Nernst branes in gauged     supergravity              Michael Haack (LMU Munich)          BSI 2011, Donji Milanovac, Augus...
Motivation Gauge/Gravity correspondence:Gauge theory at finite            Charged black branetemperature and density       ...
OutlineAdS/CFT at finite T and charge densityReview Reissner Nordström black holesExtremal black holes (T=0)Dilatonic black...
Gauge/gravity correspondence                                      Feldtheorie                                       Gauge ...
Original AdS/CFT correspondence:             [Maldacena]   N = 4, SU (N ) Yang-Mills in ‘t Hooft limit,   i.e. for large N...
Short review of AdS/CFT at finite T and charge density
Short review of AdS/CFT at finite T and charge density          AdS/CFT          dictionary          J. Maldacena          ...
CFT                      AdS    Vacuum               Empty AdS                       dr2                   ds = 2 + r2 (−d...
CFT                     AdS    Vacuum               Empty AdS                       dr2                   ds = 2 + r2 (−dt...
CFT                AdSThermal ensemble Charged black braneat temperature T with gauge fieldand charge density    At ∼      ...
RN black hole  (4D, asymptotically flat, spherical horizon)Charged black hole solution of              4 √            d x −...
RN metric can be written as              ∆ 2 r2 2        ds = − 2 dt + dr + r dΩ          2                 2   2         ...
Extremal black holes|Q| = MTH = 0r+ = r − = M               2           M                dr2ds = − 1 −  2                 ...
Near horizon geometry: Introduce r = M (1 + λ)         ds2 ∼ (−λ2 dt2 + M 2 λ−2 dλ2 ) + M 2 dΩ2to lowest order in λ       ...
Near horizon geometry: Introduce r = M (1 + λ)         ds2 ∼ (−λ2 dt2 + M 2 λ−2 dλ2 ) + M 2 dΩ2to lowest order in λ       ...
Entropy: S ∼ AH                    Horizon areaExtremal RN: AH = 4πM 2 , i.e. non-vanishing entropy!
Entropy: S ∼ AH                    Horizon areaExtremal RN: AH = 4πM 2 , i.e. non-vanishing entropy!For black branes  ds2 ...
Possible solutionsIn 5d, RN unstable at low temperature in presence ofmagnetic field and Chern-Simons term                 ...
Possible solutionsIn 5d, RN unstable at low temperature in presence ofmagnetic field and Chern-Simons term                 ...
Dilatonic black holes    (4D, asymptotically flat, spherical horizon)                             [Garfinkle, Horowitz, Stro...
StrategyLook for extremal Nernst branes (with AdS asymptotics)in N=2 supergravity with vector multiplets (i) Contains neut...
φ(r)1.0                       0.5                r = r0                                0.5    1.0     1.5                 ...
N = 2 Supergravity1                    ¯     − NIJ Dµ X I Dµ X J + 1 ImNIJ Fµν F µνJ                                    I2...
1   st                  order equationsAnsatzds2 = −e2U (r) dt2 + e−2U (r) dr2 + e2A(r) (dx2 + dy 2 )                  −1 ...
Supersymmetry preserving configurations solve:                         [Barisch, Cardoso, Haack, Nampuri, Obers]           ...
Supersymmetry preserving configurations solve:                          [Barisch, Cardoso, Haack, Nampuri, Obers]          ...
Supersymmetry preserving configurations solve:                              [Barisch, Cardoso, Haack, Nampuri, Obers]      ...
Supersymmetry preserving configurations solve:                              [Barisch, Cardoso, Haack, Nampuri, Obers]      ...
Nernst brane           [Barisch, Cardoso, Haack, Nampuri, Obers]STU-model, i.e. X I , I = 0, . . . , 3Consider Q0 , h1 , h...
−2U r→∞      Q0       1          2A r→∞      Q0        3/2e    −→                       ,   e   −→            (2r)        ...
SummaryExtremal RN black brane has finite entropy, in conflictwith the third law of thermodynamics (Nernst law)Ground state ...
