Partial Deconfinement Hiromasa Watanabe (Univ. of Tsukuba) - PowerPoint PPT Presentation

Partial Deconfinement Hiromasa Watanabe (Univ. of Tsukuba) Collaborator: M. Hanada (Univ. of Southampton), G. Ishiki (Univ. of Tsukuba) JHEP 03 (2019) 145, arXiv:1812.05494 [hep-th] 
 Strings and Fields 2019 @ YITP 2019/08/17

Contents 1, Background and Motivation - AdS/CFT correspondence 2, Partial deconfinement in certain gauge theories - One of expressions of partial deconfinement - Examples 3, Summary & Discussion � 2

Motivation Holographic principle or gauge/gravity correspondence “Equivalent” Quantum Gravity A certain QFT Black Hole in lower dimension e.g.) AdS/CFT correspondence 
 [Maldacena, 1997] SU( N ) super Yang-Mills theory Gravity theory 
 (SYM) in Anti de Sitter space • Large N We want to study it • Adjoint representation to learn about quantum gravity. � 3 � 3

Black hole in AdS 5 × S 5 <=> 4d N =4 SU( N ) SYM Strongly coupled 4d SYM / dual string theory ( E : fix ) " String gas Large BH (AdS BH) E ∼ N 0 E ∼ N 2 T 4 ! Hagedorn string Small BH E ∼ L string E ∼ N 2 T − 7 � 4

Black hole in AdS 5 × S 5 <=> 4d N =4 SU( N ) SYM Strongly coupled 4d SYM / dual string theory Gauge theory side; Large BH (AdS BH) Deconfined phase E ∼ N 2 T 4 phase transition String gas Confined phase E ∼ N 0 ( T : fix ) How about small BH or Hagedorn string? [Witten, (1998)] � 5

D-branes with open strings & BH Dp- brane : the objects that open strings can put their endpoints. High temperature T region, Classical vacua (:minima of potential) X M = diag ( x 1 M , x 2 M , ⋯ , x N M ) ( 1 1 1 ) BH 1 1 ( X M ) ii = x i ： Position of i th Dp- brane M ( X M ) ij ’s fluctuation The bound state of Configuration of ： Open strings between i th and j th Dp- brane D-branes & open strings scalar fields X M How about E ∼ M 2 = N 2 /100 ? [Hanada & Maltz, (2016)/Berkowitz, Hanada & Maltz, (2016)] M = N /10 subblock is formed M × M ( ) M 2 d.o.f. is deconfined N /10 “partial deconfinement” N � 6

Check of partial deconfinement

The order parameter of transition (review) Polyakov loop : an order parameter of confine/deconfine transition with SU( N ) adjoint fields and large N . N Tr 𝒬 exp [ − ∮ temporal e i θ j = ∫ d θ ρ ( θ ) e i θ A t ] = 1 N P = 1 ∑ N j =1 ρ ( θ ) = 1 Can be regarded as N ∑ : phase distribution δ ( θ − θ j ) continuous function in large N limit j P = 0; ρ P ≠ 0; ρ − π π − π π Confined phase Deconfined phase � 8

Partial deconfinement Polyakov loop : an order parameter of confine/deconfine transition ρ ρ ρ − π − π π π − π π confined “partially” deconfined “fully” deconfined “Gross-Witten-Wadia” transition Deconfinement transition [Gross & Witten, (1980)/ Wadia, (1980)] Partial deconfinement ( M < N ) ρ deconf ( θ ) ρ ( θ ) = N − M N ρ deconf ( θ ) = N − M ⋅ 1 ρ conf ( θ ) + M 2 π + M N ρ deconf ( θ ) N N Partial deconfinement is “the mixture.” ρ conf ( θ ) M s are in deconfined phase and N-M s are in confined phase θ j θ j � 9

Examples of partial deconfinement

Examples of partial deconfinement • 4d U( N ) Yang-Mills theory with matters on S 3 (weak coupling); [Aharony, Marsano, Minwalla, Papadodimas & Raamsdonk, (2003)/Schnitzer, (2004)] At zero ’t Hooft coupling; Z ( x ) = ∫ [ dU ]exp { ∞ z ( x ) = ∑ 1 m ( z B ( x m ) + ( − 1) m +1 z F ( x m ) ) tr( U m )tr(( U † ) m ) } ∑ x ≡ e − β , x E i i m =1 π θ i − θ j ∫ [ dU ] → ∏ ( ) , tr( U n ) → ∑ i ∫ [ d θ i ] ∏ sin 2 e in θ j 2 − π i < j j Z ( x ) = ∫ [ d θ i ] exp ( − ∑ ∞ V ( θ i − θ j ) ) 1 ∑ n ( 1 − z B ( x n ) − ( − 1) n +1 z F ( x n ) ) cos( n θ ) V ( θ ) = ln(2) + i ≠ j n =1 At small nonzero ’t Hooft coupling; Z ( β ) = ∫ [ dU ] exp [ − ( | tr( U ) | 2 ( m 2 ρ deconf ( θ ) 1 − 1 ) + b | tr( U ) | 4 / N 2 ) ] κ = 2 ( κ − 1 = u 1 (1 − m 2 When 1 − 2 bu 2 1 ), u 1 = tr( U )/ N ) b > 0 1 ( T ≤ T 1 ) 2 π κ = ∞ ρ conf ( θ ) 2 π ( 1 + 2 κ cos θ ) 1 ( T 1 < T < T 2 ) ρ ( θ ) = ( T ≥ T 2 , | θ | < 2 arcsin κ /2 ) 2 πκ cos θ 2 − sin 2 θ κ At ; “GWW transition” 2 2 � 11

