multicenter black holes with superpotentials Jan Perz Katholieke - PowerPoint PPT Presentation

Non-supersymmetric extremal multicenter black holes with superpotentials Jan Perz Katholieke Universiteit Leuven

Non-supersymmetric extremal multicenter black holes with superpotentials Jan Perz Katholieke Universiteit Leuven Based on: P . Galli, J. Perz arXiv:0909.???? [hep-th]

Single-center vs multicenter solutions ‣ superposition holds for linear systems ‣ typically not possible for black holes in GR • but: (Weyl)–Majumdar–Papapetrou solutions in Einstein–Maxwell theory —arbitrary distribution of extremally charged dust —static (as in Newtonian approximation) —described by harmonic functions

Single-center vs multicenter solutions ‣ superposition holds for linear systems ‣ typically not possible for black holes in GR • but: (Weyl)–Majumdar–Papapetrou solutions in Einstein–Maxwell theory —arbitrary distribution of extremally charged dust —static (as in Newtonian approximation) —described by harmonic functions

Single-center vs multicenter solutions ‣ superposition holds for linear systems ‣ typically not possible for black holes in GR • but: (Weyl)–Majumdar–Papapetrou solutions in Einstein–Maxwell theory —arbitrary distribution of extremally charged dust —static (as in Newtonian approximation) —described by harmonic functions

Supersymmetric black hole composites ‣ extremal multi-RN solutions are susy [Gibbons, Hull] ‣ susy (hence extremal) multicenter solutions in 4d supergravity with vector multiplets N = 2 • with identical charges [Behrndt, Lüst, Sabra]

Supersymmetric black hole composites ‣ extremal multi-RN solutions are susy [Gibbons, Hull] ‣ susy (hence extremal) multicenter solutions in 4d supergravity with vector multiplets N = 2 • with identical charges [Behrndt, Lüst, Sabra] • with arbitrary charges [Denef]

Supersymmetric black hole composites ‣ extremal multi-RN solutions are susy [Gibbons, Hull] ‣ susy (hence extremal) multicenter solutions in 4d supergravity with vector multiplets N = 2 • with identical charges [Behrndt, Lüst, Sabra] • with arbitrary charges [Denef] —relative positions of centers constrained

Supersymmetric black hole composites ‣ extremal multi-RN solutions are susy [Gibbons, Hull] ‣ susy (hence extremal) multicenter solutions in 4d supergravity with vector multiplets N = 2 • with identical charges [Behrndt, Lüst, Sabra] • with arbitrary charges [Denef] —relative positions of centers constrained —single-center solution may not exist, where a multicenter can

Non-susy extremal composites

Non-susy extremal composites ‣ via timelike dimensional reduction [Gaiotto, Li, Padi] • generate solutions (both susy and non-susy) as geodesics on augmented scalar manifold [Breitenlohner, Maison, Gibbons]

Non-susy extremal composites ‣ via timelike dimensional reduction [Gaiotto, Li, Padi] • generate solutions (both susy and non-susy) as geodesics on augmented scalar manifold [Breitenlohner, Maison, Gibbons] ‣ almost-susy [Goldstein, Katmadas] • reverse orientation of base space in 5D susy solutions

Non-susy extremal composites ‣ via timelike dimensional reduction [Gaiotto, Li, Padi] • generate solutions (both susy and non-susy) as geodesics on augmented scalar manifold [Breitenlohner, Maison, Gibbons] ‣ almost-susy [Goldstein, Katmadas] • reverse orientation of base space in 5D susy solutions ‣ here: superpotential approach

supergravity in 4 dimensions N = 2 ‣ bosonic action with vector multiplets n v � � ¯ b ( z ) d z a ∧ ⋆ d¯ b R ⋆ 1 − 2 g a ¯ z I 4D ∝ + Im N IJ ( z ) F I ∧ ⋆ F J + Re N IJ ( z ) F I ∧ F J � ‣ target space geometry: (very) special

