Michigan U.

337 views

Published on

0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
337
On SlideShare
0
From Embeds
0
Number of Embeds
4
Actions
Shares
0
Downloads
0
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

Michigan U.

  1. 1. April 18th, 2011Flow equations for AdS₄ black holes in N=2 gauged supergravity G. DallʼAgata, A.G. - arXiv:1012.3756 JHEP 1103:037,2011 Alessandra Gnecchi “G. Galilei” Physics Dept. - Padua University (Italy) & Dept. of Physics - Harvard University
  2. 2. Outline Black holes in Supergravity Duality invariance of extremal solutions BPS flow of scalar fields Black holes in gauged Supergravity The FI gauging One modulus and stu model examples Future directions 2
  3. 3. Motivations1. Gauged SupergravityThe low energy effective theories obtained from String Theorycompactifications in the presence of fluxes are gauged supergravities.2. AdS spaceThe gauging appears as a scalar potential in the four dimensional action,which plays the role of a coordinate dependent cosmological constant. We seek a systematic approach to address the problem of the destabilization of such backgrounds by the presence of a stable black hole, thus yielding new insights into the interpretation of string landscape. 3
  4. 4. MotivationsOther applications1. Gauge/gravity duality Not only the near horizon region develops an AdS geometry, but also the asymptotic space!2. AdS/CMT correspondenceBlack holes solutions with scalars are associated to physical quantities in thecondensed matter system. Building a solution with nontrivial scalar profiles, or demonstrating the non existence in a specific models reflects is a statement on the properties of the condensed matter system 4
  5. 5. Black holes in SupergravityOur approach:Start from the formalism of un-gauged supergravity and exploit thesymmetries, in particular study theories with electric-magnetic duality invariance extremal black holesWell established description in the last 15 years S. Ferrara, R. Kallosh, A. Strominger hep-th/9508072 A. Strominger hep-th/9602111 S. Ferrara, G. W. Gibbons, R. Kallosh, hep-th/9702103has lead to the classification of black hole charge orbits, multicenter solutions,split attractors, wall crossing..Cacciatori-Klemm 0911.4926: genuine black holes solutions with sphericalhorizons in N=2 Supergravity with FI electric gauging. 5
  6. 6. Black holes in SupergravityDescribe regular solutions of the classical gravitational theory which arestable: no Hawking radiation but finite horizon area they have zero temperature but finite entropy extremal solutions Already in 4d gravity, the Reissner Nordstrom solution has an extremal limit r+ − r − κ 2 r+ − r− = 2 M 2 − Q2 κ= 2 2r+ T = →0 S= πr+ 2π In 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. 6
  7. 7. 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 7
  8. 8. 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 . 2Einstein-Hilbert term 7
  9. 9. 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 . 2Einstein-Hilbert Vector fields term kinetic term 7
  10. 10. 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 . 2Einstein-Hilbert Vector fields term kinetic term Axionic coupling 7
  11. 11. 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 . 2Einstein-Hilbert Vector fields term kinetic term Axionic coupling Non-linear sigma model G M= H 7
