Introduction AdS/CFT correspondence N = 4 SYM type IIB superstring Wilson loop area of world-sheet Wilson loop + heavy local operator area of deformed world-sheet Zarembo s solution (1/2 BPS Wilson Loop) today s talk... 1/4 BPS Wilson Loop
Local Operator O J Boundary of AdS 5 Wilson Loop WC W C O J exp[ (S bulk + boundary term)]
Plan of Talk 1 Introduction 2 Calculation in Gauge Theory Operators Correlation Function 3 Calculation in String Theory Action and Solution of EOM Supersymmetry Semi-Classical Evaluation of Path Integral 4 Another Saddle Point Another Saddle Point 5 Summary and Future Works
Operators Operators 1/4 BPS Wilson loop operator W C = 1 N trp exp C (ia µ ẋ µ + Φ i Θ i ẋ ) dσ x(σ) = (r cos σ, r sin σ, 0, 0) r... radius of the loop Θ(σ) = (sin θ 0 cos σ, sin θ 0 sin σ, cos θ 0, 0, 0, 0) chiral primary operator with R-charge J O J = tr(φ 3 + iφ 4 ) J W C O J 1/8 SUSY 1/2 BPS K. Zarembo, Nucl. Phys. B643 (2002) 157-171 N. Drukker, JHEP 09 (2006) 004 G. W. Semenoff and D. Young, Phys. Lett. B643 (2006) 195-204
Correlation Function Correlation Function correlation function sum of ladder diagrams W C O J W C r J λ (l 2 + r 2 ) J J I J( λ ) I 1 ( λ ) λ λ cos θ 0 λ... t Hooft coupling I J ( λ )... J-th modified Bessel function r... radius of the loop l... distance W C O J G. W. Semenoff and K. Zarembo, Nucl. Phys. B616 (2001) 34-46 G. W. Semenoff and D. Young, Phys. Lett. B643 (2006) 195-204
Correlation Function saddle point analysis λ, J, j J/ λ fixed ( j j cos θ 0 ) I J ( λ ) = 1 +πi exp λ 2πi (cosh x j x)dx πi exp λ [ j 2 + 1 + j ln( j 2 + 1 j )] saddle points... x = ln( j 2 + 1 + j ), ln( j 2 + 1 + j ) + πi Result: W C O J r J (l 2 + r 2 ) J exp λ [ j 2 + 1 + j ln( j 2 + 1 j )] K. Zarembo, Phys. Rev. D66 (2002) 105021
Action and Solution of EOM Action and Solution of EOM string action S = 1 4πα G MN (Ẋ M Ẋ N + X M X N ) dτ E dσ AdS 5 S 5 metric ds 2 = L 2 [ cosh 2 ρdt 2 E +dρ 2 +sinh 2 ρ(dϕ 2 1 +sin 2 ϕ 1 (dϕ 2 2 +sin 2 ϕ 2 dϕ 2 3))+ + dθ 2 + sin 2 θdφ 2 + cos 2 θ(dχ 2 1 + sin 2 χ 1 (dχ 2 2 + sin 2 χ 2 dχ 2 3)) ]
Action and Solution of EOM ansatz AdS 5 : t E = t E (τ E ), ρ = ρ(τ E ), ϕ 1 = ϕ 2 = π 2, ϕ 3 = σ S 5 : θ = θ(τ E ), φ = σ, χ 1 = χ 2 = π 2, χ 3 = χ 3 (τ E ) Z 1 = L sinθ cosφ Z 2 = L sinθ sinφ Z 3 = L cosθ sinχ 3 Z 4 = L cosθ cosχ 3 L Z 2 sinθ = sinθ( ) 0 sinθ( ) Z 1 E = E 0
