Hexagonalization of Correlation Functions in N=4 SYM Shota Komatsu (Perimeter Institute) IGST 2017, Paris
Hexagonalization of Correlation Functions in N=4 SYM Shota Komatsu (Perimeter Institute) IGST 2017, Paris with (ICTP-SAIFR)
Hexagonalization of Correlation Functions in N=4 SYM Shota Komatsu (Perimeter Institute) IGST 2017, Paris with (ICTP-SAIFR) See also [Eden, Sfondrini], [Bargheer] [Basso, Coronado, SK, Lam, Vieira, Zhong]
1. Introduction 2. 3-pt functions and hexagons 3. Generalization to higher pts 4. Other related developments
Introduction • By AdS/CFT, one can potentially gain deep insight into quantum gravity. • 0 th step: Understand how the classical gravity/locality in the bulk emerges from CFT.
Introduction • By AdS/CFT, one can potentially gain deep insight into quantum gravity. • 0 th step: Understand how the classical gravity/locality in the bulk emerges from CFT. • Progress made through conformal bootstrap. [Heemskerk, Penedones, Polchinski, Sully] …[Caron -Huot], [Alday, Bissi ],… • Still many interesting questions to answer. [Maldacena, Simmons-Duffin, Zhiboedov] “Bulk - point singularity” of d+2 pt function
Introduction • By AdS/CFT, one can potentially gain deep insight into quantum gravity. • 0 th step: Understand how the classical gravity/locality in the bulk emerges from CFT. • Progress made through conformal bootstrap. [Heemskerk, Penedones, Polchinski, Sully] …[Caron -Huot], [Alday, Bissi ],… • Still many interesting questions to answer. [Maldacena, Simmons-Duffin, Zhiboedov] “Bulk - point singularity” of d+2 pt function • Would be good to have solvable examples.
N=4 Super Yang-Mills • Solvable at large N using integrability ≏ ∱ 2-pt functions ≏ ∱ 3-pt functions ≔≲≛ ⊢⊢⊢ ≝ Single-trace op. Planar surface ≏ ∲ ≏ ∳ ≏ ∲ [Minahan, Zarembo 2002]- … Nonperturbative framework proposed by … -[Gromov, Kazakov, Leurent, Volin 2013] [Basso, SK, Vieira 2015] …Eden, Sfondrini, Goncalves ,…
3+3 = 4? ≏ ∲ ≏ ∱ • We now have nonperturbative methods to study 2- and 3-pt functions. • In CFT, higher pts → 2- and 3-pt by OPE [Basso, Coronado, SK, Lam, Vieira, Zhong 2017] [Bargheer 2017] ≏ ∴ ≏ ∳ • Is this the end of the story?
Not quite! Single- trace operators are not “closed” under the OPE even at large N.
OPE at large N ≏ ∽≔≲≛ ⊢⊢⊢ ≝ ≨≏ ∨ ≸ ∱ ∩ ≏ ∨ ≸ ∲ ∩ ≏ ∨ ≸ ∳ ∩ ≏ ∨ ≸ ∴ ∩ ≩ Consider , . ≨≏≏≏≏≩ ∽ ≨≏≏≩≨≏≏≩ ∫ ≨≏≏≏≏≩ ≣≯≮ O(1/N 2 ) O(1) • At large N, ≨≏≏≏≏≩⊻ ≘ ≏ ≳ ∽≔≲≛ ⊢⊢⊢ ≝ ≃ ∲ ≏≏≏ ≳ ≆ ⊢ ≏ ≳ ∨ ≵∻≶ ∩ • Conformal block In OPE, ∫ ≘ ≏ ≤ ∽∺ ≏ ≀ ≮ ≏ ∺ ≃ ∲ ≏≏≏ ≤ ≆ ⊢ ≏ ≤ ∨ ≵∻≶ ∩ O(1/N 2 ) O(1)+O(1/N 2 ) Even at large N, we need info about double traces!
