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Scgsc10 - Paris Scgsc10 - Paris Presentation Transcript

  • 4 November 2010Dyons in U(1) gauged Supergravity Work in progress with G. DallʼAgata Alessandra Gnecchi “G. Galilei” Physics Dept. - Padua University (Italy)
  • OutlineBlack holes in SupergravityBPS flow and the superpotentialThe gauging: curved backgroundSolutions with FI gaugingGeneralizations to black hole solutions in gaugedtheories
  • MotivationsGauged supergravities arise from flux compactifications ofstring theory.This leaves a landscape of AdS4 backgrounds in whichthe cosmological constant is provided precisely by thescalar potential coming from the gauging.We want a systematic approach to address the problem ofthe destabilization of such backgrounds by the presenceof a stable black hole, thus yielding new insights into theinterpretation of string landscape. 3
  • Black Holes in Supergravity: the ungauged caseWell established description in the last 15 yearsLook for regular solution of the classical theory which are stable.They have zero temperature but finite entropy: extremal solutionsIn a gravity theory they saturates the bound M=|Q|in a Supergravity theory the charge is substituted by the central charge,and gives a BPS bound M=|Z|meaning the solution preserves some SUSY. 4
  • Black Holes in Supergravity:radial evolution and black hole dynamicsIn the SUSY variation of the fields the fermionic fields decouple.The bosonic sector of the theory is described by √ 1 1S = −g d x − R + ImNΛΓ Fµν F 4 Λ Γ, µν + √ ReNΛΓ µνρσ Fµν Fρσ + Λ Γ 2 2 −g 1 + grs (Φ)∂µ Φr ∂ µ Φs . 2 5
  • Black Holes in Supergravity:radial evolution and black hole dynamicsIn the SUSY variation of the fields the fermionic fields decouple.The bosonic sector of the theory is described by √ 1 1S = −g d x − R + ImNΛΓ Fµν F 4 Λ Γ, µν + √ ReNΛΓ µνρσ Fµν Fρσ + Λ Γ 2 2 −g 1 + grs (Φ)∂µ Φr ∂ µ Φs . 2 5
  • Black Holes in Supergravity: radial evolution and black hole dynamics In the SUSY variation of the fields the fermionic fields decouple. The bosonic sector of the theory is described by √ 1 1 S = −g d x − R + ImNΛΓ Fµν F 4 Λ Γ, µν + √ ReNΛΓ µνρσ Fµν Fρσ + Λ Γ 2 2 −g 1 + grs (Φ)∂µ Φr ∂ µ Φs . 2 Einstein-Hilbert term 5
  • Black Holes in Supergravity: radial evolution and black hole dynamics In the SUSY variation of the fields the fermionic fields decouple. The bosonic sector of the theory is described by √ 1 1 S = −g d x − R + ImNΛΓ Fµν F 4 Λ Γ, µν + √ ReNΛΓ µνρσ Fµν Fρσ + Λ Γ 2 2 −g 1 + grs (Φ)∂µ Φr ∂ µ Φs . 2 Einstein-Hilbert term 5
  • Black Holes in Supergravity: radial evolution and black hole dynamics In the SUSY variation of the fields the fermionic fields decouple. The bosonic sector of the theory is described by √ 1 1 S = −g d x − R + ImNΛΓ Fµν F 4 Λ Γ, µν + √ ReNΛΓ µνρσ Fµν Fρσ + Λ Γ 2 2 −g 1 + grs (Φ)∂µ Φr ∂ µ Φs . 2 Einstein- Vector fieldsHilbert term kinetic term 5
  • Black Holes in Supergravity: radial evolution and black hole dynamics In the SUSY variation of the fields the fermionic fields decouple. The bosonic sector of the theory is described by √ 1 1 S = −g d x − R + ImNΛΓ Fµν F 4 Λ Γ, µν + √ ReNΛΓ µνρσ Fµν Fρσ + Λ Γ 2 2 −g 1 + grs (Φ)∂µ Φr ∂ µ Φs . 2 Einstein- Vector fieldsHilbert term kinetic term 5
  • Black Holes in Supergravity: radial evolution and black hole dynamics In the SUSY variation of the fields the fermionic fields decouple. The bosonic sector of the theory is described by √ 1 1 S = −g d x − R + ImNΛΓ Fµν F 4 Λ Γ, µν + √ ReNΛΓ µνρσ Fµν Fρσ + Λ Γ 2 2 −g 1 + grs (Φ)∂µ Φr ∂ µ Φs . 2 Einstein- Vector fieldsHilbert term kinetic term Axionic coupling 5
