TESE DE DOUTORAMENTO IFT–T.002/23

Hexagonalization in AdS/CFT:
Classical Limit in AdS5/CFT4 and Mirror

Corrections in AdS3/CFT2

Matheus Augusto Fabri

Orientador

Pedro Vieira

Fevereiro de 2023



 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  
               Fabri, Matheus Augusto. 

 F124h             Hexagonalization in AdS / CFT: classical limit in AdS5 / CFT4 and mirror 
corrections in AdS3 / CFT2  /  Matheus Augusto Fabri. – São Paulo, 2023 

149 f. 
 

Tese (doutorado) – Universidade Estadual Paulista (Unesp), Instituto de 
Física Teórica (IFT), São Paulo 

Orientador: Pedro Vieira 
 

1. Física matemática. 2. Teoria quântica de campos.  3. Supersimetria. I. 
Título  

 

 

  

Sistema de geração automática de fichas catalográficas da Unesp. Biblioteca 
do Instituto de Física Teórica (IFT), São Paulo. Dados fornecidos pelo 

autor(a). 

 

 



Dedico esta tese à Adriely e a minha família.

i



Agradecimentos

No âmbito profissional há várias pessoas a quem devo agradecer por toda a assistência

dada ao longo desta jornada. Primeiramente agradeço aos meus colaboradores Carlos

Bercini, Alexandre Homrich e o Gabriel Lefundes, sem os quais este trabalho não seria

possível. Agradeço ao Gabriel pela persitência em continuarmos o trabalho mesmo após

diversas mudanças de rotas que tivemos. Agradeço também ao meu orientador Pedro

Vieira pelas oportunidades dadas neste doutorado, por tudo que aprendi com ele e pela

compreensão e paciência que ele teve ao longo dos últimos seis anos. Não poderia deixar

de agradecer também a todos os colegas do IFT com os quais passei ótimos momentos e

que sem dúvida tornaram esta jornada mais gratificante. Também gostaria de agradecer

ao Diogo H. F. de Souza e à Thais Campos Santiago por todo o trabalho que realizamos

na organização do Congresso Paulo Leal. Agradeço também aos professores do instituto

por tudo que me ensinaram e também a toda a equipe de funcionários do IFT que fazem

com que esse instituto continue em pé. Por fim agradeço também a todos os pequisadores

que me auxiliaram neste trabalho sendo estes Enrico Olivucci, Francesco Aprile, Leonardo

Rastelli, Shlomo Razamat, Madhusudhan Raman e em especial ao Alessandro Sfondrini

por todo o suporte dado.

No âmbito pessoal gostaria de agradecer a toda a minha família por todo o suporte

dado desde que inicei a minha trajetória na Física lá átras na UEM, em especial à minha

irmã Ana Paula a qual não acho palavras para agradecer por tudo o que ela fez por mim.

Ela sempre será um exemplo de dedicação e resiliência para mim. Não poderia claro de

deixar de agradecer a minha companheira Adriely que sempre me ajudou nesta longa e

díficil jornada, agradeço por toda a força que me deu nos momentos maís dificeis e por

sempre me motivar a seguir adiante.

Por fim, mas não menos importante agradeço ao suporte financeiro dado a este projeto

pelo CNPq e também pela FAPESP por meio do projeto 19/12167-3.

ii



“(. . . ) A Constituição certamente não é perfeita. Ela própria o confessa ao admitir
a reforma. Quanto a ela, discordar, sim. Divergir, sim. Descumprir, jamais.
Afrontá-la, nunca.
Traidor da Constituição é traidor da Pátria. Conhecemos o caminho maldito.
Rasgar a Constituição, trancar as portas do Parlamento, garrotear a liberdade,
mandar os patriotas para a cadeia, o exílio e o cemitério.
Quando após tantos anos de lutas e sacrifícios promulgamos o Estatuto do
Homem da Liberdade e da Democracia bradamos por imposição de sua honra.
Temos ódio à ditadura. Ódio e nojo.
Amaldiçoamos a tirania aonde quer que ela desgrace homens e nações.
Principalmente na América Latina.”

Dr. Ulysses Guimarães, presidente da Assembleia Nacional Constituinte.

iii



Resumo

Um dos principais usos da dualidade AdS/CFT é o cálculo não perturbativo
da dinâmica da teoria de campos conforme (CFT) na fronteira. Isto é possível nos
casos integráveis, como por exemplo AdS3 × S3 × T4 ou o caso mais explorado
AdS5 × S5. Neste contexto desenvolveu-se um formalismo não perturbativo de-
nominado hexagonalização para o cálculo das constantes de estruturas da CFT
dual no limite planar. Os objetos centrais nesta formulação são os chamados hexá-
gonos, os quais podem ser derivados de primeiros príncipios neste formalismo
para então se calcular as constantes de estruturas à acoplamento finito. O presente
trabalho aborda o formalismo dos hexágonos nas dualidades mencionadas ante-
riormente e este se divide em duas partes. Na Parte 1 analisamos as constantes
de estruturas a acoplamento fraco em N = 4 Yang-Mills supersimétrica. Aqui
focaremos em dois setores da teoria gerados somente por operadores escalares
ou com spin, respectivamente. Nós encontramos novas representações para os
hexágonos tal que ambos os setores estão em pé de igualdade e por meio de um
cuidadoso uso destas derivamos o limite clássico de funções de correlação nestes
setores. Nossos resultados estão de acordo com cálculos da literatura realizados
por meio de métodos indiretos. Já para a Parte 2 focaremos em hexágonos em
AdS3 × S3 × T4. Um importante problema em aberto nesta dualidade é o cálculo
da dinâmica da CFT na teoria dual. Para este fim nós estendemos a proposta de
hexagonalização dada em [B. Eden, D. l. Plat, and A. Sfondrini, J. High Energy
Phys. 08 (2021) 049]. Neste trabalho completamos esta proposta definindo as
correções mirror que permitem a descrição de constantes de estruturas para um
número finito de contrações. Além disso também provamos que operadores prote-
gidos nesta teoria não recebem estas correções. Por fim encerramos descrevendo
como utilizar hexagonalização para calcular funções de correlação com quatro
operadores nesta dualidade.

Palavras Chaves: AdS-CFT Correspondence; Integrability; Conformal Field The-
ory; Bethe Ansatz.

Áreas do conhecimento: Física matemática; Teoria quântica de campos; Super-
simetria.

iv



Abstract

One of the main uses of the AdS/CFT duality is the non-perturbative compu-
tation of the conformal field theory (CFT) data on the boundary. This is possi-
ble on integrable backgrounds like for example the AdS3 × S3 × T4 or the most
well known example of AdS5 × S5. In these contexts it was developed a non-
perturbative formalism called hexagonalization to compute structure constants
of the dual CFT in the planar limit. The main objects in this framework are the
so-called hexagons which can be bootstrapped within this formalism to then
calculate the structure constants at finite coupling. This work concerns this hexag-
onalization framework in the aforementioned backgrounds and it is divided into
two parts. In Part 1 we analyze structure constants at weak coupling in N = 4
supersymmetric Yang-Mills. We focus on two sectors of the theory spanned only
by scalar or spinning operators, respectively. We find new representations of the
hexagons that put both sectors on equal footing and by judicious use of these new
expressions we derive the classical limit of correlation functions in these sectors.
Our results match previous computations done in the literature through more
indirect means. Now Part 2 concerns hexagons in AdS3 × S3 × T4. A big open
problem in this holographic duality is to compute the CFT data of the dual theory.
For this end we extend the hexagonalization proposal made in [B. Eden, D. l. Plat,
and A. Sfondrini, J. High Energy Phys. 08 (2021) 049] for this background. In
this work we complete this picture by computing the so-called mirror corrections
that allow the description of structure constants for finite bridge lengths and as a
byproduct we prove that the protected operators in the theory do not receive these
corrections. We then close by describing how to use hexagonalization to compute
four-point functions in this background.

Keywords: AdS-CFT Correspondence; Integrability; Conformal Field Theory;
Bethe Ansatz.

Fields of knowledge: Mathematical physics; Quantum field theory; Supersymme-
try.

v



Contents

1 Introduction 1

2 Hexagons and the classical limit 13
2.1 Tailoring and double spin chain formalism . . . . . . . . . . . . . . 13

2.1.1 New expression for the structure constant . . . . . . . . . . . 21
2.2 Non-compact spin chains and hexagons . . . . . . . . . . . . . . . . 22
2.3 What is the classical limit? . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 New representations of the hexagons and classical limit . . . . . . . 31

2.4.1 SU(2) sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4.2 SL(2) sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.5 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3 Hexagonalization in AdS3/CFT2 48
3.1 Integrability in AdS3 × S3 × T4 . . . . . . . . . . . . . . . . . . . . . 48

3.1.1 From string theory to the symmetry algebra . . . . . . . . . 49
3.1.2 Particle content . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1.3 Crossing and mirror transformations . . . . . . . . . . . . . 61
3.1.4 S-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.2 Chiral ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.3 Hexagons and structure constants . . . . . . . . . . . . . . . . . . . 75

3.3.1 Fixing the hexagon . . . . . . . . . . . . . . . . . . . . . . . . 77
3.3.2 Full hexagon ansatz and partition function . . . . . . . . . . 85

3.4 Some examples of structure constants . . . . . . . . . . . . . . . . . 87
3.5 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4 Hexagons and finite length corrections 92
4.1 Mirror corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.2 Measure and transfer matrix formalism . . . . . . . . . . . . . . . . 95
4.3 Mirror corrections and the chiral ring . . . . . . . . . . . . . . . . . 101
4.4 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

vi



5 Four-point functions and hexagons 106
5.1 Four-point functions in the dual CFT2 . . . . . . . . . . . . . . . . . 107
5.2 Hexagonalization of four point functions . . . . . . . . . . . . . . . 110
5.3 Four-point function of half-BPS operators . . . . . . . . . . . . . . . 114

5.3.1 Order O(h0) correlator . . . . . . . . . . . . . . . . . . . . . . 115
5.3.2 Order O(h) correlator and some issues . . . . . . . . . . . . 117

5.4 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6 Conclusion 123

A Gaudin norm of Bethe states 129

B Bethe equations and rewriting D(αj, ᾱj) 130

C Hexagons from double spin chain 133

D Weak coupling expansions in AdS3/CFT2 hexagonalization 135

E Bound states and hexagon scalar factors 137

Bibliography 138

vii



Chapter 1

Introduction

One of most fruitful ideas of the last decades in theoretical physics was the
large-N expansion proposed by Gerardus t’Hooft in [1]. In this work it was
proposed the organization of perturbation theory in matrix models, like SU(N)

gauge theories for instance, by the topology of the Feynman diagrams instead
of the Yang-Mills coupling gYM dependence as usual. By properly defining the
t’Hooft coupling λ = Ng2

YM one sees that each graph is then accompanied by a
N2−2g factor with g being the genus of the Feynman graph. Then in the large-N
limit the usual weak coupling expansion is done in λ and the graphs organized in
a 1/N series with planar contributions (g = 0), torus contributions (g = 1), and
so on. Nonetheless this latter series looks like a genus expansion of string theory
in terms of the string coupling gs. One could heuristically say that the Feynman
diagrams are the triangulation of the string worldsheet and that somehow the
1/N expansion captures the genus expansion in gs in string theory with gs ∼ 1/N.
The concrete realization of this idea was given in the seminal work [2] by Juan
Maldacena and in the follow up by E. Witten in [3], where it was developed the
holographic principle.

