Nonplanar Correlators in N=4 SYM
From Integrability
Shota Komatsu
IASStrings 20181711.05326 [Bargheer, Caetano, Fleury, Vieira, SK]+1807.xxxxxx [Bargheer, Caetano, Fleury, Vieira, SK]AdS = CFTAdS = CFT• Deep insights into quantum gravity & interacting QFT.

• Still many things to be understood.

AdS = CFT• Deep insights into quantum gravity & interacting QFT.

• Still many things to be understood.
• (One of) the key missing link(s):Definition of quantum gravity/string theory in AdS.
Even the perturbative (calculable) definition is missing in most cases.

AdS = CFT• Deep insights into quantum gravity & interacting QFT.

• Still many things to be understood.
• (One of) the key missing link(s):Definition of quantum gravity/string theory in AdS.
Even the perturbative (calculable) definition is missing in most cases.

• Almost no checks for 1/N corrections (= loops in AdS).

AdS = CFT• Deep insights into quantum gravity & interacting QFT.

• Still many things to be understood.
• (One of) the key missing link(s):Definition of quantum gravity/string theory in AdS.
Even the perturbative (calculable) definition is missing in most cases.

• Almost no checks for 1/N corrections (= loops in AdS).
• Learn from solvable examples…?

N=4 Super Yang-Mills• Solvable at large N using integrability
2-pt :

N=4 Super Yang-Mills• Solvable at large N using integrability
2-pt :
N=4 SYM
Map between “single-trace operators”
and spin chains. [Minahan, Zarembo 2002]

Spin chain turns out to be integrable

N=4 Super Yang-Mills• Solvable at large N using integrability
2-pt :
N=4 SYM
Map between “single-trace operators”
and spin chains. [Minahan, Zarembo 2002]

Spin chain turns out to be integrable

String theory
Sigma model in AdS is integrable (classically)[Bena, Polchinski, Roiban 2003]

Integrable QFT on a cylinder : solved by
Thermodynamic Bethe Ansatz

N=4 Super Yang-Mills• Solvable at large N using integrability
2-pt :
String theory

N=4 SYM
Map between “single-trace operators”
and spin chains. [Minahan, Zarembo 2002]

Spin chain turns out to be integrable

Sigma model in AdS is integrable (classically)[Bena, Polchinski, Roiban 2003]

Integrable QFT on a cylinder : solved by
Thermodynamic Bethe Ansatz

Quantum spectral curve[Gromov, Kazakov, Leurent, Volin 2013]

Planar higher-pt functions

For simplicity, I will focus on the correlation functions of½ BPS operators.

although many things can be generalized to non-BPS operators.

3pt = a pair of pants
N=4 SYM at zero coupling[Alday et al], [Okuyama, Tseng.], [Roiban, Volovich][Escobedo et al.] and many others

Bridge length3pt = a pair of pants
N=4 SYM at zero coupling[Alday et al], [Okuyama, Tseng.], [Roiban, Volovich][Escobedo et al.] and many others

Planar surface for 3pt functions

or equivalently

Bridge length3pt = a pair of pants
Planar surface for 3pt functions

or equivalently• To use integrability, one has to consider integrable models ona pair of pants.
• Never studied before in the literature.

3pt = (Hexagon)2[Basso, SK, Vieira 2015]

triangulation of the worldsheet

3pt = (Hexagon)2Insert a complete basis
Insert a complete basis of stateson the dashed lines.

measurepropagation
factor• Similar to sewing construction of 2d CFT.
• The computation of 3pt boils down to the computation of “hexagons”.

Hexagon and integrability
Symmetry (psl(2|2)) and integrability determine the contributions from eachhexagon.
“Form factor bootstrap”

Cf. [Smirnov]

Hexagon and integrability
Symmetry (psl(2|2)) and integrability determine the contributions from eachhexagon.
“Form factor bootstrap”

Cf. [Smirnov]

Simplest 2-particle contribution:

Hexagon and integrability
Symmetry (psl(2|2)) and integrability determine the contributions from eachhexagon.
“Form factor bootstrap”

Cf. [Smirnov]

Simplest 2-particle contribution:

Factorization structure resembling Yang-Baxter.

