Shota Komatsu (Perimeter Institute) [B. Basso, P. Vieira, and S.K., - - PowerPoint PPT Presentation

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Shota Komatsu (Perimeter Institute) [B. Basso, P. Vieira, and S.K., - - PowerPoint PPT Presentation

May 05, 2023 472 likes •1.09k views

In Integra egrabl ble Bo Boots tstrap trap for Structur ructure e Co Const stants ants in N=4 =4 SYM SYM Shota Komatsu (Perimeter Institute) [B. Basso, P. Vieira, and S.K., arXiv:1505.06745] [B. Basso, V. Goncalves, P. Vieira and

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In Integra egrabl ble Bo Boots tstrap trap for Structur ructure e Co Const stants ants in N=4 =4 SYM SYM Shota Komatsu (Perimeter Institute)

[B. Basso, P. Vieira, and S.K., arXiv:1505.06745] [B. Basso, V. Goncalves, P. Vieira and S.K., arXiv:1510.01683]

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  • We have no satisfactory understanding of AdS/CFT.
  • It is important to study in detail how the building blocks
  • f the two theories are related with each other.
  • For conformal field theories,

the building blocks → 2- and 3-point functions.

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SLIDE 3

AdS5/CFT4 correspondence

N=4 U(N) SYM in 4d String Theory on AdS5×S5

Goal of this talk: Non-perturbative framework to compute 3pt-functions at finite ‘t Hooft coupling in the large N limit. How? Map the problem to 2d system and use Integrability.

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Interesting Observation:

[Eden Heslop Korchemsky Sokatchev]

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Interesting Observation:

[Eden Heslop Korchemsky Sokatchev]

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  • Why do they agree up to 1-loop?
  • Why do they start to differ at 2-loop?
  • How does zeta function come about?
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SLIDE 7
  • Why do they agree up to 1-loop?
  • Why do they start to differ at 2-loop?
  • How does zeta function come about?

Interesting physical mechanism behind!

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SLIDE 8

1. Two-point functions 2. Perturbative computation of 3pt functions 3. 3pt from Hexagons (Asymptotic Part) 4. 3pt from Hexagons (“Wrapping” Effects) 5. Outlook

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  • 1. Two-point functions
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2-point functions

anomalous dimension Single trace operators:

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Relation to spin chain

[Minahan-Zarembo]

One can efficiently compute the 1-loop anomalous dimension by solving Bethe equation.

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General spin-chain state Bethe equation (periodicity condition)

2 to 2 scattering Free propagation

Energy: Dispersion: S-matrix:

Bethe equation

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  • One can repeat the same analysis for 2-loop.
  • At higher loops, this approach is less effective

simply because the computation of the mixing matrix becomes hard.

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Use of symmetry[Beisert]

  • Consider an infinitely long BPS operator made up of Z’s
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Use of symmetry[Beisert]

  • Consider an infinitely long BPS operator made up of Z’s

spin-chain vacuum

  • Non BPS operators can be constructed by putting

magnons on top of this vacuum:

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SLIDE 16

Use of symmetry[Beisert]

  • Consider an infinitely long BPS operator made up of Z’s
  • Symmetry preserved by the “vacuum” is

spin-chain vacuum

  • Non BPS operators can be constructed by putting

magnons on top of this vacuum:

  • Magnons belong to the bifundamental irrep of PSU(2|2)2
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SLIDE 17
  • In addition, the vacuum is invariant under two central

charges:

* These generators add or subtract one unit of Z. This is the symmetry of the vacuum because the chain is infinite.

  • 2 to 2 magnon S-matrix is determined up to a phase by

this centrally-extended symmetry.

Phase can be determined by requiring the crossing symmetry of the S-matrix.

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SLIDE 18
  • Assuming the factorizability of multi-particle S-matrix,
  • ne can write down the finite-coupling version of Bethe eq,

Assumption: Only schematic (actual equations are more complicated)

Asympt ptot

  • tic

ic Bethe he Ansatz tz:

Energy: Dispersion:

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“Rapidity” parametrization:

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Rapidity torus

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Finite size correction

[Ambjorn, Janik, Kristjansen]

+ + ….

