Phase Transition of Anti-Symmetric Kazumi Okuyama Shinshu U, Japan Workshop@Michigan Kazumi Okuyama (Shinshu U, Japan) Workshop@Michigan 1 / 19 Wilson Loops in N = 4 SYM based on my recent paper [arXiv:1709.04166] Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM
AdS/CFT correspondence YM N Workshop@Michigan Kazumi Okuyama (Shinshu U, Japan) Prototypical example of AdS/CFT correspondence [Maldacena] 2 / 19 In the ’t Hooft large N limit, parameters on the bulk side are given by 4 d N = 4 SU ( N ) super Yang-Mills ⇔ type IIB string on AdS 5 × S 5 L AdS = λ 1 / 4 ℓ s , λ = g 2 ⇔ g string ∼ 1 / N We will focus on the 1/2 BPS Wilson loop in N = 4 SYM � �� �� x µ + Φ | ˙ W R = Tr R P exp ds ( iA µ ˙ x | ) Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM
3 / 19 Via AdS/CFT correspondence, various representations correspond to: Workshop@Michigan Kazumi Okuyama (Shinshu U, Japan) Z [Yamaguchi, Hartnoll-Kumar] [Drukker-Fiol] [Maldacena, Rey-Yee] Wilson loops in N = 4 SYM Expectation value of 1/2 BPS Wilson loop in N = 4 SYM is exactly given by a Gaussian matrix model [Erickson-Semenofg-Zarembo, Pestun] � √ dM e − 2 N Tr M 2 Tr R e λ M W R = 1 fundamental rep ⇔ fundamental string symmetric rep ⇔ D3-brane anti-symmetric rep ⇔ D5-brane Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM
Anti-symmetric representation and D5-brane 1 Workshop@Michigan Kazumi Okuyama (Shinshu U, Japan) 3 4 / 19 1 k th anti-symmetric representation A k ⇔ D5-brane wrapping AdS 2 × S 4 with k unit of electric fmux Leading term in 1 / N expansion matches the DBI action of D5-brane √ λ sin 3 θ k � π k � N log W A k ≈ 2 N = θ k − sin θ k cos θ k , 3 π 1 /λ corrections can be systematically computed from the low temperature expansion of Fermi distribution function [Horikoshi-KO] √ λ sin 3 θ k + π sin θ k N log W A k ≈ 2 √ + · · · 3 π λ Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM
Fermi gas picture 4 N Workshop@Michigan Kazumi Okuyama (Shinshu U, Japan) dz 8 N e 5 / 19 grand partition function of Fermi gas Z N Generating function of W A k can be thought of as a � � √ dMe − 2 N Tr M 2 det( 1 + ze W A k z k = 1 λ M ) P ( z ) = k = 0 Exact form of P ( z ) was found in [Fiol-Torrents] � � � � − λ i + zL j − i λ P ( z ) = det δ j i − 1 i , j = 1 , ··· , N W A k with fjxed k can be recovered from P ( z ) � W A k = 2 π iz k + 1 P ( z ) Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM
6 / 19 There is a discrepancy between the bulk calculation and the matrix Workshop@Michigan Kazumi Okuyama (Shinshu U, Japan) model calculation (I have nothing to say about it...) 1 / N correction 1 / N correction to W A k has been studied from the bulk side [Faraggi et al] √ λ sin 3 θ k − S D 5 = 2 N − 1 6 log sin θ k + · · · 3 π 1 / N correction on the matrix model side was recently computed [Gordon] √ λ sin 3 θ k + λ sin 4 θ k + · · · log W A k = 2 N 3 π 8 π 2 Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM
Two approaches We will follow the fjrst approach Workshop@Michigan Kazumi Okuyama (Shinshu U, Japan) connected 1 7 / 19 There are two approaches to compute 1 / N corrections to P ( z ) √ λ M ) � P ( z ) = � det( 1 + ze √ λ M ) as an operator in the Gaussian matrix model 1. Treat det( 1 + ze √ λ M ) as a part of potential 2. Treat det( 1 + ze These two approaches lead to the same result as long as λ ≪ N 2 ∞ �� � h � � √ λ M ) log P ( z ) = Tr log( 1 + ze h ! h = 1 Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM
8 / 19 z 2 calculation in the Gaussian matrix model Workshop@Michigan Kazumi Okuyama (Shinshu U, Japan) z z N Small λ expansion Small λ expansion of P ( z ) can be obtained from the perturbative � � 8 ( 1 + z ) 2 λ + z ( 1 − 4 z + z 2 ) 192 ( 1 + z ) 4 λ 2 + O ( λ 3 ) log P ( z ) = N log( 1 + z ) + 8 ( z + 1 ) 2 λ − z 2 ( 2 z − 3 ) 64 ( z + 1 ) 4 λ 2 + O ( λ 3 ) + � � � 1 − 4 z + 13 z 2 � λ 2 + O ( λ 3 ) + O ( 1 / N 2 ) + 1 384 ( z + 1 ) 4 This agrees with the small λ expansion of the exact result in [Fiol-Torrents] Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM
Topological recursion g Workshop@Michigan Kazumi Okuyama (Shinshu U, Japan) h 9 / 19 1 resolvent Tr It turns out that the 1 / N expansion of P ( z ) can be systematically computed from the topological recursion in the Gaussian matrix model Topological recursion for the genus- g , h -point function W g , h of x − M is given by [Eynard-Orantin] 4 x 1 W g , h ( x 1 , · · · , x h ) = W g − 1 , h + 1 ( x 1 , x 1 , x 2 , · · · , x h ) + 4 δ g , 0 δ h , 1 � � + W g ′ , 1 + | I 1 | ( x 1 , x I 1 ) W g − g ′ , 1 + | I 2 | ( x 1 , x I 2 ) g ′ = 0 I 1 ⊔ I 2 = { 2 , ··· , h } � W g , h − 1 ( x 1 , · · · , � x j , · · · , x h ) − W g , h − 1 ( x 2 , · · · , x h ) ∂ + , x 1 − x j ∂ x j j = 2 Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM
10 / 19 1 by replacing 1 Workshop@Michigan Kazumi Okuyama (Shinshu U, Japan) dv du 1 du The fjrst two terms are given by 1 / N corrections from topological recursion 1 / N correction J g , h of P ( z ) is obtained from W g , h √ λ u ) x − u → f ( u , z ) = log( 1 + ze ∞ ∞ � � h ! N 2 − 2 g − h J g , h ( z ) log P ( z ) = g = 0 h = 1 � 1 � J 0 , 1 = 2 1 − u 2 f ( u , z ) , π − 1 � f ( u , z ) − f ( v , z ) � 2 � 1 � 1 1 − uv � J 0 , 2 = 4 π 2 u − v ( 1 − u 2 )( 1 − v 2 ) − 1 − 1 Above J 0 , 2 reproduces the 1 / N correction found in [Gordon] Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM
Higher order corrections 1 Workshop@Michigan Kazumi Okuyama (Shinshu U, Japan) dw dv du 1 11 / 19 N topological recursion We can compute 1 / N corrections up to any desired order from the � � + · · · log P ( z ) = NJ 0 , 1 ( z ) + 1 2 J 0 , 2 ( z ) + 1 J 1 , 1 ( z ) + 1 3 ! J 0 , 3 ( z ) The order O ( 1 / N ) term is given by � 1 du 2 u 2 − 1 √ J 1 , 1 ( z ) = 1 − u 2 ∂ 2 u f ( u , z ) , 48 π − 1 � 1 � 1 � 1 u + v + w + uvw � J 0 , 3 ( z ) = 8 π 3 ( 1 − u 2 )( 1 − v 2 )( 1 − w 2 ) − 1 − 1 − 1 × ∂ u f ( u , z ) ∂ v f ( v , z ) ∂ w f ( w , z ) This reproduces the small λ expansion of the exact result Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM
Novel scaling limit N Workshop@Michigan Kazumi Okuyama (Shinshu U, Japan) Result of J. Gordon suggests that we can take a scaling limit fjxed 12 / 19 with √ λ N , λ → ∞ ξ = √ log W A k = O ( N λ ) ⇒ log W A k = O ( N 2 ξ ) In this limit, W A k admits a closed string genus expansion ∞ � N 2 − 2 g S g ( ξ ) log W A k = g = 0 The genus-zero term S 0 is given by S 0 = 2 ξ 3 π sin 3 θ k + ξ 2 ξ 3 8 π 2 sin 4 θ k + 48 π 3 sin 3 θ k + · · · Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM
13 / 19 Green dashed curve : Workshop@Michigan Kazumi Okuyama (Shinshu U, Japan) 1 exact value of Orange curve : Gray dashed curve : Plot of log W A k for N = 300 , ξ = 1 0.20 0.15 0.10 0.05 k / N 0.2 0.4 0.6 0.8 1.0 3 π sin 3 θ k 2 ξ 3 π sin 3 θ k + ξ 2 8 π 2 sin 4 θ k 2 ξ N 2 log W A k Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM
Phase transition [Yamaguchi, Lunin, D’Hoker-Estes-Gutperle] This might correspond to an exchange of dominance of two topologically difgerent geometries on the bulk side Kazumi Okuyama (Shinshu U, Japan) Workshop@Michigan 14 / 19 Closed string expansion of log W A k in this limit ⇒ D5-brane is replaced by a bubbling geometry We conjecture there is a phase transition at some ξ = ξ c ↔ one-cut phase ( ξ < ξ c ) two-cut phase ( ξ > ξ c ) Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM
Potential in the scaling limit In this scaling limit, we should take into account Workshop@Michigan Kazumi Okuyama (Shinshu U, Japan) 15 / 19 √ λ M ) the back-reaction of the operator det( 1 + ze Potential V ( w ) for eigenvalue w is shifted from the Gaussian √ V ( w ) = 2 w 2 − ξ ( w − cos θ )Θ( w − cos θ ) , z = e − λ cos θ V ( w ) develops a new minimum as we increase ξ V ( w ) V ( w ) V ( w ) 6 1.0 4 5 3 4 0.5 3 2 w 2 - 0.5 0.5 1.0 1.5 2.0 1 1 - 0.5 w w - 0.5 - 0.5 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 ξ = 2 ξ = 3 ξ = 5 Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM
One-cut solution of resolvent We fjnd the resolvent in this limit in the one-cut phase Workshop@Michigan Kazumi Okuyama (Shinshu U, Japan) Eigenvalue density can be found by taking discontinuity across the cut 16 / 19 �� � � ( x − a )( x − b ) − ξ ( x − a )( b − cos θ ) R ( x ) = 2 x − π arctan ( x − b )(cos θ − a ) Eivenvalues are distributed along the cut x ∈ [ a , b ] a , b are determined by the condition lim x →∞ R ( x ) = 1 / x �� � � ( u − a )( b − cos θ ) ( u − a )( b − u ) − ξ ρ ( u ) = 2 ( b − u )(cos θ − a ) π π 2 arctanh Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM
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