Handling Handles: Non-Planar AdS/CFT Integrability Till Bargheer Leibniz Universität Hannover & DESY Hamburg 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 + further work in progress Integrability in Gauge and String Theory Copenhagen, August 2018
Planar Limit & Genus Expansion � ’t Hooft � Gauge theory with adjoint matter in the large N c limit: 1974 ◮ Feynman diagrams are double-line diagrams. ◮ Operator traces induce a definite ordering of connecting propagators. ◮ All color lines (propagators and traces) form closed oriented loops. ◮ 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. Till Bargheer — Handling Handles — IGST 2018 Copenhagen 1 / 21
Planar Limit & Genus Expansion � ’t Hooft � Gauge theory with adjoint matter in the large N c limit: 1974 ◮ Feynman diagrams are double-line diagrams. ◮ Operator traces induce a definite ordering of connecting propagators. ◮ All color lines (propagators and traces) form closed oriented loops. ◮ 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 � = O i = Tr( Φ 1 Φ 2 . . . ) n ( 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 — IGST 2018 Copenhagen 1 / 21
Proposal Concrete and explicit realization of the general genus expansion: √ k i � n 1 i =1 � � Q 1 . . . Q n � = S ◦ × N n − 2 N 2 g ( Γ ) c c Γ ∈ Γ 2 n +4 g ( Γ ) − 4 � d ℓ b � � × d ψ b W ( ψ b ) H a . b M b a =1 b ∈ b ( Γ △ ) 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 . Goal of this talk: ◮ Explain all ingredients of the formula. ◮ Demonstrate match with known data. ◮ Show some predictions. Till Bargheer — Handling Handles — IGST 2018 Copenhagen 2 / 21
Proposal: Ingredients I √ k i � n 1 i =1 � � Q 1 . . . Q n � = S ◦ × N n − 2 N 2 g ( Γ ) c c Γ ∈ Γ 2 n +4 g ( Γ ) − 4 � d ℓ b � � × d ψ b W ( ψ b ) H a . b M b a =1 b ∈ b ( Γ △ ) Till Bargheer — Handling Handles — IGST 2018 Copenhagen 3 / 21
Proposal: Ingredients I √ k i � n 1 i =1 � � Q 1 . . . Q n � = S ◦ × N n − 2 N 2 g ( Γ ) c c Γ ∈ Γ 2 n +4 g ( Γ ) − 4 � d ℓ b � � × d ψ b W ( ψ b ) H a . b M b a =1 b ∈ b ( Γ △ ) � ( 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 . Till Bargheer — Handling Handles — IGST 2018 Copenhagen 3 / 21
Proposal: Ingredients I √ k i � n 1 i =1 � � Q 1 . . . Q n � = S ◦ × N n − 2 N 2 g ( Γ ) c c Γ ∈ Γ 2 n +4 g ( Γ ) − 4 � d ℓ b � � × d ψ b W ( ψ b ) H a . b M b a =1 b ∈ b ( Γ △ ) � ( 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 ) . Till Bargheer — Handling Handles — IGST 2018 Copenhagen 3 / 21
Proposal: Ingredients I √ k i � n 1 i =1 � � Q 1 . . . Q n � = S ◦ × N n − 2 N 2 g ( Γ ) c c Γ ∈ Γ 2 n +4 g ( Γ ) − 4 � d ℓ b � � × d ψ b W ( ψ b ) H a . b M b a =1 b ∈ b ( Γ △ ) � ( 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: Till Bargheer — Handling Handles — IGST 2018 Copenhagen 3 / 21
