Abelian hypermultiplet gaugings and BPS vacua in N = 2 supergravity - PowerPoint PPT Presentation

Introduction N = 2 supergravity Kähler geometries Kähler isometries BPS solutions Conclusion Abelian hypermultiplet gaugings and BPS vacua in N = 2 supergravity Harold Erbin LPTHE, Université Paris 6 (France) October–November 2014 Based on [1409.6310, H. E.–Halmagyi] . 1 / 44

Introduction N = 2 supergravity Kähler geometries Kähler isometries BPS solutions Conclusion Outline Introduction 1 N = 2 supergravity 2 Kähler geometries 3 Kähler isometries 4 BPS solutions 5 Conclusion 6 2 / 44

Introduction N = 2 supergravity Kähler geometries Kähler isometries BPS solutions Conclusion Outline: 1. Introduction 1 Introduction 2 N = 2 supergravity 3 Kähler geometries Kähler isometries 4 BPS solutions 5 Conclusion 6 3 / 44

Introduction N = 2 supergravity Kähler geometries Kähler isometries BPS solutions Conclusion Motivations Asymptotically AdS 4 BPS black holes (BH) are very important: BH entropy computations → need near-horizon geometries String theory and M-theory embeddings AdS /CFT correspondence Black hole: interpolation magnetic AdS (UV) → near-horizon geometry (IR) AdS 4 and near-horizon geometry → supergravity solutions 4 / 44

Introduction N = 2 supergravity Kähler geometries Kähler isometries BPS solutions Conclusion BPS equations for AdS 4 vacua BPS equations reduces to 1 �K u , V� = 0 �P , V� ∝ , R AdS P moment map, K u Killing vectors, V symplectic section But u K u = P = ω 3 ⇒ �P , V� = 0 . → no regular solution [0911.2708, Cassani et al.][1204.3893, Louis et al.] Missing piece u K u + W P = ω 3 → need to understand (special and) quaternionic isometries [de Wit–van Proeyen ’90] [hep-th/9210068, de Wit–Vanderseypen–van Proeyen] [hep-th/9310067, de Wit–van Proeyen] 5 / 44

Introduction N = 2 supergravity Kähler geometries Kähler isometries BPS solutions Conclusion Outline: 2. N = 2 supergravity 1 Introduction 2 N = 2 supergravity 3 Kähler geometries Kähler isometries 4 BPS solutions 5 Conclusion 6 6 / 44

Introduction N = 2 supergravity Kähler geometries Kähler isometries BPS solutions Conclusion N = 2 supergravity: multiplets [hep-th/9605032, Andrianopoli et al.] [ Supergravity , Freedman–van Proeyen] Gravity multiplet { g µν , ψ α µ , A 0 µ } α = 1 , 2 n v vector multiplets { A i µ , λ α i , τ i } i = 1 , . . . , n v n h hypermultiplets { ζ A , q u } u = 1 , . . . , 4 n h , A = 1 , . . . , 2 n h 7 / 44

Introduction N = 2 supergravity Kähler geometries Kähler isometries BPS solutions Conclusion N = 2 supergravity: bosonic lagrangian L bos = R 2 + 1 µν F Σ µν − 1 4 Im N ( τ ) ΛΣ F Λ 8 Re N ( τ ) ΛΣ ε µνρσ F Λ µν F Σ ρσ  − 1  ( τ ) ∂ µ τ i ∂ µ ¯ τ ¯ 2 h uv ( q ) ∂ µ q u ∂ µ q v − g i ¯ Field strengths F Λ = d A Λ , A Λ = ( A 0 , A i ) Λ = 0 , . . . , n v 8 / 44

Introduction N = 2 supergravity Kähler geometries Kähler isometries BPS solutions Conclusion Electromagnetic duality Dual (magnetic) field strengths � δ L bos � = Re N ΛΣ F Λ + Im N ΛΣ ⋆ F Λ G Λ = ⋆ δ F Λ Maxwell equations and Bianchi identities d F Λ = 0 , d G Λ = 0 invariant under symplectic transformations � � � � F Λ F Λ − → U U ∈ Sp (2 n v + 2 , R ) , G Λ G Λ The action is not invariant 9 / 44

Introduction N = 2 supergravity Kähler geometries Kähler isometries BPS solutions Conclusion Scalar geometry Non-linear sigma model: scalar fields → coordinates on target space M = M v ( τ i ) × M h ( q u ) with M v special Kähler manifold, dim R = 2 n v M h quaternionic Kähler manifold, dim R = 4 n h Isometry group (global symmetries) G ≡ ISO ( M ) , G ⊂ Sp (2 n v + 2) [hep-th/9605032, Andrianopoli et al.] 10 / 44

Introduction N = 2 supergravity Kähler geometries Kähler isometries BPS solutions Conclusion Gaugings: general case Local gauge group H H ⊂ G encoded by Killing vectors { k i Λ , k u Λ } Covariant derivatives on scalars (minimal coupling) � � Λ ( τ ) ∂ Λ ( q ) ∂ → D µ = ∂ µ − A Λ k i ∂τ i + k u ∂ µ − µ ∂ q u Generates scalar potential V ( τ, q ) → AdS 4 vacua 11 / 44

Introduction N = 2 supergravity Kähler geometries Kähler isometries BPS solutions Conclusion Outline: 3. Kähler geometries 1 Introduction 2 N = 2 supergravity 3 Kähler geometries Kähler isometries 4 BPS solutions 5 Conclusion 6 12 / 44

