NUT Black Holes in N = 2 Gauged Supergravity Harold Erbin Lpthe , Université Paris 6–Jussieu (France) 2015 Based on 1410.2602 , 1411.2909 , 1501.02188 , 1503.04686 Collaborations with Nick Halmagyi ( Lpthe ) and Lucien Heurtier ( Cpht , École Polytechnique) 1 / 35
Outline Introduction BPS equations Demiański–Janis–Newman algorithm Conclusion 2 / 35
Outline: 1. Introduction Introduction BPS equations Demiański–Janis–Newman algorithm Conclusion 3 / 35
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 a ◮ electric charge q ◮ acceleration α 4 / 35
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 a ◮ electric charge q ◮ acceleration α ◮ natural pairing as complex parameters m + in , q + ip , a + i α 4 / 35
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 a ◮ electric charge q ◮ acceleration α ◮ natural pairing as complex parameters m + in , q + ip , a + i α ◮ BPS branches [hep-th/9203018, Romans] [hep-th/9512222, Kostelecky–Perry] [hep-th/9808097, Caldarelli–Klemm] [hep-th/0003071, Alonso-Alberca–Meessen–Ortín] [1303.3119, Klemm–Nozawa] 4 / 35
Motivations Black holes ◮ sandbox for quantum gravity ◮ understand microstates from string theory ◮ adS/CFT correspondence 5 / 35
Motivations Black holes ◮ sandbox for quantum gravity ◮ understand microstates from string theory ◮ adS/CFT correspondence NUT charge for AdS/CFT ◮ gauge dual: Chern–Simons on Lens spaces S 3 / Z n [1212.4618, Martelli–Passias–Sparks] ◮ fluid/gravity: NUT charge → vorticity [1206.4351, Caldarelli et al.] 5 / 35
Roadmap Goals ◮ understand asymptotically adS 4 black holes ◮ Plebański–Demiański in N = 2 gauged supergravity with vector- and hypermultiplets 6 / 35
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 6 / 35
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 This talk: focus on NUT charge (plus mass and dyonic), no hypermultiplet 6 / 35
Fields of N = 2 supergravity ◮ Gravity multiplet and n v vector multiplets α = 1 , 2 µ , A 0 { g µν , ψ α { A i µ , λ α i , τ i } , µ } , i = 1 , . . . , n v 7 / 35
Fields of N = 2 supergravity ◮ Gravity multiplet and n v vector multiplets α = 1 , 2 µ , A 0 { g µν , ψ α { A i µ , λ α i , τ i } , µ } , i = 1 , . . . , n v ◮ Lagrangian with Fayet–Iliopoulos gaugings 2 + 1 µν F Σ µν − 1 ε µνρσ L bos = R 4 Im N ( τ ) ΛΣ F Λ √− g Re N ( τ ) ΛΣ F Λ µν F Σ ρσ 8 − V ( τ ) τ ¯ ( τ ) ∂ µ τ i ∂ µ ¯ − g i ¯ Scalars: non-linear sigma model on special Kähler manifold (prepotential F → Kähler potential K ) 7 / 35
Fields of N = 2 supergravity ◮ Gravity multiplet and n v vector multiplets α = 1 , 2 µ , A 0 { g µν , ψ α { A i µ , λ α i , τ i } , µ } , i = 1 , . . . , n v ◮ Lagrangian with Fayet–Iliopoulos gaugings 2 + 1 µν F Σ µν − 1 ε µνρσ L bos = R 4 Im N ( τ ) ΛΣ F Λ √− g Re N ( τ ) ΛΣ F Λ µν F Σ ρσ 8 − V ( τ ) τ ¯ ( τ ) ∂ µ τ i ∂ µ ¯ − g i ¯ Scalars: non-linear sigma model on special Kähler manifold (prepotential F → Kähler potential K ) ◮ 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 7 / 35
Symplectic covariance ◮ Field strength and Maxwell equations � � F Λ F = d F = 0 , G Λ Maxwell equations invariant under Sp (2 n v + 2 , R ) 8 / 35
Symplectic covariance ◮ Field strength and Maxwell equations � � F Λ F = d F = 0 , G Λ Maxwell equations invariant under Sp (2 n v + 2 , R ) ◮ Section � � τ i = L i L Λ V = , L 0 , M Λ ◮ Maxwell charges � � � 1 p Λ � Q = F = Vol Σ q Λ Σ ◮ Fayet–Iliopoulos gaugings � � g Λ G = g Λ electric/magnetic charges of ψ α µ under U (1) ⊂ SU (2) R ◮ covariant formalism for BPS equation [1012.3756, Dall’Agata–Gnecchi] 8 / 35
