Handling Handles: Non-Planar AdS/CFT Integrability Till Bargheer Leibniz University Hannover 1711.05326 : TB, J. Caetano, T. Fleury, S. Komatsu, P. Vieira 18xx.xxxxx : TB, J. Caetano, T. Fleury, S. Komatsu, P. Vieira 18xx.xxxxx : TB, F. Coronado, P. Vieira + work in progress Workshop on Holography, Gauge Theories, and Black Holes Southampton, March 2018
General Idea / Punchline In AdS 5 , string amplitudes can be cut into basic patches (rectangles, pentagons, or hexagons), which can be bootstrapped using integrability at any value of the ’t Hooft coupling . ◮ Amplitudes are given as infinite sums and integrals over intermediate states from gluing together these integrable patches. ◮ Sometimes, these sums and integrals can be re-summed, giving hints of yet-to-be uncovered structures . ◮ This holds at the planar level as well as for non-planar processes suppressed by 1 /N c . Till Bargheer — Handling Handles — Southampton — 26 March 2018 1 / 39
N = 4 SYM & The Planar Limit N = 4 super Yang–Mills: Gauge field A µ , scalars Φ I , fermions ψ αA . Gauge group: U( N c ) / SU( N c ) . Adjoint representation: All fields are N c × N c matrices. Double-line notation: Propagators: Vertices: � ∼ g 2 YM δ il δ kj = i l 1 � Φ i I j Φ k Tr( ΦΦΦΦ ) ∼ J l g 2 j k YM • Diagrams consist of color index loops ≃ oriented disks ∼ δ ii = N c • Disks are glued along propagators → oriented compact surfaces Local operators: ◮ One fewer color loop → factor 1 /N c O i = Tr( Φ . . . ) ∼ O i ◮ Surface: Hole ∼ boundary component Till Bargheer — Handling Handles — Southampton — 26 March 2018 2 / 39
Planar Limit & Genus Expansion Every diagram is associated to an oriented compact surface. Genus Expansion: Absorb one factor of N c in the ’t Hooft coupling λ = g 2 YM N c Use Euler formula V − E + F = 2 − 2 g ⇒ Correlators of single trace operators O i = Tr( Φ 1 Φ 2 . . . ) : ∞ 1 1 � �O 1 . . . O n � = G g ( λ ) N n − 2 N 2 g c c g =0 1 + 1 + 1 ∼ + . . . N 2 N 4 N 6 c c c Till Bargheer — Handling Handles — Southampton — 26 March 2018 3 / 39
Spectrum: Planar Limit Goal: Correlation functions in N = 4 SYM Step 1: Planar spectrum of single-trace local operators Tr( Φ . . . ) ◮ Spectrum of (anomalous) scaling dimensions ∆ ◮ Scale transformations represented by dilatation operator Γ ◮ Γ mixes single-trace (& multi-trace) operators ◮ Resolve mixing → Eigenstates & eigenvalues (dimensions) Planar limit: ◮ Multi-trace operators suppressed by 1 /N c ◮ Dilatation operator acts locally in color space (neighboring fields) Organize space of single-trace operators around protected states Tr Z L , Z = α I Φ I , α I α I = 0 (half-BPS, “vacuum”) . Other single-trace operators: Insert impurities { Φ I , ψ αA , D µ } into Tr Z L . Till Bargheer — Handling Handles — Southampton — 26 March 2018 4 / 39
Planar Spectrum: Integrability Initial observation: One-loop dilatation operator for scalar single-trace operators is integrable. Diagonalization by Bethe Ansatz. � Minahan � Zarembo ◮ Impurities are magnons in color space, characterized by rapidity (momentum) u and su (2 | 2) 2 flavor index . � su (2 | 2) 2 ⊂ psu (2 , 2 | 4) preserves the vacuum Tr Z L � ◮ Dynamics of magnons: integrability: → No particle production → Individual momenta preserved = → Factorized scattering ◮ Two-body ( → n -body) S-matrix completely fixed to all loops � Beisert �� Janik �� Beisert,Hernandez � 2005 2006 Lopez 2006 ⇒ Asymptotic spectrum (for L → ∞ ) solved to all loops / exactly. Till Bargheer — Handling Handles — Southampton — 26 March 2018 5 / 39
Finite-Size Effects Asymptotic spectrum solved by Bethe Ansatz. Resums ∞ Feynman diagrams that govern dynamics of ∞ strip: . . . . . . − → L → ∞ L Re-compactify: Finite-size effects. Leading effect: Momentum quantization constraint ≡ Bethe equations 1 = e ip j L � S ( p k , p j ) j � = k Moreover: Wrapping interactions. ◮ No notion of locality for dilatation operator ◮ Previous techniques (Bethe ansatz) no longer apply Till Bargheer — Handling Handles — Southampton — 26 March 2018 6 / 39
Mirror Theory � Arutyunov � Key to all-loop finite-size spectrum: Mirror map Frolov Double Wick rotation: ( σ, τ ) → ( i ˜ τ, i ˜ σ ) — exchanges space and time R L e ˜ e H L H R τ ˜ − → τ R σ ˜ L σ Magnon states: Energy and momentum interchange: ˜ E = ip , ˜ p = iE Finite size L becomes finite, periodic (discrete) time. Energy ∼ Partition function at finite temperature 1 /L , with R → ∞ . � Bombardelli ’09 �� Gromov,Kazakov �� Arutyunov � → Thermodynamic Bethe ansatz . Fioravanti,Tateo Kozak,Vieira ’09 Frolov ’09 Simplifications and refinements: ◮ Y-system (T-system, Q-system) � Gromov,Kazakov �� Arutyunov � Kozak,Vieira ’09 Frolov ’09 ◮ Quantum Spectral Curve � Gromov,Kazakov �� Gromov,Kazakov � Leurent,Volin ’13 Leurent,Volin ’14 ⇒ Scaling dimensions computable at finite coupling. Till Bargheer — Handling Handles — Southampton — 26 March 2018 7 / 39