Intuitive explanation Pictorially, ≏ ∱ ≏ ∳ ≏ ∴ ≏ ∲ Double-trace Single-trace We need double traces*! But we don’t know how to study double traces using integrability … (*See, however, [Caron-Huot], [Alday, Bissi])
A way out ≏ ∱ ≏ ∲ ≏ ∴ ≏ ∳ Decompose the correlator into “hexagons”!
3-pt functions and hexagons ≏ ∱ ≏ ∲ ≏ ∳
3pt = a pair of pants Planar surface for 3pt functions ≏ ∱ ≏ ∱ or equivalently ≏ ∲ ≏ ∲ ≏ ∳ ≏ ∳ Tree level : Wick contractions ≏ ∱ [Alday et al], [Okuyama et al.], [Escobedo et al.] Bridge length and many others ≏ ∲ ≏ ∳
3pt = (Hexagon) 2 ⊻ [Basso, SK, Vieira] ≏ ∱ triangulation of the worldsheet ≏ ∲ ≏ ∳
3pt as a sum over partitions ≏ ∱ [Basso, SK Vieira] ∫ ≏ ∲ ≏ ∳ ∽ ∫ ∫ • 3pt can be decomposed into hexagons! • Contributions from each hexagon are fixed by symmetry (+ integrability)!
Finite size corrections Insert a complete basis of states Insert a complete basis on the dashed lines. Mirror magnons. ∫ ⊢⊢⊢ propagation measure factor • Finite size corrections can also be computed by integrability! • Exponentially suppressed for long operators due to propagation factors. • Propagation factors “Wick rotation”
Generalization to higher pt ≏ ∴ ≠ ∱∴ ≏ ∱ ≠ ∱∳ ≠ ∱∲ ≏ ∲ ≏ ∳
Q: How do the cross-ratios appear in the formulae?
Simple exercise at tree level Set-up: Pick 2d plane inside 4d.
Simple exercise at tree level Set-up: At tree level, this can be computed by 1) List up all possible planar graphs. ≘ 2) Sum over the positions of the derivative in each graph. ≏ ∴ ≠ ∱∴ ≏ ∱ 4 hexagons! ≠ ≩≪ ≠ ∱∳ ≠ ∱∲ ≏ ∲ ≏ ∳ Cf. Triangulation of a 4 punctured sphere
Simple exercise at tree level ≏ ∴ ≠ ∱∴ ≏ ∱ ≠ ∱∳ ≠ ∱∲ ≏ ∲ ≏ ∳ Result:
Simple exercise at tree level ≏ ∱ ≏ ∴ ≠ ∱∴ ≏ ∱ ≠ ∱∳ ≠ ∱∲ ≏ ∲ ≏ ∳ Edge and cross ratios: ≏ ≩ ≏ ≪ c.f. Fock coordinate for Teichmuller space. ≏ ≫ ≏ ≬
Simple exercise at tree level Lesson: Cross ratios appear as a weight factor for physical magnons ≏ ∱ It suggest that cross ratios couple also to mirror magnons …
Hexagonalization (for 4 BPS operators)
Proposal for 4 BPS correlators Basic Idea : Decorate the tree-level diagrams with (mirror) magnons.
