Handling Handles: Non-Planar AdS/CFT Integrability T ILL B ARGHEER Leibniz Universität Hannover & DESY Hamburg 1711.05326 : TB, J. Caetano, T. Fleury, S. Komatsu, P. Vieira 1809.09145 : TB, J. Caetano, T. Fleury, S. Komatsu, P. Vieira 18xx.xxxxx : TB, F. Coronado, P. Vieira 18xx.xxxxx : TB, F. Coronado, V. Gonçalves, P. Vieira + further work in progress ENS/S ACLAY I NTEGRABILITY M EETING S ACLAY , O CTOBER 2018
Invitation String Theory: String amplitudes are integrals over the moduli space of Riemann surfaces of various genus. Large- N c Gauge Theory: Correlation functions are sums over double-line Feynman (ribbon) graphs of various genus. AdS/CFT: These two quantities/concepts should be the same. Question: How does the continuous worldsheet moduli space integration emerge from the discrete sum over Feynman graphs? Answering this questions is crucial for understanding the nature of holography. Till Bargheer — Handling Handles — ENS/Saclay Seminar — 12 October 2018 1 / 33
Invitation String Theory: String amplitudes are integrals over the moduli space of Riemann surfaces of various genus. Large- N c Gauge Theory: Correlation functions are sums over double-line Feynman (ribbon) graphs of various genus. AdS/CFT: These two quantities/concepts should be the same. Question: How does the continuous worldsheet moduli space integration emerge from the discrete sum over Feynman graphs? Answering this questions is crucial for understanding the nature of holography. [ TB, Caetano, Fleury Komatsu, Vieira ’17 ][ Komatsu, Vieira ’18 ] TB, Caetano, Fleury This Talk: Provide one concrete realization. Initial motivation: Compute non-planar corrections to correlators using [ Basso,Komatsu Vieira ’15 ][ Fleury ’16 Komatsu ] hexagon form factors, building on planar methods/results. Along the way understood that the necessary sum over worldsheet tessellations quantizes the string moduli space integration. [ ’03 ’04 ’04 ’05 ][ Gopakumar Aharony, Komargodski ] In this respect, a finite-coupling extension of Razamat ’06 [ Aharony, David, Gopakumar Komargodski, Razamat ’07 ][ Razamat ’08 ] ideas by Gopakumar, Razamat et al. Till Bargheer — Handling Handles — ENS/Saclay Seminar — 12 October 2018 1 / 33
Illustration Four strings scattering: Moduli space ↔ Strebel graphs. Discretization. Till Bargheer — Handling Handles — ENS/Saclay Seminar — 12 October 2018 2 / 33
Planar Limit & Genus Expansion [ 1974 ] Gauge theory with adjoint matter in the large N c limit: ’t Hooft ◮ Feynman diagrams are double-line diagrams. ◮ All color lines (propagators and traces) form closed oriented loops. ◮ Can unambiguously assign an oriented disk (face) to each loop. ◮ Obtain a compact oriented surface (operators form boundaries). ◮ The genus of the surface defines the genus of the diagram. Correlators of single-trace operators: Count powers of N c and g 2 YM for propagators ( ∼ g 2 YM ), vertices ( ∼ 1/ g 2 YM ), and faces ( ∼ N c ), define λ = g 2 YM N c , use Euler formula: ∞ 1 1 G ( g ) ∑ �O 1 . . . O n � = n ( λ ) O i = Tr ( Φ 1 Φ 2 . . . ) N n − 2 N 2 g c g = 0 c 1 + 1 + 1 ∼ + . . . N 2 N 4 N 6 c c c Till Bargheer — Handling Handles — ENS/Saclay Seminar — 12 October 2018 3 / 33
Proposal Concrete and explicit realization [ TB, Caetano, Fleury Komatsu, Vieira ’17 ][ TB, Caetano, Fleury Komatsu, Vieira ’18 ] of the general genus expansion: √ k i � Q 1 . . . Q n � = ∏ n 1 S ◦ ∑ i = 1 × N n − 2 N 2 g ( Γ ) c Γ ∈ Γ c 2 n + 4 g ( Γ ) − 4 ∏ � d ℓ b ∏ × d ψ b W ( ψ b ) H a . b M b b ∈ b ( Γ △ ) a = 1 Remarkable fact: For N = 4 SYM, all ingredients of the formula are well-defined and explicitly known as functions of the ’t Hooft coupling λ . This talk: ◮ Explain all ingredients of the formula. ◮ Demonstrate match with known data. ◮ Show (moderate) predictions. Till Bargheer — Handling Handles — ENS/Saclay Seminar — 12 October 2018 4 / 33