The holographic principle, also called gauge-gravity or AdS/CFT duality,
basically tell us that a specific quantum gravity theory on a d-dimensional anti-
de Sitter space (AdSd) is dual to a conformal quantum field theory on (d − 1)-
dimensional spacetime (CFTd−1). By these theories being dual we mean that there
is a dictionary mapping dynamical quantities in the AdSd bulk to the conformal
boundary, like for example energies of states in AdSd to scaling dimensions of the
dual CFTd−1. We represent the holographic duality in Figure 1.1. We can hardly
overestimate the impact of this principle in theoretical high-energy physics. For
instance it yields to us a non-perturbative definition of quantum gravity on anti-de
Sitter spacetimes in terms of (relatively) mundane CFTs on flat space. However
this principle remains at the moment a conjecture, meaning that at the moment its
underlying mechanics are not well understood even though there are many tests
of its validity.

1



Chapter 1. Introduction 2

Figure 1.1: Holographic duality. Here we show an example of how some calcu-
lation on AdS/CFT would work. We represent the AdSd space on the left by a
cylinder and some calculation on the bulk gets translated to a calculation on a
CFTd−1 calculation on the right. One can think of the CFTd−1 as living on the
conformal boundary of AdSd thus the name holographic, since all d-dimensional
quantum gravity information would be coded into d− 1 dimensions.

In this work we will focus on the use of specific instances of this duality to
compute quantities of the dual CFT and remarkably in the examples we will
discuss this can be done at finite coupling. Let us detail these examples. The most
well known instance of gauge-gravity duality is:

The Type IIB superstring theory in AdS5 × S5, which can be seen as the
near horizon limit of a system with N D3-branes, is dual to N = 4 super
Yang-Mills (SYM) in a flat four-dimensional spacetime with gauge group
SU(N). The string tension T, the t‘Hooft coupling λ and the AdS5 radius R
are related by

TR2 =
√

λ.

The quantum gravity theory in this case is a type of string theory on the product
space AdS5 × S5 and as G. t’Hooft told us gs ∼ 1/N, with N being the number
of colors in the gauge theory1. Thus the large-N limit picks only the sphere con-
tributions in string perturbation theory which are related to the planar diagrams
(g = 0) in the dual CFT4, then large-N limit is called planar limit.

The dual CFT4 in this case is a non-abelian Yang-Mills theory with N = 4
supersymmetry with all matter fields in the adjoint representation of the gauge
group, that is, they are in the same supermultiplet of the gauge fields. Also the
R-charge group is simply SU(4) with the states organizing in representations
of this group. The R-charge is related to the isometry group of the compact S5.
This CFT at first looks as complicated as any usual quantum field theory (QFT)

1For a review on the string and CFT sides of the duality and the duality itself see [4, 5, 6, 7].



Chapter 1. Introduction 3

since even at the planar limit one has to employ the usual perturbation theory to
compute observables. This is where we can resort to the AdS/CFT duality. Note
that it is a strong-weak coupling duality, which means that weak coupling in the
CFT4 side (λ → 0) is related to strong coupling computations in the string side
(T → 0 in units of AdS5 radius) and vice versa. Therefore to analyze complicated
computations at strong coupling in one side we look at the easier weak coupling
analysis in the other side of the duality.

Another duality that has been of great interest in the last couple of years and it
will also be central for us is the following [2]:

The Type IIB superstring theory in AdS3 × S3 × T4 with Neveu–Schwarz
(NSNS) flux κ and Ramond-Ramond (RR) flux h, which can be seen as
the near horizon limit of a F1-NS5-D1-D5 brane system, is dual to a two-
dimensional N = (4, 4) superconformal CFT.

The reader probably noted that in this case we are not as precise we were in the
AdS5/CFT4 duality. The reason is that for most of the moduli space of the string
theory, the dual CFT2 is not known. Note that the moduli space is actually higher-
dimensional for this duality, however only the fluxes h and κ are relevant for the
spectrum [8]. The recent interest for this duality is due to the fact that for κ = 1
and h = 0 one can actually prove the duality [9, 10, 11], and with this get hints
of how it works and maybe analyze the higher-dimensional cases which remain
unproven.

Even though we do not now exactly the dual CFT2 we have some information
of it. First it has N = (4, 4) supersymmetry with SU(2)L ⊗ SU(2)R R-charge,
which as before it is the isometry of the compact S3. For κ = 0 we are in the pure
RR limit and here the duality behaves much like the AdS5 × S5 case with h being
the analog of the t’Hooft coupling. However the CFT2 dual here is completely
unknown with some proposals given in [12, 13]. It turns out that for the pure NSNS
limit with h = 0 and κ = 1 the dual CFT2 is known and it is simply the SymN(T4)

[10]. This theory is defined by taking N copies of free bosons and fermions and
then identifying these by the permutation group SN. This identification work as
the gauge symmetry in N = 4 SYM and N is the analog of the number of colors
in the gauge group, thus as before the planar limit is the large-N limit2. Some

2However the notion of planarity works a little different here and we will describe in the
Conclusion (Chapter 6) how a computation of some observable works and the planarity count for
SymN(T4).



Chapter 1. Introduction 4

proposal for the dual CFT2 for κ ≥ 2 and h = 0 are given in [14, 15], however for
mixed flux (generic h and κ) this landscape remains uncharted.

In this work we will focus on the two dualities discussed above, however
these are not the only examples known3. Another well known example is the
AdS4/CFT3 duality. Here we have type IIA string on AdS4 × CP3 dual to three-
dimensional N = 6 superconformal Chern-Simons with gauge group U(N)×
U(N) with levels k and −k, respectively [17]. Now that we have these dualities
what we want to do with them? It can be used in two ways. First as a non-
perturbative definition of quantum gravity in AdS spacetimes. Then we compute
dynamical quantities in the CFTd and map them to gravity properties in the bulk.
Another way to use it is to do computations on the string side of the duality and
extract information of the CFTd. It is the latter procedure we use here, that is,
using AdS/CFT duality we try to “solve” the dual conformal field theory.

Let us define what we mean by solving a CFT. From the conformal bootstrap
program it is well known that the entire dynamical content of a CFT is coded into
the so-called CFT data [18]. For instance, to build representations of the conformal
algebra we consider primary operators Oj which are just the highest weight states
in this algebra. Thus the CFT data is given by {∆j, Cijk} where ∆j is the scaling
dimension of a primary and Cijk are the structure constants of three primaries,
id est, constants that say how three operators interact. Then to solve the CFT we
must compute these data, the idea then is to use the holographic duality to extract
them. However usual computations in the string side are challenging since string
theory on AdS spacetimes are not well understood. Although some advances
were recently made in [19], computations of desired quantities remain for the
moment out of reach. But there is one technique that comes to our rescue: quantum
integrability.

Integrable models or exactly solved models are a class of nontrivial systems
that have an exact solution, which means that they can be solved without resorting
to perturbation theory. This concept seems kind of vague, however in classical
mechanics it can be precisely defined. Indeed, if we have a system defined in a 2n-
dimensional phase space and it possess n linearly independent conserved charges
we can construct the action-angle variables and then integrate the equations of
motion, thus solving it [20, 21]. This is the so-called Liouville integrability. Some
examples of classical integrable models are the Euler, Lagrange and Kowalevski
tops, the Kepler problem and the Neumann model.

3For a list with known examples AdS/CFT dualities see [16].



Chapter 1. Introduction 5

This concept of integrability can be extended to the realm of quantum mechan-
ics. Here we do not have a precise meaning of it as in the classical case4, however
if a system possess sufficient conserved charges it can be exactly solved. Therefore
solvability is tied to the existence of conserved charges beyond the usual ones
like energy and momentum. For quantum mechanical systems, being integrable
means that we can extract the eigenvalues and eigenvectors of the hamiltonian and
other quantities of interest, like correlation functions for instance. Some examples
of integrable models are XXX, XXZ and XYZ one-dimensional Heisenberg spin
chains, Hubbard model, one-dimensional Bose Gas, and so on. These are solved
using a technique called quantum inverse scattering method which leads to the
famous algebraic Bethe ansatz (ABA) and through it we can extract the physical
quantities of interest. For some reviews in quantum integrable models see [23, 24].

Another realm we can apply these techniques is statistical mechanics. Here
integrability works remarkably similar to the quantum mechanics case and one
can use it to compute for example the partition function and from it extract
all the thermodynamic content of the desired model. This was essential to the
understanding of critical phenomena (see the reviews [25, 26]). Some examples
of integrable systems in statistical mechanics are two-dimensional Ising model
and six- and eight-vertex models. Here we see a common feature of all these
integrable models in quantum and statistical mechanics, they are restricted to 1+ 1
dimensions.

We can see the origin of their restriction to two-dimensional models through
the example of integrable quantum field theories (IQFTs). This class of QFTs have
the special property of possessing factorized S-matrices and an infinite number of
conserved charges, they first appeared in the seminal work [27] by Zamolodchikov
and Zamolodchikov. Consider for example a model with some global symmetry,
like O(N) for instance. Then we can use it to restrict scattering to O(N)-invariant
channels and that is it, we can not go beyond that. We thus use the usual QFT
methods to compute the scattering matrix. Turns out that some models, like
the O(N)-invariant QFT in two dimensions, have an infinite tower of conserved
charges that do not commute with the Lorentz group, that is, they are momentum
dependent. We can use these to constrain the S-matrix and they imply no particle
production [28]. Also the S-matrix is factorized, which means that 3→ 3 scattering
decomposes into a sequence of 2→ 2 events [29]. This is the famous Yang-Baxter
equation, which is a cubic constrain for the 2 → 2 S-matrix and by using it we

4See for example the discussion in the first two chapters of [22].



Chapter 1. Introduction 6

Figure 1.2: Yang-Baxter equation. Here we have represent graphically the Yang-
Baxter equation. Each small yellow dot represent a 2→ 2 scattering event. Then
this equation tell us that the two possible ways of decomposing a 3→ 3 event as a
sequence of 2→ 2 scatterings are equal. In a similar way we can decompose any
n→ n event.

can completely fix the O(N) S-matrix for example (after imposing some physical
conditions like the spectrum of bound states and unitarity). The Yang-Baxter
equation is represented in Figure 1.2. Then integrable quantum field theories have
factorized scattering and due to it the S-matrix is fixed and the spectrum can be
determined at finite volume using the thermodynamic Bethe ansatz [30]. Some
examples of IQFTs are the O(N) model, sine-Gordon model, massive Thirring
model and so on. The reason for the restriction of integrable models to two-
dimensional QFTs is due to the Coleman-Mandula theorem5 [31]. With this brief
detour into IQFTs let us head back to holographic dualities.

This story of integrable models is very interesting and all, however how it
links with AdSd+1/CFTd dualities if it is restricted to two-dimensional models?
Note that the underlying theory describing quantum gravity in the bulk is a string
theory which can be seen as a two-dimensional QFT in the worldsheet. Turns
out that for some backgrounds this QFT is integrable and when this happens we
have an integrable background. The previous discussed backgrounds AdS5/CFT4,
AdS4/CFT3 and AdS3/CFT2 are integrable [32, 33]. This means that we can use
all the tools from quantum integrability to find dynamical quantities in the string
theory and then use the AdS/CFT dictionary to find their dual in the CFT side.
All of this is done at at finite coupling, hence the power integrability in AdS/CFT.

As said before, in this work we will focus on the AdS5× S5 and AdS3× S3× T4

dualities. The first use of integrability in AdS/CFT was in the computation of

5Basically if higher-dimensional QFTs had this infinite tower of conserved charges the theorem
says that the scattering would be trivial and thus the QFT would be simply a free field theory.