Proposal: Ingredients I √ k i � n 1 i =1 � � Q 1 . . . Q n � = S ◦ × N n − 2 N 2 g ( Γ ) c c Γ ∈ Γ 2 n +4 g ( Γ ) − 4 � d ℓ b � � × d ψ b W ( ψ b ) H a . b M b a =1 b ∈ b ( Γ △ ) � ( 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 . Till Bargheer — Handling Handles — IGST 2018 Copenhagen 3 / 21
Proposal: Ingredients I √ k i � n 1 i =1 � � Q 1 . . . Q n � = S ◦ × N n − 2 N 2 g ( Γ ) c c Γ ∈ Γ 2 n +4 g ( Γ ) − 4 � d ℓ b � � × d ψ b W ( ψ b ) H a . b M b a =1 b ∈ b ( Γ △ ) � ( 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 . ◮ Subdivide all faces into triangles by inserting zero-length bridges. Till Bargheer — Handling Handles — IGST 2018 Copenhagen 3 / 21
Proposal: Ingredients I √ k i � n 1 i =1 � � Q 1 . . . Q n � = S ◦ × N n − 2 N 2 g ( Γ ) c c Γ ∈ Γ 2 n +4 g ( Γ ) − 4 � d ℓ b � � × d ψ b W ( ψ b ) H a . b M b a =1 b ∈ b ( Γ △ ) � ( 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 . ◮ Subdivide all faces into triangles by inserting zero-length bridges. Set of all bridges: b ( Γ △ ) . Till Bargheer — Handling Handles — IGST 2018 Copenhagen 3 / 21
Proposal: Ingredients II √ k i � n 1 i =1 � � Q 1 . . . Q n � = S ◦ × N n − 2 N 2 g ( Γ ) c c Γ ∈ Γ 2 n +4 g ( Γ ) − 4 � d ℓ b � � × d ψ b W ( ψ b ) H a . b M b a =1 b ∈ b ( Γ △ ) Till Bargheer — Handling Handles — IGST 2018 Copenhagen 4 / 21
Proposal: Ingredients II √ k i � n 1 i =1 � � Q 1 . . . Q n � = S ◦ × N n − 2 N 2 g ( Γ ) c c Γ ∈ Γ 2 n +4 g ( Γ ) − 4 � d ℓ b � � × d ψ b W ( ψ b ) H a . b M b a =1 b ∈ b ( Γ △ ) On each bridge b ∈ b ( Γ △ ) : Till Bargheer — Handling Handles — IGST 2018 Copenhagen 4 / 21
Proposal: Ingredients II √ k i � n 1 i =1 � � Q 1 . . . Q n � = S ◦ × N n − 2 N 2 g ( Γ ) c c Γ ∈ Γ 2 n +4 g ( Γ ) − 4 � d ℓ b � � × d ψ b W ( ψ b ) H a . b M b a =1 b ∈ b ( Γ △ ) On each bridge b ∈ b ( Γ △ ) : Sum/integrate over states ψ b ∈ M b of the mirror theory on the bridge b . Insert a kinematical weight factor W ( ψ b ) , it depends on the cross ratios. Till Bargheer — Handling Handles — IGST 2018 Copenhagen 4 / 21
Proposal: Ingredients II √ k i � n 1 i =1 � � Q 1 . . . Q n � = S ◦ × N n − 2 N 2 g ( Γ ) c c Γ ∈ Γ 2 n +4 g ( Γ ) − 4 � d ℓ b � � × d ψ b W ( ψ b ) H a . b M b a =1 b ∈ b ( Γ △ ) On each bridge b ∈ b ( Γ △ ) : Sum/integrate over states ψ b ∈ M b of the mirror theory on the bridge b . Insert a kinematical weight factor W ( ψ b ) , it depends on the cross ratios. △ contains 2 n + 4 g ( Γ ) − 4 faces. By Euler, the triangulation Γ Till Bargheer — Handling Handles — IGST 2018 Copenhagen 4 / 21
Proposal: Ingredients II √ k i � n 1 i =1 � � Q 1 . . . Q n � = S ◦ × N n − 2 N 2 g ( Γ ) c c Γ ∈ Γ 2 n +4 g ( Γ ) − 4 � d ℓ b � � × d ψ b W ( ψ b ) H a . b M b a =1 b ∈ b ( Γ △ ) On each bridge b ∈ b ( Γ △ ) : Sum/integrate over states ψ b ∈ M b of the mirror theory on the bridge b . Insert a kinematical weight factor W ( ψ b ) , it depends on the cross ratios. △ 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. Till Bargheer — Handling Handles — IGST 2018 Copenhagen 4 / 21
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