’t Hooft Expansion of 1 / 2 BPS Wilson Loop Kazumi Okuyama JHEP 0609 (2006) 007 – p. 1/ ??
Stringy Geometry? Target space picture of string theory at g s � = 0 might be quite different from that at g s = 0 In general, it is extremely difficult to study string theory at finite coupling Exceptions: topological string, c ≤ 1 string All order results in g s expansion are available Toplological string is reformulated as a statistical mechanics of melting “Calabi-Yau crystal” lattice spacing = inverse temperature = g s Q: What is the spacetime picture of (non-topological) string theory? – p. 2/ ??
AdS/CFT at finite g s Type IIB string on AdS 5 × S 5 ⇔ N = 4 SU ( N ) SYM on R 4 g s 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 – p. 3/ ??
1/2 BPS Wilson Loops 1 / 2 BPS Wilson loop has the form �� � x µ + φ I θ I | ˙ W = Tr P exp dt iA µ ˙ x | C 1 / 2 BPS − → C is a straight line or a circle θ I is a constant unit vector Expectation value � W � � W � = 1 straight line : � W � is a non-trivial function of circle : ’t Hooft coupling λ = g 2 YM N – p. 4/ ??
1/2 BPS Circular Loop Perturbative Calculation of � W circle � (Erickson-Semenoff-Zarembo) Propagator is a constant (independent of two end points) x ( s ) · ˙ x ( t ) − | ˙ x ( s ) || ˙ x ( t ) | ˙ = − 1 x ( t ) | x ( s ) − x ( t ) | 2 2 x ( s ) conjecture: diagrams with internal vertices vanish = 0 – p. 5/ ??
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 �� � � W circle � = 1 � dMe − Tr M 2 1 λ N Tr exp 2 N M Z – p. 6/ ??
1/2 BPS Loop at Finite g s 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 g YM dependence only comes from the anomaly conformal transform: strainght line − → circle Finite N result is given by a Laguerre polynomial � W circle � = 1 N e g 2 YM / 8 L 1 N − 1 ( − g 2 YM / 4) What does the ’t Hooft expansion of � W circle � look like? – p. 7/ ??
’t Hooft Expansion Large N expansion of Yang-Mills ⇒ triangulated worldsheet 1 � Tr F 2 µν + · · · S = 2 g 2 YM propagator ( P ): g 2 YM vertex ( V ): g − 2 χ = V − P + h = 2 − 2 g YM hole ( h ) : N N h = g 2( P − V − h ) YM N ) h = g 2 g − 2 g 2( P − V ) ( g 2 λ h s YM YM We define string coupling and ’t Hooft coupling as g s = g 2 λ = g 2 YM / 4 , YM N – p. 8/ ??
String Loop Expansion W = � W circle � is written as a contour integral � λ � � 2 z + g s W = 2 dz exp 2 coth( g s z ) This is expanded in Buchholtz polynomials and modified Bessel functions √ ∞ I n +1 ( λ ) � λ ) n +1 g n √ W = 2 s p n ( g s ) ( n =0 Buchholtz polynomial p n is defined by ∞ � a � �� coth x − 1 � x n p n ( a ) = exp 2 x n =0 – p. 9/ ??
Expansion in terms of Number of Holes It is interesting to reorganize W as an expansion in number of holes ∞ � N h F h ( g s ) W = h =0 This is easily found from the expression k ∞ g k � � 1 + N � � W = e − gs s 2 ( k + 1)! j j =1 k =0 For instance, the h = 0 term is F 0 ( g s ) = 2 � g s � sinh g s 2 – p. 10/ ??
Curious Observation (I) Remarkably, we find that the number of holes increases by one when convolving the h = 0 term g s F h +1 = ( g s F 0 ) ∗ F h One can show from this relation that F h ( g s ) is analytic in g s The physical origin of this recursion relation in not clear.... – p. 11/ ??
Curious Observation (II) We can “turn on” the string coupling g s from g s = 0 � � �� ∂ W ( λ, g s ) = exp g s H 2 g s W ( λ, g s = 0) ∂λ � � H ( x ) = 1 coth x − 1 2 x √ W ( λ, g s = 0) = 2 I 1 ( λ ) √ λ There is an analogous relation in topological string ⇒ next section – p. 12/ ??
Topological String on Conifold Gauge/String duality in topological theory Chern-Simons theory on S 3 ⇔ topological string on conifold Partition function ∞ � (1 − e − t − ng s ) n Z = n =0 t = g s N = Kahler moduli of CP 1 Free energy (genus ≥ 1 part) ∞ B 2 g � g 2 g − 2 2 g (2 g − 2)!Li 3 − 2 g ( e − t ) F ( t, g s ) = log Z = s g =1 – p. 13/ ??
Topological String on Conifold One can easily show that F ( t, g s ) is obtained from the genus one term F 1 ( t ) F ( t, g s ) = K ( g s ∂ t ) F 1 ( t ) F 1 ( t ) = − 1 12 log(1 − e − t ) � ∞ p K ( x ) = 24 dp e 2 πp − 1 cos( xp ) 0 The above expression is written as an integral of F 1 ( t ) with shifted ’t Hooft parameter � ∞ p � � F ( t, g s ) = 12 dp F 1 ( t + ig s p ) + F 1 ( t − ig s p ) e 2 πp − 1 0 – p. 14/ ??
Origin of Calabi-Yau Crystal? Topological A-model has a melting crystal description � e − g s E Z = 3D partition As a consequence, Z admits a q -expansion with q = e − g s 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 ) √ λ For the Wilson loop case, worldsheet instanton factor is e ⇒ W doesn’t have a q -expansion – p. 15/ ??
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 g s by applying a differential operator of ’t Hooft coupling This implies that the g s dependence is closely tied to the ’t Hooft coupling dependence of g s = 0 term, especially the form of worldsheet instanton √ λ : e − S inst = e N = 4 SYM e − S inst = e − t : Chern-Simons theory on S 3 – p. 16/ ??
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