supergravity in 4 dimensions N = 2 ‣ bosonic action with vector multiplets n v I = ( 0, a ) � � ¯ b ( z ) d z a ∧ ⋆ d¯ b a = 1, . . . , n v R ⋆ 1 − 2 g a ¯ z I 4D ∝ + Im N IJ ( z ) F I ∧ ⋆ F J + Re N IJ ( z ) F I ∧ F J � ‣ target space geometry: (very) special

supergravity in 4 dimensions N = 2 ‣ bosonic action with vector multiplets n v I = ( 0, a ) � � ¯ b ( z ) d z a ∧ ⋆ d¯ b a = 1, . . . , n v R ⋆ 1 − 2 g a ¯ z I 4D ∝ + Im N IJ ( z ) F I ∧ ⋆ F J + Re N IJ ( z ) F I ∧ F J � ‣ target space geometry: (very) special z a = X a X a X b X c F = − 1 6 D abc X 0 X 0

supergravity in 4 dimensions N = 2 ‣ bosonic action with vector multiplets n v I = ( 0, a ) � � ¯ b ( z ) d z a ∧ ⋆ d¯ b a = 1, . . . , n v R ⋆ 1 − 2 g a ¯ z I 4D ∝ + Im N IJ ( z ) F I ∧ ⋆ F J + Re N IJ ( z ) F I ∧ F J � ‣ target space geometry: (very) special z a = X a X a X b X c F = − 1 6 D abc X 0 X 0 g a ¯ b = ∂ z a ∂ ¯ b K z ¯

supergravity in 4 dimensions N = 2 ‣ bosonic action with vector multiplets n v I = ( 0, a ) � � ¯ b ( z ) d z a ∧ ⋆ d¯ b a = 1, . . . , n v R ⋆ 1 − 2 g a ¯ z I 4D ∝ + Im N IJ ( z ) F I ∧ ⋆ F J + Re N IJ ( z ) F I ∧ F J � ‣ target space geometry: (very) special z a = X a X a X b X c F = − 1 6 D abc X 0 X 0 g a ¯ b = ∂ z a ∂ ¯ b K z ¯ � �� X I � � � � 0 − 1 X I � K = − ln i ∂ I F 1 0 ∂ I F

Black holes in 4d supergravity N = 2 ‣ bosonic action with vector multiplets n v � � ¯ b ( z ) d z a ∧ ⋆ d¯ b R ⋆ 1 − 2 g a ¯ z I 4D ∝ + Im N IJ ( z ) F I ∧ ⋆ F J + Re N IJ ( z ) F I ∧ F J � ‣ static, spherically symmetric ansatz (1 center) τ = 1 d s 2 = − e 2 U ( τ ) d t 2 + e − 2 U ( τ ) δ ij d x i d x j | x | ‣ charged solution p I � � ∂ L � � p I ∝ F I q I ∝ = : Q ∂ F I q I S 2 S 2 ∞ ∞

Black hole potential (single-center) � � ¯ b ( z ) d z a ∧ ⋆ d¯ b R ⋆ 1 − 2 g a ¯ z I 4D ∝ + Im N IJ ( z ) F I ∧ ⋆ F J + Re N IJ ( z ) F I ∧ F J �

Black hole potential (single-center) � ˙ � ˙ = d ¯ b ( z ) d z a ∧ ⋆ d¯ U 2 b d τ − 2 g a ¯ z I eff ∝ d τ + Im N IJ ( z ) F I ∧ ⋆ F J + Re N IJ ( z ) F I ∧ F J �

Black hole potential (single-center) � ˙ � ˙ = d ¯ z a ˙ U 2 b d τ + g a ¯ z I eff ∝ b ˙ ¯ d τ + Im N IJ ( z ) F I ∧ ⋆ F J + Re N IJ ( z ) F I ∧ F J �

Black hole potential (single-center) ‣ action with effective potential [Ferrara, Gibbons, Kallosh] � ˙ � ˙ = d ¯ z a ˙ U 2 b + e 2 U V BH � d τ + g a ¯ z I eff ∝ b ˙ ¯ d τ V BH = | Z | 2 + 4 g a ¯ b ∂ a | Z | ∂ ¯ b | Z |