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M. Haack - Nernst Branes in Gauged Supergravity

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The SEENET-MTP Workshop BW2011
Particle Physics from TeV to Plank Scale
28 August – 1 September 2011, Donji Milanovac, Serbia

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M. Haack - Nernst Branes in Gauged Supergravity

  1. 1. Nernst branes in gauged supergravity Michael Haack (LMU Munich) BSI 2011, Donji Milanovac, August 301108.0296 (with S. Barisch, G.L. Cardoso, S. Nampuri, N.A. Obers)
  2. 2. Motivation Gauge/Gravity correspondence:Gauge theory at finite Charged black branetemperature and density in AdS Mainly studied Reissner-Nordström (RN) black branesProblem: RN black branes have finite entropy densityat T=0, in conflict with the 3rd law of thermodynamics(Nernst law)Task: Look for black branes with vanishing entropydensity at T=0 (Nernst branes)
  3. 3. OutlineAdS/CFT at finite T and charge densityReview Reissner Nordström black holesExtremal black holes (T=0)Dilatonic black holesNernst brane solution
  4. 4. Gauge/gravity correspondence Feldtheorie Gauge theory Supergravity Stringtheorie AdS Locally: dr2 ds2 = + r2 (−dt2 + dx2 ) r2 r
  5. 5. Original AdS/CFT correspondence: [Maldacena] N = 4, SU (N ) Yang-Mills in ‘t Hooft limit, i.e. for large N, and large λ‘tHooft = gY M N 2 ≡ Supergravity in the space AdS5Generalizes to other dimensions and other gaugetheories
  6. 6. Short review of AdS/CFT at finite T and charge density
  7. 7. Short review of AdS/CFT at finite T and charge density AdS/CFT dictionary J. Maldacena (Editor)
  8. 8. CFT AdS Vacuum Empty AdS dr2 ds = 2 + r2 (−dt2 + dx2 ) 2 rThermal ensemble AdS black braneat temperature T r0 r→∞
  9. 9. CFT AdS Vacuum Empty AdS dr2 ds = 2 + r2 (−dt2 + dx2 ) 2 rThermal ensemble dr2 ds2 = 2 + r2 (−f dt2 + dx2 )at temperature T r f 2M f =1− rD−1 T = TH ∼ r0 ∼ M 1/(D−1)
  10. 10. CFT AdSThermal ensemble Charged black braneat temperature T with gauge fieldand charge density At ∼ ρρ rD−3 Usually Reissner- Nordström (RN): 2M Q2 f = 1 − D−1 + 2(D−2) r r with Q ∼ ρ
  11. 11. RN black hole (4D, asymptotically flat, spherical horizon)Charged black hole solution of 4 √ d x −g[R − Fµν F ] µν 2M Q2 dr2ds2 = − 1 − + 2 dt2 + + r2 dΩ2 r r Q2 1− 2M r + r2 QA = dt r
  12. 12. RN metric can be written as ∆ 2 r2 2 ds = − 2 dt + dr + r dΩ 2 2 2 r ∆with ∆ =( r − r+ )(r − r− )where r± = M ± M 2 − Q2 r+ : event horizonBlack hole for M ≥ |Q| M 2 − Q2 |Q|→MTH = −→ 0 2M r+2
  13. 13. Extremal black holes|Q| = MTH = 0r+ = r − = M 2 M dr2ds = − 1 − 2 dt2 + + r2 dΩ2 r M 2 1− rDistance to the horizon at r = M R dr →0 −→ ∞ M+ 1− r M
  14. 14. Near horizon geometry: Introduce r = M (1 + λ) ds2 ∼ (−λ2 dt2 + M 2 λ−2 dλ2 ) + M 2 dΩ2to lowest order in λ AdS2 × S 2
  15. 15. Near horizon geometry: Introduce r = M (1 + λ) ds2 ∼ (−λ2 dt2 + M 2 λ−2 dλ2 ) + M 2 dΩ2to lowest order in λ AdS2 × S 2 Infinite throat:[From: Townsend “Black Holes”]
  16. 16. Entropy: S ∼ AH Horizon areaExtremal RN: AH = 4πM 2 , i.e. non-vanishing entropy!
  17. 17. Entropy: S ∼ AH Horizon areaExtremal RN: AH = 4πM 2 , i.e. non-vanishing entropy!For black branes ds2 = −e2U (r) dt2 + e−2U (r) dr2 + e2A(r) dx2entropy density 2A(r0 ) s∼e Horizon radius
  18. 18. Possible solutionsIn 5d, RN unstable at low temperature in presence ofmagnetic field and Chern-Simons term [D’Hoker, Kraus]In general dimensions, RN not the zero temperatureground state when coupled to (i) charged scalars (holographic superconductors) [Gubser; Hartnoll, Herzog, Horowitz](ii) neutral scalars (dilatonic black holes) [Goldstein, Kachru, Prakash, Trivedi; Cadoni, D’Appollonio, Pani]