Chapter 1. Introduction 7

the spectrum of the dual CFT. Let us start with the AdS5/CFT4 case. One of
the first signs of integrability was that the string model at large tension T, that
is classical string theory, is classically integrable for this background. Now on
the CFT4 side, one of the first hints of integrable structures was the remarkable
observation that we can recast the one-loop dilatation operator of N = 4 SYM as
the hamiltonian of an integrable spin chain, for the SU(2) sector of the theory this
is just the one-dimensional XXX Heisenberg spin chain [34]. In this language single
trace operators are dual to eigenstates of the chain, also called Bethe states, and
the eigenvalues of this hamiltonian are the one-loop anomalous dimensions. The
spectrum and thus the anomalous dimensions can be found using the algebraic
Bethe ansatz. Then in a series of works this result was extended to higher and
higher loops where the hamiltonian one obtains becomes of long-range. Over the
last two decades, many techniques have been taken from exactly solved models
and applied to the analysis of a plethora of observables in N = 4 SYM at weak,
strong, and even at finite coupling. An example of outstanding success is the
computation of the spectrum of single trace operators to all loops in perturbation
theory that revealed rich integrable structures such as the quantum spectral curve
[35]. For reviews on these topics see [32, 36, 37].

For AdS3 × S3 × T4 the integrable techniques for the spectrum are similar to
the AdS5/CFT4 case. As before, this string background is integrable and and
the spectral problem has been analyzed in depth [33, 38, 39, 40, 41, 42, 43]. Even
though we do not have the dual CFT2 for mixed flux, the spectral problem can still
be analyzed in terms of a spin chain. However there are some glaring differences
with respect to AdS5 × S5 integrability. One of them is that the IQFT in the
worldsheet has massless excitations in contrast with its higher dimensional cousin
where all excitations are massive. These massless modes make the comparison
with the string theory a little tricky and introduces novel features to the integrable
structure6. Nonetheless quantum integrability can still be applied to analyze the
spectral problem in this background, and more recently even a quantum spectral
curve formalism, as in AdS5/CFT4, was developed7 [45, 46]. In this work we focus
on the pure RR case whose integrable structure is very similar to the AdS5 × S5

6For example, usually the ABA yields the spectrum of asymptotically large operators. Finite
length corrections are given by the so-called wrapping corrections that are suppressed for large
operators. However here this is not exactly true because massless modes’ wrapping corrections to
the spectrum enters at the same order of the ABA ones [44].

7However here the story still not complete here since the massless modes still elusive in this
formalism.



Chapter 1. Introduction 8

one. Nonetheless it is worth noticing that the pure NSNS case is also integrable
and in this exceptional situation the string theory is exactly given by a quantum
integrable spin chain without any corrections [42]. The model can also be solved
by worldsheet CFT techniques [47]. Note that for mixed flux the spectral problem
remains open due to some technical problems [48, 49, 38].

One can see the spectral problem in AdS/CFT as the problem of defining an
IQFT in the cylinder. This is due to the fact that a closed string moving in space
defines a worldsheet that is topologically equivalent to a cylinder. However one
can try to analyze an IQFT on more complicated topologies, like a pair of pants or
a sphere with four punctures. These more complicated geometries are related to
the computation of structure constants (pair of pants) and of four-point functions
(sphere with four punctures) through a procedure called hexagonalization. This
formalism is the central theme in this work and we will describe it now.

After the integrability for the spectral problem was worked out in details, the
natural next step was the computation of the other half of the CFT data: the struc-
ture constants. With this in mind the tailoring procedure for AdS5× S5 emerged in
a series of works [50, 51, 52, 53, 54, 55, 56]. It allowed to find three-point functions
of single trace operators at weak coupling, the main idea was to map each operator
to a Bethe state and cut these states into subchains. Then we join these subchains
through inner products, computed using quantum integrability techniques, in
a way that reproduces Wick contractions and then sum over all possible ways
of cutting. The next link in the chain of developments was the hexagonalization
formalism, which was introduced in [57] and it enabled the calculation of struc-
ture constants of N = 4 SYM. Unlike tailoring, hexagonalization works at finite
coupling for asymptotically large operators8.

Let us describe hexagonalization in more detail. Basically it consists in taking
the pair of pants topology that describes the structure constant of the CFT4 in the
string side and cut it in half. Each side is the so-called hexagon form factor. When
we cut the pair of pants, the excitations of each operator can be in the front or back
hexagons and we sum over all the possible distributions of the physical excitations
in each form factor. Note here the resemblance with the tailoring formalism. Also
in moving an excitation from the front to the back hexagon we gain a propagation
factor that we need to take into account. Each hexagon has three physical edges,
where the operators are cut, and three mirror edges corresponding to where we cut
the pair of pants. To obtain the structure constant we then glue back the hexagons

8A recent proposal for finite length operators was made in [58].



Chapter 1. Introduction 9

Figure 1.3: Hexagonalization of a structure constant. On the left we have a pair of
pants representing a structure constant and on the right its hexagonalization. We
cut the pair of pants into two hexagons. Each hexagon has physical and mirror
edges denoted by the solid and dashed lines, respectively. Note that we sum over
all possible ways of distributing the excitations in the back and front hexagons.
The first contribution correspond to the asymptotic hexagon. The remaining ones
are the mirror corrections where we add a single mirror particle (square) in a
mirror edge, two mirror particles, and so on.

by inserting a complete basis of states in the mirror edges. The first contribution
is the insertion of the vacuum which yields the so-called asymptotic hexagon. In
this limit the structure constant is just the product of the two hexagons with the
physical excitations distributed in both of them. The contributions corresponding
to adding excitations in the mirror edges are the mirror corrections. These are
exponentially suppressed in the large bridge length limit, therefore if all the bridge
lengths are large the structure constant is given only by the asymptotic piece. The
hexagonalization procedure is shown in Figure 1.3.

The only piece remaining is the hexagon itself. To fix it we bootstrap them by
using four axioms that are inspired by the form factor program in IQFTs for local
operators (see the reviews [29, 59]) and for the non-local operators introduced in
[60]. We can understand these axioms as a way of defining QFT in nonstandard
spacetimes. This hexagonalization formalism was then improved and tested in
[61, 62, 63, 64, 65, 66, 67] and it passed in all the tests. Later the same idea was
applied to more complicated topologies like the sphere with four or five punctures
that correspond to four- and five-point functions in N = 4 SYM, respectively
[68, 69, 70, 71, 72, 73]. Hexagonalization works in a similar way for these, we
make more cuts and sum over all the possible ways of cutting the worldsheet with
some extra ingredients that we shall explain in this work. Although we discussed
hexagonalization only for AdS5 × S5 duality, it was recently introduced in the



Chapter 1. Introduction 10

AdS3/CFT2 context in [74]. It works remarkably similar as the higher dimensional
case, however only the asymptotic hexagon was described in that work and no
attempt for the hexagonalization of higher point functions was made.

With this introduction into the subject matter made, let us describe the content
of this work. This thesis is divided into two parts. The first part covers hexagonal-
ization in AdS5 × S5 and the classical limit. It is an expansion of the work done
in collaboration with Gabriel Lefundes in [75]. This part is described in Chapter
2. The second part covers hexagonalization in AdS3 × S3 × T4, mirror corrections
and four-point functions. This is an extension of the work done in [76] and it is in
detailed in Chapters 3, 4 and 5. Below we expand on the content of each part.

Part 1: Hexagons and the classical limit

In the first part we are interested in structure constants of N = 4 SYM at weak
coupling. The tailoring and hexagonalization formalisms should compute the
same structure constants at tree level in the SU(2) sector of N = 4 SYM, however
their equivalence was not proven in full generality. In Chapter 2 we amend this
by proving that they yield the same result at tree level for three excited operators,
thus closing this hole in the hexagon literature. In this process we work out some
analytic properties of the hexagons and find new representations for them at tree
level in both the compact SU(2) and non-compact SL(2) sectors. These could hint
to new representations of the hexagon form factor at finite coupling. An advantage
of these new expressions is that they are computationally less costly, since we
do not need to do multiple tensor contractions to compute them as the original
definition of the hexagon in [57].

Another aspect we also consider in AdS5/CFT4 hexagonalization is the classical
limit of structure constants. This is a central concept at strong coupling in N = 4
SYM. As discussed before, in this regime the string side of the duality is described
by classical strings. Indeed we can map three-point functions in this limit to the
scattering of closed strings in AdS5× S5. The non-linear sigma model on the string
dual is classically integrable and for strings with large charges the correlation
functions can be evaluated by the saddle-point method. These “heavy” string
states correspond in N = 4 SYM to operators with large charges [77].

Nonetheless we are interested in the weak coupling regime and here the clas-
sical limit also plays a role. An example of this is the famous Frolov-Tseytlin
limit, where at strong coupling and asymptotically large charges we find the same
results at weak coupling under the expansion in a appropriate parameter [78].



Chapter 1. Introduction 11

Although it does not work at all loops, indeed there are some observables where
this duality breaks at some loop order with one example described in [79]. The
classical and quantum integrability descriptions appearing at weak and strong
coupling also show some similarities when taking the classical limit of the spin
chain associated with heavy single trace operators. In this limit, we have a large
number of Bethe roots that condense into macroscopic branch cuts in the complex
plane. These resemble the finite-gap solutions of the classical string sigma model
described in [80, 81]. For operators in the SU(2) sector of N = 4 SYM the classical
limit of spin chains is described by the Landau-Lifschitz model, which makes the
relationship with the strong coupling framework explicit as shown in [82].

Here we compute the classical limit of a special class of structure constants
in the SU(2) sector. From the tailoring formalism we know that three-point
functions belong to two classes: the type I-I-I and type I-I-II. Each correspond
to specific choices of the external operators. The type I-I-II are simpler and they
are in the same complexity class as structure constants with only two excited
operators. Their classical limit was found [55, 82]. Now the type I-I-I correlators
are more involved and the derivation of their classical limit from the tailoring
formulas was an open problem, we then compute this classical limit in Chapter
2. Our derivation is done directly from the microscopic formulas using recursion
relations we derived for the hexagons analogous to the ones in [67]. We use the
same methods to also compute the classical limit of SL(2) three-point functions
for some fixed polarizations of the external operators.

Part 2: Hexagonalization AdS3 × S3 × T4 and mirror corrections

As described before the dual CFT2 for the AdS3 × S3 × T4 is unknown beyond
the pure NSNS case with κ = 1, therefore any method of extracting information
about this CFT2 is relevant. Due to this, the hexagonalization of structure constants
for this background, introduced in [74], yields valuable information. In Chapter
3 we introduce the necessary integrability background and the hexagonalization
formalism for this duality. We will work here in the pure RR limit. Also we
compute some examples of structure constants.

Note however that in [74] it was introduced only the asymptotic part of the
hexagon which computes structure constants for large bridge lengths. Thus it
remained open the problem of defining the mirror corrections for finite length
bridges. We attack this problem in Chapter 4, there we introduce the mirror
corrections and demonstrate how they behave. These corrections are in general



Chapter 1. Introduction 12

coupling dependent. Note however that there is a set of operators in AdS3 × S3 ×
T4 whose CFT data are protected. These operators form the so-called chiral ring.
We then prove that all the chiral ring structure constants do not receive mirror
corrections, which is a nontrivial fact in this model due to the intricate structure of
the chiral ring.

The role of mirror corrections go beyond yielding the finite length corrections
for structure constants, they are indeed paramount to the hexagonalization of
n-point functions. The main reason is that independent of the choices made for the
external operators we will always have small bridge lengths for correlators with
more than three insertions. Therefore mirror corrections are weakly suppressed
in the hexagonalization of n-point functions. In Chapter 5 we provide a first
step on computing n-point correlation functions for the pure RR dual CFT2 and
we discuss the problems which appear when computing four-point functions by
hexagonalization in this background.

