The document discusses finding black brane solutions in gauged supergravity that satisfy the Nernst law of thermodynamics by having vanishing entropy at zero temperature. It reviews how extremal Reissner-Nordström black branes have non-zero entropy at zero temperature. The paper then presents a solution for a "Nernst brane" in STU supergravity that has vanishing entropy density as temperature goes to zero and satisfies the first-order equations from supersymmetry. Finding such Nernst brane solutions could provide insights into the gauge/gravity correspondence at finite temperature and charge density.

1. Nernst branes in gauged
supergravity
Michael Haack (LMU Munich)
BSI 2011, Donji Milanovac, August 30
1108.0296 (with S. Barisch, G.L. Cardoso, S. Nampuri, N.A. Obers)

2. Motivation
Gauge/Gravity correspondence:
Gauge theory at finite Charged black brane
temperature and density in AdS
Mainly studied Reissner-Nordström (RN) black branes
Problem: RN black branes have finite entropy density
at T=0, in conflict with the 3rd law of thermodynamics
(Nernst law)
Task: Look for black branes with vanishing entropy
density at T=0 (Nernst branes)

3. Outline
AdS/CFT at finite T and charge density
Review Reissner Nordström black holes
Extremal black holes (T=0)
Dilatonic black holes
Nernst brane solution

4. Gauge/gravity correspondence
Feldtheorie
Gauge theory
Supergravity
Stringtheorie
AdS
Locally:
dr2
ds2 = + r2 (−dt2 + dx2 )
r2
r

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 AdS5
Generalizes to other dimensions and other gauge
theories

6. Short review of AdS/CFT at
finite T and charge density

7. Short review of AdS/CFT at
finite T and charge density
AdS/CFT
dictionary
J. Maldacena
(Editor)

8. CFT AdS
Vacuum Empty AdS
dr2
ds = 2 + r2 (−dt2 + dx2 )
2
r
Thermal ensemble AdS black brane
at temperature T
r0 r→∞

9. CFT AdS
Vacuum Empty AdS
dr2
ds = 2 + r2 (−dt2 + dx2 )
2
r
Thermal 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. CFT AdS
Thermal ensemble Charged black brane
at temperature T with gauge field
and charge density At ∼
ρ
ρ rD−3
Usually Reissner-
Nordström (RN):
2M Q2
f = 1 − D−1 + 2(D−2)
r r
with Q ∼ ρ

11. RN black hole
(4D, asymptotically flat, spherical horizon)
Charged black hole solution of
4 √
d x −g[R − Fµν F ] µν
2M Q2 dr2
ds2 = − 1 − + 2 dt2 + + r2 dΩ2
r r Q2
1− 2M
r + r2
Q
A = dt
r

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 horizon
Black hole for M ≥ |Q|
M 2 − Q2 |Q|→M
TH = −→ 0
2M r+2

13. Extremal black holes
|Q| = M
TH = 0
r+ = r − = M
2
M dr2
ds = − 1 −
2
dt2 + + r2 dΩ2
r M 2
1− r
Distance to the horizon at r = M
R
dr →0
−→ ∞
M+ 1− r
M

14. Near horizon geometry: Introduce r = M (1 + λ)
ds2 ∼ (−λ2 dt2 + M 2 λ−2 dλ2 ) + M 2 dΩ2
to lowest order in λ
AdS2 × S 2

15. Near horizon geometry: Introduce r = M (1 + λ)
ds2 ∼ (−λ2 dt2 + M 2 λ−2 dλ2 ) + M 2 dΩ2
to lowest order in λ
AdS2 × S 2
Infinite throat:
[From: Townsend “Black Holes”]

16. Entropy: S ∼ AH
Horizon area
Extremal RN: AH = 4πM 2 , i.e. non-vanishing entropy!

17. Entropy: S ∼ AH
Horizon area
Extremal RN: AH = 4πM 2 , i.e. non-vanishing entropy!
For black branes
ds2 = −e2U (r) dt2 + e−2U (r) dr2 + e2A(r) dx2
entropy density
2A(r0 )
s∼e
Horizon radius

18. Possible solutions
In 5d, RN unstable at low temperature in presence of
magnetic field and Chern-Simons term
[D’Hoker, Kraus]
In general dimensions, RN not the zero temperature
ground 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. Possible solutions
In 5d, RN unstable at low temperature in presence of
magnetic field and Chern-Simons term
[D’Hoker, Kraus]
In general dimensions, RN not the zero temperature
ground 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. 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 →0
For α < 1 : S −→ 0 [Preskill, Schwarz, Shapere,
Trivedi, Wilczek]

