every ν -BPS background of type I and heterotic supergravities with ν > 1 2 is homogeneous. JMF+Hackett-Jones+Moutsopoulos (2007) JMF+Hustler (2012) Every ν -BPS background of six-dimensional ( 1, 0 ) and ( 2, 0 ) supergravities with ν > 1 2 is homogeneous. JMF + Hustler (2013) The theorems actually prove the strong version of the conjecture: that the symmetries which are generated from the supersymmetries already act (locally) transitively. Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 14 / 26

  68. Idea of proof The proof consists of two steps: Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 15 / 26

  69. Idea of proof The proof consists of two steps: One shows the existence of the Killing superalgebra 1 k = k 0 ⊕ k 1 Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 15 / 26

  70. Idea of proof The proof consists of two steps: One shows the existence of the Killing superalgebra 1 k = k 0 ⊕ k 1 One shows that for all p ∈ M , ev p : k 0 → T p M is surjective 2 whenever dim k 1 > 1 2 rank S Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 15 / 26

  71. Idea of proof The proof consists of two steps: One shows the existence of the Killing superalgebra 1 k = k 0 ⊕ k 1 One shows that for all p ∈ M , ev p : k 0 → T p M is surjective 2 whenever dim k 1 > 1 2 rank S This actually only shows local homogeneity. Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 15 / 26

  72. What good is it? The homogeneity theorem implies that classifying homogeneous supergravity backgrounds also classifies ν -BPS backgrounds for ν > 1 2 . Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 16 / 26

  73. What good is it? The homogeneity theorem implies that classifying homogeneous supergravity backgrounds also classifies ν -BPS backgrounds for ν > 1 2 . This is good because the supergravity field equations for homogeneous backgrounds are algebraic and hence simpler to solve than PDEs Jos´ e Figueroa-O’Farrill Homogeneous supergravity backgrounds 16 / 26

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