STAG RESEARCH RESEARCH C E N CENTER T E R R C E N T E - PowerPoint PPT Presentation

Localised thermal phases in holography STAG RESEARCH RESEARCH C E N CENTER T E R R C E N T E Óscar Dias Ernest Rutherford Based on: OD, Jorge Santos, Benson Way, 1605.04911 & 1501.06574 & 1702.07718 SHEP, Southampton May 2019

Outline : 1. Non-uniform & Localized AdS 5 xS 5 black holes of sugra IIB ( thermal phases of SYM 1+3 ) Case where localised states only dominate microcanonical ensemble 2.Non-uniform & Localized thermal phases of SYM 1+1 on a circle Case where localised states dominate both (micro)canonical ensembles

➙ Recalling the primordial days: AdS 5 / CFT 4 Type IIB supergravity on AdS 5 × S 5 with radius of curvature L and N units of flux F (5) on S 5 is equivalent to Large N and strong t’Hooft coupling λ = g YM 2 N limit of N = 4 SYM theory in R 1,3 with gauge group SU( N ) and YM coupling g YM • Type IIB supergravity ( only with g and F (5) ): G MN ⌘ R MN � 1 48 F MPQRS F N PQRS = 0 , r M F MPQRS = 0 , F (5) = ? F (5) s): any soln of Einstein-AdS 5 can be oxidised to 10D via: • Freund-Rubin (80’ d s 2 = g µ ν d x µ d x ν + L 2 d Ω 2 5 , F (5) = Vol AdS 5 + Vol S 5

➙ Recalling the primordial days: AdS 5 / CFT 4 • AdS 5 xS 5 is a solution: d s 2 = − f ( r )d t 2 + d r 2 f ( r ) + r 2 d Ω 2 3 + L 2 d Ω 2 F µ νρστ = ✏ µ νρστ , F abcde = ✏ abcde 5 , f ( r ) = 1 + r 2 L 2 • Schwarzschild-AdS 5 xS 5 is also a solution: L 2 − r 2 ✓ r 2 f ( r ) = 1 + r 2 ◆ + + L 2 + 1 r 2

➙ Thermal Phases of AdS 5 xS 5 and their competition Microcanonical ensemble: (fixed E) Canonical ensemble: (fixed T) S / N 2 Δ F / N 2 ( regular cusp: dF =—S d T ) C V Small BH Large BH E / N 2 Thermal T HP AdS Δ F = 0 Confinement / deconfinement

➙ Are these 2 the only solutions with AdS 5 xS 5 asymptotics ? … Two scales: horizon radius r + and S 5 radius L r + L x Horizon topology S 3 × S 5 But we can have hierarchy of scales :

• Recall Gregory-Laflamme instability on a black string Mink 4 x S 1 with r + << L L r + Horizon topology S 3 Horizon topology S 2 × S 1 Time • Hierarchy of scales => GL instability => new phases: S / S us C 1 A B logy S E / E GL 1

• Recall Gregory-Laflamme instability on a black string Mink 4 x S 1 with r + << L L r + Horizon topology S 3 Horizon topology S 2 × S 1 • Expect that for r + << L Schwarzschild-AdS 5 xS 5 : Horizon topology S 8 Horizon topology S 3 × S 5 Localised BHs s H B y p m u L

➙ Complete Phase diagram ( Microcanonical ensemble ): ����� ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ����� ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ����� ● Δ � / � � ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ����� ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● l= 1 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ����� ● ● ● ● ● ■ ◆ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ���� ���� ���� ���� ���� � � / � �

➙ Update: Thermal Phases of AdS 5 xS 5 & their competition Microcanonical ensemble: (fixed E) Canonical ensemble: S / N 2 (fixed T) B Δ F / N 2 C V A B A E / N 2 T HP

➙ CFT dual interpretation ? • Bosonic sector of N =4 SYM contains 6 real scalars X i ( in the vector representation of SO(6) ) and a gauge field Α μ : 0 1 1 D µ X i D µ X i + 1 F µ ⌫ F µ ⌫ � X i , X j ⇤ 2 L (bosonic) X X 2 g 2 ⇥ = Tr @ � YM A SYM 2 g 2 YM i i,j • Localisation < = > Spontaneous symmetry breaking of 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

➙ CFT dual interpretation ? Spontaneous symmetry breaking => condensation of an infinite tower of scalar operators with increasing Δ . Lowest has Δ = 2 and vev: r h O 2 i = � N 2 1 5 3 � 2 , ⇡ 2 8 ������ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ������ ● ● ● ● ● ● ● ● ● ● ● ● ������ ● ● ● ● ● ● ● ● ● ● ● ● ������ ● ● 〈 � � 〉 / � � ● n o ● i t a ● t u p ● m o ● c ? ������ ● l a u ● d ● ● ● ● e ● c i t ● t a ● l ������ ● e h ● t ● n o ● ● ● ● ● ● ● ● ● ● ������ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ������ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ���� ���� ���� ���� � � / � �