  • Black Holes in Supergravity: radial evolution and black hole dynamics In the SUSY variation of the fields the fermionic fields decouple. The bosonic sector of the theory is described by √ 1 1 S = −g d x − R + ImNΛΓ Fµν F 4 Λ Γ, µν + √ ReNΛΓ µνρσ Fµν Fρσ + Λ Γ 2 2 −g 1 + grs (Φ)∂µ Φr ∂ µ Φs . 2 Einstein- Vector fieldsHilbert term kinetic term Axionic coupling 5
  • Black Holes in Supergravity: radial evolution and black hole dynamics In the SUSY variation of the fields the fermionic fields decouple. The bosonic sector of the theory is described by √ 1 1 S = −g d x − R + ImNΛΓ Fµν F 4 Λ Γ, µν + √ ReNΛΓ µνρσ Fµν Fρσ + Λ Γ 2 2 −g 1 + grs (Φ)∂µ Φr ∂ µ Φs . 2 Einstein- Vector fieldsHilbert term kinetic term Axionic coupling Non-linear sigma model 5
  • Black Holes in Supergravity: radial evolution and black hole dynamics In the SUSY variation of the fields the fermionic fields decouple. The bosonic sector of the theory is described by √ 1 1 S = −g d x − R + ImNΛΓ Fµν F 4 Λ Γ, µν + √ ReNΛΓ µνρσ Fµν Fρσ + Λ Γ 2 2 −g 1 + grs (Φ)∂µ Φr ∂ µ Φs . 2 Einstein- Vector fieldsHilbert term kinetic term Axionic coupling Non-linear sigma model G M= H 5
  • Black Holes in Supergravity:radial evolution and black hole dynamicsIn the SUSY variation of the fields the fermionic fields decouple.The bosonic sector of the theory is described by √ 1 1S = −g d x − R + ImNΛΓ Fµν F 4 Λ Γ, µν + √ ReNΛΓ µνρσ Fµν Fρσ + Λ Γ 2 2 −g 1 + grs (Φ)∂µ Φr ∂ µ Φs . 2The equations for the scalar fields are geodesic equations d2 φ(τ ) dφj dφk dφi dφj 2 + Γi (φ) jk =0, Gij (φ) = 2c2 , dτ dτ dτ dτ dτwhere c2 = 4S 2 T 2 .The effective one dimensional Lagrangian is 2 dU dφa dφb L= + Gab + e2U VBH − c2 dτ dτ dτ 6
  • Black Holes in Supergravity: radial evolution and black hole dynamicsWe can write the black hole potential in a manifestly symplectic way 1 TΛ VBH = Q MΛΣ QΣ , 2 where µ + νµ−1 ν νµ−1 pΛ M= . Q =Λ , µ−1 ν µ−1 qΛ d2 U = 2e2U VBH (φ, p, q),Equations of motions: dτ 2 Dφa ∂VBH 2 = e2U a , Dτ ∂φRegularity of the scalar dφaconfiguration at the Gij ∂m φi ∂n φj γ mn < ∞ =0 dωhorizon: 1 ω = log ρ , ρ=− , τAttractor behaviour: 2π ∂VBH ∂VBH a φ ≈ φa H + log τ =0 A ∂φa ∂φa hor 7
  • Black Holes in Supergravity: first order formalism 1 AB IRewrite the black hole potential VBH = ZAB Z + ZI Z through 2a real function W.From the ansatz ˙ U = eU W(φ(τ ))the scalar field equation follows: ˙ φa = 2eU g rs ∂s WIn this description, the extremum condition on the potential is given by ∂a VBH = 2∂b W(Wδa + 2Gbc b a ∂c W) =0thus the attractor eq.s are equally expressed as a critical point of V or W.For N=2 Supergravity W = |Z| A first order description is possible not only for BPS but also for non- BPS extremal solutions: fake superpotential [Ceresole-DallʼAgata ʻ07] 8
  • The gauging [Ceresole-DʼAuria-Ferrara ʻ95]Let gi¯ be the Kähler metric of a Kähler manifold M. If it has a a non trivial group of continuous isometries G generated byKilling vectors, then the kinetic Lagrangian admits G as agroup of global space-time symmetries.The holomorphic Killing vectors, which are defined by thevariation of the fields δz i = Λ kΛ (z) are defined by the iequations i kj + j ki =0; ¯kj ı + j k¯ ı =0This are identically satisfied once we can write kΛ = ig i¯∂¯PΛ , PΛ = PΛ i   ∗thus defining a momentum map, which also preserves theKähler structure of the scalar manifold.The momentum map construction applies to all manifolds witha symplectic structure, in particular to Kähler, HyperKählerand Quaternionic manifolds. 9