Outline

This thesis is organized in the following way. As said before, Part 1 is covered
in Chapter 2. In Sections 2.1, 2.2 and 2.3 we review the tailoring formalism,
hexagonalization in AdS5 × S5 and the classical limit, respectively. In Section 2.4
we then describe new representations for the hexagons and the classical limit
of structure constants in the SU(2) and SL(2) sectors. Now Part 2 is covered in
Chapters 3, 4 and 5. Section 3.1 is a review of integrability in AdS3 × S3 × T4. In
Section 3.2 we present the chiral ring and we end the review part of the chapter in
Section 3.3 by presenting the hexagonalization formalism in this background. Then
we show some simple structure constants that can be found using hexagonalization
in Section 3.4. We move to mirror corrections in Section 4.1 and describe the
transfer matrix formalism and mirror measure in Section 4.2. The cancellation
of mirror corrections in the chiral ring is described in Section 4.3. We close by
discussing hexagonalization of four-point functions in Chapter 5. In Section 5.1 we
review the properties of the dual CFT2 and the hexagonalization itself is described
in Section 5.2. Then in Section 5.3 we end up by discussing an example of a simple
correlator and some problems that appeared in the analysis. All chapters end with
a summary of what was discussed in it.



Chapter 2

Hexagons and the classical limit

As described in the Introduction (Chapter 1), the spectrum of single trace
operators in N = 4 SYM at the planar level was described non-perturbatively
using integrability techniques. This was achieved by writing the operator as a
eigenstate (also called Bethe state) of a quantum mechanical integrable spin chain
and then one diagonalizes this operator. Using this spin chain description, one
them attempts to compute the structure constants.

In this chapter we will describe structure constants of N = 4 SYM at weak
coupling in the so-called classical limit in both SU(2) and SL(2) sectors. For this
we introduce the double spin chain formalism for the SU(2) sector in Section
2.1. Then we describe the spinning hexagons for the SL(2) sector in Section 2.2.
Following this we recall the properties of the classical limit in Section 2.3. Finally
in Section 2.4 we close the chapter by computing the classical limit of a class of
structure constants and as a byproduct we also give new representations of the
hexagons in both sectors.

2.1 Tailoring and double spin chain formalism

An integrability framework to compute structure constants at tree level was
introduced in [50, 51, 52, 53, 54] and was dubbed tailoring. The main idea is to take
the Bethe states describing the three operators in a structure constant, cut and glue
them in a way that reproduces the Wick contractions. This formalism worked well
at tree level and was extended to one-loop using a technique called θ-morphism
[53, 83]. The main drawbacks of this formalism are that it describes only the SU(2)
sector of N = 4 SYM and it is limited to one-loop only.

The first problem was partially solved in [55] by another procedure called
double spin chain formalism, which we will briefly describe below. However
both of them, in some limits, were fully solved by hexagonalization. This latter
approach will be quickly discussed in Section 2.2 for AdS5 × S5 and fully detailed

13



Chapter 2. Hexagons and the classical limit 14

in Chapter 3 for AdS3× S3× T4. In the double spin chain formalism the operators
are mapped to Bethe states and then cut and contracted as in tailoring. However
the main conceptual change here is that these are mapped to double spin chain
states. This allows the description of the SO(4) sector of the theory and contains
the previous description as a special case. Let us describe this formalism in more
details now.

In this approach, we define a double spin chain using the decomposition of
the fundamental SO(4) representation as a tensor product of two doublet SU(2)
representations. This allows us to describe correlators with operators built on top
of distinct vacua. More explicitly we can map each scalar in the SO(4) sector to a
double spin chain state:

Z → | ↑⟩ ⊗ | ↑⟩, Z̄ → | ↓⟩ ⊗ | ↓⟩,
Y → | ↑⟩ ⊗ | ↓⟩, −Ȳ → | ↓⟩ ⊗ | ↑⟩.

(2.1)

Then to build the excited states (or operators) we put excitations on one sector or
the other of the double spin chain. We consider | ↑ · · · ↑⟩ as the vacuum state and
excitations (also called magnons) are down spins | ↓⟩ moving in the chain as a
plane wave with amplitudes given by the scattering matrices [23].

Note that we can not excite both sectors at once since this would give an
operator outside the SO(4) sector. This yields two classes of operators: if we excite
the left spin chain we have a type I operator, otherwise we have a type II operator.
In the field language the vacuum of lenght ℓ is simply tr(Zℓ) (which is the vacuum
considered in [50]). However by applying a SU(2)L × SU(2)R R-charge rotation
we can go to general vacua, id est, polarized operators. This allows the description
of more general correlators.

After defining the correspondence of spin chains and operators one then has
to cut and then contract these states to build the structure constant. Then we first
cut the spin chain state into left and right part as in [50]. An excited (non-BPS)
operator in N = 4 SYM with M excitations is given by a Bethe state described
by the roots u = {u1, · · · , uM}. Consider for example the left spin chain of some
operator with this set of Bethe roots in the usual vacuum of ℓ up states which we
denote by |u, ↑ℓ⟩L. Then its cutting is given by

|u, ↑ℓ⟩L = ∑
αl∪αr=u

Hℓ(αl, αr)|αl, ↑ℓl⟩ ⊗ |αr, ↑ℓr⟩, (2.2)

where ℓl and ℓr are the lengths of the left and right subchains, respectively. Here



Chapter 2. Hexagons and the classical limit 15

we divided the set of Bethe roots into two disjoint subsets αl and αr. Clearly the
number of terms in the sum is 2M, which is just the number of partitions of u. The
explicit form of the cutting factor Hℓ(αl, αr) is

Hℓ(αl, αr) = ∏
u∈αl

∏
v∈αr

u− v + i
u− v

(
u− i

2

)ℓr (
v +

i
2

)ℓl

. (2.3)

This cutting factor can be understood physically as a propagating factor of roots
from one subchain to another1. Therefore in the final form of the structure constant
one has three sums over partitions of the operators with Hℓ(αl, αr) weighting the
terms.

So far we constructed Bethe states on top of the usual vacuum of up spins
| ↑ · · · ↑⟩. Now we rotate this state to build a generalized vacuum. We have that
half-BPS states in N = 4 SYM in the SO(4) sector can be defined in terms of a
polarization vector P ∈ C4:

P · Φ = PaȧΦaȧ with Φ =

(
Z Y
−Ȳ Z̄

)
, (2.4)

with a, ȧ ∈ {1, 2}. We note that this vector must be null to describe a half-BPS state,
then by representation theory of SO(4) we can write it as a product of two spinors

Paȧ = nañȧ. (2.5)

For example, na = ñȧ = (1, 0) gives us the Z field. The SO(4) rotation of the scalars
act on the spinors as SU(2)L⊗ SU(2)R, respectively. Clearly the spinors transform
in the spin 1/2 representation of SU(2) and there are many parametrizations we
can consider for its action. Here we pick the coherent state representation, id est,
let g ∈ SU(2)L then

g = ezS−e− log(1+|z|2)S3ez̄S+ , (2.6)

where S−, S+ and S3 are the usual lowering, raising and spin operators in SU(2),
respectively. Then z, z̄ ∈ C parametrize the rotation and for SU(2)R we use the

1Due to this there are two ways of define this quantity depending on the normalization. Basically
one can define states in the algebraic Bethe ansatz normalization or in the coordinate Bethe ansatz
one. The only change on one or another is an overall function of the roots. We work here on the
algebraic normalization.



Chapter 2. Hexagons and the classical limit 16

analogous variables z̃, ˜̄z ∈ C. Therefore n and ñ are

n = g

(
1
0

)
=

1√
1 + |z|2

(
1
z

)
and ñ =

1√
1 + |z̃|2

(
1
z̃

)
. (2.7)

Thus to build an excited state on top of a generalized vacuum |u, nℓ⟩L,R we simply
rotate |u, ↑ℓ⟩L,R:

|u, nℓ⟩L =
ezS−

(1 + |z|2)ℓ−M/2 |u, ↑ℓ⟩L and |u, ñℓ⟩R =
ez̃S−

(1 + |z̃|2)ℓ−M/2 |u, ↑ℓ⟩R. (2.8)

To derive this we used the fact that the Bethe state is |u, ↑ℓ⟩ is a highest weight
with spin ℓ−M/2. We can then apply the cutting (2.2) in the same way to the
polarized Bethe state as

|u, nℓ⟩L =
1

(1 + |z|2)ℓ−M/2 ∑
αl∪αr=u

Hℓ(αl, αr)
(

ezSl
− |αl, ↑ℓl⟩

)
⊗
(

ezSr
− |αr, ↑ℓr⟩

)
,

(2.9)
where we just divided the lowering operator for the right and left subchains using
ezSr
− and ezSl

− , respectively. In the tailoring procedure of [50] one constructed the
Bethe states only on top of the tr(Zℓ) vacuum, which restricted the analysis to the
SU(2) sector of the theory. However, as advertised earlier, the double spin chain
formalism allows the description of the larger SO(4) sector.

We described the first step of the double spin chain formalism, the second step
is the gluing or in field language the Wick contractions. The Wick contractions act
on conjugated fields and we can see this action in terms of the double spin chain
as an antisymmetric pairing. Indeed, consider two fields that are described by
polarizations P1 and P2, then their contraction P1 · P2 is simply given by the spinor
product

P1 · P2 = (ϵabn1,an2,b)(ϵ
ȧḃñ1,ȧñ2,ḃ) = ⟨n1, n2⟩⟨ñ1, ñ2⟩. (2.10)

Where we see the antisymmetric pairing due to the ϵ-tensor. In the spin chain
language we define a antisymmetric singlet state ⟨112| such that its action on two
spin chain states mimics the Wick contraction of the corresponding operators.
Indeed, let us define an antisymmetric pairing in the left spin chain as

⟨|n1⟩, |n2⟩⟩ = ⟨1|(|n1⟩ ⊗ |n2⟩) = ⟨n1, n2⟩, where ⟨1| = ϵab⟨a| ⊗ ⟨b|. (2.11)



Chapter 2. Hexagons and the classical limit 17

Figure 2.1: Representation of the singlet state. In (a) we have a schematic repre-
sentation of how the singlet state acts on a two-point function and in (b) how it
acts on three-point function. Here connecting two sites of chains denotes a Wick
contraction. Note that that way the antisymmetric pairing is defined we have
planarity of the three-point function.

We can do this for general spin chain states by defining ⟨112|:

⟨|Ψ1⟩, |Ψ2⟩⟩ = ⟨112|(|Ψ1⟩ ⊗ |Ψ2⟩) =
l

∏
m=1
⟨1k,ℓ−k+1|(|Ψ1⟩ ⊗ |Ψ2⟩), (2.12)

where ⟨1k,ℓ−k+1| is the antisymmetric pairing between the site k of |Ψ1⟩ and ℓ− k +
1 of |Ψ2⟩. Which is done in this way such that we get only planar diagrams, which
is fine since we are working in the large-N limit. This is represented schematically
in Figure 2.1.