  12. 12. 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 The geodesic equations of free scalar d2 φi (τ ) dφj dφk dφi dφj i + Γjk (φ) = 0, Gij (φ) = 2c2 , dτ 2 dτ dτ dτ dτ where c2 = 4S 2 T 2 . are modified by the abelian field strengths through a black hole potential, appearing in the effective one dimensional Lagrangian 2 dU dφa dφb L= + Gab + e2U VBH − c2 dτ dτ dτ 8
  13. 13. Radial evolution and black hole dynamics We can write the black hole potential in a manifestly symplectic way 1 TΛ VBH = Q MΛΣ QΣ , 2 Λ where µ + νµ ν νµ −1 −1 p M = . Q = Λ , µ−1 ν µ−1 qΛ d2 U = 2e2U VBH (φ, p, q), Equations of motions: dτ 2 D2 φa 2U ∂VBH 2 = e a , Dτ ∂φ Regularity of the scalar dφa configuration at the Gij ∂m φi ∂n φj γ mn ∞ =0 dω horizon: 1 ω = log ρ , ρ=− , τ Attractor behaviour: 2π ∂VBH ∂VBH a φ ≈ φa H + log τ =0 A ∂φa ∂φa hor 9
  14. 14. 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 ∇a ∂c W) = 0 b thus the attractor equations are equally expressed as a critical point of V or W. For N=2 Supergravity W = |Z|the attractor point condition thus relates the spacetime dynamics with theflow on moduli space. 10
  15. 15. BPS flow, rotating and non SUSY solutions Extension to rotating solution [Denef, hep-th/0005049] Exemples of multicenter configurations N ¯ ω ¯ F, V = −e2iα [ζ + i (˜ ∧ ζ)] ζ= Z(Qi )dτi i=1 Attractor equations describe also extremal, non supersymmetric black holes, that can be built as intersecting branes systems from type IIA string theory [Kallosh-Sivanandam-Soroush, hep-th/0602005] [Gimon-Larsen-Simon, 0710.4967] The first order description generalizes to the non-BPS case by introducing a fake superpotential , built out of invariants of symplectic geometry [Ceresole-DallʼAgata hep-th/0702088] Extremal non-BPS solutions can be decomposed as threshold states of BPS constituents, thus revealing the existence of multicenter extremal non supersymmetric configurations, that one has to take into account when counting the degeneracy of the black holes states. [Gimon-Larsen-Simon, 0903.0719] [Bena-DallʼAgata-Giusto-Ruef-Warner, 0902.4526] 11
  16. 16. The gaugingMomentum map procedure [Ceresole-DʼAuria-Ferrara, hep-th/9509160] Let gi¯ be the Kähler metric of a Kähler manifold M. If it has a  a non trivial group of continuous isometries G generated by Killing vectors, then the kinetic Lagrangian admits G as a group of global space-time symmetries. The holomorphic Killing vectors, which are defined by the variation of the fields δz i = Λ kΛ (z) are defined by the i equations ∇i kj + ∇j ki = 0 ; ∇¯kj + ∇j k¯ = 0 ı ı This are identically satisfied once we can write kΛ = ig i¯∂¯PΛ , i   PΛ = PΛ ∗ thus defining a momentum map, which also preserves the Kähler structure of the scalar manifold. The momentum map construction applies to all manifolds with a symplectic structure, in particular to Kähler, HyperKähler and Quaternionic manifolds. 12
  17. 17. The gauging [Ceresole-DʼAuria-Ferrara ʻ95] Gauging involving hypermultiplets: Triholomorphic momentum map that leaves invariant the hyperkahler structure up to SU(2) rotations. In N=2 theories the same group of isometries G acts both on the SpecialKähler and HyperKähler manifolds: ˆ Λ = k i ∂i + k¯ ∂¯ + k u ∂u ı k Λ Λ ı Λ Fayet-Iliopoulos gauging = constant prepotential PΛ = ξΛ x x 13
  18. 18. The gauging [Ceresole-DʼAuria-Ferrara ʻ95] Gauging involving hypermultiplets: Triholomorphic momentum map that leaves invariant the hyperkahler structure up to SU(2) rotations. In N=2 theories the same group of isometries G acts both on the SpecialKähler and HyperKähler manifolds: ˆ Λ = k i ∂i + k¯ ∂¯ + k u ∂u ı k Λ Λ ı Λ Fayet-Iliopoulos gauging = constant prepotential PΛ = ξΛ x x 13
