PhD defense Black holes in N = 2 supergravity Harold Erbin LPTHE, - PowerPoint PPT Presentation

PhD defense Black holes in N = 2 supergravity Harold Erbin LPTHE, Université Pierre et Marie Curie (France) Wednesday 23rd September 2015 1 / 36

Papers ◮ H. Erbin and N. Halmagyi. “Abelian Hypermultiplet Gaugings and BPS Vacua in N = 2 Supergravity”. JHEP 2015.5 (May 2015), 1409.6310. ◮ H. Erbin and N. Halmagyi. “Quarter-BPS Black Holes in AdS 4 -NUT from N = 2 Gauged Supergravity”. Accepted in JHEP (Mar. 2015), 1503.04686. ◮ H. Erbin. “Janis-Newman algorithm: simplifications and gauge field transformation”. General Relativity and Gravitation 47.3 (Mar. 2015), 1410.2602. ◮ H. Erbin and L. Heurtier. “Five-dimensional Janis-Newman algorithm”. Classical and Quantum Gravity 32.16 (July 2015), p. 165004, 1411.2030. ◮ H. Erbin. “Deciphering and generalizing Demiański-Janis-Newman algorithm”. Submitted to Classical and Quantum Gravity (Nov. 2014), 1411.2909 ◮ H. Erbin and L. Heurtier. “Supergravity, complex parameters and the Janis-Newman algorithm”. Classical and Quantum Gravity 32.16 (July 2015), p. 165005, 1501.02188. 2 / 36

Outline Introduction Motivations Supergravity and BPS solutions Demiański–Janis–Newman algorithm Conclusion 3 / 36

Outline: 1. Introduction Introduction Motivations Supergravity and BPS solutions Demiański–Janis–Newman algorithm Conclusion 4 / 36

Modèle standard et relativité générale Modèle standard : ◮ interactions entre particules élémentaires ◮ trois forces (électromagnétisme, faible, forte) ◮ théorie quantique 5 / 36

Modèle standard et relativité générale Modèle standard : ◮ interactions entre particules élémentaires ◮ trois forces (électromagnétisme, faible, forte) ◮ théorie quantique Relativité générale ◮ force gravitationnelle = déformation de l’espace-temps ◮ nécessaire si vitesse/gravité élevées ◮ théorie classique 5 / 36

Modèle standard et relativité générale Modèle standard : ◮ interactions entre particules élémentaires ◮ trois forces (électromagnétisme, faible, forte) ◮ théorie quantique Relativité générale ◮ force gravitationnelle = déformation de l’espace-temps ◮ nécessaire si vitesse/gravité élevées ◮ théorie classique Objectifs de la physique moderne : ◮ quantifier la gravité ◮ décrire ensemble le modèle standard et la gravité → théorie des cordes 5 / 36

Supersymétrie Deux types de particules : ◮ les bosons : transmettent les forces (e.g. le photon) ◮ les fermions : constituent la matière (e.g. l’électron) 6 / 36

Supersymétrie Deux types de particules : ◮ les bosons : transmettent les forces (e.g. le photon) ◮ les fermions : constituent la matière (e.g. l’électron) Supersymétrie Q susy | boson � = | fermion � , Q susy | fermion � = | boson � 6 / 36

Supersymétrie Deux types de particules : ◮ les bosons : transmettent les forces (e.g. le photon) ◮ les fermions : constituent la matière (e.g. l’électron) Supersymétrie Q susy | boson � = | fermion � , Q susy | fermion � = | boson � Supergravité relativité générale + supersymétrie ◮ limite de la théorie des cordes ◮ unification interactions/gravité ◮ meilleur comportement quantique N : nombre de Q susy différents Choix : N = 2 (compromis liberté/simplicité) 6 / 36

Trous noirs ◮ champ gravitationnel extrême ◮ horizon : limite au-delà de laquelle rien ne peut s’échapper ◮ centre = singularité (gravité infinie) ◮ description complète : nécessite une gravité quantique 7 / 36

Trous noirs ◮ champ gravitationnel extrême ◮ horizon : limite au-delà de laquelle rien ne peut s’échapper ◮ centre = singularité (gravité infinie) ◮ description complète : nécessite une gravité quantique ◮ bac à sable pour tester les théories de gravité quantique ◮ peu de paramètres : ressemble à une particule 7 / 36

Outline: 2. Motivations Introduction Motivations Supergravity and BPS solutions Demiański–Janis–Newman algorithm Conclusion 8 / 36

Plebański–Demiański solution (’76) Most general black hole solution [Plebański–Demiański ’76] ◮ Einstein–Maxwell theory with cosmological constant Λ ◮ equivalently pure N = 2 gauged supergravity ◮ 6 parameters ◮ mass m ◮ magnetic charge p ◮ NUT charge n ◮ rotation j ◮ electric charge q ◮ acceleration a ◮ natural pairing as complex parameters m + in , q + ip , j + ia 9 / 36

Motivations (AdS) black holes ◮ sandbox for quantum gravity ◮ understand microstates from string theory ◮ adS/CFT correspondence 10 / 36

Motivations (AdS) black holes ◮ sandbox for quantum gravity ◮ understand microstates from string theory ◮ adS/CFT correspondence Black hole: interpolation magnetic adS (UV) → near-horizon geometry (IR) AdS 4 and near-horizon geometry → supergravity solutions 10 / 36

Motivations (AdS) black holes ◮ sandbox for quantum gravity ◮ understand microstates from string theory ◮ adS/CFT correspondence Black hole: interpolation magnetic adS (UV) → near-horizon geometry (IR) AdS 4 and near-horizon geometry → supergravity solutions 10 / 36