Black hole potential (single-center) ‣ action with effective potential [Ferrara, Gibbons, Kallosh] � ˙ � ˙ = d ¯ z a ˙ U 2 b + e 2 U V BH � d τ + g a ¯ z I eff ∝ b ˙ ¯ d τ V BH = | Z | 2 + 4 g a ¯ b ∂ a | Z | ∂ ¯ b | Z | X I � � � � � 0 − 1 Z = e K /2 � p I q I 1 0 ∂ I F

Black hole potential (single-center) ‣ action with effective potential [Ferrara, Gibbons, Kallosh] � ˙ � ˙ = d ¯ z a ˙ U 2 b + e 2 U V BH � d τ + g a ¯ z I eff ∝ b ˙ ¯ d τ V BH = | Z | 2 + 4 g a ¯ b ∂ a | Z | ∂ ¯ b | Z | ‣ rewriting not unique [Ceresole, Dall’Agata] V BH = Q T M Q = Q T S T M SQ S T M S = M

Black hole potential (single-center) ‣ action with effective potential [Ferrara, Gibbons, Kallosh] � ˙ � ˙ = d ¯ z a ˙ U 2 b + e 2 U V BH � d τ + g a ¯ z I eff ∝ b ˙ ¯ d τ V BH = | Z | 2 + 4 g a ¯ b ∂ a | Z | ∂ ¯ b | Z | ‣ rewriting not unique [Ceresole, Dall’Agata] V BH = Q T M Q = Q T S T M SQ S T M S = M ‣ ‘superpotential’ not necessarily equal to | Z | W V BH = W 2 + 4 g a ¯ b ∂ a W ∂ ¯ b W

Flow equations ‣ effective Lagrangian as a sum of squares U 2 + g a ¯ b + e 2 U ( W 2 + 4 g a ¯ ¯ z a ˙ b ∂ a W ∂ ¯ L eff ∝ ˙ b W ) b ˙ z ¯ ‣ first-order gradient flow, equivalent to EOM ‣ when : non-supersymmetric

Flow equations ‣ effective Lagrangian as a sum of squares U 2 + g a ¯ b + e 2 U ( W 2 + 4 g a ¯ ¯ z a ˙ b ∂ a W ∂ ¯ L eff ∝ ˙ b W ) b ˙ z ¯ � 2 2 � � � z a + 2e U g a ¯ U + e U W b ∂ ¯ ˙ b W � ˙ + � � ∝ � ‣ first-order gradient flow, equivalent to EOM ‣ when : non-supersymmetric

Flow equations ‣ effective Lagrangian as a sum of squares U 2 + g a ¯ b + e 2 U ( W 2 + 4 g a ¯ ¯ z a ˙ b ∂ a W ∂ ¯ L eff ∝ ˙ b W ) b ˙ z ¯ � 2 2 � � � z a + 2e U g a ¯ U + e U W b ∂ ¯ ˙ b W � ˙ + � � ∝ � ‣ first-order gradient flow, equivalent to EOM U = − e U W ˙ z a = − 2e U g a ¯ b ∂ ¯ b W ˙

Flow equations ‣ effective Lagrangian as a sum of squares U 2 + g a ¯ b + e 2 U ( W 2 + 4 g a ¯ ¯ z a ˙ b ∂ a W ∂ ¯ L eff ∝ ˙ b W ) b ˙ z ¯ � 2 2 � � � z a + 2e U g a ¯ U + e U W b ∂ ¯ ˙ b W � ˙ + � � ∝ � ‣ first-order gradient flow, equivalent to EOM U = − e U W ˙ z a = − 2e U g a ¯ b ∂ ¯ b W ˙ ‣ when : non-supersymmetric W � = | Z |

Geometrical perspective ‣ IIA string theory compactified on a CY 3-fold X � D a : basis of H 2 ( X , Z ) D abc = X D a ∧ D b ∧ D c ‣ scalars: in the normalized period vector ‣ charges: branes wrapping even cycles of ‣ central charge