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 ′ 9 / 35
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] [0902.3973, Cerchiai et al.] ∂ i I 4 ( A ) = 0 9 / 35
Quartic function: some identities Arbitrary space I 4 (Re V ) = I 4 (Im V ) = 1 I 4 ( V ) = 0 , 16 Re V = 2 I ′ 4 (Im V ) Symmetric space 4 ( I ′ 4 ( A ) , A , A ) = − 8 I 4 ( A ) A I ′ I ′ 4 ( I ′ 4 ( A ) , I ′ 4 ( A ) , A ) = 8 I 4 ( A ) I ′ 4 ( A ) 4 ( A )) = − 16 I 4 ( A ) 2 A I ′ 4 ( I ′ 10 / 35
Quartic invariants ◮ cubic prepotential (symmetric very special Kähler manifold) F = − D ijk τ i τ j τ k quartic invariant I 4 ( Q ) = − ( q Λ p Λ ) 2 + 1 16 p 0 ˆ D ijk q i q j q k − 4 q 0 D ijk p i p j p k + 9 ˆ D ijk D k ℓ m q i q j p ℓ p m 16 11 / 35
Quartic invariants ◮ cubic prepotential (symmetric very special Kähler manifold) F = − D ijk τ i τ j τ k quartic invariant I 4 ( Q ) = − ( q Λ p Λ ) 2 + 1 16 p 0 ˆ D ijk q i q j q k − 4 q 0 D ijk p i p j p k + 9 ˆ D ijk D k ℓ m q i q j p ℓ p m 16 ◮ quadratic prepotential F = i 2 η ΛΣ X Λ X Σ quartic invariant � i � 2 2 η ΛΣ p Λ p Σ + i 2 η ΛΣ q Λ q Σ I 4 ( Q ) = 11 / 35
Outline: 2. BPS equations Introduction BPS equations Demiański–Janis–Newman algorithm Conclusion 12 / 35
Ansatz AdS–NUT dyonic black hole d s 2 = − e 2 U � d t + 2 n H ( θ ) d φ � 2 + e − 2 U d r 2 + e 2( V − U ) d Σ 2 g � + ˜ � d t + 2 n H ( θ ) d φ A Λ = ˜ q Λ ( r ) p Λ ( r ) H ( θ ) d φ τ i = τ i ( r ) U = U ( r ), V = V ( r ); Riemann surface Σ g of genus g − cos θ κ = 1 g = d θ 2 + H ′ ( θ ) 2 d φ 2 , d Σ 2 H ( θ ) = κ = 0 θ cosh θ κ = − 1 with curvature κ = sign(1 − g ) NUT charge: preserves SO (3) isometry 13 / 35
Root structure In general, e 2 V : quartic polynomial e 2 V = v 0 + v 1 r + v 2 r 2 + v 3 r 3 + v 4 r 4 ◮ naked singularity: pair of complex conjugate roots, v 3 = 0 → no horizon ◮ black hole: two real roots, v 0 = 0 → horizon and finite temperature ◮ extremal black hole: real double root, v 0 = v 1 = 0 → two coincident horizons, vanishing temperature, near-horizon adS 2 × Σ g ◮ double extremal black hole: pair of real double roots, v 0 = v 1 = 0 and v 3 = √ v 2 v 4 ◮ ultracold black hole: real triple root, v 0 = v 1 = v 2 = 0 [hep-th/9203018, Romans] 14 / 35
Root structure In general, e 2 V : quartic polynomial e 2 V = v 0 + v 1 r + v 2 r 2 + v 3 r 3 + v 4 r 4 ◮ naked singularity: pair of complex conjugate roots, v 3 = 0 → no horizon ◮ extremal black hole: real double root, v 0 = v 1 = 0 → two coincident horizons, vanishing temperature, near-horizon adS 2 × Σ g ◮ double extremal black hole: pair of real double roots, v 0 = v 1 = 0 and v 3 = √ v 2 v 4 14 / 35
Constant scalar black hole Minimal gauged supergravity ( n v = 0 , Λ = − 3 g 2 ) e 2 V = g 2 ( r 2 + n 2 ) 2 + ( κ + 4 g 2 n 2 )( r 2 − n 2 ) − 2 mr + P 2 + Q 2 q = Qr − nP e 2( V − U ) = r 2 + n 2 , ˜ ˜ p = P r 2 + n 2 , 15 / 35
Constant scalar black hole Minimal gauged supergravity ( n v = 0 , Λ = − 3 g 2 ) e 2 V = g 2 ( r 2 + n 2 ) 2 + ( κ + 4 g 2 n 2 )( r 2 − n 2 ) − 2 mr + P 2 + Q 2 q = Qr − nP e 2( V − U ) = r 2 + n 2 , ˜ ˜ p = P r 2 + n 2 , 1 / 4-BPS conditions [hep-th/0003071, Alonso-Alberca–Meessen–Ortín] P = ± κ + 4 g 2 n 2 m = | 2 gnQ | , 2 g Two pairs of complex conjugate roots 15 / 35
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