Examples of partial deconfinement • 4d U( N ) Yang-Mills theory with matters on S 3 (weak coupling); [Aharony, Marsano, Minwalla, Papadodimas & Raamsdonk, (2003)/Schnitzer, (2004)] ρ deconf ( θ ) 1 ( T ≤ T 1 ) M N = 2 κ = 2 2 π κ 2 π ( 1 + 2 κ cos θ ) 1 ( T 1 < T < T 2 ) ρ ( θ ) = ( T ≥ T 2 , | θ | < 2 arcsin κ /2 ) 2 πκ cos θ 2 − sin 2 θ κ 2 2 κ = ∞ ρ conf ( θ ) κ cos θ ) = ( 1 − 2 2 π ( 1 + 2 κ ) ⋅ 1 ρ ( θ ) = 1 2 π + 2 κ ⋅ 1 2 π (1 + cos θ ) At ; “GWW transition” Partial deconfinement ( M < N ) ρ ( θ ) = N − M ρ conf ( θ ) + M N ρ deconf ( θ ) = N − M ⋅ 1 2 π + M N ρ deconf ( θ ) N N • 4d =4 SYM on S 3 (weak coupling) ; [Sundborg, (2000), Aharony et al, (2003)] 𝒪 • Free vector model , etc… � 12

Examples of partial deconfinement • The bosonic part of plane wave matrix model (PWMM or BMN matrix model) = the mass deform. of (0+1)d SYM / Matrix quantum mechanics. 9 9 3 9 3 2 − μ 2 i − μ 2 1 2 + 1 ∑ ∑ ∑ ∑ ∑ ( D t X I ) [ X I , X J ] X 2 X 2 μϵ ijk X i X j X k L = N Tr a − i 2 4 2 8 I =1 I , J =1 i =1 a =4 i , j , k =1 Assumed GWW transition; Check by Monte Carlo simulation; • Hysteresis ( T 2 ≤ T 1 ) 1 2 π • Phase distribution 2 π ( 1 + 2 κ cos θ ) 1 ρ ( θ ) = πκ cos θ 2 2 − sin 2 θ κ 2 2 0.35 Fitting 0.3 0.25 0.2 ρ ( θ ) Plotting phase distribution 0.15 0.1 N = 128 ( 5 ) 0.05 μ = 0 − 3 − 2 − 1 0 1 2 3 θ � 13

Summary & Discussion • We proposed the partial deconfinement which implies a part of color d.o.f. is deconfined in theory. • It’s relating to small BH in dual gravity via holography • We demonstrated the existence of partial deconfinement in several SU( N ) gauge theories. • This should happen in which does not have the center symmetry. • How about finite N case such as real world QCD? • Can we apply it to the unstable black hole? � 14

Backup Slides

Examples of partial deconfinement • 4d U( N ) Yang-Mills theory with matters on S 3 (weak coupling); [Aharony, Marsano, Minwalla, Papadodimas & Raamsdonk, (2003)/Schnitzer, (2004)] Z ( β ) = ∫ [ dU ] exp [ − ( | tr( U ) | 2 ( m 2 1 − 1 ) + b | tr( U ) | 4 / N 2 ) ] ( κ − 1 = u 1 (1 − m 2 When b < 0 1 − 2 bu 2 1 ), u 1 = tr( U )/ N ) 2 π ( 1 + 2 κ cos θ ) 1 ( κ ≥ 2) ρ ( θ ) = πκ cos θ 2 2 − sin 2 θ κ ( κ ≤ 2) 2 2 From [Aharony, Marsano, Minwalla, Papadodimas & Raamsdonk, (2003)] � 16

Ant model and partial deconfinement N trail ↔ N BH p ↔ T Ant trail : ants bound by pheromone BH : D-branes bound by open strings p : pheromone from each ant T : index for excitation of open string dN trail sN trail saddle point = ( α + pN trail )( N − N trail ) − s + N trail dt dN trail = 0, x ≡ N trail Inflow effect dt N Outflow effect [Beekman, Sumpter & Ratnieks, (2001)] small s large s � 17