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)
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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)
MotivationGauge/Gravity correspondence:
Gauge theory at finite temperature and density
Charged black branein 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)
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
Gauge/gravity correspondence
r
Locally:
ds2 =dr2
r2+ r2(!dt2 + d!x2)
AdS
Stringtheorie
Feldtheorie
Supergravity
Gauge theory
Generalizes to other dimensions and other gauge theories
!
N = 4, SU(N) Yang-Mills in ‘t Hooft limit,
Supergravity in the space AdS5
i.e. for large , and large N !‘tHooft = g2Y MN
Original AdS/CFT correspondence: [Maldacena]
Short review of AdS/CFT at finite T and charge density
Short review of AdS/CFT at finite T and charge density
AdS/CFTdictionary
J. Maldacena
(Editor)
AdSCFT
Vacuum Empty AdS
ds2 =dr2
r2+ r2(!dt2 + d!x2)
Thermal ensembleat temperature T
AdS black brane
r !"r0
AdSCFT
Vacuum Empty AdS
ds2 =dr2
r2+ r2(!dt2 + d!x2)
Thermal ensembleat temperature
f = 1! 2M
rD!1
T
T = TH ! r0 !M1/(D!1)
ds2 =dr2
r2f+ r2(!fdt2 + d!x2)
AdSCFT
Thermal ensembleat temperature T
Charged black branewith gauge field
At !!
rD!3and charge density!
Usually Reissner-Nordström (RN):
with
f = 1! 2M
rD!1+
Q2
r2(D!2)
Q ! !
RN black hole(4D, asymptotically flat, spherical horizon)
Charged black hole solution of
ds2 = !!
1! 2M
r+
Q2
r2
"dt2 +
dr2
#1! 2M
r + Q2
r2
$ + r2d!2
A =Q
rdt
!d4x!"g[R" Fµ!Fµ! ]
RN metric can be written as
ds2 = !!r2
dt2 +r2
!dr2 + r2d"2
with
! =( r ! r+)(r ! r!)
where
r± = M ±!
M2 !Q2
r+ : event horizon
Black hole for M ! |Q|
TH =!
M2 !Q2
2Mr2+
|Q|!M!" 0
Extremal black holes|Q| = M
TH = 0
r+ = r! = M
ds2 = !!
1! M
r
"2
dt2 +dr2
#1! M
r
$2 + r2d!2
Distance to the horizon at r = M! R
M+!
dr
1! Mr
!!0!"#
Near horizon geometry: r = M(1 + !)Introduce
ds2 ! ("!2dt2 + M2!!2d!2) + M2d!2
AdS2 ! S2to lowest order in !
Near horizon geometry: r = M(1 + !)Introduce
ds2 ! ("!2dt2 + M2!!2d!2) + M2d!2
AdS2 ! S2
Infinite throat:
[From: Townsend “Black Holes”]
to lowest order in !
Entropy: S ! AH
Horizon area
Extremal RN: AH = 4!M2 , i.e. non-vanishing entropy!
Entropy: S ! AH
Horizon area
Extremal RN: AH = 4!M2 , i.e. non-vanishing entropy!
For black branes
ds2 = !e2U(r)dt2 + e!2U(r)dr2 + e2A(r)d!x2
entropy density
Horizon radius
s ! e2A(r0)
Possible solutionsIn 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]
Possible solutionsIn 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
Dilatonic black holes(4D, asymptotically flat, spherical horizon)
Charged black hole solution of !
d4x!"g[R" !µ"!µ"" e!2!"Fµ#Fµ# ]
E.g. ! = 1
[Garfinkle, Horowitz, Strominger]
F = P sin(!) d! ! d"e!2! = e!2!0 ! P 2
Mr
ds2 = !!
