Phase Transition of Anti-Symmetric Kazumi Okuyama Shinshu U, Japan - - PowerPoint PPT Presentation

Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM

Kazumi Okuyama

Shinshu U, Japan

Workshop@Michigan based on my recent paper [arXiv:1709.04166]

Kazumi Okuyama (Shinshu U, Japan) Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM Workshop@Michigan 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 × S5 In the ’t Hooft large N limit, parameters on the bulk side are given by λ = g2

YMN

⇔ LAdS = λ1/4ℓs, gstring ∼ 1/N We will focus on the 1/2 BPS Wilson loop in N = 4 SYM WR =

• TrR P exp
• ds(iAµ ˙

xµ + Φ|˙ x|)

• Kazumi Okuyama (Shinshu U, Japan)

Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM Workshop@Michigan 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-Semenofg-Zarembo, Pestun] WR = 1 Z

• dM e−2NTrM2 TrRe

√ λM

Via AdS/CFT correspondence, various representations correspond to: fundamental rep ⇔ fundamental string

[Maldacena, Rey-Yee]

symmetric rep ⇔ D3-brane

[Drukker-Fiol]

anti-symmetric rep ⇔ D5-brane

[Yamaguchi, Hartnoll-Kumar] Kazumi Okuyama (Shinshu U, Japan) Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM Workshop@Michigan 3 / 19

Anti-symmetric representation and D5-brane

kth anti-symmetric representation Ak ⇔ D5-brane wrapping AdS2 × S4 with k unit of electric fmux Leading term in 1/N expansion matches the DBI action of D5-brane 1 N log WAk ≈ 2 √ λ sin3 θk 3π , πk N = θk − sin θk cos θk

• 1/λ corrections can be systematically computed from

the low temperature expansion of Fermi distribution function [Horikoshi-KO] 1 N log WAk ≈ 2 √ λ sin3 θk 3π + π sin θk 3 √ λ + · · ·

Kazumi Okuyama (Shinshu U, Japan) Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM Workshop@Michigan 4 / 19

Fermi gas picture

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

N

• k=0

WAkzk = 1 Z

• dMe−2NTrM2 det(1 + ze

√ λM)

Exact form of P(z) was found in [Fiol-Torrents] P(z) = det

• δj

i + zLj−i i−1

• − λ

4N

• e

λ 8N

• i,j=1,··· ,N

WAk with fjxed k can be recovered from P(z) WAk =

• dz

2πizk+1 P(z)

Kazumi Okuyama (Shinshu U, Japan) Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM Workshop@Michigan 5 / 19

1/N correction

1/N correction to WAk has been studied from the bulk side [Faraggi et al] −SD5 = 2N √ λ sin3 θk 3π −1 6 log sin θk + · · · 1/N correction on the matrix model side was recently computed [Gordon] log WAk = 2N √ λ sin3 θk 3π + λ sin4 θk 8π2 + · · · 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 N = 4 SYM Workshop@Michigan 6 / 19

Two approaches

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

√ λM)

• 1. Treat det(1 + ze

√ λM) as an operator in the Gaussian matrix model

• 2. Treat det(1 + ze

√ λM) as a part of potential

These two approaches lead to the same result as long as λ ≪ N2 We will follow the fjrst approach log P(z) =

∞

• h=1

1 h!

• Tr log(1 + ze

√ λM)

h

connected

Kazumi Okuyama (Shinshu U, Japan) Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM Workshop@Michigan 7 / 19

Small λ expansion

Small λ expansion of P(z) can be obtained from the perturbative calculation in the Gaussian matrix model log P(z) = N

• log(1 + z) +

z 8(1 + z)2 λ + z(1 − 4z + z2) 192(1 + z)4 λ2 + O(λ3)

• +

z2 8(z + 1)2 λ − z2(2z − 3) 64(z + 1)4 λ2 + O(λ3) + 1 N

• z
• 1 − 4z + 13z2

384(z + 1)4 λ2 + O(λ3)

• + O(1/N2)

This agrees with the small λ expansion of the exact result in [Fiol-Torrents]

Kazumi Okuyama (Shinshu U, Japan) Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM Workshop@Michigan 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 resolvent Tr

1 x−M is given by [Eynard-Orantin]

4x1Wg,h(x1, · · · , xh) =Wg−1,h+1(x1, x1, x2, · · · , xh) + 4δg,0δh,1 +

• I1⊔I2={2,··· ,h}

g

• g′=0

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

h

• j=2

∂ ∂xj Wg,h−1(x1, · · · , xj, · · · , xh) − Wg,h−1(x2, · · · , xh) x1 − xj ,

Kazumi Okuyama (Shinshu U, Japan) Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM Workshop@Michigan 9 / 19