Analogously to the two states case we can define the three states contractions.
Thus for the L spin chain we have

⟨|O1⟩, |O2⟩, |O3⟩⟩L = ∑⟨|O1⟩l, |O3⟩r⟩L⟨|O3⟩l, |O2⟩r⟩L⟨|O2⟩l, |O1⟩r⟩L, (2.13)

where the upper scripts l and r represent the left or right subchain of the |Oj⟩L state
and we sum here over all partitions of the roots for each operator as stated before.
We have similar contributions for the R spin chain. The ordering of contractions is
such that the corresponding fatgraph contribution is planar as shown in Figure
2.1. Note that the length of the subchains are fixed by the bridge lengths, id est, the



Chapter 2. Hexagons and the classical limit 18

number of contractions between operators. These are:

ℓl1 = ℓr3 = ℓ13 =
ℓ1 + ℓ3 − ℓ2

2
, (2.14)

ℓl3 = ℓr2 = ℓ23 =
ℓ2 + ℓ3 − ℓ1

2
, (2.15)

ℓl2 = ℓr1 = ℓ12 =
ℓ1 + ℓ2 − ℓ3

2
. (2.16)

In the tailoring formalism, the same result was achieved for the SU(2) sector,
however the authors did not introduce the singlet state. They defined the so-called
flipping and gluing operations which in the end just yield the good old Wick
contractions [50]. Another approach to contracting the fields is the so-called spin
vertex introduced in [84]. It is similar in nature to the singlet state and it extends
the analysis to all sectors of N = 4 SYM by using an oscillator representation
of the psu(2, 2|4) algebra. However it has some problems that were amended in
[85], where the authors introduced a singlet state for the full psu(2, 2|4), but no
application to the computation of structure constants was done and they restricted
to the analysis of two-point functions.

We will not detail how to construct ⟨|Ψ1⟩, |Ψ2⟩⟩ here since it involves the tools
of algebraic Bethe ansatz. We simply state the final result:

⟨|Ψ1⟩, |Ψ2⟩⟩ = (−1)M1⟨↓ℓ |e(z1−z2)S− |u1 ∪ u2, ↑ℓ⟩, (2.17)

where Mj is the number of roots and zj the polarization parametrization for |Ψj⟩.
This scalar product on the right can be written as

⟨↓ℓ |ezS− |u, ↑ℓ⟩ = zℓ−M

(ℓ−M)!
⟨↓ℓ |Sℓ−M

− |u, ↑ℓ⟩ = zℓ−MZp(u|ℓ). (2.18)

Note that we must have S3 spin conservation, then the only non-zero term is the
one with ℓ−M descendant operators. The Zp(u|ℓ) is the so-called partial domain
wall partition function (pDWPF) [55, 86, 87]. This is given by

Zp(u|ℓ) =
∏
j
(uj − i/2)ℓ

∏
i<j

(ui − uj)
× det

(
ua−1

b −
(

ub + i/2
ub − i/2

)ℓ

(ub − i)a−1

)
. (2.19)

In the case at hand ℓ will be the bridge length of the contractions where the roots
u lie. A physical interpretation for the pDWPF is that it is the usual domain wall



Chapter 2. Hexagons and the classical limit 19

partition function when some rapidities are taken to infinity, id est, it is the scalar
product of a Bethe state and a descendant of a Bethe state which is exactly what
we have in (2.18).

With all these ingredients we can write the structure constant of three non-BPS
operators in the SO(4) of N = 4 SYM at the planar level. We need simply to join
the left and right contractions of the states and normalize accordingly. Then the
structure constant is given by2

C•••123 =

√
ℓ1ℓ2ℓ3

N N1N2N3
⟨|O1⟩, |O2⟩, |O3⟩⟩L⟨|O1⟩, |O2⟩, |O3⟩⟩R, (2.20)

where Nj is the normalization of each operator that is given by the Gaudin norm
(see Appendix A). The left and right spin chain contributions, are given as a sum
over partitions as aforementioned. For the L contribution

⟨|O1⟩, |O2⟩, |O3⟩⟩L =
3

∏
i=1

(
1

1 + |zi|2

)ℓi/2−Mi

∑
αk∪ᾱk=uk

zℓ12−|α2|−|ᾱ1|
12 zℓ13−|α1|−|ᾱ3|

31

zℓ23−|α3|−|ᾱ2|
23 D(αj, ᾱj), (2.21)

where

D(αj, ᾱj) = (−1)|α1|+|α2|+|α3|
3

∏
i=1

Hℓi(αi, ᾱi)Zp(α1 ∪ ᾱ3|ℓ13)Zp(α2 ∪ ᾱ1|ℓ12)×

× Zp(α3 ∪ ᾱ2|ℓ23). (2.22)

Also ⟨|O1⟩, |O2⟩, |O3⟩⟩R is basically the same with the appropriate right sector
variables. Here zij = zi − zj and |α| denotes the length of the set α. This is the
final form of the structure constant in the SO(4) sector and we can separate them
into two classes, if all operators are of the same type we have a type I-I-I or pure
correlator. Otherwise if one of the operators is of type II we have a mixed or type I-
I-II correlation function. Examples of representatives of these classes of correlators
are given in Figure 2.2.

There are two technical details that will serve for our purposes later: the zi

dependence factorizes and the order of the roots uk when you partition them does
not matter. The first is expected since it is known that in the computation of the

2Here we use the common notation of C•••123 indicating the structure constant of three non-BPS
operators, C••◦123 the one with only two non-BPS, and so on.



Chapter 2. Hexagons and the classical limit 20

Figure 2.2: Examples of type I-I-II and I-I-I correlators. In (a) we have an example
of a type I-I-II correlator analyzed in [52]. Now in (b) we have a type I-I-I correlator
where Z̃ = Z + Z̄ + Y− Ȳ and Ỹ = Y + Z̄. This structure constant was analyzed
in [57]. Note that to describe (b) we have to rotate the vacuum to Z̃ and Ỹ, which
was not possible in the tailoring formalism.

three-point function of scalar operators the polarizations factorizes [55]. This is
not true for spinning operators (see Section 2.4.2). The independence of the root’s
ordering comes from the fact that any ordering define the same spin chain state
in the algebraic Bethe ansatz. Both of these states are related to the fact that for
physical operators the roots satisfy the Bethe equations, id est, they are on-shell.
As proven in [55] this implies that then the zij dependence factorizes as

⟨|O1⟩, |O2⟩, |O3⟩⟩L =
3

∏
i=1

(
1

1 + |zi|2

)ℓi/2−Mi

zℓ12−M12
12 zℓ23−M23

23 zℓ13−M13
31 G(L)

123 , (2.23)

where Mij = Mi + Mj −Mk and G(L)
123 is a function of the rapidities only. We have

a similar function G(R)
123 for the R spin chain. These functions were known, prior to

this work, only for type I-I-II correlators [55, 87]. Indeed, by simply comparing
the powers of zij in (2.21) and (2.23) one can derive for type I-I-II correlators

G(L)
123 = (−1)M2

M1

∏
j=1

(
uj +

i
2

)ℓ13 M2

∏
j=1

(
vj −

i
2

)ℓ23

Zp (u∪ v|ℓ12) , (2.24)

G(R)
123 =

M3

∏
j=1

(
wj +

i
2

)ℓ23

Zp (w|ℓ13) . (2.25)

Where u, v and w are the roots of the first, second and third operator, respectively.
However for the more complicated case of type I-I-I it was unknown how to
derive these functions from the double spin chain prior to this work. The main
complication is that all three operators interact upon contracting. As will be
detailed in Section 2.2, finding this function is equivalent to derive the hexagon



Chapter 2. Hexagons and the classical limit 21

formalism at tree level from the double spin chain formalism, which was an open
problem [57].

2.1.1 New expression for the structure constant

In Section 2.4 we will derive the classical limit of pure correlators. However
first we need to put it on an appropriate form, that is, we need to eliminate the
dependence on the polarization as in (2.23). Since we will work only with type
I-I-I correlators, we have only the L part of these ⟨|O1⟩, |O2⟩, |O3⟩⟩L. Then we
drop the labels on G(L)

123 to make the notation lighter:

⟨|O1⟩, |O2⟩, |O3⟩⟩L =
3

∏
i=1

(
1

1 + |zi|2

)ℓi/2−Mi

zℓ12−M12
12 zℓ23−M23

23 zℓ13−M13
31 G. (2.26)

As explained before, if the roots satisfy the Bethe equations then G is polarization
independent. However here we will use the Bethe equations to rewrite the struc-
ture constant, then we find G and we take the roots in the final expression to be
off-shell. Thus we write

G = ∑
αk∪ᾱk=uk

zM12−|α2|−|ᾱ1|
12 zM23−|α3|−|ᾱ2|

23 zM31−|α1|−|ᾱ3|
31 D̃(αj, ᾱj), (2.27)

where the new D̃(αj, ᾱj) assumes the following convoluted form

D̃(αj, ᾱj) = (−1)|ᾱ1|+|ᾱ2|+|ᾱ3| ∑
β1∪β̄1=ᾱ1∪α2
β2∪β̄2=ᾱ2∪α3
β3∪β̄3=ᾱ3∪α1

(−1)|β1|+|β2|+|β3|

× aℓ12(β̄1 ∩ α2)aℓ23(β̄2 ∩ α3)aℓ31(β̄3 ∩ α1)aℓ31(β1 ∩ ᾱ1)aℓ12(β2 ∩ ᾱ2)aℓ23(β3 ∩ ᾱ3)

× S (u1, β1 ∩ ᾱ1) S (u2, β2 ∩ ᾱ2) S (u3, β3 ∩ ᾱ3)

× f+(ᾱ1, α1) f+(ᾱ2, α2) f+(ᾱ3, α3) f+(β1, β̄1) f+(β2, β̄2) f+(β3, β̄3), (2.28)



Chapter 2. Hexagons and the classical limit 22

and

f+(x, y) = ∏
u∈x

∏
v∈y

(
u− v + i

u− v

)
, (2.29)

S(x, y) =
f (x, y)
f (y, x)

= ∏
u∈x

∏
v∈y

(
u− v + i
u− v− i

)
, (2.30)

aℓ(x) = ∏
u∈x

(
u + i/2
u− i/2

)ℓ

. (2.31)

Physically, S(u, v) is the SU(2) spin chain S-matrix, aℓ is eiPℓ, where P is the total
momentum of the excitations, and f+(u, v) can be seen as the hexagon scalar factor
at tree level. The derivation of these formulas is done in details in Appendix B.

The reader can complain that we claimed G independence from the polariza-
tions, however they remain present in (2.27). But if one takes this expression
evaluate it for a few examples, it can be seen that indeed all the polarizations
cancel. We will see explicitly this cancelling in Section 2.4 for the classical limit.
With the description of the SU(2) laid out, let us move it to the description of
correlators in the SL(2) sector.

2.2 Non-compact spin chains and hexagons

Now we will briefly review how the computation of structure constants of
spinning operators was done in [67]. An operator in the so-called SL(2) sector in
N = 4 SYM is given simply by

OJ(x, L, R) = Lα1 Rα̇1 · · · LαJ Rα̇J O
α1α̇1···αJ α̇J
J (x), (2.32)

where Oα1α̇1···αSα̇S
J (x) is a BPS vacuum with J covariant derivatives Dαα̇ applied

on it with α, α̇ ∈ {1, 2} being spinor indices for each derivative. The polarization
spinors Lα and Rα̇ play the same role as the polarization vectors in the SU(2)
sector by characterizing each operator. Due to SL(2) invariance, the final structure
constant must contain only the following spinor contractions

Hij = ⟨Li, Rj⟩⟨Lj, Ri⟩ and Vi = ⟨Li, Ri⟩ with ⟨A, B⟩ = ϵαα̇ AαBα̇,

where ϵαα̇ = −ϵα̇α is an invariant SL(2) tensor. Up to now the story is similar to the
SU(2) sector, however, unlike the case of scalars operators, the spinning structure



Chapter 2. Hexagons and the classical limit 23

for the three-point function is not fixed by SL(2) invariance [88]. Consider for
example a correlator with J1 = 2, J2 = 4 and J3 = 6, then the normalized structure
constant at tree level is given in [89] and it can be written as

C•••123
C◦◦◦123

=
V2

1 V4
2 V6

3

84
√

55

(
t13t23 960 + t2

12 15− t4
23t13 2007− t2

13t2
23 7680 + · · ·

)
, (2.33)

where we defined the quantity

tij =
Hij

ViVj
.

