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Localised thermal phases in holography

Óscar Dias

STAG

RESEARCH C E N T E R RESEARCH C E N T E R CENTER

Based on:

OD, Jorge Santos, Benson Way, 1605.04911 & 1501.06574 & 1702.07718

Ernest Rutherford

SHEP, Southampton May 2019

Outline:

• 1. Non-uniform & Localized AdS5 xS5 black holes of sugra IIB

( thermal phases of SYM1+3 ) 2.Non-uniform & Localized thermal phases of SYM1+1 on a circle

Case where localised states only dominate microcanonical ensemble Case where localised states dominate both (micro)canonical ensembles

➙ Recalling the primordial days: AdS5 / CFT4

Type IIB supergravity on AdS5 × S5 with radius of curvature L and N units of flux F(5) on S5 is equivalent to Large N and strong t’Hooft coupling λ= gYM2N limit of N = 4 SYM theory in R1,3 with gauge group SU(N) and YM coupling gYM

• Type IIB supergravity ( only with g and F(5) ):

GMN ⌘ RMN 1 48FMPQRSFN PQRS = 0, rMF MPQRS = 0 , F(5) = ?F(5)

• Freund-Rubin (80’

s): any soln of Einstein-AdS5 can be oxidised to 10D via:

ds2 = gµνdxµdxν + L2dΩ2

5 ,

F(5) = VolAdS5 + VolS5

• AdS5xS5 is a solution:

ds2 = −f(r)dt2 + dr2 f(r) + r2dΩ2

3 + L2dΩ2 5 ,

Fµνρστ = ✏µνρστ, Fabcde = ✏abcde

f(r) = 1 + r2 L2 f(r) = 1 + r2 L2 − r2

+

r2 ✓r2

+

L2 + 1 ◆

• Schwarzschild-AdS5xS5 is also a solution:

➙ Recalling the primordial days: AdS5 / CFT4

Microcanonical ensemble: (fixed E) Canonical ensemble: (fixed T)

➙ Thermal Phases of AdS5xS5 and their competition

ΔF / N2 T

HP Large BH Small BH CV Thermal AdS ΔF = 0

S / N2 E / N2

( regular cusp: dF =—S dT )

Confinement / deconfinement

… Two scales: horizon radius r+ and S5 radius L

But we can have hierarchy of scales:

L

x

r+

Horizon topology S3 × S5

➙ Are these 2 the only solutions with AdS5xS5 asymptotics ?

• Recall Gregory-Laflamme instability on a black string Mink 4 x S1 with r+ << L
• Hierarchy of scales => GL instability => new phases:

Horizon topology S2 × S1 Horizon topology S3

L

r+

E / EGL S / Sus A B C

1 1

logy S

Time

• Recall Gregory-Laflamme instability on a black string Mink 4 x S1 with r+ << L

Horizon topology S3 × S5 Horizon topology S8

• Expect that for r+ << L Schwarzschild-AdS5 xS5:

Horizon topology S2 × S1 Horizon topology S3

L

r+

L u m p y B H s Localised BHs

• ■

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• /

Δ/ ■ ◆

➙ Complete Phase diagram (Microcanonical ensemble):

l=1

Microcanonical ensemble: (fixed E) Canonical ensemble: (fixed T)

➙ Update: Thermal Phases of AdS5xS5 & their competition

S / N2 E / N2 A B ΔF / N2 T

HP CV

B A

➙ CFT dual interpretation ?

• Localisation < = > Spontaneous symmetry breaking
• f the SO(6) R-symmetry of the scalar sector of N=4 SYM down to SO(5).

=> condensation of an infinite tower of scalar operators with increasing Δ. Lowest is Δ = 2:

N

L(bosonic)

SYM

= Tr @ 1 2 g2

YM

Fµ⌫F µ⌫ X

i

DµXiDµXi + 1 2 g2

YM

X

i,j

⇥ Xi, Xj⇤2 1 A

• Bosonic sector of N=4 SYM contains 6 real scalars X i (in the vector representation of SO(6))

and a gauge field Αμ :

• /

〈〉/

hO2i = N 2 ⇡2 1 8 r 5 3 2, ➙ CFT dual interpretation ? Spontaneous symmetry breaking

=> condensation of an infinite tower of scalar operators with increasing Δ.

Lowest has Δ = 2 and vev:

d u a l c

• m

p u t a t i

• n
• n

t h e l a t t i c e

?

• ■

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• /

Δ/ ■ ◆

⇢EL N 2 , S N 2

• ≈ {0.225, 0.374}

d u a l c

• m

p u t a t i

• n
• n

t h e l a t t i c e

?

➙ CFT dual interpretation ? Spontaneous symmetry breaking

• f the SO(6) R-symmetry of the scalar sector of N=4 SYM down to SO(5).

➙ Precision Holography on AdS5xS5 — SYM3+1 ?

• Can we reproduce AdS5xS5 localisation phase transition on dual CFT ?
• … seems very hard to find this phase transition on a lattice computation:

1) lattice computations best adapted to the canonical ensemble;

• ur localised AdS5xS5 phase is sub-dominant in this ensemble

2) QFT lives in a curved background … no such lattice computations available

To have dual QFT (lattice) computations … look for precision holography of localisation somewhere else

➙ Precision Holography on Near-Horizon limit of D1 branes

OD, J. Santos, B. Way, 1702.07718

➙ Precision Holography on Near-Horizon limit of D1 branes

• Decoupling limit of stack of N non-extremal D1 branes is dual to SYM1+1

Itzhaki, Maldacena, Sonnenschein, Yankielowicz, [hep-th/9802042]

• Compactify spatial worlvolume (1D) of D1 brane on a circle length L

Dual to SU(N) SYM1+1 on S1 with length L , temperature T and t’Hooft coupling λ

• Alike original black string: two scales L & T that might have hierarchy of scales

⇒ GL instability at TGL L2 ~ 2.243 λ-1/2

➙ What is phase diagram of static BHs in this theory?

• Lumpy BHs with nontrivial dependence on S1 and horizon topology S 7 x S1
• Localised BHs with nontrivial dependence on on S1 and horizon topology S 8.
• λε /

λ/Δσ / ◆

• λ/τ

λΔ / ◆

Localised Localised

Lumpy Lumpy

phase transition

p h a s e t r a n s i t i

• n

Microcanonical ensemble: Canonical ensemble:

OD, Santos, Way, 1702.07718

Uniform

L

➙ What is phase diagram of static BHs in this theory?

• Localised BHs dominate the micro-canonical (for small E) & canonical (for small T) ensembles
• There is a first order phase transition at TL2 ~ 2.440 λ-1/2
• Lattice computation is available and its critical T for phase transition is compatible

TL2 ~ 3.5 λ-1/2 ——> TL2 ~ 2.47 λ-1/2 (after fixing factor of 21/2) Caterrall, Joseph, Wiseman [1008.4964] Catterall,Jha,Schaich,Wiseman [1709.07025]


• λε /

λ/Δσ / ◆

• λ/τ

λΔ / ◆

Localised Localised

Lumpy Lumpy

p h a s e t r a n s i t i

• n

p h a s e t r a n s i t i

• n

➙ Conclusions

• 1. Constructed Localized AdS5 xS5 BHs

and localized collection of D0 branes

• 2. Localised thermal phases can dominate (over the smeared phase)

the micro-canonical & canonical ensembles

• 3. First order phase transitions do occur at critical E or T
• 4. Precision holography:

some of these phase transitions are (might be) reproduced by Lattice computations