Action and Solution of EOM λ S = 2 ( cosh 2 ρ t E 2 + ρ 2 + sinh 2 ρ + θ 2 + sin 2 θ + cos 2 θ χ 2 3 ) dτ E EOM (j J/ λ) Virasoro constraint d (cosh 2 ρṫ E ) = 0 cosh 2 ρṫ E = j dτ E ρ sinh ρ cosh ρ(ṫe 2 1) = 0 θ + sin θ cos θ( χ 2 3 1) = 0 d dτ E (cos 2 θ χ 3 ) = 0 cos 2 θ χ 3 = ij cosh 2 ρ t E 2 + ρ 2 + θ 2 + cos 2 θ χ 2 3 = sinh 2 ρ + sin 2 θ
Action and Solution of EOM solution of EOM t E = jτ E 1 2 ln cosh( j 2 + 1τ E + ξ) cosh( j 2 + 1τ E ξ) j 2 + 1 sinh ρ = sinh( j 2 + 1τ E ) j 2 + 1 sin θ = cosh( j 2 + 1(τ E + τ 0 )) χ 3 = ijτ E + i 2 ln sinh( j 2 + 1(τ E + τ 0 ) ξ) sinh( j 2 + 1(τ E + τ 0 ) + ξ) + i 2 ( ξ ln( j 2 + 1 + j) ) sinh( j 2 + 1τ 0 + ξ) ln sinh( j 2 + 1τ 0 ξ)
Supersymmetry Supersymmetry Killing spinor equation for AdS 5 S 5 D M... covariant derivative ( D M + ε ) ( ) 2L Γ ɛ1 ˆΓ M = 0 ɛ 1, ɛ 2... 32-components Majorana-Weyl spinors ˆΓ M... 10-dimensional gamma matrices Γ... a product of all gamma matrices in AdS ( ) ( ) 5 ɛ1 ɛ2 ε... ε = iσ 2 : ε = ɛ 2 ɛ 1 ɛ 2
Supersymmetry solution of Killing spinor equation ( ɛ1e ɛ 2E ) ( ) ( ) ( ) ( 1 1 1 = exp 2 ργ Γ 1 ε exp 2 t EΓ Γ E ε exp 2 ϕ 3Γ 14 exp 1 ) 2 θγ 1γ ε ( ) ( 1 exp 2 φγ 12 exp 1 )( ) 2 χ η1e 3γ 5 γ ε η 2E Γ µ (µ = E, 1, 2, 3, 4)... gamma matrices in AdS 5 γ m (m = 1, 2, 3, 4, 5)... gamma matrices in S 5 γ... a product of all gamma matrices in S 5 η 1E, η 2E... constant spinors
Supersymmetry projection ( ) i τe X M σ X N e a ˆΓ det g M a en b ˆΓ ɛ1e b σ 3 = g... induced metric on world-sheet em a... vielbein ( ) ɛ1e σ 3... σ 3 : σ 3 ɛ 2E = ( ɛ1e ɛ 2E ) ɛ 2E ( ɛ1e ɛ 2E )
Supersymmetry three projections (Γ E + iγ 5 ) (Γ 14 + γ 12 ) ( η1e η 2E ( η1e η 2E ( η1e (sin θ 0 γ 1 Γ 4 + cos θ 0 Γ 14 ) η 2E ) = 0 ) = 0 ) = i Our solution is 1/8 BPS. ( η1e η 2E )
Semi-Classical Evaluation of Path Integral Semi-Classical Evaluation of Path Integral Local Operator O J Boundary of AdS 5 Wilson Loop WC W C O J exp[ (S bulk + boundary term)]
Semi-Classical Evaluation of Path Integral Coordinates transformations, global Poincaré: Using isometry: Ỹ = re t E cosh ρ = Y, ret E tanh ρ = R l 2 + r 2 l 2 + R(τ E ) 2 + Y (τ E ) 2 Y (τ E) ( X 1, X 2 ) = X 3 = 0 l 2 + r 2 l 2 + R(τ E ) 2 + Y (τ E ) 2 R(τ E)(cos σ, sin σ) X 4 (l 2 + r 2 )l = l 2 + R(τ E ) 2 + Y (τ E ) 2 + l
Semi-Classical Evaluation of Path Integral Table: Boundaries τ E 0 τ E Ỹ (τ E ) 0 0 ( X 1 (τ E ), X 2 (τ E )) r(cos σ, sin σ) 0 X 3 0 0 X 4 (τ E ) 0 l r... radius of the Wilson loop l... distance W C O J