With this redefinition we see that the structure constant in the SL(2) sector is a
polynomial in the variables tij with all possible tensor structures contributing and
not just a single term like in the SU(2) sector.

To compute these structure constants, the authors in [67] used the hexagonal-
ization approach. To avoid repeating ourselves later we will restrain here to just
briefly describe the expressions for these correlators and later in Chapter 3 we
will show the origin of hexagonalization in the context of AdS3/CFT2 duality.
As described in Chapter 1, in the hexagon formalism the three-point function in
N = 4 SYM can be described as

C•••123
C◦◦◦123

= ∑
δ1∪δ̄1=u1
δ2∪δ̄2=u2
δ3∪δ̄3=u3

(−1)|δ1|+|δ2|+|δ3| aℓ31(δ1)S<(δ̄1, δ1)× aℓ12(δ2)S<(δ̄2, δ2)

× aℓ23(δ3)S<(δ̄3, δ3)Z(δ1, δ3, δ2)Z(δ̄1, δ̄2, δ̄3). (2.34)

Where aℓℓ(z) is a momentum propagation factor eip(z)ℓ from an hexagon to another
and the form factors Z(δ1, δ3, δ2) are the so-called hexagons. The remaining factor
S<(δ̄j, δj) is an S-matrix factor gained from scattering excitations in one hexagon
to bring them to the other hexagon. These are given by

S<(δ̄j, δj) = ∏
um∈δ̄j, un∈δj

m<n

S(um, un), (2.35)

with S(u, v) being the SL(2) scattering matrix. In (2.34) we are considering only
the asymptotic hexagon, thus it works at finite coupling and for asymptotically
large operators and bridge lengths.



Chapter 2. Hexagons and the classical limit 24

The momentum factor and the SL(2) S-matrix are well known from the spectral
problem. We have only to define the hexagon form factor now. As explained in
[57], it is defined as tensor contractions of the Beisert S-matrix, that is, the su(2|2)
S-matrix of N = 4 SYM. However the main insight of [67] was to note that this
tensor contraction can be reinterpreted as a partition function with the Boltzmann
weights being the su(2|2) S-matrix. In this language the polarizations spinors
defining the operators enter as boundary conditions of this partition function such
that in the end we obtain a structure constant that is polynomial in the tij variables.
The full computation of the hexagon at finite coupling for a generic configuration,
even in this new reinterpretation, remains costly nonetheless. However we can
restrict ourselves to tree level where this partition function has a much simpler
analytical structure and can be efficiently computed.

At tree level the hexagon form factor is a rational function of the rapidities and
it has poles when the roots of two operators coincide. This is due to the decoupling
property of the hexagon and it implies recursion relations for them. Consider for
example coinciding roots on the first two operators

Resvj→uiZ(u, v, w) = i⟨L1, R2⟩⟨L2, R1⟩
i−1

∏
i′=1

f−(ui, ui′)

f−(ui′ , ui)

J2

∏
j′=j+1

f−(vj′ , ui)

f−(ui, vj′)
×

×Z(u/{ui}, v/{vj}, w), (2.36)

where J1, J2 and J3 are the number of excitations on each operator and u/{x}
denote set difference. Here f−(u, v) is

f−(u, v) =
u− v− i

u− v
. (2.37)

Note here the sign change in f−(u, v) with respect to the previously defined
f+(u, v) for SU(2). This is due to the fact that the SL(2) S-matrix is the inverse of
the SU(2) one. Also we have that when a root is taken to infinity that

lim
ui→∞

Z(u, v, w) = (−1)J1+J2+J3+1⟨L1, R1⟩ × Z(u/{ui}, v, w). (2.38)

Since the hexagons are rational functions of the rapidities and we known the poles
and asymptotic conditions at infinity we can totally fix them. This will be done in
Section 2.4.2 where we find a new representation for this partition function at tree
level.



Chapter 2. Hexagons and the classical limit 25

For our convenience, we will rewrite the hexagon to match with our notation
for the SU(2) sector and as a byproduct the recursion relations will simplify
considerably. We then work withH(u, v, w), which is defined by

H(u, v, w) = (−1)
(J1+J2+J3)(J1+J2+J3−1)

2 Z(u, v, w)
f−< (u1) f−< (u2) f−< (u3)

V J1
1 V J2

2 V J3
3

, (2.39)

where
f−< (u) = ∏

i<j
f−(ui, uj). (2.40)

Using this redefinition in the recursion relation (2.36) we obtain

Resvj→uiH(u, v, w)

f−< (u1) f−< (u2/{vj}) f−< (u3)

j−1

∏
j′=1

1
f−(vj′ , ui)

J2

∏
j′=j+1

1
f−(ui, vj′)

=

− it12

i−1

∏
i′=1

f−(ui, ui′)

f−(ui′ , ui)

J2

∏
j′=j+1

f−(vj′ , ui)

f−(ui, vj′)

H(u/{ui}, v/{vj}, w)

f−< (u1/{ui}) f−< (u2/{vj}) f−< (u3)
, (2.41)

which, after simplifications, become simply

Resy→x H(u1 ∪ {x}, u2 ∪ {y}, u3) = −it12 f−(x, u1) f−(u2, x)H(u1, u2, u3),

(2.42)

Resy→x H(u1, u2 ∪ {x}, u3 ∪ {y}) = −it23 f−(x, u2) f−(u3, x)H(u1, u2, u3),
(2.43)

Resy→x H(u1 ∪ {y}, u2, u3 ∪ {x}) = −it31 f−(x, u3) f−(u1, x)H(u1, u2, u3).
(2.44)

Here we wrote all the possible recursion relations from decoupling. Also we can
simplify the second recursion relation (2.38) and obtain

limx→∞H(u1 ∪ {x}, u2, u3) = H(u1, u2, u3), (2.45)

limx→∞H(u1, u2 ∪ {x}, u3) = H(u1, u2, u3), (2.46)

limx→∞H(u1, u2, u3 ∪ {x}) = H(u1, u2, u3). (2.47)

As previously said, in Section 2.4.2 we found a solution for these and thus a new
representation for the hexagon at tree level.

We can also rewrite the full structure constant and for it we find the following



Chapter 2. Hexagons and the classical limit 26

simplification of (2.34):

G(u1, u2, u3) = ∑
δi∪δ̄i=ui

(−1)|δ1|+|δ2|+|δ3| f−(δ̄1, δ1) f−(δ̄2, δ2) f−(δ̄3, δ3)

× aℓ31(δ1)aℓ12(δ2)aℓ23(δ3)H(δ1, δ3, δ2)H(δ̄1, δ̄2, δ̄3). (2.48)

With all these tools from the structure constant in the SU(2) and SL(2) sectors
discussed, let us end the review part by discussing the classical limit.

2.3 What is the classical limit?

There are many limits that one can take on the observables of N = 4 SYM
that may be of interest depending on the context: weak coupling, strong coupling,
lightcone limit, and so on. One of them is the so-called classical limit. In the
simplest terms, in this regime we consider heavy operators with many excitations
on top of it. This limit is natural when one considers the strong coupling limit of
N = 4 SYM, where it can be analyzed in terms of classical string integrability in
AdS5 × S5 [90]. In this semiclassical regime the operators one observe correspond
to string states with asymptotically large quantum numbers, id est, semiclassical
string states [77]. Nonetheless this regime is also interesting at weak coupling, here
similar integrability structures between the weak and strong coupling regimes
start to emerge. One instance of this is the classical limit of structure constants
in the SU(2) sector at weak and strong coupling analyzed in [82], where the
same classical integrability tools where used in both of regimes. Another instance
is the famous Frolov-Tseytlin limit, which is defined by λ, Ji → ∞ such that
λ/J2

i ≪ 1 with λ and Ji being the t’Hooft coupling and some charge, respectively
[78]. Remarkably for some observables we have that an expansion in λ/J2

i is equal
to the weak coupling expansion [51, 91], although some mismatches may happen
at higher loops as explored in [79]. Therefore the classical limit is an important
tool to understand the AdS/CFT duality.

Let us then detail this limit. We consider here the classical limit at weak
coupling described in [80]. The classical limit is defined as limit of asymptotically
large charges for the operators or Bethe states. Consider an operator in the SU(2)
sector with M Bethe roots and length ℓ, then this corresponds to

ℓ ∼ M→ ∞.



Chapter 2. Hexagons and the classical limit 27

Figure 2.3: Condensation of roots. In the classical limit, the roots condense into
cuts and we describe the set of roots ui by a root density ρi in the complex plane.

In this limit we have that the Bethe roots are asymptotically large, i.e., ui ∼ ℓ.
However their separation remain of order O(1). This means that if one reescales
the roots as ui → ui/ℓ, then their separation is of order O(1/ℓ) and the roots
condense into a continuum, which is represented schematically in Figure 2.3.

This root condensation generates a cut in the density of roots. Indeed we can
see this clearly by taking the classical limit of the Bethe equations. These in the
SU(2) sector are given by

(
uj + i/2
uj − i/2

)ℓ

= ∏
k ̸=j

uj − uk + i
uj − uk − i

. (2.49)

We can take the logarithm of these equations, substitute uj = ℓxj and then consider
the large ℓ limit:

1
xj

=
2
ℓ ∑

k ̸=j

1
xj − xk

− 2πnj. (2.50)

Here nj ∈ Z is an integer that signals which branch of the logarithm one is
considering. Note that we are taking a sum over all roots in the right hand side. In
the classical limit, not only ℓ goes to infinity but also M. Since the root separation
goes to zero we have a continuum as said before and the sum becomes

1
ℓ ∑

k
· · · =

∫
C

dz ρ(z) · · · . (2.51)

With ρ(z) being the density of roots defined in the contour C = ∪jCj where the
roots condense. Note that each integer nj that specifies the solution leads to a cut
and the union of all of them describe the full operator. This means that the Bethe



Chapter 2. Hexagons and the classical limit 28

equations in the classical limit become

G(z + i0) + G(z− i0)
2

= /G(z) =
1
2z

+ πnp, where z ∈ Cp. (2.52)

With /G(z) being the principal value of G(z). Here we used the resolvent function:

G(z) =
∫
C

dx
ρ(x)
z− x

. (2.53)

Note that since we are integrating on the rapidity cut C then if z ∈ Cp the function
diverges. That is why in (2.52) one introduces the ±i0 prescription which yields
the principal value. This has its microscopic origin in the k ̸= j prescription in
(2.50). Therefore the roots condense into cuts of the resolvent function, this fact
will play a role later in the derivation of the classical limit of the structure constants
in Section 2.4.

Up to now we discussed the classical limit for the SU(2) sector. Nonetheless
for the non-compact SL(2) spin chain, in specific cases, it works very similarly
to the compact case [81]. If we consider the large spin limit S → ∞ and large
length ℓ→ ∞, with ℓ ∼ S, the discussion follows exactly as in the compact case.
However this is not the only large spin limit one can consider for SL(2) chains.
Since here the spin S is not bounded by the length we can keep finite ℓ and take
only S→ ∞. In this case we can not apply the same arguments as before for the
condensation of roots. Here we will work strictly in the classical limit defined
with both spin and length being large. There are also some notable differences
between them. The SL(2) case can be seen as the analytic continuation in spin of
the compact SU(2) one. This simple modification make the Bethe roots all real,
id est, the cuts are real. At strong coupling these large spin states corresponds to
folded string solutions (or GKP strings) and in this limit their structure function is
known for specific polarizations [92, 93].

