[email protected] based on my recent paper [arXiv:1709.04166]

Kazumi Okuyama (Shinshu U, Japan)

Phase Transition of Anti-Symmetric Wilson Loops in [email protected] N = 4 SYM

1 / 19

AdS/CFT correspondence Prototypical example of AdS/CFT correspondence

[Maldacena]

4d N = 4 SU(N) super Yang-Mills ⇔ type IIB string on AdS5 × S 5 In the ’t Hooft large N limit, parameters on the bulk side are given by 2 λ = gYM N

⇔

LAdS = λ1/4 `s ,

gstring ∼ 1/N

We will focus on the 1/2 BPS Wilson loop in N = 4 SYM WR =

Kazumi Okuyama (Shinshu U, Japan)

TrR P exp

I

ds(iAµ x˙ + Φ|x|) ˙ µ

Phase Transition of Anti-Symmetric Wilson Loops in [email protected] N = 4 SYM

2 / 19

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-Semenoff-Zarembo, Pestun] Z √ 1 2 WR = dM e −2NTrM TrR e λM Z Via AdS/CFT correspondence, various representations correspond to: fundamental rep ⇔ fundamental string symmetric rep ⇔ D3-brane anti-symmetric rep ⇔ D5-brane

Kazumi Okuyama (Shinshu U, Japan)

[Maldacena, Rey-Yee]

[Drukker-Fiol] [Yamaguchi, Hartnoll-Kumar]

Phase Transition of Anti-Symmetric Wilson Loops in [email protected] N = 4 SYM

3 / 19

Anti-symmetric representation and D5-brane k th anti-symmetric representation Ak ⇔ D5-brane wrapping AdS2 × S 4 with k unit of electric flux Leading term in 1/N expansion matches the DBI action of D5-brane √ πk 1 2 λ sin3 θk log WAk ≈ , = θk − sin θk cos θk N 3π N 1/λ corrections can be systematically computed from the low temperature expansion of Fermi distribution function √ 1 2 λ sin3 θk π sin θk log WAk ≈ + √ + ··· N 3π 3 λ

Kazumi Okuyama (Shinshu U, Japan)

[Horikoshi-KO]

Phase Transition of Anti-Symmetric Wilson Loops in [email protected] N = 4 SYM

4 / 19

Fermi gas picture Generating function of WAk can be thought of as a grand partition function of Fermi gas P(z) =

N X k=0

WAk z k =

1 Z

Z

Exact form of P(z) was found in " P(z) = det

δij

+

2

dMe −2NTrM det(1 + ze

√

λM

)

[Fiol-Torrents]

zLj−i i−1

λ λ − e 8N 4N

# i,j=1,··· ,N

WAk with fixed k can be recovered from P(z) I dz WAk = P(z) 2πiz k+1 Kazumi Okuyama (Shinshu U, Japan)

Phase Transition of Anti-Symmetric Wilson Loops in [email protected] N = 4 SYM

5 / 19

1/N correction 1/N correction to WAk has been studied from the bulk side √ 2N λ sin3 θk 1 −SD5 = − log sin θk + · · · 3π 6

[Faraggi et al]

1/N correction on the matrix model side was recently computed √ 2N λ sin3 θk λ sin4 θk log WAk = + + ··· 3π 8π 2

[Gordon]

There is a discrepancy between the bulk calculation and the matrix model calculation (I have nothing to say about it...)

Kazumi Okuyama (Shinshu U, Japan)

Phase Transition of Anti-Symmetric Wilson Loops in [email protected] N = 4 SYM

6 / 19

Two approaches There are two approaches to compute 1/N corrections to P(z) P(z) = hdet(1 + ze 1. Treat det(1 + ze

√

2. Treat det(1 + ze

√

√

λM

)i

λM

) as an operator in the Gaussian matrix model

λM

) as a part of potential

These two approaches lead to the same result as long as λ N 2 We will follow the first approach log P(z) =

∞ ih E √ X 1 Dh Tr log(1 + ze λM ) h! connected h=1

Kazumi Okuyama (Shinshu U, Japan)