Proposal for 4 BPS correlators
Proposal for 4 BPS correlators Tree-level: 1) List up all possible planar graphs. 2) Compute contributions from each graph. ≏ ∱ ≘ ≠ ∱∴ ≏ ∳ ≏ ∲ ≦ ≠ ≩≪ ≧ ≏ ∴
Proposal for 4 BPS correlators Tree-level: Idea : Decorate the tree-level diagrams with (mirror) magnons. ≏ ∱ ≘ ≠ ∱∴ ≏ ∳ ≏ ∲ ≦ ≠ ≩≪ ≧ ≏ ∴ Insert a complete basis
Proposal for 4 BPS correlators Tree-level: Idea : Decorate the tree-level ≈ ∳ diagrams with (mirror) magnons. ≈ ∴ ≏ ∱ ≠ ∱∴ ≈ ∲ ≈ ∱ ≏ ∳ ≏ ∲ ≏ ∴ a complete basis
Proposal for 4 BPS correlators Proposal for finite coupling: Tree level Idea : Decorate the tree-level ≈ ∳ diagrams with (mirror) magnons. ≈ ∴ ≏ ∱ ≠ ∱∴ ≈ ∲ ≈ ∱ ≏ ∳ ≏ ∲ ≏ ∴ a complete basis
Proposal for 4 BPS correlators Proposal for finite coupling: propagation measure factor Tree level Idea : Decorate the tree-level ≈ ∳ diagrams with (mirror) magnons. ≈ ∴ ≏ ∱ ≠ ∱∴ ≈ ∲ ≈ ∱ ≏ ∳ ≏ ∲ ≏ ∴ a complete basis
Proposal for 4 BPS correlators Proposal for finite coupling: propagation hexagons measure factor Tree level Idea : Decorate the tree-level ≈ ∳ diagrams with (mirror) magnons. ≈ ∴ ≏ ∱ ≠ ∱∴ ≈ ∲ ≈ ∱ ≏ ∳ ≏ ∲ ≏ ∴ a complete basis
Proposal for 4 BPS correlators Proposal for finite coupling: Weight factor (cross-ratio dependent) New! propagation hexagons measure factor Tree level Idea : Decorate the tree-level ≈ ∳ diagrams with (mirror) magnons. ≈ ∴ ≏ ∱ ≠ ∱∴ ≈ ∲ ≈ ∱ ≏ ∳ ≏ ∲ ≏ ∴ a complete basis
Weight factor from symmetry ≏ ∱ • Consider gluing of the edge 14: ≈ ∲ ≈ ∱ ≏ ∳ ≏ ∲ ≏ ∴ • ≏ ∲ Perform the conformal transformation to map it to ≏ ∱ ≏ ∴ ≏ ∳ ∱ ∰ ∱ • In this frame, canonical “rotated”
Weight factor from symmetry ≏ ∱ canonical ≈ ∲ ≈ ∱ “rotated” ≏ ∳ ≏ ∲ ≏ ∴ ∨ ≺∻ ⊹ ≺ ∩ “Rotation”: ≥ ≩ ≌ ⋁ ∱ ∰ ∱ ≪ ≺ ≪ ≥ ⊡ ≄ ≬≯≧ ≪ ≺ ≪ Gluing: Weight factor (cross-ratio dependent) (Similar story for the R-symmetry part)
Testing the proposal at one-loop
Four-point functions of length-2 operators 1. List all tree-level diagrams ∴ ∱ ∲ ∱ ∲ ∱ ∴ ∴ ∳ ∳ ∳ ∲ propagators
Four-point functions of length-2 operators 1. List all tree-level diagrams, and cut them into hexagons. ∴ ∱ ∲ ∱ ∲ ∱ ∴ ∴ ∳ ∳ ∳ ∲ length zero
Four-point functions of length-2 operators 1. List all tree-level diagrams, and cut them into hexagons. ∴ ∱ ∲ ∱ ∲ ∱ ∴ ∴ ∳ ∳ ∳ ∲ ∲ 2. Decorate them with magnons. ∱ ≶ ∴ ∳ N-particle At one loop, only 1 particle on zero-length channel!
Four-point functions of length-2 operators ∲ ∱ ≶ • 1-particle state: derivative, scalar, fermion rapidity bound states ∴ ∳ etc. (KK modes) • flavor sum = character a-th anti-sym rep. su(2|2)
Four-point functions of length-2 operators ∲ ∱ • Full 1-particle integral ≶ ∴ ∳
Four-point functions of length-2 operators ∲ ∱ • Full 1-particle integral ≶ ∴ ∳ • Leading order at weak coupling ∱ ∲ ∳ ∴ 1-loop conformal integral
Four-point functions of length-2 operators ∲ ∱ • Full 1-particle integral ≶ ∴ ∳ • Leading order at weak coupling ∱ ∲ ∳ ∴ 1-loop conformal integral • ∲ ∱ ∴ ∱ Sum over 3 graphs ∴ ∲ ∳ ∳ supersymmetry Complete agreement with perturbative result!
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