Roadmap Goals ◮ understand asymptotically adS 4 black holes ◮ Plebański–Demiański in N = 2 gauged supergravity with vector- and hypermultiplets 11 / 36

Roadmap Goals ◮ understand asymptotically adS 4 black holes ◮ Plebański–Demiański in N = 2 gauged supergravity with vector- and hypermultiplets Two strategies ◮ study simpler solution classes → BPS equations ◮ use a solution generating technique → Janis–Newman algorithm 11 / 36

BPS equations ◮ BPS equations fermions = 0 , δ Q (fermions) = 0 ◮ background preserves part of supersymmetry ◮ first order differential equations on bosonic fields ◮ imply (most of) the equations of motion N = 2: give Einstein and scalar equations, but not Maxwell [1005.3650, Hristov–Looyestijn–Vandoren] 12 / 36

Outline: 3. Supergravity and BPS solutions Introduction Motivations Supergravity and BPS solutions Demiański–Janis–Newman algorithm Conclusion 13 / 36

N = 2 supergravity Algebra � Q β � Q α , ¯ ∼ δ β α P , [ J , Q α ] ∼ γ · Q α , [ R , Q α ] ∼ U β { Q α , Q β } ∼ ε αβ Z , α Q β P momentum, Z central charge, J angular momentum automorphism U , R-symmetry U (2) R 14 / 36

N = 2 supergravity Algebra � Q β � Q α , ¯ ∼ δ β α P , [ J , Q α ] ∼ γ · Q α , [ R , Q α ] ∼ U β { Q α , Q β } ∼ ε αβ Z , α Q β P momentum, Z central charge, J angular momentum automorphism U , R-symmetry U (2) R Field content ◮ gravity multiplet µ , A 0 { g µν , ψ α µ } , α = 1 , 2 ◮ n v vector multiplets { A i µ , λ α i , τ i } , i = 1 , . . . , n v ◮ n h hypermultiplets u = 1 , . . . , 4 n h , { ζ A , q u } , A = 1 , . . . , 2 n h 14 / 36

Bosonic Lagrangian ε µνρσ L bos = R 2 + 1 µν F Σ µν − 1 4 Im N ( τ ) ΛΣ F Λ √− g Re N ( τ ) ΛΣ F Λ µν F Σ ρσ 8  − 1 2 h uv ( q ) D µ q u D µ q v − V ( τ, q ) τ ¯  ( τ ) ∂ µ τ i ∂ µ ¯ − g i ¯ Electric and magnetic field strengths F Λ = d A Λ , Λ = 0 , . . . , n v , � δ L bos � = Re N ΛΣ F Λ + Im N ΛΣ ⋆ F Λ G Λ = ⋆ δ F Λ 15 / 36

Scalar geometry Non-linear sigma model: scalar fields = coordinates on target space M = M v ( τ i ) × M h ( q u ) ◮ M v special Kähler manifold, dim R = 2 n v , U (1) bundle ◮ M h quaternionic manifold, dim R = 4 n h , SU (2) bundle Consequence of R-symmetry group U (2) R = SU (2) R × U (1) R 16 / 36

Gaugings Isometry group G (global symmetries) and local gauge group K G ≡ ISO ( M ) , K ⊂ G Here K = U (1) n v +1 , two simpler possibilities: ◮ Fayet–Iliopoulos (FI): n h = 0, ψ α µ charged under U (1) ⊂ SU (2) R ◮ quaternionic gauging: Killing vectors k u Λ Λ = θ A k u Λ k u A , [ k Λ , k Σ ] = 0 k u A generates iso ( M h ), θ A Λ gauging parameters A = 1 , . . . , dim ISO ( M h ) 17 / 36

Symplectic covariance ◮ Field strength and Maxwell–Bianchi equations � � F Λ F = , d F = 0 G Λ Maxwell–Bianchi equations invariant under Sp (2 n v + 2 , R ) 18 / 36

Symplectic covariance ◮ Field strength and Maxwell–Bianchi equations � � F Λ F = , d F = 0 G Λ Maxwell–Bianchi equations invariant under Sp (2 n v + 2 , R ) ◮ Section � � L Λ τ i = L i V = , L 0 , M Λ ◮ Maxwell charges � � � p Λ 1 Q = F = q Λ Vol Σ Σ ◮ Killing vectors, prepotentials and compensators � � � � k u Λ P x Λ K u = P x = K u ω x u + W x = , k u P x Λ Λ FI: P 3 = cst , EM charges of ψ α µ ◮ covariant formalism for BPS equation [1012.3756, Dall’Agata–Gnecchi] 18 / 36

Quartic function Symplectic vector A : order-4 homogeneous polynomial I 4 = I 4 ( A , τ i ) Define symmetric 4-tensor ∂ 4 I 4 ( A ) t MNPQ = ∂ A M ∂ A N ∂ A P ∂ A Q Different arguments and gradient I 4 ( A , B , C , D ) = t MNPQ A M B N C P D Q 4 ( A , B , C ) M = Ω MR t RNPQ A N B P C Q I ′ 19 / 36

Quartic function Symplectic vector A : order-4 homogeneous polynomial I 4 = I 4 ( A , τ i ) Define symmetric 4-tensor ∂ 4 I 4 ( A ) t MNPQ = ∂ A M ∂ A N ∂ A P ∂ A Q Different arguments and gradient I 4 ( A , B , C , D ) = t MNPQ A M B N C P D Q 4 ( A , B , C ) M = Ω MR t RNPQ A N B P C Q I ′ Quartic invariant Symmetric space [hep-th/9210068, de Wit–Vanderseypen–Van Proeyen] ∂ i I 4 ( A ) = 0 19 / 36