Semi-Classical Evaluation of Path Integral ~ Y Local Operator ~ 4 X l r X ~ 2 Wilson loop r X ~ 2 X ~ 1 X ~ 1
Semi-Classical Evaluation of Path Integral ~ Y cut-off X ~ 2 X ~ 1
Semi-Classical Evaluation of Path Integral evaluation of bulk action λ τ+ S bulk = ( cosh 2 2 ρt E + ρ 2 + sinh 2 ρ + 2 θ 2 + sin 2 θ + cos 2 θ χ 2 3 ) dτ E τ ( 1 λ ) j τ 2 + cos 2 θ 0 τ, τ +... cut-off
Semi-Classical Evaluation of Path Integral boundary term Drukker-Gross-Ooguri prescription L u u λ = τ=τ τ ( u 1 ) Ỹ L... Lagrangian N. Drukker, D. Gross, and H. Ooguri, Phys. Rev. D60 (1999) 125006
Semi-Classical Evaluation of Path Integral vertex operator O J Ỹ J ln Ỹ 2 + R l 2 τ+ = J[ ln(l 2 + r 2 ) + jτ + + ln r + ln( j 2 + 1 j)], J ln(cos θe iχ3 ) τ+ = J[ jτ + + ln( j 2 + 1 j ) ln( j 2 + 1 j)] ( j j ), R2 cos θ l ( X 1 ) 2 + ( X 2 ) 2 + ( X 3 ) 2 + ( X 4 l) 2 0 R. Roiban and A. A. Tseytlin, Phys. Rev. D82 (2010) 106011
Semi-Classical Evaluation of Path Integral Result: r J (l 2 + r 2 ) J exp λ [ j 2 + 1 + j ln( j 2 + 1 j )] Zarembo s result: exp λ[ j 2 + 1 + j ln( j 2 + 1 j)] ( λ λ cos θ 0, j j ) cos θ 0 K. Zarembo, Phys. Rev. D66 (2002) 105021 D. Berenstein, R. Corrado, W. Fischler, and J. Maldacena, Phys. Rev. D59 (1999) 105023
Another Saddle Point Another Saddle Point another solution for EOM j 2 + 1 sin θ = cosh j 2 + 1( τ E + τ 0 ) χ 3 = ijτ E + i 2 ln sinh( j 2 + 1( τ E + τ 0 ) ξ) sinh( j 2 + 1( τ E + τ 0 ) + ξ) + i 2 ( ξ ln( j 2 + 1 + j) ) sinh( j 2 + 1τ 0 + ξ) ln sinh( j 2 + 1τ 0 ξ)
Another Saddle Point evaluation of bulk action vertex operator S bulk ( 1 λ + ) j τ 2 + cos 2 θ 0 J ln(cos θe iχ3 ) τ+ = πij + J[ jτ + + ln( j 2 + 1 + j ) ln( j 2 + 1 j)] Result: r J (l 2 + r 2 ) J ( 1)J exp λ [ j 2 + 1 + j ln( j 2 + 1 + j )]
Another Saddle Point behavior of W C O J I J ( λ ) = 1 +πi exp λ 2πi (cosh x j x)dx πi Another saddle point does not exist on the stationary phase contour: Result: x = ln( j 2 + 1 + j ) + πi. r J (l 2 + r 2 ) J ( 1)J exp λ [ j 2 + 1 + j ln( j 2 + 1 + j )]
Summary and Future Works Summary gravity solution for correlation function supersymmetry... coincide the solution reproduces the large-r behavior of the correlation function another gravity solution another saddle point of Bessel function Future Works D-brane version exact calculation in gauge theory