Download - 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]

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

"



Y I = XIeA


"! e"U Im

!XI hI

"[Barisch, Cardoso, Haack, Nampuri, Obers]

Y !I = eA N IJ qJ

A! = !Re!XI qI

"



Y I = XIeA


"! 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

"


Constraint:

QIhI ! P IhI = 0


Y I = XIeA


"! e"U Im

!XI hI

"

QI = QI ! FIJP JhI = hI ! FIJhJ,

qI = e!U!2A(QI ! ie2AhI)


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


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?

Download - M. Haack - Nernst Branes in Gauged Supergravity

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