Proposal: Ingredients I √ k i � Q 1 . . . Q n � = ∏ n 1 S ◦ ∑ i = 1 × N n − 2 N 2 g ( Γ ) c Γ ∈ Γ c 2 n + 4 g ( Γ ) − 4 ∏ � d ℓ b ∏ × d ψ b W ( ψ b ) H a . b M b b ∈ b ( Γ △ ) a = 1 � ( α i · Φ ( x i )) k i � α 2 Half-BPS operators: Q i = Q ( α i , x i , k i ) = Tr , i = 0 . Internal polarizations α i , positions x i , weights (charges) k i . Set of all Wick contractions Γ ∈ Γ of the free theory, genus g ( Γ ) . Promote each Γ to a triangulation Γ △ of the surface in two steps: ◮ Collect homotopically equivalent lines in “bridges” → skeleton graph. The number of lines in a bridge b is the bridge length (width) ℓ b . ◮ All faces are triangles or higher polygons. Subdivide all faces into triangles by inserting zero-length ℓ b = 0 bridges. Set of all bridges: b ( Γ △ ) . On each bridge: Propagator factor d ℓ b b . Till Bargheer — Handling Handles — ENS/Saclay Seminar — 12 October 2018 5 / 33
Proposal: Ingredients II √ k i � Q 1 . . . Q n � = ∏ n 1 i = 1 S ◦ ∑ × N n − 2 N 2 g ( Γ ) c Γ ∈ Γ c 2 n + 4 g ( Γ ) − 4 ∏ � d ℓ b ∏ × d ψ b W ( ψ b ) H a . b M b b ∈ b ( Γ △ ) a = 1 On each bridge b ∈ b ( Γ △ ) : Sum/integrate over states ψ b ∈ M b of the mirror theory on the bridge b , with a kinematical weight factor W ( ψ b ) that depends on the cross ratios defined by the surrounding four operators. △ contains 2 n + 4 g ( Γ ) − 4 faces. By Euler, the triangulation Γ For each face a , insert one hexagon form factor H a : Accounts for interactions among three operators and three mirror states ψ b . Finally, S : Stratification. Sum over graphs quantizes the integration over the string moduli space. S accounts for contributions at the boundaries. Till Bargheer — Handling Handles — ENS/Saclay Seminar — 12 October 2018 6 / 33
The Sum over Mirror States On each bridge lives a mirror theory, which is obtained from the physical worldsheet theory by an analytic continuation, a double-Wick (90 degree) rotation ( σ , τ ) → ( i ˜ τ , i ˜ σ ) that exchanges space and time: R L e ˜ e H L H R ˜ τ − → τ R ˜ σ L σ In all computations, the volume R can be treated as infinite. ⇒ Mirror states are free multi-magnon Bethe states, characterized by rapidities u i , bound state indices a i , and flavor indices ( A i , ˙ A i ) . The mirror integration therefore expands to � ∞ ∞ ∞ m � u i = − ∞ d u i µ a i ( u i ) e − ˜ E ai ( u i ) ℓ b . ∑ ∏ a i = 1 ∑ ∑ d ψ b = M b m = 0 i = 1 A i , ˙ A i µ a i : measure factor, ˜ E : mirror energy, ℓ b : length of bridge b (discrete “time”). Till Bargheer — Handling Handles — ENS/Saclay Seminar — 12 October 2018 7 / 33
The Hexagon Form Factors Hexagon = Amplitude that measures the overlap between three mirror and three physical off-shell Bethe states. Worldsheet branching operator that [ Basso,Komatsu Vieira ’15 ] creates an excess angle of π . Explicitly: H ( χ A 1 χ ˙ A 1 χ A 2 χ ˙ A 2 . . . χ A n χ ˙ A n ) � � � χ A 1 χ A 2 . . . χ A n | S | χ ˙ A n . . . χ ˙ A 2 χ ˙ = ( − 1 ) F A 1 � ∏ h ij i < j ◮ χ A , χ ˙ A : Left/Right su ( 2 | 2 ) fundamental magnons ◮ F : Fermion number operator ◮ S : Beisert S-matrix x − i − x − x + j − 1/ x − x ± ( u ) = x ( u ± i u g = x + 1 1 2 ) , j i x ◮ h ij = , x − i − x + x + 2 − 1/ x + σ ij σ ij : BES dressing phase j 1 Example: ⊗ = Two magnons ( , ) S S Till Bargheer — Handling Handles — ENS/Saclay Seminar — 12 October 2018 8 / 33
Frames & Weight Factors Hexagon depends on positions x i and polarizations α i of the three half-BPS operators O i = Tr [( α i · Φ ( x i )) k ] . These preserve a diagonal su ( 2 | 2 ) that defines the state basis and S-matrix of excitations on the hexagon. Two neighboring hexagons share two operators, but the third/fourth operator may not be identical. ⇒ The two hexagon frames are misaligned. In order to consistently sum over mirror states, need to align the two frames by a PSU ( 2, 2 | 4 ) transformation g that maps O 3 onto O 2 : ( z, ¯ z ) O 2 g = e − D log | z | e i φ L · 1 2 · e J log | α | e i θ R , e iφL H 2 0 1 ∞ e 2 i φ = z /¯ H 1 z , O 1 O 3 O 4 e − D log | z | e 2 i θ = α /¯ 3 4 α . H is canonical, and H 2 = g − 1 ˆ Hexagon H 1 = ˆ H g . [ Fleury ’16 Komatsu ] Sum over states in mirror channel: µ ( ψ ) � g − 1 ˆ µ ( ψ ) �H 2 | ψ �� ψ |H 1 � = ∑ H| ψ �� ψ | g | ψ �� ψ | ˆ ∑ H� ψ ψ Weight factor: W ( ψ ) = � ψ | g | ψ � = e − 2 i ˜ p ψ log | z | e J ψ ϕ e i φ L ψ e i θ R ψ , i ˜ p = ( D − J ) /2. → Contains all non-trivial dependence on cross ratios z , ¯ z and α , ¯ α . Till Bargheer — Handling Handles — ENS/Saclay Seminar — 12 October 2018 9 / 33
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