21. Strategy
Look 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 mechanism
Attractor mechanism: Values of scalar fields at the
horizon are fixed, independent of their asymptotic values

22. φ(r)1.0
0.5
r = r0
0.5 1.0 1.5
0.5
r→∞
First found for N=2 supersymmetric, asymptotically
flat black holes in 4D [Ferrara, Kallosh, Strominger]
Consequence of infinite throat of extremal BH
Half the number of d.o.f. =⇒ 1st order equations
Compare domain walls in fake supergravity
[Freedmann, Nunez, Schnabl, Skenderis; Celi, Ceresole, Dall’Agata, Van Proyen,
Zagermann; Zagermann; Skenderis, Townsend]

23. N = 2 Supergravity
1 ¯
− NIJ Dµ X I Dµ X J + 1 ImNIJ Fµν F µνJ
I
2R 4
I ˜ ¯
− 1 ReNIJ Fµν F µνJ − V (X, X)
4
NIJ , NIJ , V can be expressed in terms of holomorphic
prepotential 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. 1 st
order equations
Ansatz
ds2 = −e2U (r) dt2 + e−2U (r) dr2 + e2A(r) (dx2 + dy 2 )
−1 IJ
I
Ftr ∼ (Im N ) QJ , Fxy ∼ P I
I
Plug into action and rewrite it as sum of squares
2
S1d = dr φα − fα (φ) + total derivative
α
with {φα } = {U, A, X I }
φα − fα (φ) = 0 =⇒ δS1d = 0

25. Supersymmetry preserving configurations solve:
[Barisch, Cardoso, Haack, Nampuri, Obers]
ˆ ˆ
U = e−U −2A Re X I QI − e−U Im X I hI
A = −Re X qI I
[compare also: Gnecchi, Dall’Agata]
Y I
=e N
A IJ
¯
qJ

26. Supersymmetry preserving configurations solve:
[Barisch, Cardoso, Haack, Nampuri, Obers]
ˆ ˆ
U = e−U −2A Re X I QI − e−U Im X I hI
A = −Re X qI I
[compare also: Gnecchi, Dall’Agata]
Y I
=e N
A IJ
¯
qJ
Y I = X I eA

27. Supersymmetry preserving configurations solve:
[Barisch, Cardoso, Haack, Nampuri, Obers]
ˆ ˆ
U = e−U −2A Re X I QI − e−U Im X I hI
A = −Re X qI I
[compare also: Gnecchi, Dall’Agata]
Y I
=e N
A IJ
¯
qJ
Y I = X I eA
ˆ
QI = QI − FIJ P J ˆ
hI = hI − FIJ hJ
,
ˆ ˆ
qI = e−U −2A (QI − ie2A hI )

28. Supersymmetry preserving configurations solve:
[Barisch, Cardoso, Haack, Nampuri, Obers]
ˆ ˆ
U = e−U −2A Re X I QI − e−U Im X I hI
A = −Re X qI I
[compare also: Gnecchi, Dall’Agata]
Y I
=e N
A IJ
¯
qJ
Y I = X I eA
ˆ
QI = QI − FIJ P J ˆ
hI = hI − FIJ hJ
,
ˆ ˆ
qI = e−U −2A (QI − ie2A hI )
Constraint:
QI hI − P I hI = 0

29. Nernst brane
[Barisch, Cardoso, Haack, Nampuri, Obers]
STU-model, i.e. X I , I = 0, . . . , 3
Consider Q0 , h1 , h2 , h3 = 0
ds2 = −e2U (r) dt2 + e−2U (r) dr2 + e2A(r) dx2
−2U r→0 Q0 1 2A r→0 Q0 1/2
e −→ , e −→ (2r)
h1 h2 h3 (2r)5/2 h1 h2 h3
infinite throat s→0

30. −2U r→∞ Q0 1 2A r→∞ Q0 3/2
e −→ , e −→ (2r)
h1 h2 h3 (2r)3/2 h1 h2 h3
Not asymptotically AdS

31. Summary
Extremal RN black brane has finite entropy, in conflict
with the third law of thermodynamics (Nernst law)
Ground state might be a different extremal black brane
with vanishing entropy Nernst brane
Nernst brane in N=2 supergravity with AdS
asymptotics?
Applications to gauge/gravity correspondence?