in N=4 SYM Shota Komatsu (Perimeter Institute) IGST 2017, Paris - - PowerPoint PPT Presentation

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)

• 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.

• 0th 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.

• 0th step: Understand how the classical gravity/locality in the

bulk emerges from CFT.

• Progress made through conformal bootstrap.
• Still many interesting questions to answer.

Introduction

• By AdS/CFT, one can potentially gain deep insight into

quantum gravity.

• 0th step: Understand how the classical gravity/locality in the

bulk emerges from CFT.

• Progress made through conformal bootstrap.
• Still many interesting questions to answer.
• Would be good to have solvable examples.

N=4 Super Yang-Mills

• Solvable at large N using integrability

≏∳

≏∱ ≏∲

Planar surface

≏∱ ≏∲

≔≲≛⊢⊢⊢≝

2-pt functions 3-pt functions

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
• 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 , .

• At large N,
• In OPE,

≏ ∽≔≲≛⊢⊢⊢≝

≨≏∨≸∱∩≏∨≸∲∩≏∨≸∳∩≏∨≸∴∩≩

≨≏≏≏≏≩∽≨≏≏≩≨≏≏≩∫≨≏≏≏≏≩≣≯≮

O(1/N2) O(1)

≨≏≏≏≏≩⊻ ≘

≏≳∽≔≲≛⊢⊢⊢≝≃∲ ≏≏≏≳≆⊢≏≳∨≵∻≶∩

∫ ≘

≏≤∽∺≏≀≮≏∺≃∲ ≏≏≏≤≆⊢≏≤∨≵∻≶∩

Even at large N, we need info about double traces!

Pictorially,

Intuitive explanation

≏∲ ≏∳ ≏∴

≏∱

Single-trace Double-trace We need double traces*! But we don’t know how to study double traces using integrability…

≏∱ ≏∲ ≏∳ ≏∴

A way out

Decompose the correlator into “hexagons”!

3-pt functions and hexagons

≏∳

≏∱ ≏∲

3pt = a pair of pants

≏∳

≏∱ ≏∲

≏∱ ≏∲

≏∳

Planar surface for 3pt functions

• r equivalently

Tree level : Wick contractions

≏∱ ≏∲ ≏∳

⊻

3pt = (Hexagon)2

≏∳

≏∱ ≏∲

triangulation of the worldsheet

∽

∫ ∫ ∫

3pt as a sum over partitions

• 3pt can be decomposed into hexagons!
• Contributions from each hexagon are fixed by symmetry (+ integrability)!

≏∱ ≏∲ ≏∳

Finite size corrections

Insert a complete basis Insert a complete basis of states

• n the dashed lines.

Mirror magnons.

∫⊢⊢⊢

• Finite size corrections can also be computed by integrability!
• Exponentially suppressed for long operators due to propagation factors.
• Propagation factors

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

At tree level, this can be computed by Set-up: 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:

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)

Basic Idea: Decorate the tree-level diagrams with (mirror) magnons. Proposal for 4 BPS correlators

Proposal for 4 BPS correlators

Proposal for 4 BPS correlators

Tree-level:

≏∳ ≏∱ ≏∲ ≏∴

≠∱∴

≘

≦≠≩≪≧

Proposal for 4 BPS correlators

Tree-level:

≏∳ ≏∱ ≏∲ ≏∴

≠∱∴

≘

≦≠≩≪≧

Insert a complete basis Idea: Decorate the tree-level diagrams with (mirror) magnons.

Proposal for 4 BPS correlators

Tree-level: Idea: Decorate the tree-level diagrams with (mirror) magnons.

≏∳ ≏∱ ≏∲ ≏∴

≈∱ ≈∲

≈∳ ≈∴

≠∱∴

a complete basis

Proposal for finite coupling:

Proposal for 4 BPS correlators

≏∳ ≏∱ ≏∲ ≏∴

≈∱ ≈∲

≈∳ ≈∴

≠∱∴

Idea: Decorate the tree-level diagrams with (mirror) magnons. a complete basis

Proposal for finite coupling:

Proposal for 4 BPS correlators

≏∳ ≏∱ ≏∲ ≏∴

≈∱ ≈∲

≈∳ ≈∴

≠∱∴

Idea: Decorate the tree-level diagrams with (mirror) magnons. a complete basis

Proposal for finite coupling:

Proposal for 4 BPS correlators

≏∳ ≏∱ ≏∲ ≏∴

≈∱ ≈∲

≈∳ ≈∴

≠∱∴

Idea: Decorate the tree-level diagrams with (mirror) magnons. a complete basis

Proposal for finite coupling:

Proposal for 4 BPS correlators

≏∳ ≏∱ ≏∲ ≏∴

≈∱ ≈∲

≈∳ ≈∴

≠∱∴

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,

≏∲

Weight factor from symmetry

≏∳ ≏∱ ≏∲ ≏∴

≈∱ ≈∲

∨≺∻⊹ ≺∩ ≥≩≌⋁

∰ ∱

≪≺≪

∱

≥⊡≄≬≯≧≪≺≪

“Rotation”:

Gluing:

(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

∱ ∲ ∳ ∴ ∱ ∲ ∳ ∴ ∱ ∲ ∳ ∴

• 1. List all tree-level diagrams, and cut them into hexagons.

length zero

Four-point functions of length-2 operators

Four-point functions of length-2 operators

∱ ∲ ∳ ∴ ∱ ∲ ∳ ∴ ∱ ∲ ∳ ∴ ∱ ∲ ∳ ∴

≶

• 2. Decorate them with magnons.
• 1. List all tree-level diagrams, and cut them into hexagons.

At one loop, only 1 particle on zero-length channel!

Four-point functions of length-2 operators

∱ ∲ ∳ ∴

≶

• 1-particle state:

• flavor sum = character

• Full 1-particle integral

Four-point functions of length-2 operators

∱ ∲ ∳ ∴

≶

• Full 1-particle integral

Four-point functions of length-2 operators

• Leading order at weak coupling

1-loop conformal integral

∱ ∲ ∳ ∴ ∱ ∲ ∳ ∴

≶

• Full 1-particle integral

Four-point functions of length-2 operators

• Leading order at weak coupling

1-loop conformal integral

∱ ∲ ∳ ∴

• Sum over 3 graphs

∱ ∲ ∳ ∴

≶

∱ ∲ ∳ ∴ ∱ ∲ ∳ ∴

Complete agreement with perturbative result!

Higher loops

• Full 1-particle integral

∱ ∲ ∳ ∴

≶

L-th subleading term L-loop conformal ladder integral One can expand this integral at arbitrary order at weak coupling

∱ ∲ ∳ ∴ ∲ ∳ ∱ ∴

Full 1-particle integral resums all the ladder integrals!

Some related developments

Fishnet = multi-particle integrals

• m particle states on the length n-m bridge ∱

∲ ∳ ∴

[Basso, Dixon 2017]

Mellin = GKP ?

• The same diagram can be computed by Pentagon OPE for null polygons.

[Basso, Dixon 2017]

• 1-loop conformal integral

≃

Generalization to open strings

• Three-point function on the straightline BPS Wilson loop Tree-level: [Kim, Kiryu 2017]

• Zero length operators in the ladders limit

• Cf. [Correa, Maldacena, Sever, Henn]

(A nice testing ground for multi-mirror-particle corrections and resummation.)

• cf. [Mamroud, Torrents]
• Four-point function = (Hexagon)2

• Cf. Kazakov et al. Caetano’s talk

Conclusion & Prospects

• Proposed a non-perturbative framework to study correlation

functions in N=4 SYM.

• Checked for 4 BPS, and 3 BPS + Konishi at one loop.
• Higher loops, higher points, loops in AdS…
• Various limits (Bulk point, Regge limit, strong coupling etc.)
• 2d CFT correlators from triangulation

Conclusion

• “String field theory for open mirror strings”?

≏≩ ≏≪ ≏≫ ≏≬ ≏≳

Speculation?

• Open-closed-open duality (Gopakumar)

N=4 SYM (open string) Closed SFT in AdS

• pen mirror SFT??

• Cf. [Kazakov 2000]:

General lessons for string theory / SFT? (incl. flat space??)

終

Equivalence of different cuttings

∱ ∲ ∳ ∴ ∱ ∲ ∳ ∴

We can glue two hexagons in 2 different ways. This guarantees the crossing symmetry of the final result!

Flip invariance