  19. 19. Possible solutionsIn 5d, RN unstable at low temperature in presence ofmagnetic field and Chern-Simons term [D’Hoker, Kraus]In general dimensions, RN not the zero temperatureground state when coupled to (i) charged scalars (holographic superconductors) [Gubser; Hartnoll, Herzog, Horowitz](ii) neutral scalars (dilatonic black holes) [Goldstein, Kachru, Prakash, Trivedi; Cadoni, D’Appollonio, Pani]no embedding into string theory
  20. 20. Dilatonic black holes (4D, asymptotically flat, spherical horizon) [Garfinkle, Horowitz, Strominger]Charged black hole solution of 4 √ d x −g[R − ∂µ φ∂ µ φ − e−2αφ Fµν F µν ]E.g. α = 1 −1 2M 2M P 2 e2φ0 ds = − 1 − 2 dt + 1 − 2 dr2 + r r − dΩ2 r r M P2 e−2φ = e−2φ0 − , F = P sin(θ) dθ ∧ dϕ Mr T →0For α < 1 : S −→ 0 [Preskill, Schwarz, Shapere, Trivedi, Wilczek]
  21. 21. StrategyLook for extremal Nernst branes (with AdS asymptotics)in N=2 supergravity with vector multiplets (i) Contains neutral scalars (ii) Straightforward embedding into string theory (iii) Attractor mechanismAttractor mechanism: Values of scalar fields at thehorizon are fixed, independent of their asymptotic values
  22. 22. φ(r)1.0 0.5 r = r0 0.5 1.0 1.5 0.5 r→∞First found for N=2 supersymmetric, asymptoticallyflat black holes in 4D [Ferrara, Kallosh, Strominger] Consequence of infinite throat of extremal BHHalf the number of d.o.f. =⇒ 1st order equationsCompare domain walls in fake supergravity[Freedmann, Nunez, Schnabl, Skenderis; Celi, Ceresole, Dall’Agata, Van Proyen,Zagermann; Zagermann; Skenderis, Townsend]
  23. 23. N = 2 Supergravity1 ¯ − NIJ Dµ X I Dµ X J + 1 ImNIJ Fµν F µνJ I2R 4 I ˜ ¯ − 1 ReNIJ Fµν F µνJ − V (X, X) 4NIJ , NIJ , V can be expressed in terms of holomorphicprepotential F (X)E.g. ¯ ¯V (X, X) = N IJ − 2X I X J hK FKI − hI ¯ hK FKJ − hJ ∂2F ∂X I ∂X K
  24. 24. 1 st order equationsAnsatzds2 = −e2U (r) dt2 + e−2U (r) dr2 + e2A(r) (dx2 + dy 2 ) −1 IJ IFtr ∼ (Im N ) QJ , Fxy ∼ P I IPlug into action and rewrite it as sum of squares 2S1d = dr φα − fα (φ) + total derivative α with {φα } = {U, A, X I } φα − fα (φ) = 0 =⇒ δS1d = 0
  25. 25. Supersymmetry preserving configurations solve: [Barisch, Cardoso, Haack, Nampuri, Obers] ˆ ˆU = e−U −2A Re X I QI − e−U Im X I hIA = −Re X qI I [compare also: Gnecchi, Dall’Agata]Y I =e N A IJ ¯ qJ
  26. 26. Supersymmetry preserving configurations solve: [Barisch, Cardoso, Haack, Nampuri, Obers] ˆ ˆU = e−U −2A Re X I QI − e−U Im X I hIA = −Re X qI I [compare also: Gnecchi, Dall’Agata]Y I =e N A IJ ¯ qJ Y I = X I eA
  27. 27. Supersymmetry preserving configurations solve: [Barisch, Cardoso, Haack, Nampuri, Obers] ˆ ˆU = e−U −2A Re X I QI − e−U Im X I hIA = −Re X qI I [compare also: Gnecchi, Dall’Agata]Y I =e N A IJ ¯ qJ Y I = X I eAˆQI = QI − FIJ P J ˆ hI = hI − FIJ hJ , ˆ ˆqI = e−U −2A (QI − ie2A hI )
  28. 28. Supersymmetry preserving configurations solve: [Barisch, Cardoso, Haack, Nampuri, Obers] ˆ ˆU = e−U −2A Re X I QI − e−U Im X I hIA = −Re X qI I [compare also: Gnecchi, Dall’Agata]Y I =e N A IJ ¯ qJ Y I = X I eAˆQI = QI − FIJ P J ˆ hI = hI − FIJ hJ , ˆ ˆqI = e−U −2A (QI − ie2A hI )Constraint:QI hI − P I hI = 0
  29. 29. Nernst brane [Barisch, Cardoso, Haack, Nampuri, Obers]STU-model, i.e. X I , I = 0, . . . , 3Consider Q0 , h1 , h2 , h3 = 0ds2 = −e2U (r) dt2 + e−2U (r) dr2 + e2A(r) dx2 −2U r→0 Q0 1 2A r→0 Q0 1/2e −→ , e −→ (2r) h1 h2 h3 (2r)5/2 h1 h2 h3 infinite throat s→0
  30. 30. −2U r→∞ Q0 1 2A r→∞ Q0 3/2e −→ , e −→ (2r) h1 h2 h3 (2r)3/2 h1 h2 h3Not asymptotically AdS
  31. 31. SummaryExtremal RN black brane has finite entropy, in conflictwith the third law of thermodynamics (Nernst law)Ground state might be a different extremal black branewith vanishing entropy Nernst braneNernst brane in N=2 supergravity with AdSasymptotics?Applications to gauge/gravity correspondence?
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