  • The gauging [Ceresole-DʼAuria-Ferrara ʻ95]Gauging involving hypermultiplets:Triholomorphic momentum map that leaves invariant thehyperkahler structure up to SU(2) rotations.In N=2 theories the same group of isometries G acts both onthe SpecialKähler and HyperKähler manifolds: ˆ kΛ = kΛ ∂i + k¯ ∂¯ + kΛ ∂u i ı Λ ı uFayet-Iliopoulos gauging = constant prepotential PΛ = ξΛ x x 10
  • The gauging [Ceresole-DʼAuria-Ferrara ʻ95]Gauging involving hypermultiplets:Triholomorphic momentum map that leaves invariant thehyperkahler structure up to SU(2) rotations.In N=2 theories the same group of isometries G acts both onthe SpecialKähler and HyperKähler manifolds: ˆ kΛ = kΛ ∂i + k¯ ∂¯ + kΛ ∂u i ı Λ ı uFayet-Iliopoulos gauging = constant prepotential PΛ = ξΛ x x 10
  • The gauging [Ceresole-DʼAuria-Ferrara ʻ95]Gauging involving hypermultiplets:Triholomorphic momentum map that leaves invariant thehyperkahler structure up to SU(2) rotations.In N=2 theories the same group of isometries G acts both onthe SpecialKähler and HyperKähler manifolds: ˆ kΛ = kΛ ∂i + k¯ ∂¯ + kΛ ∂u i ı Λ ı uFayet-Iliopoulos gauging = constant prepotential PΛ = ξΛ x x Non-trivial gauging! 10
  • N=2 Supergravity with FI gauging [Ceresole-DʼAuria-Ferrara ʻ95]Consider the scalar potential for an N=2 theory.Due to the fact that all the relevant quantities are derived fromthe Kähler vectors and prepotential, this can be written in ageometrical way as ¯ ¯ V = (kΛ , kΣ )LΛ LΣ + (U ΛΣ − 3LΛ LΣ )(PΛ PΣ − PΛ PΣ ) x xThus, one easily sees that for an abelian theory this potentialcan still be nonzero, as long as the prepotentials are taken asconstants, PΛ = ξΛ leading to the form of V on which we will x xfocus: ¯ VF I = (U ΛΣ − 3LΛ LΣ )ξΛ ξΣ x x 11
  • N=2 Supergravity with FI gaugingThe action of the theory is R 1 S= d4 x − + gi¯∂µ z i ∂ µ z¯ + NΛΣ Fµν F Λ µν + √ NΛΣ  ¯ Λ µνρσ Λ Σ Fµν Fρσ − Vg 2 2 −gThe gauging is encoded in the potential Vg = g i¯Di LD¯L − 3|L|2   L = G, Vit extends the electric gauging to include magnetic gaugecharges, it is constructed only in terms of symplectic sectionsand symplectic vector of charges V = eK/2 (X Λ (z), FΛ (z)) G = (˜Λ , gΛ ) g 12
  • Static dyonic black holes with FI gaugingAnsatz for the spacetime background ds2 = −e2U (r) dt2 + e−2U (r) (dr2 + e2ψ(r) dΩ2 )A second warp factor provides the deviation from the ansatz forasymptotically flat configurations.It compensates for the additional contribution to Einsteinequations due to the non-trivial cosmological constant.The effective action for a static spherically configurationbecomes S1d = dr{e2ψ(r) [U 2 ˙ − ψ 2 + gi¯z i z¯ + e2U −4ψ VBH + e−2U Vg ] − 1} ˙ ¯ 13
  • Static dyonic black holes with FI gaugingWe can rearrange the terms in the action in a BPS-like form, as asum of squares containing the equations of motions, that become U = −eU −2ψ Re(e−iα Z) + e−U Im(e−iα L) ψ = 2e−U Im(e−iα L) ¯ ¯ ¯ ¯ z i = −eiα g i¯(eU −2ψ D¯Z + ie−U D¯L) ˙  α + Ar = −2e−U Re(e−iα L)we also get the constraints G, Q = −1 , e2U −2ψ Im(e−iα Z) = Re(e−iα L) Notice: the ungauged limit of the same metric ansatz has to be performed taking a BPS rewriting of the action −(eψ ψ − 1)2 → eψ(r) = r 14