No virtual particles

1 virtual particles

  • For a finite-size operator, there are corrections coming

from virtual particles going around the chain and scattering physical particles.

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Virtual particle from “mirror transformation”

Mirror dispersion is obtained by the analytic continuation in the u-space

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Mirror dispersion is obtained by the analytic continuation in the u-space

Virtual particle from “mirror transformation”

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Lessons from 2pt functions

  • First study infinitely long operators.
  • Make use of (centrally extended) symmetry.
  • Finite size corrections from virtual particles.
  • One can move a particle from one edge to the other by the

“mirror transformation”.

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  • 2. Perturbative computation of

3pt functions

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3-point functions

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Perturbative computation

[Escobedo-Gromov-Sever-Vieira] [Foda] [Kazama-Nishimura-S.K.]

Tree-level:

[Okuyama-Tseng] [Roiban-Volovich][Alday-David-Narain-Gava]

bridge length

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square-root of S-matrix

Perturbative computation

Result (tree-level SL(2) sector):

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1-loop:

Perturbative computation

The actual computation is much more complicated.

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Perturbative computation

Result (one-loop SL(2) sector):

[Vieira-Wang]

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Lessons from perturbative 3pt

  • 3pt = sum over partition of magnons
  • Building block = “square-root” of the S-matrix.
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  • 3. 3pt from Hexagons
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3pt

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3pt = Hexagon2

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propagation scattering

More precisely…

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propagation scattering

More precisely…

Sum m ove ver r parti rtiti tions ns!

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Building block = Hexagon form factor Severely constrained by the symmetry (+ Integrable bootstrap equations)

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Use of symmetry

BPS 3pt = Residual symmetry =

Twisted translation

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One magnon form factor

Left Right

SU(2|2)2 exicitation

Use of symmetry

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SU(2|2)2 exicitations SU(2|2) S-matrix

“square-root” of S-matrix

Two magnon form factor

Use of symmetry

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SU(2|2) S-matrices

Multi-magnon form factor

Use of symmetry

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Bootstrap eq. for h12

Watson eq. :

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Bootstrap eq. for h12

Crossing eq. :

Particle-antiparticle pair

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Solution: (Not unique but this choice is the simplest and correctly reproduces the weak-coupling result.)

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All-loop prediction

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Matches with the OPE decomposition of 4pt functions of BPS ops. Bridge-length 2

[Sokacthev et al.]

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Bridge-length 1 Perturbation result contains a zeta-function part, which cannot be reproduced by the sum over partitions.

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  • 4. Finite size correction to

Hexagons

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Finite size correction

In addition to sum over partitions, we should include the virtual- particle corrections

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Virtual particle corrections

Suppression coming from the propagation of the virtual particles

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Virtual particle corrections from mirror transformation

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Virtual particle corrections from mirror transformation

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Virtual particle corrections from mirror transformation

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Full expression for the integrand

Measure: Transfer matrix: (comes from matrix structure)

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Virtual particle corrections

Tree-level and 1-loop: 2-loop:

Zeta function indeed comes from the mirror particle.

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Virtual particle corrections

Tree-level and 1-loop: 2-loop: 3-loop:

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Virtual particle corrections

Tree-level and 1-loop: 2-loop: No new contributions. 3-loop:

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All our predictions agree with the recent 3-loop results

[Chicherin, Drummond, Heslop, Sokatchev] [Eden] (see also [Eden, Sfondrini])

Virtual particle corrections

Tree-level and 1-loop: 3-loop: 2-loop: No new contributions.

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SLIDE 59

Summary

  • Non-perturbative approach to study 3pt functions:

3pt = Hexagon2

  • Complete agreement with 3-loop data.
  • Agreement with the strong coupling result (minimal surface

in AdS) [Kazama, SK]

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SLIDE 60
  • 1. 4-loop
  • 2. 4-point function from hexagons?
  • 3. Resumming virtual particles?

Reproducing the strong coupling result? TBA, QSC for 3pt?

  • 4. Other theories?

ABJM? 4d N=2? [Pomoni, Mitev]

Future directions