Thermodynamics of the BMN matrix model at strong coupling Miguel S. - PowerPoint PPT Presentation

Thermodynamics of the BMN matrix model at strong coupling Miguel S. Costa Faculdade de Ciências da Universidade do Porto Works with L. Greenspan, J. Penedones and J. Santos Crete Center for Theoretical Physics - Heraklion, May 2014

Motivation • Gauge/gravity duality as definition of quantum gravity in AdS Dual CFT is renormalizable and unitary. Problem: how to decode the hologram? Unfortunately field theory very difficult in region of interest for quantum gravity (strong coupled; classical gravity , expansion loop expansion). N → ∞ 1 /N ≡ • Would like example where computations in both sides are within reach Test and understand the gauge/gravity duality with observables that are not protected by SUSY or can be computed using integrability. How does gravitation phenomena, like black holes, emerge from gauge theory side? Idea: Study thermodynamics of black holes dual to Matrix Quantum Mechanics that can be simulated on a computer using Monte-Carlo methods.

The case of D0-branes • Closed strings interact with D0-branes in flat space D0-brane Closed string Open string ⇔ • Closed strings interact with geometry produced by D0-branes Closed string D0-brane geometry [Itzhaki, Maldacena, Sonnenschein, Yankielowicz ´98]

D0-branes: field theoretic description (matrix quantum mechanics)  � ( D t X i ) 2 + Ψ α D t Ψ α + 1 S D 0 = N Z X i , X j ⇤ 2 + i Ψ α γ j αβ [ Ψ β , X j ] ⇥ dt Tr 2 λ 2 X i ≡ SU ( N ) bosonic matrices ( i = 1 , . . . , 9) Ψ ≡ SU ( N ) fermionic matrices (16 real components) SO(9) global symmetry D t = ∂ t − i [ A, ] ≡ covariant derivative γ i ≡ SO (9) gamma matrices γ i , γ j = 2 δ ij � �� • ‘t Hooft coupling is dimensionfull (relevant) E → ∞ ( UV ) ≡ weak coupling λ eff = λ g s N λ = g 2 ≡ mass 3 Y M N = E 3 E → 0 ( IR ) ≡ strong coupling (2 π ) 2 l 3 s • Dual 10D gravitational coupling = (2 π ) 11 ( λ l 3 s ) 2 16 π G N l − 8 s N 2

• Theory on Euclidean time circle with periodicity β = 1 /T Z β  � S D 0 = N dt Tr · · · 2 λ 0 T Low temperatures is strong coupling Dimensionless temperature τ = λ 1 / 3 • Can put theory on a computer using Monte Carlo simulations, accessing in particular strong coupled region. ✏ E Dimensionless mean energy N 2 = N 2 � 1 / 3 [Catterall, Wiseman ´07,´08,´09; Anagnostopoulos et al ´07; Hanada et al ´08,´13]

D0-branes: gravitational description • 11D SUGRA solution (near horizon geometry of non-extremal D0-brane) ◆ 7 ds 2 = dr 2 ✓ R ✓ ◆ ⇣ r 0 ⌘ 7 dz 2 + f ( r ) dt f ( r ) + r 2 d Ω 2 8 + 2 dz − dt r R ✓ R ◆ 7 ◆ 5 = 120 ⇡ 2 ✓ r 0 ⇣ r 0 ⌘ 7 5 3 ⌧ 2 = 60 ⇡ 3 g s N , f ( r ) = 1 − (2 ⇡ g s N ) , 49 r ` s ` s • Classical gravity domain (at horizon) τ = T/ λ 1 / 3 Horizon at G-L instability scale l s l 2 s R ( r 0 ) ⌧ 1 ) τ ⌧ 1 11D SUGRA IIA g s e φ ( r 0 ) ⌧ 1 ) τ � N − 10 N − 5 N − 5 N − 10 τ 1 21 6 9 21 Horizon at scale l P

