M. Haack - Nernst Branes in Gauged Supergravity
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Transcript of M. Haack - Nernst Branes in Gauged Supergravity
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]
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!
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]
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?