Phase Transition of Anti-Symmetric Wilson Loops in [email protected] N = 4 SYM

7 / 19

Small λ expansion Small λ expansion of P(z) can be obtained from the perturbative calculation in the Gaussian matrix model z z(1 − 4z + z 2 ) 2 3 log P(z) = N log(1 + z) + λ+ λ + O(λ ) 8(1 + z)2 192(1 + z)4 z2 z 2 (2z − 3) 2 + λ − λ + O(λ3 ) 8(z + 1)2 64(z + 1)4 " # 1 z 1 − 4z + 13z 2 2 3 + λ + O(λ ) + O(1/N 2 ) N 384(z + 1)4

This agrees with the small λ expansion of the exact result in

Kazumi Okuyama (Shinshu U, Japan)

[Fiol-Torrents]

Phase Transition of Anti-Symmetric Wilson Loops in [email protected] N = 4 SYM

8 / 19

Topological recursion 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 Wg,h of 1 resolvent Tr x−M is given by [Eynard-Orantin] 4x1 Wg,h (x1 , · · · , xh )

=Wg−1,h+1 (x1 , x1 , x2 , · · · , xh ) + 4δg,0 δh,1 X

g X

I1 tI2 ={2,··· ,h}

g 0 =0

+

Wg 0 ,1+|I1 | (x1 , xI1 )Wg−g 0 ,1+|I2 | (x1 , xI2 )

h X ∂ Wg,h−1 (x1 , · · · , xbj , · · · , xh ) − Wg,h−1 (x2 , · · · , xh ) + , ∂xj x1 − xj j=2

Kazumi Okuyama (Shinshu U, Japan)

Phase Transition of Anti-Symmetric Wilson Loops in [email protected] N = 4 SYM

9 / 19

1/N corrections from topological recursion 1/N correction Jg,h of P(z) is obtained from√ Wg,h 1 by replacing x−u → f (u, z) = log(1 + ze λu ) ∞ X ∞ X 1 2−2g−h log P(z) = N Jg,h (z) h! h=1 g=0

The first two terms are given by J0,1

2 = π

J0,2

1 = 2 4π

Z

1

du

p

1 − u 2 f (u, z),

du

Z

−1

Z

1

−1

1

−1

1 − uv

dv p (1 − u 2 )(1 − v 2 )

f (u, z) − f (v, z) u−v

Above J0,2 reproduces the 1/N correction found in

Kazumi Okuyama (Shinshu U, Japan)

2

[Gordon]

Phase Transition of Anti-Symmetric Wilson Loops [email protected] N = 4 SYM

10 / 19

Higher order corrections We can compute 1/N corrections up to any desired order from the topological recursion 1 1 1 log P(z) = NJ0,1 (z) + J0,2 (z) + J1,1 (z) + J0,3 (z) + · · · 2 N 3! The order O(1/N) term is given by 1

2u 2 − 1 2 du √ ∂u f (u, z), 1 − u2 −1 Z 1 Z 1 Z 1 1 u + v + w + uvw J0,3 (z) = 3 du dv dw p 8π −1 (1 − u 2 )(1 − v 2 )(1 − w 2 ) −1 −1 J1,1 (z) =

1 48π

Z

× ∂u f (u, z)∂v f (v, z)∂w f (w, z) This reproduces the small λ expansion of the exact result Kazumi Okuyama (Shinshu U, Japan)

Phase Transition of Anti-Symmetric Wilson Loops [email protected] N = 4 SYM

11 / 19

Novel scaling limit Result of J. Gordon suggests that we can take a scaling limit √ λ N, λ → ∞ with ξ = fixed N √ log WAk = O(N λ) ⇒ log WAk = O(N 2 ξ) In this limit, WAk admits a closed string genus expansion log WAk =

∞ X g=0

N 2−2g Sg (ξ)

The genus-zero term S0 is given by S0 =

2ξ ξ2 ξ3 sin3 θk + 2 sin4 θk + sin3 θk + · · · 3π 8π 48π 3

Kazumi Okuyama (Shinshu U, Japan)