• Standard gravitation thermodynamics S = A H ⌘ 14 9 ⇣ π = d 1 N 2 τ 13 2 5 5 15 d 1 = 4 5 5 4 G N 7 c 1 = 9 ✏ 14 N 2 = c 1 ⌧ 14 d 1 (because dE = TdS ) 5 • corrections give next term in expansion, at large N α 0 [Hanada et al´08] τ Z S 1 only fixes next power d 10 x √− g e � 2 φ ⇣ ⌘ ⇣ ⌘ R + · · · + α 0 3 R 4 + . . . 9 9 N 2 = d 1 τ 1 + d 2 τ ⇒ 5 5 in expansion 16 π G N τ • corrections to 11D SUGRA give 1/N correction (solution purely gravitational) l P Z S 1 5 + 1 fixes both coefficient and d 11 x √− g ⇣ P R 4 ⌘ 9 N 2 d 3 τ − 3 R + l 6 N 2 = d 1 τ ⇒ 5 1 /N 2 power of correction 16 π G 11 [Hanada et al´13]

• Low temperature expansion predicted from gravity + 1 h i h i ✏ 14 23 2 5 + c 2 ⌧ 5 + . . . 5 + . . . c 1 ⌧ c 3 ⌧ + . . . N 2 = N 2 3.0 N=17, ! =6 Monte-Carlo N=17, ! =8 2.5 7.41T 2.8 simulation 7.41T 2.8 -5.58T 4.6 of MQM SUGRA 2.0 ✏ /N 2 negligible E/N 2 1.5 checked SUGRA + 1.0 α 0 corrections predicted 0.5 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 [Hanada, Hyakutake, Nishimura, Takeuchi ´08] τ T

• Low temperature expansion predicted from quantum gravity + 1 h i h i ✏ 14 23 2 5 + c 2 ⌧ 5 + . . . 5 + . . . c 1 ⌧ c 3 ⌧ + . . . N 2 = N 2 Monte-Carlo simulation of MQM ✏ /N 2 negligible checked [Hyakutake ´13] [Hanada, Hyakutake, Ishiki, Nishimura ´13] τ

Today’s talk is not about D0-brane matrix model • Caveat: canonical ensemble ill defined - IR divergences from flat directions in D0-brane moduli space. This is suppressed at large N (metastable state), but it is a source of tension in Monte Carlo simulations [Catterall, Wiseman ´09] F ( T, r ) ∼ F finite ( T ) + 9 N ln r N 2 • Instability corresponds to Hawking radiation of D0-branes. At large N this is suppressed and black hole is stable (positive specific heat). • Today’s talk is about BMN matrix model [Berenstein, Maldacena, Nastase ´02] Mass deformation resolves IR divergence - canonical ensemble well defined. Much richer thermodynamics with a 1st order phase transition (at large N there are two dimensionless parameters).

BMN matrix model  µ 2 3 2 ( X i ) 2 + µ 2 � αβ Ψ β + i 2 µ Z S = S D 0 − N 6 2 ( X a ) 2 + µ 4 Ψ α � � 123 � 3 ✏ ijk X i X j X k dt Tr 2 � Massive deformation of D0-brane MQM. Preserves SUSY but breaks SO (9) → SO (6) × SO (3) a = 4 , . . . , 9 i = 1 , 2 , 3 λ = g 2 YM N In large N ‘t Hooft limit dimensionless coupling constant µ 3 X i = µ X a = 0 3 J i Many vacua [ J i , J j ] = i � ijk J k    M5 − brane vacua ≡ m → ∞ , n fixed N = mn n × n 0 . . . 0   D2 − brane vacua ≡ n → ∞ , m fixed (decoupled) 0 n × n . . . 0  X i ∼   m times   . . . . . . . . . . . .    Focus on trivial vacuum (single M5-brane) that  0 0 . . . n × n  is SO(9) invariant X i = X a = 0 Canonical ensemble is well defined. Still “easy” to simulate on a computer.