The idea now is to apply this classical limit to the structure constant. One
sees that microscopically the characterization of the operator depends heavily on
their Bethe roots and similarly for the structure constants. However as we have
seen for the classical limit, the microscopic details about the Bethe roots become
unimportant and the entire operator is described by a single density satisfying the
functional equation (2.52). Similarly, we expect that such microscopic details at
the classical limit for the expression (2.27) gets averaged out and we obtain then a
clean expression for the structure constants as a function of the densities only. So



Chapter 2. Hexagons and the classical limit 29

we need the classical limit of quantities appearing in (2.28). By directly taking the
classical limit we have

f±(z, u) = e±iG(z) and aℓ(z) = eiℓ/z. (2.54)

Another useful function to introduce is the quasi-momentum:

p(z) = G(z)− ℓ

2z
. (2.55)

Turns out that the classical limit of the structure constants depends on the densities
only through the quasi-momenta. Note that the quasi-momenta are meromorphic
functions with singularities at the origin and branch cuts at C.

Up to now we considered the classical limit of the summand only, however we
need to see how the sum over partitions is transformed in this limit. Since in the
classical limit each set of Bethe roots is associated with a density, to each partition
of the roots we also associate a density. Then given u = α ∪ ᾱ associated with ρ(z)
we have

α→ ρα(z) and ᾱ→ ρᾱ(z), with the constraint ρ(z) = ρα(z) + ρᾱ(z).

Naturally the sum over partitions then turns into a functional integral over the
densities ρα(z) and ρᾱ(z). Indeed:

∑
αk∪ᾱk=uk

· · · →
∫
[dραk ][dρᾱk(z)] · · · , (2.56)

and in the summand we write everything as a function of the densities. In the
end we can then compute this functional integral by saddle point method with
the care of taking the constraint for the densities into account. This approach was
developed in [52] and used to compute the classical limit of type I-I-II correlators
for a single non-BPS operator. We will further detail the subtleties of this method
for the type I-I-I correlators in Section 2.4. Note that for the mixed correlators this
is not necessary since as shown in (2.24) and (2.25) the correlator can be written
without the sum over partitions and the classical limit can be taken directly as it
was did in [55].

Later we will compute the classical limit for type I-I-I correlators directly from
the microscopic expression (2.27). However some results were already known in
the literature. The classical limit of type I-I-II correlators was computed by many



Chapter 2. Hexagons and the classical limit 30

methods and it is given by

log
(

C•••123
C◦◦◦123

)
= pols+

∮
C1∪C2

dz
2π

Li2
(

eip1+ip2+iℓ3/2z
)
+

∮
C3

dz
2π

Li2
(

eip3+i(ℓ2−ℓ1)/2z
)
− 1

2

3

∑
j=1

∮
Cj

dz
2π

Li2
(

e2ipj
)

, (2.57)

where pols is just all the terms containing the polarizations, which are fixed by
SU(2) invariance

pols = M1 log(1 + |z1|2) + M2 log(1 + |z2|2) + M3 log(1 + |z̃3|2)+
− (M1 + M2) log(z12) + (M1 −M2) log(z23) + (M2 −M1) log(z31)+

+ M3 log(z̃12)−M3 log(z̃23)−M3 log(z̃31), (2.58)

and C◦◦◦123 is the structure constant of three BPS operators with the same lengths as
the excited operators. Note that such configuration is considerably simpler than
I-I-I, since each sector has at most two interacting operators. Indeed this class of
correlators is equivalent in complexity to the two non-BPS correlators since in each
sector at most two operators interact. To truly uncover the three-point function
integrability one must also tackle the pure correlators, which we do here.

The classical limit for mixed corelators was first found using the aforemen-
tioned functional integral method for a single non-BPS in [52]. Then later in [94]
this was extended for the full three non-BPS correlator by considering an expan-
sion on aℓ(u) and then taking the classical limit on a specific ansatz. Both of these
works were situated in the weak coupling regime, however in [95] using hexago-
nalization the authors were able to derive this result, but now at finite coupling. It
still retains the same structure as in (2.57) with the quasi-momentum p(u) being
the finite coupling one, but they could explore only the type I-I-II correlators. This
situation was partially remedied in [82], here for the first time the classical limit of
the pure and mixed correlators were found at weak and strong coupling. However
the method used was quite indirect and no limit of (2.27) was ever taken. The



Chapter 2. Hexagons and the classical limit 31

classical limit of the type I-I-I correlator found in [82] is

log
(

C•••123
C◦◦◦123

)
= pols+

1
2 ∑
{i,j,k}=cperm{1,2,3}

∮
Ci∪Cj

dz
2π

Li2
(

eipi+ipj−ipk
)
+

− 1
2

3

∑
j=1

∮
Cj

dz
2π

Li2
(

e2ipj
)

. (2.59)

Where the polarization term is given by:

pols =
3

∑
j=1

Mj log(1 + |zj|2)− ∑
{i,j,k}=cperm{1,2,3}

Mij log(zij). (2.60)

Here cperm{1, 2, 3} means all cyclic permutations of the set {1, 2, 3}. Note that
unlike (2.57) we have that all the operators interact at once and not in a two-by-
two fashion. In this work we attest the validity of this expression by finding the
classical limit of pure correlators directly from the double spin chain formula. In
the next section, we present such derivation.

Some results are also known in the SL(2) sector. As said before we have two
parameters to take the asymptotic limit which are ℓ and S. By taking both to
infinity, in [91] it was computed the classical limit of SL(2) structure constants
with two non-BPS operators consisting of lightcone derivatives and a BPS one.
This means that the derivatives lie in the plane and we are restricted to a specific
polarization setting. Also it was observed that at leading order, it satisfied the
weak/strong coupling relation in the Frolov-Tseytlin limit. With the review finally
done, let us then discuss the classical limits of SU(2) and SL(2) correlators and
the new hexagon representations in the next section.

2.4 New representations of the hexagons and classical

limit

In this section we will compute the classical limit for type I-I-I correlators and
of the hexagons for some choices of polarizations in the non-compact sector. A
consequence of this is that we derive new representation of the hexagons.



Chapter 2. Hexagons and the classical limit 32

2.4.1 SU(2) sector

We will first derive our first new result, a new definition of the hexagon for the
SU(2) sector direct from the sum (2.27). Then we consider this formula and take
the classical limit of pure correlators to derive (2.59) directly from the microscopic
sum. We also derive recursion relations analogous to (2.42) - (2.47), but now for
the SU(2) case.

Deriving the hexagons from double spin chains

In the end we want to explore the classical limit of (2.27). Before doing this we
will derive the tree level hexagon directly from the microscopic sum. In [94] it was
introduced a method for computing the classical limit of the structure constants
and we will use it here to prove this hexagon relation. The main idea of [94]
consists in expanding G as a series on aℓ. Clearly, the first term of this series does
not contain any aℓ and it is given by the following constraints:

β̄1 ∩ α2 = β̄2 ∩ α3 = β̄3 ∩ α1 = β1 ∩ ᾱ1 = β2 ∩ ᾱ2 = β3 ∩ ᾱ3 = ∅. (2.61)

These conditions fix the subsets β j and β̄ j of the subpartitions in (2.28) to be

β1 = α2, β2 = α3, β3 = α1, β̄1 = ᾱ1, β̄2 = ᾱ2, β̄3 = ᾱ3. (2.62)

Therefore, plugging these back in (2.27), we get the zeroth order contribution that
we denote by

H(u1, u2, u3) = (−1)M1+M2+M3 ∑
α1∪ᾱ1=u1
α2∪ᾱ2=u2
α3∪ᾱ3=u3

zM2−M3−|α2|+|α1|
12 zM3−M1−|α3|+|α2|

23 ×

× zM1−M2−|α1|+|α3|
31 f+(ᾱ1, α1) f+(ᾱ2, α2) f+(ᾱ3, α3)

f+(α2, ᾱ1) f+(α3, ᾱ2) f+(α1, ᾱ3). (2.63)

Here by naming it asH(u1, u2, u3) we anticipate that in the end it will be equivalent
to the hexagon form factor. This is independent of the polarizations too, indeed
we can consider for example the hexagon with two excitations on a operator and



Chapter 2. Hexagons and the classical limit 33

only one on the other two. We then find:

H(u, v, w) =
i

w1 − v1
− (u1 − v1 − i) (u2 − v1 − i)

(u1 − v1) (v1 − u2)
+

+ i
(u1 − u2 − i) (u2 − v1 − i)

(u1 − u2) (u2 − v1) (u1 − w1)
+ i

(u1 − u2 + i) (u1 − v1 − i)
(u1 − u2) (u1 − v1) (u2 − w1)

. (2.64)

For more complicated cases we can still check this independence on the polariza-
tions, however we will not show this here because the expressions are too large
and not illuminating. Note that H(u1, u2, u3) has a certain structure where we
have weights f+(ᾱj, αj) associated with each operator and other weigths f+(αi, ᾱj)

related to interactions between “neighbor” partitions. Remarkably the same kind
of structure remains for the for SL(2) correlators as it will be shown in Section
2.4.2.

Now we need to analyze the rest of the terms in the G expansion and rewrite it
in terms ofH(u1, u2, u3). Let us partition each root set as ui = δi ∪ δ̄i and consider
the term in the expansion proportional to

aℓ31(δ1)aℓ12(δ2)aℓ23(δ3). (2.65)

The partitions δi and δ̄i are defined such that given βi we define γi ∪ γ̄i = δi with
β̄i ∩ αj = γ̄j and βi ∩ ᾱi = γi. These conditions fix the βi and β̄i partitions to be

βi = αj/γ̄j ∪ γi and β̄i = ᾱi/γi ∪ γ̄j. (2.66)

We then insert these back into (2.27) and do some convoluted algebraic manipula-
tions and groupings (detailed in Appendix C if the reader is interested) to obtain
in the end the following expression for the structure constant

G(u1, u2, u3) = ∑
δ1∪δ̄1=u1
δ2∪δ̄2=u2
δ3∪δ̄3=u3

(−1)|δ1|+|δ2|+|δ3| f+(δ̄1, δ1) f+(δ̄2, δ2) f+(δ̄3, δ3)

× aℓ31(δ1)aℓ12(δ2)aℓ23(δ3)H(δ1, δ3, δ2)H(δ̄1, δ̄2, δ̄3). (2.67)

This expression has exactly the same form as hexagonalization expression, as
shown in Section 2.2, and we indeed identifyH(u1, u2, u3) with the hexagon form
factor at tree level3. This completes the proof in Appendix L of [57], where there it

3Note that in the hexagon formula we have the S-matrix S(δ̄1, δ1) and not f+(δ̄1, δ1) for each



Chapter 2. Hexagons and the classical limit 34

was derived the hexagon form factors from the double spin chain for at most one
excitation in each operator. Here we worked it out for general excited operators
thus proving the equality of these two formalisms at tree level. Turns out that the
expression (2.63) lies at the heart of our derivation for the classical limit of type
I-I-I correlator.

