t Hooft Expansion of 1 / 2 BPS Wilson Loop Kazumi Okuyama JHEP - - PowerPoint PPT Presentation

’t Hooft Expansion

• f

1/2 BPS Wilson Loop Kazumi Okuyama JHEP 0609 (2006) 007

– p. 1/??

Stringy Geometry?

Target space picture of string theory at gs = 0 might be quite different from that at gs = 0 In general, it is extremely difficult to study string theory at finite coupling Exceptions: topological string, c ≤ 1 string All order results in gs expansion are available Toplological string is reformulated as a statistical mechanics of melting “Calabi-Yau crystal” lattice spacing = inverse temperature = gs Q: What is the spacetime picture of (non-topological) string theory?

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AdS/CFT at finite gs

Type IIB string on AdS5 × S5 ⇔ N = 4 SU(N) SYM on R4 gs expansion ⇔ ’t Hooft expansion (1/N expansion) Q: Is there any exact finite N result in N = 4 SYM ? A: Yes. Expectation value of 1/2 BPS Wilson loop (Erickson-Semenoff-Zarembo, Drukker-Gross) String worldsheet Σ with ∂Σ = C ⇔ Wilson loop W(C) Σ C

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1/2 BPS Wilson Loops

1/2 BPS Wilson loop has the form W = TrP exp

• C

dt iAµ ˙ xµ + φIθI| ˙ x|

• 1/2 BPS −

→ C is a straight line or a circle θI is a constant unit vector Expectation value W straight line : W = 1 circle : W is a non-trivial function of ’t Hooft coupling λ = g2

YMN

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1/2 BPS Circular Loop

Perturbative Calculation of Wcircle (Erickson-Semenoff-Zarembo) Propagator is a constant (independent of two end points) x(t) x(s) ˙ x(s) · ˙ x(t) − | ˙ x(s)|| ˙ x(t)| |x(s) − x(t)|2 = −1 2 conjecture: diagrams with internal vertices vanish

= 0

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1/2 BPS Circuler Loop

Only ladder diagrams contribute at the planar level Since propagator is constant, summation of ladder diagrams is reproduced by a Gaussian matrix model Wcircle = 1 Z

• dMe−TrM 2 1

N Tr exp

• λ

2N M

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1/2 BPS Loop at Finite gs

Drukker and Gross argued that the above matrix model result is exact at finite N Evidence The above argument for the reduction to matrix model applies also for the non-planar diagrams gYM dependence only comes from the anomaly conformal transform: strainght line − → circle Finite N result is given by a Laguerre polynomial Wcircle = 1 N eg2

YM/8L1

N−1(−g2 YM/4)

What does the ’t Hooft expansion of Wcircle look like?

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’t Hooft Expansion

Large N expansion of Yang-Mills ⇒ triangulated worldsheet S = 1 2g2

YM

• Tr F 2

µν + · · ·

propagator (P): g2

YM

vertex (V ): g−2

YM

χ = V − P + h = 2 − 2g hole (h) : N g2(P−V )

YM

N h = g2(P−V −h)

YM

(g2

YMN)h = g2g−2 s

λh We define string coupling and ’t Hooft coupling as gs = g2

YM/4,

λ = g2

YMN

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String Loop Expansion

W = Wcircle is written as a contour integral W = 2

• dz exp

λ 2z + gs 2 coth(gsz)

• This is expanded in Buchholtz polynomials and modified

Bessel functions W = 2

∞

• n=0

In+1( √ λ) ( √ λ)n+1 gn

s pn(gs)

Buchholtz polynomial pn is defined by

∞

• n=0

xnpn(a) = exp a 2

• coth x − 1

x

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Expansion in terms of Number of Holes

It is interesting to reorganize W as an expansion in number of holes W =

∞

• h=0

N hFh(gs) This is easily found from the expression W = e− gs

2

∞

• k=0

gk

s

(k + 1)!

k

• j=1
• 1 + N

j

• For instance, the h = 0 term is

F0(gs) = 2 gs sinh gs 2

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Curious Observation (I)

Remarkably, we find that the number of holes increases by

• ne when convolving the h = 0 term

gsFh+1 = (gsF0) ∗ Fh One can show from this relation that Fh(gs) is analytic in gs The physical origin of this recursion relation in not clear....

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Curious Observation (II)

We can “turn on” the string coupling gs from gs = 0 W(λ, gs) = exp

• gsH
• 2gs

∂ ∂λ

• W(λ, gs = 0)

H(x) = 1 2

• coth x − 1

x

• W(λ, gs = 0) = 2I1(

√ λ) √ λ There is an analogous relation in topological string ⇒ next section

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Topological String on Conifold

Gauge/String duality in topological theory Chern-Simons theory on S3 ⇔ topological string on conifold Partition function Z =

∞

• n=0

(1 − e−t−ngs)n t = gsN = Kahler moduli of CP1 Free energy (genus≥ 1 part) F(t, gs) = log Z =

∞

• g=1

g2g−2

s

B2g 2g(2g − 2)!Li3−2g(e−t)

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Topological String on Conifold

One can easily show that F(t, gs) is obtained from the genus

• ne term F1(t)

F(t, gs) = K(gs∂t)F1(t) F1(t) = − 1 12 log(1 − e−t) K(x) = 24 ∞ dp p e2πp − 1 cos(xp) The above expression is written as an integral of F1(t) with shifted ’t Hooft parameter F(t, gs) = 12 ∞ dp p e2πp − 1

• F1(t + igsp) + F1(t − igsp)
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Origin of Calabi-Yau Crystal?

Topological A-model has a melting crystal description Z =

• 3D partition

e−gsE As a consequence, Z admits a q-expansion with q = e−gs From our perspective, this is related to the following two facts Worldsheet instanton factor is e−t The p-integral has poles at p = in (n ∈ Z) For the Wilson loop case, worldsheet instanton factor is e

√ λ

⇒ W doesn’t have a q-expansion

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Summary

1/2 BPS circular Wilson loop in N = 4 SYM is solved exactly by a matrix model ’t Hooft expansion of W has curious properties ⇒ it is better to understand the physical meaning from the string theory side One can turn on gs by applying a differential operator of ’t Hooft coupling This implies that the gs dependence is closely tied to the ’t Hooft coupling dependence of gs = 0 term, especially the form of worldsheet instanton e−Sinst = e

√ λ :

N = 4 SYM e−Sinst = e−t : Chern-Simons theory on S3

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