Introduction N = 2 supergravity Kähler geometries Kähler isometries BPS solutions Conclusion Kähler manifold: definition Manifold ( M , g ) with Hermitian metric d s 2 = 2 g i ¯  d τ i d ¯ τ ¯  , i = 1 , . . . , n Complex structure J 2 = − 1 , J g J t = g Fundamental 2-form  d τ i ∧ d ¯ τ ¯  , vol = Ω n Ω = − 2 J i ¯ Kähler condition d Ω = 0 13 / 44

Introduction N = 2 supergravity Kähler geometries Kähler isometries BPS solutions Conclusion Kähler manifold: Kähler potential Metric given by Kähler potential K ( τ, ¯ τ ) ∂ g i ¯  = ∂ i ∂ ¯  K , ∂ i ≡ ∂τ i Metric invariant under Kähler transformations τ ) + f ( τ ) + ¯ τ ) − → K ( τ, ¯ K ( τ, ¯ f (¯ τ ) 14 / 44

Introduction N = 2 supergravity Kähler geometries Kähler isometries BPS solutions Conclusion Special Kähler manifold: definition Kähler manifold with bundle Sp (2 n v + 2 , R ) and section � � X Λ F Λ = ∂ F v = , ∂ X Λ F Λ (assuming F exists) Prepotential F F ( λ X ) = λ 2 F ( X ) Homogeneous coordinates X Λ , special coordinates τ i = X i X 0 (common choice: X 0 = 1) 15 / 44

Introduction N = 2 supergravity Kähler geometries Kähler isometries BPS solutions Conclusion Special Kähler manifold: symplectic structure Symplectic inner product � � 0 1 � A , B � = A t Ω B = A Λ B Λ − A Λ B Λ , Ω = − 1 0 Kähler potential � X Λ ¯ X Λ � e − K = − i � v , ¯ F Λ − F Λ ¯ v � = − i Covariant section � � X Λ K K 2 v = e V = e 2 F Λ Covariant Kähler derivative � � ∂ i + 1 U i ≡ D i V = 2 ∂ i K V 16 / 44

Introduction N = 2 supergravity Kähler geometries Kähler isometries BPS solutions Conclusion Special Kähler manifold: structure Gauge coupling matrix N (built from F ) F Λ = N ΛΣ X Σ Complex structure M on the bundle (built from N ) MV = − i V , M D i V = i D i V 17 / 44

Introduction N = 2 supergravity Kähler geometries Kähler isometries BPS solutions Conclusion Special Kähler manifold: prepotentials Usual examples and Calabi–Yau have: cubic X i X j X k F = − D ijk X 0 quadratic F = i 2 η ΛΣ X Λ X Σ η = diag( − 1 , 1 , . . . , 1) 18 / 44

Introduction N = 2 supergravity Kähler geometries Kähler isometries BPS solutions Conclusion Quaternionic manifold: definition Metric d s 2 = h uv d q u d q v , u = 1 , . . . , 4 n h holonomy SU (2) × Sp ( n h ) Complex structure triplet J x , x = 1 , 2 , 3 J x h ( J x ) t = h ∀ x : SU (2) algebra J x J y = − δ xy + ε xyz J z Hyperkähler forms uv d q u ∧ d q v K x = J x 19 / 44

Introduction N = 2 supergravity Kähler geometries Kähler isometries BPS solutions Conclusion Quaternionic manifold: SU (2) properties SU (2) connection ω x = ω x u d q u Curvature 2-form Ω x = ∇ ω x = d ω x + 1 2 ε xyz ω y ∧ ω z with Ω x = λ K x , λ ∈ R Supersymmetry → λ = − 1 Fundamental 4-form Ω = Ω x ∧ Ω x , vol = Ω n d Ω = 0 , 20 / 44

Introduction N = 2 supergravity Kähler geometries Kähler isometries BPS solutions Conclusion Quaternionic manifold: c-map construction � � 2 b + e 4 φ − e 2 φ d σ + 1 d s 2 = d φ 2 +2 g a ¯ ¯ b d z a d ¯ 2 ξ t C d ξ 4 d ξ t CM d ξ z 4 a = 1 , . . . , n h − 1 , A = (0 , a ) Heisenberg fibers: Dilaton φ , axion σ , Ramond pseudo-scalars ξ = ( ξ A , ˜ ξ A ) Base special Kähler M z : Prepotential G , Kähler potential K Ω , metric g a ¯ b Symplectic vector Z = ( Z A , G A ) C symplectic metric, M complex structure 21 / 44

Introduction N = 2 supergravity Kähler geometries Kähler isometries BPS solutions Conclusion Outline: 4. Kähler isometries 1 Introduction 2 N = 2 supergravity 3 Kähler geometries Kähler isometries 4 BPS solutions 5 Conclusion 6 22 / 44

Introduction N = 2 supergravity Kähler geometries Kähler isometries BPS solutions Conclusion Killing vectors and Lie derivative Isometry of spacetime ( M , g ): transformation that preserved distance (i.e. the metric) Acts with Lie derivative, generated by Killing vector k L k g = 0 Set of Killing vectors → Lie algebra c [ k a , k b ] = f ab k c , a = 1 , . . . , dim ISO ( M ) 23 / 44

Introduction N = 2 supergravity Kähler geometries Kähler isometries BPS solutions Conclusion Kähler manifolds: moment maps Holomorphic Killing vectors → preserve complex structure L k J = 0 Killing vectors given by moment maps ı = − i ∂ ¯ k i = i ∂ i P , k ¯ ı P τ ¯ P ( τ i , ¯ ı ) ∈ R Gives the coupling of the gravitini ψ α µ to A Λ µ Kähler potential invariant up to Kähler transformation L k K = 2 Re f 24 / 44