  19. 19. The gauging [Ceresole-DʼAuria-Ferrara ʻ95] Gauging involving hypermultiplets: Triholomorphic momentum map that leaves invariant the hyperkahler structure up to SU(2) rotations. In N=2 theories the same group of isometries G acts both on the SpecialKähler and HyperKähler manifolds: ˆ Λ = k i ∂i + k¯ ∂¯ + k u ∂u ı k Λ Λ ı Λ Fayet-Iliopoulos gauging = constant prepotential PΛ = ξΛ x x Non-trivial gauging! 13
  20. 20. 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 from the Kähler vectors and prepotential, this can be written in a geometrical way as ¯ ¯ V = (kΛ , kΣ )LΛ LΣ + (U ΛΣ − 3LΛ LΣ )(PΛ PΣ − PΛ PΣ ) x x Thus, one easily sees that for an abelian theory this potential can still be nonzero, as long as the prepotentials are taken as constants, PΛ = ξΛ leading to the form of V on which we will x x focus: VF I = (U ΛΣ ¯ Λ LΣ )ξΛ ξΣ − 3L x x 14
  21. 21. N=2 Supergravity with FI gaugingDuality invariant theoryThe action of the theory becomes R 1S= d x − + gi¯∂µ z ∂ z + NΛΣ Fµν F 4  ¯ i µ ¯  Λ Λ µν + √ NΛΣ µνρσ Fµν Fρσ − Vg Λ Σ 2 2 −gThe gauging is encoded in the potential Vg = g Di LD¯L − 3|L| i¯   2 where L = G, V = eK/2 Λ X gΛ − FΛ g Λit extends the electric gauging to include magnetic gauge charges, it isconstructed only in terms of symplectic sections and symplectic vector ofcharges V = eK/2 (X Λ (z), FΛ (z)) G = (˜Λ , gΛ ) ganalogously to the central charge used to define the black hole potential Z ≡ Q, V VBH = |DZ|2 + |Z|2 15
  22. 22. Static dyonic black holes Ansatz for the space-time 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 Einstein equationsdue to the non-trivial cosmological constant. In general the existence of BPS solutions only constrains the threedimensional base to be a space ds2 = dz 2 + e2Φ dwdw 3 ¯with U(1) holonomy and torsion. 16
  23. 23. Static dyonic black holesThe effective action for a static spherically configuration becomes 2ψ 2 2 i  ¯ 2U −4ψ −2U S1d = dr e U − ψ + gi¯z z + e  ¯ VBH + e Vg − 1 d 2ψ + dr e (2ψ − U ) dr 17
  24. 24. Static dyonic black holesThe effective action for a static spherically configuration becomes 2ψ 2 2 i  ¯ 2U −4ψ −2U S1d = dr e U − ψ + gi¯z z + e  ¯ VBH + e Vg − 1 d 2ψ + dr e (2ψ − U ) dr Possible squaring? 17
  25. 25. Static dyonic black holesThe effective action for a static spherically configuration becomes 2ψ 2 2 i  ¯ 2U −4ψ −2U S1d = dr e U − ψ + gi¯z z + e  ¯ VBH + e Vg − 1 d 2ψ + dr e (2ψ − U ) dr 17
  26. 26. Static dyonic black holesThe effective action for a static spherically configuration becomes 2ψ 2 2 i  ¯ 2U −4ψ −2U S1d = dr e U − ψ + gi¯z z + e  ¯ VBH + e Vg − 1 d 2ψ + dr e (2ψ − U ) drThe same action can be written 1 2(U −ψ) T 2ψ −U −iα 2 S1d = dr − e E ME − e (α + Ar ) + 2e Re(e L) 2 2ψ −U −iα 2 −e ψ − 2e Im(e L) − (1 + G, Q) d 2ψ−U −iα U −iα −2 e Im(e L) + e Re(e Z) drthis, together with −U −iα TE ≡ 2e 2ψ e Im(e V) T − e2(ψ−U ) G T ΩM−1 + 4e−U (α + Ar )Re(e−iα V)T + QTgives the BPS equations 17
  27. 27. Static dyonic black holesProjecting the E vector on the sections, we get the equations of motions 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 18
  28. 28. Static dyonic black holesProjecting the E vector on the sections, we get the equations of motions 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 A new branch ofthere is no smooth limit to the un-gauged case solitonic solutions 18