Dimensionless coupling ≡ λ = g 2 Y M N Start • Thermodynamics ( ) µ 3 N → ∞ here Dimensionless temperature ≡ T µ T F = O ( N 0 ) ? W S µ Deconfined phase T E Today: strongly coupled limit R A F = O ( N 2 ) O K N T G T H µ → 0 , µ fixed and large C O C µ Confined phase O U Dual geometry is SO(9) invariant U P P non-extremal D0-brane with L L deformation turned on I I N N G G λ = g 2 YM N µ 3 Exponential growth of spectrum with energy Hagedorn transition → 1 + 2 6 5  � 1 T c 3 4 λ − c λ 2 + O ( λ 3 ) First-order phase transition at µ = ≈ 0 . 076 + O ( λ ) 12 log 3 [Hadizadeh, Ramadanovic, Semenoff, Young ’04]

Gravitational dual • The different vacua of BMN matrix model correspond to the Lin-Maldacena geometries and asymptote to the M-theory plane wave solution ✓ µ 2 3 2 x i x i + µ 2 ◆ ds 2 = dx i dx i + dx a dx a + 2 dtdz − 6 2 x a x a dt 2 dC = µ dt ∧ dx 1 ∧ dx 2 ∧ dx 3 Non-normalizable mode responsible for massive deformation • At strong coupling, for large temperature, dual geometry is SO(9) invariant and is approximately the non-extremal D0-brane solution ds 2 = dr 2 8 + R 7 2 dz − r 7 Need back-reaction to decrease ✓ ◆ r 7 dz 2 + f ( r ) dt 0 f ( r ) + r 2 d Ω 2 R 7 dt temperature and study phase transition at strong coupling. In particular, dC = µ dt ∧ dx 1 ∧ dx 2 ∧ dx 3 SO (9) → SO (6) × SO (3)

pole S 2 • Ansatz for 11D SUGRA x = 0 � 2 ds 2 = − A (1 − y 7 ) d ζ + Ω (1 − y 7 ) d η  d η 2 + T 4 y 7 x = 1 equator y 7 y 7 S 5 " # B ( dy + Fdx ) 2 4 dx 2 + 1 2 − x 2 + T 2 x 2 (2 − x 2 ) d Ω 2 2 + T 3 (1 − x 2 ) 2 d Ω 2 + T 1 5 y 2 (1 − y 7 ) y 2 y = 0 y = 1 | {z } S 8 × S 1 S 8 × S 1 × S 1 d Ω 2 if T 1 = T 2 = T 3 =1 boundary horizon 8 C = ( M d η + L d ζ ) ∧ d 2 Ω 2 M-theory circle ζ ∼ ζ + 2 π x S 8 is a angular coordinate on compact 8-dimensional space with topology x S 2 collapses 1 y is a radial coordinate from boundary ( ) to horizon ( ) y = 1 y = 0 are functions of and Horizon A, B, F, T 1 , T 2 , T 3 , T 4 , Ω , M, L x y ∞ Tailored to numerical implementation (domain of unknown is the unit square; everything dimensionless) collapses S 5 y 0 1

pole S 2 • Ansatz for 11D SUGRA x = 0 � 2 ds 2 = − A (1 − y 7 ) d ζ + Ω (1 − y 7 ) d η  d η 2 + T 4 y 7 x = 1 equator y 7 y 7 S 5 " # B ( dy + Fdx ) 2 4 dx 2 + 1 2 − x 2 + T 2 x 2 (2 − x 2 ) d Ω 2 2 + T 3 (1 − x 2 ) 2 d Ω 2 + T 1 5 y 2 (1 − y 7 ) y 2 y = 0 y = 1 | {z } S 8 × S 1 S 8 × S 1 × S 1 d Ω 2 if T 1 = T 2 = T 3 =1 boundary horizon 8 C = ( M d η + L d ζ ) ∧ d 2 Ω 2 Non-extremal D0-brane solution corresponds to β = 4 π A = B = T 1 = T 2 = T 3 = T 4 = Ω = 1 , F = M = L = 0 , 7 (Euclidean time circle) and need to use scaling symmetry of 11D SUGRA action g µ ν → s 2 g µ ν , C µ νρ → s 3 C µ νρ I → s 9 I s = r 0 ⇒ with ✓ R ◆ 7 2 g s ` s ζ ∼ ζ + 2 π → ζ ∼ ζ + 2 π s 0 I → s 0 I s 0 = ⇒ r 0 r 0 This scaling symmetry will be important later...