Phase Transition of Anti-Symmetric Wilson Loops [email protected] N = 4 SYM

12 / 19

Plot of log WAk for N = 300, ξ = 1

0.20

0.15

0.10

0.05

0.2

0.4

0.6

0.8

1.0

k /N

2ξ 3 3π sin θk 2ξ ξ2 3 4 3π sin θk + 8π 2 sin θk exact value of N12 log WAk

Green dashed curve : Gray dashed curve : Orange curve : Kazumi Okuyama (Shinshu U, Japan)

Phase Transition of Anti-Symmetric Wilson Loops [email protected] N = 4 SYM

13 / 19

Phase transition Closed string expansion of log WAk in this limit ⇒ D5-brane is replaced by a bubbling geometry [Yamaguchi, Lunin, D’Hoker-Estes-Gutperle]

We conjecture there is a phase transition at some ξ = ξc one-cut phase (ξ < ξc ) ↔ two-cut phase (ξ > ξc ) This might correspond to an exchange of dominance of two topologically different geometries on the bulk side

Kazumi Okuyama (Shinshu U, Japan)

Phase Transition of Anti-Symmetric Wilson Loops [email protected] N = 4 SYM

14 / 19

Potential in the scaling limit In this scaling limit, we should take into account √ the back-reaction of the operator det(1 + ze λM ) Potential V (w) for eigenvalue w is shifted from the Gaussian √

V (w) = 2w 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 2

-0.5

0.5

1.0

1.5

2.0

w

1 1 -0.5 -0.5

0.5

1.0

1.5

2.0

ξ=2

Kazumi Okuyama (Shinshu U, Japan)

w

-0.5

0.5

1.0

ξ=3

1.5

2.0

w

ξ=5

Phase Transition of Anti-Symmetric Wilson Loops [email protected] N = 4 SYM

15 / 19

One-cut solution of resolvent We find the resolvent in this limit in the one-cut phase s ! p ξ (x − a)(b − cos θ) R(x) = 2x − (x − a)(x − b) − arctan π (x − b)(cos θ − a) Eivenvalues are distributed along the cut x ∈ [a, b] a, b are determined by the condition lim R(x) = 1/x x→∞

Eigenvalue density can be found by taking discontinuity across the cut s ! 2p ξ (u − a)(b − cos θ) ρ(u) = (u − a)(b − u) − 2 arctanh π π (b − u)(cos θ − a)

Kazumi Okuyama (Shinshu U, Japan)

Phase Transition of Anti-Symmetric Wilson Loops [email protected] N = 4 SYM

16 / 19

Eigenvalue density Plot of the eigenvalue density ρ(u) (we set cos θ = 1/2 in this plot) ρ(u)

ρ(u)

ρ(u)

0.6

0.6

0.6

0.5

0.5

0.5

0.4

0.4

0.4

0.3

0.3

0.3

0.2

0.2

0.2

0.1

0.1

0.1

u

u -1.0

-0.5

0.5

ξ = 0.5

1.0

-1.0

-0.5

0.5

ξ=1

1.0

u -1.0

-0.5

0.5

1.0

ξ=2

One can imagine that the support of eigenvalue density splits into two above some critical value ξ > ξc It would be interesting to find a two-cut solution and compare with the bubbling geometry Kazumi Okuyama (Shinshu U, Japan)

Phase Transition of Anti-Symmetric Wilson Loops [email protected] N = 4 SYM

17 / 19

Summary 1/N corrections to WAk can be computed systematically from the topological recursion There is a discrepancy in the 1/N correction between bulk and boundary computations In the scaling limit with ξ = genus expansion

√

λ/N fixed, WAk admits a closed sting

We conjecture that there is a phase transition between one-cut phase and two-cut phase

Kazumi Okuyama (Shinshu U, Japan)

Phase Transition of Anti-Symmetric Wilson Loops [email protected] N = 4 SYM

18 / 19

Future directions Exact results from localization allow us to study AdS/CFT correspondence beyond the planar limit 1/N corrections provide us with valuable information of quantum gravity effects in AdS We have not fully explored the quantum gravity regime even in this simple example of 1/2 BPS Wilson loops We should work hard to extract more information from the exact results!

Kazumi Okuyama (Shinshu U, Japan)

Phase Transition of Anti-Symmetric Wilson Loops [email protected] N = 4 SYM

19 / 19