  29. 29. (more than) A glance at Supersymmetry Supersymmetry − i δψµ A = Dµ A − εAB Tµν γ − L δAB γ ν ηµν B ν B 2variations for general i gauging δλiA = −i ∂µ z i γ µ A − G−i γ µν εAB B + D L δ AB B µν The covariant derivative is 1 ab i Dµ A ≡ ∂µ A − ωµ γab A + Aµ A + gΛ AΛ δAC εCB B µ 4 2Choice of the projectors γ 0 A = i eiα εAB B γ 1 A = eiα δAB BRecover the equations of motion and the constraints eU −2ψ Im(e−iα Z) = e−U Re(e−iα L) AΛ gΛ = 2 eU Re(e−iα L) t G, Q + 1 = 0 19
  30. 30. (more than) A glance at Supersymmetry Supersymmetry − i δψµ A = Dµ A − εAB Tµν γ − L δAB γ ν ηµν B ν B 2variations for general i gauging δλiA = −i ∂µ z i γ µ A − G−i γ µν εAB B + D L δ AB B µν The covariant derivative is 1 ab i Dµ A ≡ ∂µ A − ωµ γab A + Aµ A + gΛ AΛ δAC εCB B µ 4 2Choice of the projectors γ 0 A = i eiα εAB B γ 1 A = eiα δAB BTwo projections required 1/4 - BPS solutions!Recover the equations of motion and the constraints eU −2ψ Im(e−iα Z) = e−U Re(e−iα L) AΛ gΛ = 2 eU Re(e−iα L) t G, Q + 1 = 0 19
  31. 31. The phase of the superpotential As for the un-gauged solution, the phase appears in the projection of the SUSY transformation parameter. We had e−iα Z = |Z| Solving the constraint e2U −2ψ Im(e−iα Z) = Re(e−iα L) for the phase we get Z − ie2(ψ−U ) L e2iα = ¯ ¯ Z + ie2(ψ−U ) L An additional request of positivity for the gauge charges may prevent from finding regular BPS solutions! Also notice, itʼs no more α + Ar = 0 (It will be recovered at the horizon) 20
  32. 32. The phase of the superpotential The flow can be expressed in terms of a single real function   U = −g U U ∂U W ψ = −g ψψ ∂ψ W  z i = −2˜i¯∂¯W ˙ g  gU U = −gψψ = e2ψ , gi¯ = e2ψ gi¯ ˜  for a superpotential W = eU |Z − ie2(ψ−U ) L| the flow stops at the horizon for the scalar fields and the combination of warp factors A=ψ−U At the attractor point ∂i W|h = 0 , W|h = 0 21
  33. 33. Near horizon geometryExtremal four dimensional near horizon geometry AdS₂x S² r 2 2 RA 2 2 ds2 hor = 2 dt − 2 dr − RS (dθ2 + sin2 θdφ2 ) 2 RA rrequires the warp factors behavior r rRS U ∼ log ψ ∼ log A = log RS RA RAattractor mechanism requires the scalars to be constant at thehorizon, thus completing the set of equations ∂i |Z − i e2A L| = 0 ⇔ Di Z − i e−2A Di L = 0 |Z − i e2A L| = 0 22
  34. 34. Attractor equations The BPS attractors for U(1) gauged supergravity are Q + e2A ΩMG = −2Im(ZV) + 2 e2A Re(LV) 2A Z 2 e = −i = RS L If one project these equations on the black hole charges or gauging charges, they give 2A 2 2 e = 2 |Di Z| − |Z| −2A 2 2 e = 2 |Di L| − |L| which are related to the second symplectic invariant 2 2 1 I2 (Q) = |Z| − |Di Z| = − QM(F )Q 2 23
  35. 35. Solutions with constant scalars Asymptotic AdS background : Di L = 0 Equal radii would imply vanishing potential at the horizon R S = RA → Vg = 0 [Bellucci-Ferrara-Marrani-Yeranyan ʻ08] The form of the gauge potential: Vg = −3|L|2 + |DL|2 A configuration with constant scalars along the flow has |L| = 0 In general, for constant scalars, the attractor equations imply 2A Im(ZL) 2A 1 G, Q e =− e = |L|2 2 |L|2 which is inconsistent for spherical horizons for which G, Q = −1 0 24
  36. 36. Exemple of dyonic solutionsOne modulus case Quadratic model F = iX 0 X 1 with Kähler metric K = − log 2(z + z ) ¯ Rez 0 AdS vacuum fixes the asymptotic modulus at g0 g1 + g 0 g 1 + i (g0 g 0 − g1 g 1 ) z= (g1 )2 + (g 0 )2 Attractor equations are I2 (G) = |G|2 − |Di G|2 = g0 g1 + g 0 g 1 e−2A = −I2 (G) thus requiring g0 g1 + g 0 g 1 0 25