1! 2M
r
"dt2 +
!1! 2M
r
"!1
dr2 + r
!r ! P 2e2!0
M
"d!2
,
For ! < 1 : ST!0!" 0 [Preskill, Schwarz, Shapere,
Trivedi, Wilczek]
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
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. =! order equations1st
Compare domain walls in fake supergravity[Freedmann, Nunez, Schnabl, Skenderis; Celi, Ceresole, Dall’Agata, Van Proyen, Zagermann; Zagermann; Skenderis, Townsend]
0.5 1.0 1.5
!0.5
0.5
1.0
r !"r = r0
!(r)
SupergravityN = 2
NIJ ,NIJ , V can be expressed in terms of holomorphic
prepotential F (X)
V (X, X) =!N IJ ! 2XIXJ
" #hK FKI ! hI
$ #hK FKJ ! hJ
$E.g.
!2F
!XI!XK
12R!NIJ DµXI DµXJ + 1
4 ImNIJ F Iµ! Fµ!J
! 14ReNIJ F I
µ! Fµ!J ! V (X, X)
order equations1st
Ansatz
ds2 = !e2U(r)dt2 + e!2U(r)dr2 + e2A(r)(dx2 + dy2)
F Itr !
!(ImN )!1
"IJQJ F I
xy ! P I,
Plug into action and rewrite it as sum of squares
S1d =!
dr"
!
#!!
! ! f!(!)$2
+ total derivative
{!!} = {U, A,XI}with
!!! ! f!(!) = 0 =! !S1d = 0
Y !I = eA N IJ qJ
A! = !Re!XI qI
"
Supersymmetry preserving configurations solve:
[compare also: Gnecchi, Dall’Agata]
U ! = e"U"2A Re!XI QI
"! e"U Im
!XI hI
"[Barisch, Cardoso, Haack, Nampuri, Obers]
Y !I = eA N IJ qJ
A! = !Re!XI qI
"
Supersymmetry preserving configurations solve:
[compare also: Gnecchi, Dall’Agata]
Y I = XIeA
U ! = e"U"2A Re!XI QI
"! e"U Im
!XI hI
"[Barisch, Cardoso, Haack, Nampuri, Obers]
Y !I = eA N IJ qJ
A! = !Re!XI qI
"
Supersymmetry preserving configurations solve:
[compare also: Gnecchi, Dall’Agata]
Y I = XIeA
U ! = e"U"2A Re!XI QI
"! e"U Im
!XI hI
"
QI = QI ! FIJP JhI = hI ! FIJhJ,
qI = e!U!2A(QI ! ie2AhI)
[Barisch, Cardoso, Haack, Nampuri, Obers]
Y !I = eA N IJ qJ
A! = !Re!XI qI
"
Supersymmetry preserving configurations solve:
Constraint:
QIhI ! P IhI = 0
[compare also: Gnecchi, Dall’Agata]
Y I = XIeA
U ! = e"U"2A Re!XI QI
"! e"U Im
!XI hI
"
QI = QI ! FIJP JhI = hI ! FIJhJ,
qI = e!U!2A(QI ! ie2AhI)
[Barisch, Cardoso, Haack, Nampuri, Obers]
Nernst brane
STU-model, i.e. XI , I = 0, . . . , 3
ds2 = !e2U(r)dt2 + e!2U(r)dr2 + e2A(r)d!x2
Consider Q0, h1, h2, h3 != 0
e!2U r"0!"!
Q0
h1 h2 h3
1(2r)5/2
, e2A r"0!"!
Q0
h1 h2 h3(2r)1/2
infinite throat s! 0
[Barisch, Cardoso, Haack, Nampuri, Obers]
e!2U r"#!"!
Q0
h1 h2 h3
1(2r)3/2
, e2A r"#!"!
Q0
h1 h2 h3(2r)3/2
Not asymptotically AdS
Summary
Nernst brane in N=2 supergravity with AdS asymptotics?
Extremal RN black brane has finite entropy, in conflictwith the third law of thermodynamics (Nernst law)
Ground state might be a different extremal black brane with vanishing entropy Nernst brane
Applications to gauge/gravity correspondence?
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