1/N corrections from topological recursion

1/N correction Jg,h of P(z) is obtained from Wg,h by replacing

1 x−u → f (u, z) = log(1 + ze √ λu)

log P(z) =

∞

• h=1

∞

• g=0

1 h!N2−2g−hJg,h(z) The fjrst two terms are given by J0,1 = 2 π 1

−1

du

• 1 − u2f (u, z),

J0,2 = 1 4π2 1

−1

du 1

−1

dv 1 − uv

• (1 − u2)(1 − v2)

f (u, z) − f (v, z) u − v 2 Above J0,2 reproduces the 1/N correction found in [Gordon]

Kazumi Okuyama (Shinshu U, Japan) Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM Workshop@Michigan 10 / 19

Higher order corrections

We can compute 1/N corrections up to any desired order from the topological recursion log P(z) = NJ0,1(z) + 1 2J0,2(z) + 1 N

• J1,1(z) + 1

3!J0,3(z)

• + · · ·

The order O(1/N) term is given by J1,1(z) = 1 48π 1

−1

du 2u2 − 1 √ 1 − u2 ∂2

uf (u, z),

J0,3(z) = 1 8π3 1

−1

du 1

−1

dv 1

−1

dw u + v + w + uvw

• (1 − u2)(1 − v2)(1 − w2)

× ∂uf (u, z)∂vf (v, z)∂wf (w, z) This reproduces the small λ expansion of the exact result

Kazumi Okuyama (Shinshu U, Japan) Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM Workshop@Michigan 11 / 19

Novel scaling limit

Result of J. Gordon suggests that we can take a scaling limit N, λ → ∞ with ξ = √ λ N fjxed log WAk = O(N √ λ) ⇒ log WAk = O(N2ξ) In this limit, WAk admits a closed string genus expansion log WAk =

∞

• g=0

N2−2gSg(ξ) The genus-zero term S0 is given by S0 = 2ξ 3π sin3 θk + ξ2 8π2 sin4 θk + ξ3 48π3 sin3 θk + · · ·

Kazumi Okuyama (Shinshu U, Japan) Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM Workshop@Michigan 12 / 19

Plot of log WAk for N = 300, ξ = 1

0.2 0.4 0.6 0.8 1.0 k /N 0.05 0.10 0.15 0.20

Green dashed curve :

2ξ 3π sin3 θk

Gray dashed curve :

2ξ 3π sin3 θk + ξ2 8π2 sin4 θk

Orange curve : exact value of

1 N2 log WAk

Kazumi Okuyama (Shinshu U, Japan) Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM Workshop@Michigan 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

• ne-cut phase (ξ < ξc)

↔ two-cut phase (ξ > ξc) This might correspond to an exchange of dominance of two topologically difgerent geometries on the bulk side

Kazumi Okuyama (Shinshu U, Japan) Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM Workshop@Michigan 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) = 2w2 − ξ(w − cos θ)Θ(w − cos θ), z = e−

√ λ cos θ

V (w) develops a new minimum as we increase ξ

• 0.5

0.5 1.0 1.5 2.0 w 1 2 3 4 5 6 V(w)

ξ = 2

• 0.5

0.5 1.0 1.5 2.0 w 1 2 3 4 V(w)

ξ = 3

• 0.5

0.5 1.0 1.5 2.0 w

• 0.5

0.5 1.0 V(w)

ξ = 5

Kazumi Okuyama (Shinshu U, Japan) Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM Workshop@Michigan 15 / 19

One-cut solution of resolvent

We fjnd the resolvent in this limit in the one-cut phase R(x) = 2x −

• (x − a)(x − b) − ξ

πarctan

• (x − a)(b − cos θ)

(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

Eigenvalue density can be found by taking discontinuity across the cut ρ(u) = 2 π

• (u − a)(b − u) − ξ

π2 arctanh

• (u − a)(b − cos θ)

(b − u)(cos θ − a)

• Kazumi Okuyama (Shinshu U, Japan)

Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM Workshop@Michigan 16 / 19

Eigenvalue density

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

• 1.0
• 0.5

0.5 1.0 u 0.1 0.2 0.3 0.4 0.5 0.6 ρ(u)

ξ = 0.5

• 1.0
• 0.5

0.5 1.0 u 0.1 0.2 0.3 0.4 0.5 0.6 ρ(u)

ξ = 1

• 1.0
• 0.5

0.5 1.0 u 0.1 0.2 0.3 0.4 0.5 0.6 ρ(u)

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 fjnd a two-cut solution and compare with the bubbling geometry

Kazumi Okuyama (Shinshu U, Japan) Phase Transition of Anti-Symmetric Wilson Loops in N = 4 SYM Workshop@Michigan 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 = √ λ/N fjxed, WAk admits a closed sting genus expansion 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 in N = 4 SYM Workshop@Michigan 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 efgects 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 in N = 4 SYM Workshop@Michigan 19 / 19