  37. 37. Exemple of dyonic solutionsOne modulus case Quadratic model F = iX 0 X 1 with Kähler metric K = − log 2(z + z ) ¯ Rez 0 AdS vacuum fixes the asymptotic modulus at g0 g1 + g 0 g 1 + i (g0 g 0 − g1 g 1 ) z= (g1 )2 + (g 0 )2 Attractor equations are I2 (G) = |G|2 − |Di G|2 = g0 g1 + g 0 g 1 e−2A = −I2 (G) thus requiring g0 g1 + g 0 g 1 0 !! sis tent I ncon 25
  38. 38. Exemple of dyonic solutionsThe stu model X 1X 2X 3 STU model with prepotential F =− : the potential of the X0 gauging has no critical point no asymptotic AdS configurations. √ STU model with prepotential F = −i X 0 X 1 X 2 X 3 admits regular solutions with spherical horizon for magnetic charges [Cacciatori-Klemm 0911.4926] the duality invariant setup allow us to build a genuine dyonic solution by rotation of both electromagnetic and gauging charges VCK = eK/2 (1, −tu, −su, −st, −stu, s, t, u)T   1 K/2 T  −1  V=e (1, s, t, u, −stu, tu, su, st)   −1      −1  S=     1    VCK = SV G = S −1 GCK   1 1   −1 1 Q = S QCK 26
  39. 39. Exemple of dyonic solutionsThe stu model ChargesKahler potential Q = (p0 , 0, 0, 0, 0, q1 , q2 , q3 )T ¯ ¯K = − log[−i(s − s)(t − t)(u − u)] ¯ G = (0, g 1 , g 2 , g 3 , g0 , 0, 0, 0)TSuperpotentialW = eK/2 |q1 s + q2 t + q3 u + p0 stu − ie2A (g0 − g 1 tu − g 2 su − g 3 st)|No axion solution Re s = Re t = Re u = 0 The case where all the scalars are identified can be solved analitically; the attractor values of the fields are g0 −1 + 6gq + 1 − 16gq + 48g 2 q 2 y= 0 2g 1 − 3gq 2A 1 1 + 2(1 − 4gq) 1 − 16gq + 48g 2 q 2 − 3(1 − 4gq)2 e = 4 g0 g 3 27
  40. 40. Exemple of dyonic solutionsThe stu model 3 SU (1, 1) M= U (1) SU(1,1)³ is broken to U(1) by the gauging, consider the U(1) ⊂ SU(1,1) action cos θi z i + sin θi zi → i + cos θ . − sin θi z i 28
  41. 41. Exemple of dyonic solutionsThe stu model 3 SU (1, 1) M= U (1) SU(1,1)³ is broken to U(1) by the gauging, consider the U(1) ⊂ SU(1,1) action cos θi z i + sin θi zi → i + cos θ . − sin θi z i 28
  42. 42. Exemple of dyonic solutionsThe stu model 3 SU (1, 1) M= U (1) SU(1,1)³ is broken to U(1) by the gauging, consider the U(1) ⊂ SU(1,1) action cos θi z i + sin θi zi → i + cos θ . − sin θi z i Generate non zero axions! 28
  43. 43. The entropy 2A Zh 2 At the horizon e = −i = RS Lh 2A 2 2 −2A 2 2 e = 2 |Di Z| − |Z| e = 2 |Di L| − |L| Zh thus the entropy is proportional to S∼ Lh New dependence on the charges! The analytically solved example does not provide a check whether the entropy assumes integer values 2A 1 1 + 2(1 − 4gq) 1 − 16gq + 48g 2 q 2 − 3(1 − 4gq)2 e = 4 g0 g 3 29
  44. 44. The entropy 2A Zh 2 At the horizon e = −i = RS Lh 2A 2 2 −2A 2 2 e = 2 |Di Z| − |Z| e = 2 |Di L| − |L| Zh thus the entropy is proportional to S∼ Lh New dependence on the charges! The analytically solved example does not provide a check whether the entropy assumes integer values 2A 1 1 + 2(1 − 4gq) 1 − 16gq + 48g 2 q 2 − 3(1 − 4gq)2 e = 4 g0 g 3 Need for more examples!! 29
  45. 45. Future developments 1. Flow equationsWhat’s the geometric meaning of the gauging? Multiplying by the symplectic operator MΩ + i , the attractor equations can be expanded to give −iα −U 2A Q + e ΩMG = −2e 2A+2U Im −iα −U ∂r + i(α + Ar − 2Re(e ˙ e L)) (e e V) confront them with the un-gauged flow −iα −U Q = −2Im (∂r + i(α + Ar ) (e e V) ˙ Need for an interpretation of the “gauging section” What happens to the harmonic functions? 30
