M. Haack - Nernst Branes in Gauged Supergravity
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M. Haack - Nernst Branes in Gauged Supergravity

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The SEENET-MTP Workshop BW2011

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 M. Haack - Nernst Branes in Gauged Supergravity Presentation Transcript

  • 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)
  • 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)
  • OutlineAdS/CFT at finite T and charge densityReview Reissner Nordström black holesExtremal black holes (T=0)Dilatonic black holesNernst brane solution
  • Gauge/gravity correspondence Feldtheorie Gauge theory Supergravity Stringtheorie AdS Locally: dr2 ds2 = + r2 (−dt2 + dx2 ) r2 r
  • 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
  • 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 (Editor)
  • CFT AdS Vacuum Empty AdS dr2 ds = 2 + r2 (−dt2 + dx2 ) 2 rThermal ensemble AdS black braneat temperature T r0 r→∞
  • 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)
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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”]
  • 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 = −e2U (r) dt2 + e−2U (r) dr2 + e2A(r) dx2entropy density 2A(r0 ) s∼e Horizon radius
  • 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]
  • 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
  • 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]
  • 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
  • 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]
  • 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
  • 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
  • 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
  • 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
  • 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 )
  • 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
  • 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
  • −2U r→∞ Q0 1 2A r→∞ Q0 3/2e −→ , e −→ (2r) h1 h2 h3 (2r)3/2 h1 h2 h3Not asymptotically AdS
  • 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?