  • The phaseThe solution is 1/4-BPS, and the preserved supersymmetry isselected by the projections γ0 A = ieiα εAB B γ1 A = eiα δAB BThe constraint can be solved to give Z − ie2(ψ−U ) L e2iα = ¯ ¯ Z + ie2(ψ−U ) LRemark: the degrees of freedom. As for the ungauged case, theBPS equation for the phase is identically satisfied once we assumethe equations for the scalar fields and the warp factors hold.However, in the ungauged case α + Ar = 0 15
  • The superpotentialIt is possible to express the BPS equations of motion through a realfunction, as happens in the ungauged case, namely we have   U = −g U U ∂U W ψ = −g ψψ ∂ψ W  z i = −2˜i¯∂¯W ˙ g  gU U = −gψψ = e2ψ , gi¯ = e2ψ gi¯ ˜ with the superpotential given by W = eU |Z − ie2(ψ−U ) L|and obtain the attractor equations as ∂i W|h = 0 , W|h = 0 16
  • Near horizon geometry and existence of spherical horizonsExtremal 4-dimensional black hole near horizon geometry is the productspace AdS₂x S² r 2 2 RA 2 2 ds2 hor = 2 dt − 2 dr − RS (dθ2 + sin2 θdφ2 ) 2 RA rit is obtained for the near horizon behaviour of the warp factors r rRS U ∼ log ψ ∼ log RA RAthe scalars are at a critical point ˙ ˙ z i = z¯ = 0 ¯ıBPS equations at the horizons for the scalar fields yield to the conditions π h αZ = h αL + → αh = 0 2and |Z|h e2ψh −2Uh = |L|h 17
  • Solutions with constant scalarsLook for a system in which ˙ ˙ z i (r) = z¯(r) = 0 , solution of the ¯ıequations   2 RS e−U U = eU −2ψ + 2RA 2RA  e−U ψ = 2RAthe general form of the warp factors eψ eU = ec , 2RA r + d ec 2 d e = ψ r + r + (d2 − RS )e−c . 2 4RA2 RAwith r > −2dRA e−cThen choose the integration constants to recover a metric whichinterpolates between AdS₄ and AdS₂x S. 18
  • Solutions with constant scalarsAn exemple: d = −RS , ec = 4RA 2 2 r 2 r−σ 2r − σ 2 R A 2 RS 2 2 ds = 2 2 − dr − 2 (2r − σ) dΩ2 RA 2r − σ r−σ r2 σwith σ = RS .Near horizon and asymptotic behaviour ρ2 2 dρ2 n.h. 2 ds2 ∼ 2 dt − RA 2 − RS dΩ2 2 RA ρ r2 dr2 ds2 ∼ ∞ dt2 − 4RA 2 − 4RA r2 dΩ2 2 2 4RA2 r 19
  • Solutions with constant scalarsAn exemple: d = −RS , ec = 4RA 2 2 r 2 r−σ 2r − σ 2 R A 2 RS 2 2 ds = 2 2 − dr − 2 (2r − σ) dΩ2 RA 2r − σ r−σ r2 σwith σ = RS .Near horizon and asymptotic behaviour ρ2 2 dρ2 n.h. 2 ds2 ∼ 2 dt − RA 2 − RS dΩ2 2 RA ρ r2 dr2 ds2 ∼ ∞ dt2 − 4RA 2 − 4RA r2 dΩ2 2 2 4RA2 r A solution with the same AdS₂ and S² radius is not allowed! 19
  • Solutions with constant scalarsEqual radii would imply vanishing potential at the horizon R S = RA → Vg = 0 [Bellucci-Ferrara-Marrani-Yeranyan ʻ08]Asymptotic AdS background : Di L = 0The form of the gauge potential: Vg = −3|L|2 + |DL|2A configuration with constant scalars along the flow has |L| = 0Don’t forget the constraint G, Q = −1 !!(Not always compatible with the model) 20
  • Existence of solutionsQuadratic model F = iX 0 X 1 : attractor equations ∂i W|h = 0 , W|h = 0are incompatible for any choice of charges, also in the case of nonconstant scalars no black hole solutions with spherical horizon. X 1X 2X 3STU model with prepotential F =− 0 : the potential of the Xgauging has no critical point no asymptotic AdS configurations. √STU model with prepotential F = −i X 0 X 1 X 2 X 3 admits regularsolutions, with spherical horizon and satisfying the constraint G, Q = −1 21
  • ConclusionsAsymptotically non flat solutions have been studied using thegeometric description of duality invariant supergravities.Although in the standard lore static and supersymmetric solutions aresingular, many regular solutions are found, for charges satisfying theconstraint G, Q = −1Uncomplete enhancement of SUSY at the horizon: 1/4-BPSsolutions.Extend this study to other supergravity models and obtain asystematic description of black holes in AdS curved background. 22