Localization, Wilson Loops and Precision Tests Guillermo A Silva - PowerPoint PPT Presentation

Localization, Wilson Loops and Precision Tests Guillermo A Silva IFLP-CONICET & Departamento de Fisica, FCE, Universidad Nacional de La Plata In collaboration with J Aguilera-Damia, D Correa, A Faraggi, L Pando-Zayas, V Rathee and D Trancanelli Based on JHEP 1406 (2014) 139, JHEP 1503 (2015) 002, JHEP 1604 (2016) 053, arXiv:1802.03016, JHEP 1806 (2018) 007, 1805.00859 Workshop on Supersymmetric Localization and Holography: Black Hole Entropy and Wilson Loops ICTP, Trieste, July 9th, 2018

— AIM Make precision tests of AdS/CFT Gain insights into string perturbation theory — WHAT DO WE MEAN BY AdS/CFT? The dual pairs, N = 4 SYM in d = 4 IIB string theory ← → on AdS 5 × S 5 G = U ( N ) gauge group N = 6 SCSM in d = 3 → IIA string theory G = U k ( N ) × U − k ( N ) gauge group ← on AdS 4 × CP 3 —

Outline 1. Exact results: Susy Localization 2. Gauge theory Wilson loops bestiary: Straight lines, circles and cusps 3. AdS/CFT parameters 4. String theory and semiclassical expansion: String duals to Wilson loops. Classical worldsheet. Semiclassical expansion of string partition function Zero modes: CKV,... 1001 Determinants: Zeta function, Gelfand-Yaglom, . . . 5. Comparison of String/Gauge, Summary & Open Questions

1 Exact results in gauge theory: Susy Localization Exact results in QFT are rare and difficult to obtain, but... The situation improves if enough symmetry is present, and this is the case for N =4 SYM and N =6 SCSM. On the gauge theory side two techniques have been exploited to obtain exact results: • Integrability: exact results in planar level limit and beyond see recent work by P Vieira 4pt • Localization: exact results ∀ λ, N , but only applicable for susy objects.

2 Wilson loops bestiary: straight lines, circles, ... Susy Wilson loops require additional couplings to scalars and fermions. N =4 SYM WLs depend on two data: C ,� n �� � 1 x µ + | ˙ x | � W R [ C ,� dim [ R ] Tr R P exp φ · � n ] = ( iA µ ˙ n ) ds x µ ( s ) curve on spacetime. timelike/spacelike/null (additional i ′ s required) C : � n : n ( s ) maps every point in C to R 6 . Dictates scalar field coupling along C . � representation for charged particle. A µ , φ i in adjoint of U ( N ) , e.g.: A µ = A a R: µ T a R ( a = 1 , . . . N 2 ) n 2 = 1 ( � n ∈ S 5 ) and a relation Supersymmetry demands � x µ ( s ) ˙ ← → � n ( s ) [Maldacena, Rey-Yee, Drukker-Gross-Ooguri, Erickson-Semenoff-Zarembo, Drukker-Giombi-Ricci-Trancanelli, . . . ]

N = 4 SYM WLs 1 • C : straight line, ⇒ � n = � n 0 constant 2 - BPS (16 susies) � W R ( line ,� n 0 ) � = 1 , independent of λ, N , R When computing vev perturbatively, the result arises from an exact cancelation between gauge field and scalar propagators. Localization technique allows to compute exact vevs of some WLs: 1 • C : circle, � n = � n 0 constant ⇒ 2 - BPS . Answer is non-trivial: For R = ✷ � λ � � � 1 − λ N L 1 � W ✷ ( circle ,� n 0 ) � = exp , ∀ λ, N N − 1 4 N 8 N √ √ λ 2 2 λ √ = I 1 ( λ ) + 48 N 2 I 2 ( λ ) + 1280 N 4 I 4 ( λ ) + . . . N ≫ 1 √ λ √ �� � λ = e π − 3 2 1 λ 1 / 2 − 15 1 1 1 √ √ N = ∞ , λ ≫ 1 λ + ... , λ 3 / 4 4 64 2 π 2 π Now scalar and gauge propagators add up to a constant, summation of planar ladder diagrams give I 1 Bessel [Eriksson-Semenoff-Zarembo 99, Drukker-Gross 00, Pestun 07]

1 • C : circle, ⇒ � n θ 0 ( s ) = (0 , 0 , 0 , sin θ 0 cos s, sin θ 0 sin s, cos θ 0 ) 4 - BPS (8 susies) The WL vev was conjectured to be given by the 1 2 - expression above by making the replacement λ → λ ′ = λ cos 2 θ 0 [Drukker] √ 2 √ � W ✷ ( circle ,� n θ 0 ) � = I 1 ( λ cos θ 0 ) , N = ∞ λ cos θ 0 � √ � √ λ cos θ 0 − 3 λ − 3 2 ln cos θ 0 + 1 2 ln 2 π − . . . λ ≫ 1 = exp 2 ln , This is the result we want to reproduce from string theory computations. Equatorial loop on S 5 : � W ✷ ( circle ,� n π/ 2 ) � = 1

3 AdS/CFT parameters N = 4 SYM in d = 4 → IIB string theory G = U ( N ) gauge group ← on AdS 5 × S 5 L 2 ( g YM , N ) → λ = g 2 1 YM N , T eff = 2 πα ′ , g s N The parameters are related as � L 2 � 2 λ = α ′ λ 4 πN = g s t’ Hooft limit : keep λ fixed and take N → ∞ . When doing so the perturbative expansion in terms of λ reorganizes as planar and non-planar graphs ↔ sphere/disk and higher genus (handles) string amplits