Three-Point Functions: Hexagons Differences: Topology: Pair of pants instead of cylinder Non-vanishing for three generic operators (two-point: diagonal) ⇒ Previous techniques not directly applicable Observation: The green parts are similar to two-point functions: O 1 Two segments of physical operators joined by parallel propagators (“bridges”, ℓ ij = ( L i + L j − L k ) / 2 ). The red part is new: “Worldsheet splitting”, O 2 O 3 “three-point vertex” (open strings) Take this serious → cut worldsheet along “bridges”: � Basso,Komatsu � Vieira ’15 O 1 − → ⊗ O 2 O 3 Till Bargheer — Handling Handles — Southampton — 26 March 2018 8 / 39
Hexagons & Gluing O 1 − → ⊗ O 2 O 3 �� Basso,Goncalves ⊗ Glue hexagons along three mirror channels : � Basso,Komatsu � Vieira ’15 Komatsu,Vieira ’15 ◮ Sum over complete state basis (magnons) in the mirror theory ◮ Mirror magnons: Boltzmann weight exp( − ˜ E ij ℓ ij ) , ˜ E ij = O ( g 2 ) → mirror excitations are strongly suppressed. Hexagonal worldsheet patches (form factors): ◮ Function of rapidities u and su (2 | 2) 2 labels ( A, ˙ A ) of all magnons. ◮ Conjectured exact expression, based on diagonal su (2 | 2) symmetry as well as form factor axioms. � Basso,Komatsu � Vieira ’15 Finite-coupling hexagon proposal: Supported by very non-trivial matches. Till Bargheer — Handling Handles — Southampton — 26 March 2018 9 / 39
Planar Four-Point Functions: Hexagonalization Move on to planar four-point functions: One way to cut (now that three-point is understood): OPE cut Problem: Sum over physical states! ◮ No loop suppression, all states contrib. ◮ Double-trace operators. � Fleury ’16 �� Eden ’16 � Instead: Cut along propagator bridges Komatsu Sfondrini − → Benefits: ◮ Mirror states highly suppressed in g . ◮ No double traces. Till Bargheer — Handling Handles — Southampton — 26 March 2018 10 / 39
Hexagonalization: Formula H 4 H 3 1 2 H 2 ← → H 1 3 4 � � � = � d ℓ c � � O 1 O 2 O 3 H 1 ( ψ 1 , ψ 2 , ψ 3 ) H 2 ( ψ 1 , ψ 2 , ψ 3 ) µ ( ψ c ) c channels ψ c c ∈{ 1 , 2 , 3 } � � � = � � d ℓ c � � O 1 O 2 O 3 O 4 µ ( ψ c ) H 1 H 2 H 3 H 4 c planar channels ψ c c ∈{ 1 ,..., 6 } prop. graphs New Features: � Fleury ’16 � Komatsu ◮ Bridge lengths vary, may go to zero ⇒ Mirror corrections at one loop ◮ Hexagons are in different “frames” ⇒ Weight factors Till Bargheer — Handling Handles — Southampton — 26 March 2018 11 / 39
Hexagonalization: Frames Hexagon depends on positions x i and polarizations α i of the three half-BPS “vacuum” operators O i = Tr[( α i · Φ ( x i )) L ] . Any three x i and α i preserve a diagonal su (2 | 2) that defines the state basis and S-matrix of excitations on the hexagon. Three-point function: Both hexagons connect to the same three operators, so their frames ( su (2 | 2) and state basis) are identical. 1 2 Higher-point function: Two neighboring H 2 hexagons always share two operators, but the third/fourth operator may not be identical. H 1 ⇒ The two hexagon frames are misaligned. 3 4 In order to consistently sum over mirror states, need to align the two frames by a PSU(2 , 2 | 4) transformation that maps O 3 onto O 2 . Till Bargheer — Handling Handles — Southampton — 26 March 2018 12 / 39
Hexagonalization: Weight Factors By conformal and R-symmetry transformation, bring O 1 , O 2 , and O 4 to canonical configuration: � Fleury ’16 � Komatsu ( z, ¯ z ) O 2 1 2 e iφL H 2 0 1 ∞ H 1 O 1 O 3 O 4 e − D log | z | 3 4 Transformation that maps O 3 to O 2 : g = e − D log | z | e iφL e J log | α | e iθR , where e 2 iφ = z/ ¯ z , e 2 iθ = α/ ¯ α ) is the R-coordinate of O 3 . α , and ( α, ¯ H is canonical, and H 2 = g − 1 ˆ Hexagon H 1 = ˆ H g . � Fleury ’16 � Sum over states in mirror channel: Komatsu µ ( ψ ) � g − 1 ˆ H| ψ �� ψ | g | ψ �� ψ | ˆ � � µ ( ψ ) �H 2 | ψ �� ψ |H 1 � = H� ψ ψ Weight factor: � ψ | g | ψ � = e − 2 i ˜ p ψ log | z | e J ψ ϕ e iφL ψ e iθR ψ , i ˜ p = ( D − J ) / 2 . → Contains all non-trivial dependence on cross ratios z, ¯ z and α, ¯ α . Till Bargheer — Handling Handles — Southampton — 26 March 2018 13 / 39
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