Hexagon and integrability
Symmetry (psl(2|2)) and integrability determine the contributions from eachhexagon.
“Form factor bootstrap”

Cf. [Smirnov]

Simplest 2-particle contribution:

The result beautifully reproduces the perturbative computation(checked up to four loops and partially at strong coupling)[Jiang, Kostov, Serban, SK], [Eden, Sfondrini][Basso, Goncalves, Vieira SK], [Basso, Goncalves, SK]

Generalization to higher pt
All these can be generalized to higher point functions.
1) List up all possible
planar graphs.
2) Compute contributions
from each graph.

Generalization to higher pt
All these can be generalized to higher point functions.
Idea: Insert a complete basis andcut it into hexagons.

Insert a complete basis

Generalization to higher pt
All these can be generalized to higher point functions.

“Moduli integral”

Generalization to higher pt
All these can be generalized to higher point functions.

Generalization to higher pt
All these can be generalized to higher point functions.

Weight factor(purely kinematical)

Generalization to higher pt
All these can be generalized to higher point functions.

Weight factor(purely kinematical)

Reproduces the perturbative computation (1-loop four pt, 1-loop five pt,[Fleury, SK 2016, 2017], [Fleury, Goncalves, SK in progress]
higher loops in special kinematics)
See also [Eden, Sfondrini 2016]

Can we generalize this tononplanar surfaces?

Decomposing the surface with hexagons
First guessDecomposing the surface with hexagons
First guess[Bargheer, Caetano, Fleury, Vieira, SK] also [Eden, Jiang, Sfondrini]

Cut the surface into planar
hexagons, insert a complete basis,and sum over the graph.

Decomposing the surface with hexagons
First guess[Bargheer, Caetano, Fleury, Vieira, SK] also [Eden, Jiang, Sfondrini]

Cut the surface into planar
hexagons, insert a complete basis,and sum over the graph.
8 hexagons, complicated...

Simplification for large-length operators
Combinatorial enhancement:
When the operators are very long,the maximally connected graphs will be combinatorially dominant.
# of ways to split L propagators to n groups =Simplification for large-length operators
Combinatorial enhancement:
When the operators are very long,the maximally connected graphs will be combinatorially dominant.
# of ways to split L propagators to n groups =Dynamical suppression:
If allare large, the contributions are exponentially suppressed.

Simplification for large-length operators

Submaximal graphs, obtained from the maximal graphs by erasing a few
connections, will dominate!
(Locally look the same as planar four-point function.)match!

prediction

Finite L’s
At, we can also try to study the correlators of finite-length operators.

Puzzle:• Our basic strategy : Start from tree-level graphs, cut it into hexagons
and dress them with magnons.
• Should we include• Not at tree level. But we cannot simply throw it away since this caninclude genuine nonplanar contributions (when dressed with magnons).

Resolution: Use the analogy between the sum over graphs
and the moduli space of Riemann surface.

“Stratification”• The graphs that are secretly planar are the analogues of thedegenerate Riemann surfaces.

6-pt with 2 “holes”.

• They correspond to the boundaries of the torus moduli space.

“Stratification”• Prescription: Sum all the graphs and subtract the “boundaries of moduli”.
[Deligne-Mumford][Chekhov]

All the tree-level graphs on atorus

All the tree-level graphs on a sphere with twoholes.

“Stratification”• Prescription: Sum all the graphs and subtract the “boundaries of moduli”.
[Deligne-Mumford][Chekhov]

All the tree-level graphs on atorus

All the tree-level graphs on a sphere with twoholes.

• No dynamics at the holes. They are just the holes of the double-line
Feynman diagram.
Same as adding probe D3’s

“Stratification”• Prescription: Sum all the graphs and subtract the “boundaries of moduli”.
[Deligne-Mumford][Chekhov]

All the tree-level graphs on atorus

All the tree-level graphs on a sphere with twoholes.

• No dynamics at the holes. They are just the holes of the double-line
Feynman diagram.
Same as adding probe D3’s

• Correctly reproduces the finite L answers at

Summary and outlook
Summary:• One can use integrability to study non-planar quantities.
• Sum over graphs, cut them into planar hexagons.

• Finite L→ “Stratification”.

Outlook:• Better understand the relation between the graphs and themoduli space, and the stratification. Cf. [Gopakumar 2003] [Razamat 2008][Gopakumar, Pius 2012]• Try to resum 1/N series in some kinematics?
Cf. [Gross Mende] [Mende Ooguri]• Find ways to efficiently resum magnons. Quantum spectral
curve?