4 String Theory The main importance of the AdS/CFT toolkit is that the gauge theory strong coupling λ ≫ 1 regime is easily studied using perturbative string theory. Why? Wilson loop gauge observable in fundamental representation relates to the string parti- tion function with the WL contour C being the boundary condition for the string � [ D g D X D Ψ] e − S string [ g,X, Ψ] � W ( C ,� n ) � = ∂X = C ,� n here � dτdσ √ g S string = T s g αβ G MN ( X ) ∂ α X M ∂ β X N + ¯ � � Ψ D Ψ 2 The crucial point is that the action becomes weighted by an effective string tension √ T eff = L 2 1 λ ↔ 1 ← L 2 : AdS radius T s = 2 πα ′ → 2 πα ′ = 2 π � At strong coupling λ ≫ 1 we perform a semiclassical expansion of the path integral. Genus expansion is weighted by g s ∼ 1 N . In t’ Hooft limit the leading contribution comes from disk topology with no handles.

Parallel lines ↔ V q ¯ q . Cusped line ↔ Bremsstrahlung. Circular loop ↔ AdS/CFT test 4 - BPS loop: non trivial embedding in S 2 ⊂ S 5 1 2 - BPS loop: constant position in S 5 . 1 N = 4 Wilson loop disk amplitude of IIB superstring ↔ with RR flux in AdS 5 × S 5 at large N

Circular Loop in Fundamental Representation Expanding the exact planar result for N = 4 in the strong coupling λ ≫ 1 regime √ √ λ − 3 λ + 1 2 ln 2 π − 3 1 ln � W 1 / 2 ( circle ) � = √ 2 ln + . . . 8 λ √ √ λ cos θ 0 − 3 λ − 3 2 ln cos θ 0 + 1 2 ln 2 ln � W 1 / 4 ( circle , θ 0 ) � = π − . . . , λ ≫ 1 , N = ∞ 2 ln On general grounds we expect: √ • λ : should arise from classical worldsheet area, once properly renormalized S ren . √ • ln λ : correction is typical of zero modes. Drukker-Gross suggested its origin can be traced to the Fadeed-Popov diffeo fixing determinant. The FP determinant has 3 z.m. in disk topology hence 3 × log λ 1 / 4 is found. Recall each zero mode contributes with � 1 / 2 . • 1 2 ln 2 π should come from measure factor of semiclassical partition function + fluctuation determinants over classical string solution. To avoid the tricky (topological) issue related to FP ghosts the natural thing to do is to compare WL with same topology. The natural observable is then √ ln � W 1 / 4 ( circle , θ 0 ) � − 3 + O ( 1 λ (cos θ 0 − 1) √ = 2 ln cos θ 0 ) � W 1 / 2 ( circle ) � λ � �� � � �� � leading 1-loop

STRING PARTITION FUNCTION TO 1-LOOP ORDER About the expression S [ X cl ] det 1 / 2 O F √ 2 π ˜ λ Z string ≈ C e − det O FP det 1 / 2 O B C : Normalization factor of the path integral (measure) (eliminated when computing W 1 / 4 W 1 / 2 ) X cl : Classical string worldsheet above which we fluctuate ( classical fermions Ψ cl = 0 ) ˜ S [ X cl ] : Action evaluated on classical solution = Area. O F : fermionic fluctuations O B : bosonic fluctuations. O FP : FP diffeo fixing + κ - fixing. ∃ worldsheet CKV = ⇒ zero modes (topological) ———

CLASSICAL 1 4 -BPS STRING SOLUTION ds 2 = L 2 � � ds 2 AdS 5 + d Ω 2 Background : 5 . AdS 5 in H 2 × S 2 foliation → EAdS 5 in H 2 × S 2 foliation via u → iu and ϑ → iϑ EAdS 5 = du 2 + cosh 2 u dρ 2 + sinh 2 ρ dψ 2 � + sinh 2 u � � dϑ 2 + sin 2 ϑ dϕ 2 � ds 2 . S 5 in S 3 × S 1 foliation: 5 = dθ 2 + sin 2 θ dφ 2 + cos 2 θ dξ 2 + cos 2 ξ dα 2 1 + sin 2 ξ dα 2 � � d Ω 2 , 2 � �� � Ω 3 ——— String embedding : depends on latitude θ 0 and position Ω ( 0 ) in S 3 sinh ρ sin θ 0 ρ = σ, Ω 3 = Ω ( 0 ) = cte, sin θ ( ρ ) = u = 0 , ψ = φ = τ, cosh ρ + cos θ 0 Homogeneity of S 3 implies independence of Ω ( 0 ) in all physical quentities.

(from now on we set L 2 = 1 ) Induced geometry : is asymptotic to EAdS 2 ∀ θ 0 sin 2 θ 0 ds 2 = M ( ρ )( dρ 2 + sinh 2 ρ dφ 2 ) , M ( ρ ) = 1 + (cosh ρ + cos θ 0 ) 2 Worldsheets have disk topology ⇒ 3 CKV ∀ slns. On shell action ≡ Worldsheet area is divergent � R sinh 2 R e R − cos θ 0 + O ( e − R ) � � ˜ cos θ 0 + cosh R ≈ 2 π S = 2 π sinh ρ dρ M ( ρ ) = 2 π 0 � 1 Adding boundary Euler χ b = ds κ g (or performing Legendre transform), effectively 2 π eliminates the divergent piece leaving a negative regularized area: S reg = ˜ ˜ S − χ b = − 2 π cos θ 0 ——— This result successfully matches the leading order localization result √ � W 1 / 4 � √ � S reg [0]) = e 2 π ( ˜ λ S reg [ θ 0 ] − ˜ leading ≈ e − λ (cos θ 0 − 1) � � � W 1 / 2 