  46. 46. Future developments 1. Flow equationsWhat’s the geometric meaning of the gauging? Multiplying by the symplectic operator MΩ + i , the attractor equations can be expanded to give −iα −U 2A Q + e ΩMG = −2e 2A+2U Im −iα −U ∂r + i(α + Ar − 2Re(e ˙ e L)) (e e V) confront them with the un-gauged flow −iα −U Q = −2Im (∂r + i(α + Ar ) (e e V) ˙ Need for an interpretation of the “gauging section” What happens to the harmonic functions?Possible insights from a higher dimensional construction! 30
  47. 47. Future developments 2. M-theory embedding Reductions from 10 or 11 dimensions on spheres preserve to many supersymmetries. Additional truncations are possible, leading to N=2 U(1) gauged supergravity [Cvetič-Duff-Hoxha-Liu-Lü-Lu-Martinez Acosta-Pope- Sati-Tran, hep-th/9903214] M-theory reductions give in this cases only magnetic charges The magnetic field mixes internal angles and 4dim angular variables. This would require the presence of topological charges in the low energy configuration, but such monopoles might break all the supersymmetries [Vandoren-Hristov, 1012.4314] 31
  48. 48. Future developments 3. Rotating BHs D. Klemm arxiv:1103.4699, the solutions have an enhancement of supersymmetry at the horizon: 1/2-BPS black holes How does the attractor equations get modified for these solutions? The metric ansatz in the rotating case can be modified introducing the fibration ds2 = −e2U (dt + ω)2 + e−2U (dr2 + e−2ψ dΩ2 ) keeping the three base space conformally flat do multicenter solutions also exist? Does the generalization of the symplectic section defining the prepotential govern the dynamics in the rotating case? 32
  49. 49. Future developments4. More general gaugings - Adding Hypermultiplets Hypermultiplets are always present in theories obtained from flux compactifications Gauging of non-abelian isometries requires nontrivial scalar charge, what happens to the attractor mechanism?5. Extend these solutions out of extremality Duff-Liu, hep-th/9901149: “merging” of the gauging and the out-of- extremality contribution in metric functions dr2 ds2 = −e2A f dt2 + e2B ( + r2 dΩ2 ) f k 2 2 f = 1 − + 2g r (H1 H2 H3 H4 ) r Interesting phenomena might be described from an holographic perpective, once the finite temperature system is known. 33
  50. 50. Conclusions Asymptotically non flat solutions have been studied using the geometric formulation of duality invariant supergravities. Very close analogies have been found to the un-gauged case, and an easy generalization obtained for the superpotential of N=2 supergravity Although in the standard lore static and supersymmetric solutions are singular, many regular solutions are found, representing a new solitonic branch, for charges satisfying the constraint G, Q = −1 There is however an incomplete enhancement of SUSY at the horizon: 1/4-BPS solutions. 34
  51. 51. Conclusions Asymptotically non flat solutions have been studied using the geometric formulation of duality invariant supergravities. Very close analogies have been found to the un-gauged case, and an easy generalization obtained for the superpotential of N=2 supergravity Although in the standard lore static and supersymmetric solutions are singular, many regular solutions are found, representing a new solitonic branch, for charges satisfying the constraint G, Q = −1 There is however an incomplete enhancement of SUSY at the horizon: 1/4-BPS solutions. Definitely more to come! 34

×