Decoupling and the classical limit

We saw in Section 2.2 that the hexagon form factors for the SL(2) sector satisfy
recursion relations related to the decoupling property. We will derive exactly
the same relations but for the hexagon expression (2.63) in the SU(2) sector. To
derive these we look at (2.63) and note that it is a rational function with poles at
coincident roots. There are two potential sources for these poles: f+(ᾱi, αi) and
f+(αi, ᾱj). Note that if we take coincident roots x and y on the same operator there
are two contributions for the residue x ∈ αi and y ∈ ᾱi or x ∈ ᾱi and y ∈ αi. Since
the function f (u, v) is antisymmetric on the pole these two contributions cancel
each other. Then we can have poles only for coincident roots on distinct operators.
If we consider x ∈ ui and y ∈ uj we have the following equations

Resy→x H(u1 ∪ {x}, u2 ∪ {y}, u3) = i f+(x, u1) f+(u2, x)H(u1, u2, u3), (2.68)

Resy→x H(u1 ∪ {x}, u2, u3 ∪ {y}) = i f+(x, u3) f+(u1, x)H(u1, u2, u3), (2.69)

Resy→x H(u1, u2 ∪ {x}, u3 ∪ {y}) = i f+(x, u2) f+(u3, x)H(u1, u2, u3). (2.70)

Another decoupling condition we could derive is when a root goes to infinity, id
est, the analogous of (2.45) - (2.47). It can be seen that these are

lim
x→∞
H(u1 ∪ {x}, u2, u3) = H(u1, u2, u3), (2.71)

lim
x→∞
H(u1, u2 ∪ {x}, u3) = H(u1, u2, u3), (2.72)

lim
x→∞
H(u1, u2, u3 ∪ {x}) = H(u1, u2, u3). (2.73)

Since at tree level the hexagon is fully rational, it can be fixed totally by these
relations. In general it is hard to reduce the sum over partitions (2.63) to a single
function, however in some cases it can be done. For example for one and two

operator. This is explained by the fact that G is equal to the hexagonalization expression up to
some overall factor of f+(ui, uj). This doesn’t affect the result since it is an overall factor and it
was also observed in [96].



Chapter 2. Hexagons and the classical limit 35

non-BPS operators it is simple to check that

H(u1, ∅, ∅) = 1, (2.74)

H(u1, u2, ∅) = f+(u2, u1), (2.75)

solve all these decoupling relations. These cases when inserted on G yields the
mixed correlators (2.24) and (2.25).

Just for completeness, we can check that (2.63) indeed satisfies the recursion
relations (2.68) - (2.70). For example look at the first one, as said before the only
contribution for the residues comes when x ∈ ᾱ1 and y ∈ α2. Let Mi = |ui| and
redefine the partitions in (2.63) to be

α1 = γ1

ᾱ1 = γ̄1 ∪ {x}
,

α2 = γ2 ∪ {x}
ᾱ2 = γ̄2

, and
α3 = γ3

ᾱ3 = γ̄3
,

then we obtain

Resy→x H(u1 ∪ {x}, u2 ∪ {y}, u3) = ∑
γ1∪γ̄1=u1
γ2∪γ̄2=u2
γ3∪γ̄3=u3

zM2+1−M3−|γ2|−1+|γ1|
12

zM3−M1−1−|γ3|+|γ2|+1
23 zM1+1−M2−1−|γ1|+|γ3|

31 f+(γ̄1 ∪ {x}, γ1) f+(γ̄2, γ2 ∪ {x})
f+(γ̄3, γ3) f+(γ3, γ̄2) f+(γ1, γ̄3) f+(γ2, γ̄1) f+(γ2, {x})

f+({x}, γ̄1)Resy→x f+(y, x), (2.76)

which after simplifications yield us exactly the recursion relation (2.68), with
the factor of i coming from the residue of f+(y, x). Therefore all the recurrence
relations (2.68) - (2.70) are satisfied and for the (2.71) - (2.73) the proof is direct.
Note the cancelling of the extra roots’ contribution in the polarizations, a step that
does not happen in the SL(2) case as we will show later.

We note here the power of the expression (2.63). As described before, in [67]
the hexagon was described as a partition function for the SL(2) sector. The same
rewriting of the hexagon could be done in the SU(2) sector by exchanging spinors
to polarization vectors. Therefore, even at tree level the hexagon is a complicated
partition function on a Kagome lattice. Here we shown that this complicated
partition function can be written at tree level as a simple sum over partitions.

Up to now we did not relate these decoupling relations with the classical limit.
Turns out that these are essential to our derivation of the hexagon’s classical limit.



Chapter 2. Hexagons and the classical limit 36

Based on the pole structure and the two non-BPS result, a natural guess for the
hexagon for the full three non-BPS operators is

Hansatz(u1, u2, u3) = f+(u2, u1) f+(u3, u2) f+(u1, u3). (2.77)

Despite its simplicity, the decoupling relations are not satisfied. Indeed for (2.68)
we find

Resy→x Hansatz(u1 ∪ {x}, u2 ∪ {y}, u3) = i f+(x, u1) f+(u2, x) f+(u3, x) f+(x, u3)

Hansatz(u1, u2, u3). (2.78)

It is now that the classical limit enters. Note that in this regime the Bethe roots are
large and the roots of distinct operators are macroscopically distinct. This means
that in the classical limit

f+(u3, x) f+(x, u3) ≈ 1. (2.79)

Note that this is only valid because x /∈ u3, otherwise we would have roots with
separation of O(1/ℓ) and this argument would fail. Using this fact we derive that
the classical limit of the hexagon satisfy the decoupling relations. Therefore using
the decoupling conditions we can establish that our ansatz (2.77) is indeed the
hexagon, however only in the classical limit.

Now we have all the tools to compute the classical limit of the pure correlators.
Inserting our ansatz for the hexagon at the classical limit in (2.67) we derive the
following expression for the structure constant:

G(u1, u2, u3)

H(u1, u2, u3)
= ∏
{i,j,k}=cperm{1,2,3}

(
∑

αi∪ᾱi=ui

(−1)|αi|aℓki
(αi)

f+(ᾱi, αi)

f+(αi, uk) f+(uj, αi)

)
,

(2.80)

Remarkably the sum over partitions is decoupled now and the classical limit
becomes much easier to take, indeed we will simply reuse the techniques of [52].
We will do it step-by-step below.

First we need to turn the sum over partitions in (2.80) into a functional integral



Chapter 2. Hexagons and the classical limit 37

over the root’s densities as aforementioned in Section 2.3. We then find

∑
αi∪ᾱi=ui

(−1)|αi| aℓki
(αi) f+(ᾱi, αi)

f+(αi, uk) f+(uj, αi)
→
∫
[Dραi ] exp

(∫
Ci

du F [ραi ] + iραi /q i

)
.

(2.81)

Here we already solved the constraint and integrate over ραi only. Also we recom-
bined the classical limits of aℓki

(αi) and f+(αi, uk) f+(uj, αi) to obtain qi(z) which
contains the quasi-momenta and it is given by

qi(z) = π + pj(z)− pk(z)− pi(z). (2.82)

and /q i(z) denotes the principal value prescription for the quasi-momenta eval-
uated at the cut Ci. The missing ingredient here is F [ραi ], which is called the
stochastic anomaly.

When passing from the sum over partitions to the functional integral, there are
many contributions one has to take into account. The classical limit of f+(u, v)
for roots belonging to distinct operators was given in (2.54). However if the roots
belong to the same operator as in f+(ᾱi, αi) we have a problem. Basically there
are roots with separation of order O(1/ℓ) such that the final contribution in the
product is of order O(1). These contributions are harder to compute the classical
limit, nonetheless this was achieved in [52] and encompassed in the so-called
stochastic anomaly:

F [ραi ] = F(ρi)− F(ραi)− F(ρᾱi) with F(ρ) =
∫ ρ

0
dx log sinh(πx). (2.83)

The specialist reader may have missed the entropy contribution. This extra term
arises from the fact that multiple microscopic contributions describe the same
macroscopic configurations. The same term also appears in the formulation of
Thermodynamic Bethe Ansatz [30]. However this contribution is already included
in the stochastic anomaly, therefore we have the full classical limit of the structure
constant and it remains just to compute these functional integrals.

To calculate the functional integrals (2.81) we use the saddle point method. The
saddle point equations for ραi(z) are easily derivable and given by

log

(
sinh(πραj)

sinh(π(ρj − ραj))

)
= i/q j. (2.84)



Chapter 2. Hexagons and the classical limit 38

These are very similar to the saddle point equations for a single non-BPS operator
derived in [52]. We can invert these equations for ραi(z) and we find the following
simple result after we substitute in the saddle point

log
G(u1, u2, u3)

H(u1, u2, u3)
= − ∑

{i,j,k}=cperm{1,2,3}

∮
Ci

du
2π

Li2
(

e−ipk−ipi+ipj
)

. (2.85)

These are different from the expected result (2.59) due to the presence of single
contour terms and the hexagon prefactor. Now we convert this expression into
the previously known result.

To find the same as in [82] we have to join the contours Ci in (2.85). For this
end we will use the Bethe equations and some judicious contour manipulations.
We can rewrite the classical limit of the Bethe equations (2.52) in terms of the
quasi-momenta as

pj(u + i0) + pj(u− i0) = 2πn(k)
j for u ∈ C(k)j . (2.86)

Where as before ∪kC
(k)
j = Cj and n(k)

j ∈ Z. The nice thing about this form of Bethe
equations is that we can use it to manipulate p(u). It tell us that whenever one
crosses a rapidity cut the following changes occur: p(u) → −p(u) modulo 2π

and Cj → −Cj. So any crossing of a cut changes the orientation of the integration
contour and flips the exponential of the quasi-momenta.

Let us do these manipulations then. Consider for example the first term in
(2.85), we start by splitting it in the following way

∮
C1

du
2π

Li2
(

e−ip3−ip1+ip2
)
= −1

2

∮
C1

du
2π

Li2
(

eip1+ip2−ip3
)
+

+
1
2

∮
C1

du
2π

Li2
(

e−ip3−ip1+ip2
)

. (2.87)

In the first term we used the Bethe equations to go to the second sheet, then the
sign of the integral and p1 changed. To deal with the second term we consider the
following dilogarithm identity

Li2(x) = −Li2(−1/x)− π2

6
− log2(1/x)

2
. (2.88)



Chapter 2. Hexagons and the classical limit 39

With this we invert the pj in the second term of (2.87) and find:

∮
C1

du
2π

Li2
(

e−ip3−ip1+ip2
)
= −1

2

∮
C1

du
2π

Li2
(

eip1+ip2−ip3
)
+

− 1
2

∮
C1

du
2π

Li2
(

eip1+ip3−ip2
)
− 1

4

∮
C1

du
2π

log2
(
−eip3+ip1−ip2

)
. (2.89)

We can do similar manipulations in all the other terms of (2.85) and we find exactly
the contributions that recombine to almost yield us the desired result

log
G(u1, u2, u3)

H(u1, u2, u3)
= rest+

1
2 ∑
{i,j,k}=cperm{1,2,3}

∮
Ci∪Cj

dz
2π

Li2
(

eipi+ipj−ipk
)

.

(2.90)
Here rest are all the terms obtained when flipping the dilogarithms and are
simply

rest = ∑
{i,j,k}=cperm{1,2,3}

1
4

∮
Ci

du
2π

log2
(
−eipi−ipj+ipk

)
. (2.91)

After somewhat extensive, but simple, simple algebra one can derive that the
hexagon prefactor logH(u1, u2, u3) exactly cancels the term rest up to phases
that can be reabsorbed by operator redefinitions.

Therefore we find our final result for the classical limit of type I-I-I correlators

log
(

C•••123
C◦◦◦123

)
= pols+

1
2 ∑
{i,j,k}=cperm{1,2,3}

∮
Ci∪Cj

dz
2π

Li2
(

eipi+ipj−ipk
)
+

− 1
2

3

∑
j=1

∮
Cj

dz
2π

Li2
(

e2ipj
)

. (2.92)

Which is the same as (2.59). Note here that we included the polarizations’ contribu-
tions and the classical limit of the determinant part of the operator’s norm which
are the single contour terms (see Appendix A). The main point of this derivation
is that we used the decoupling relations and with them the classical limit’s compu-
tation of this structure constant directly from the microscopic formula becomes
straightforward. This then validates the result of [82] which were obtained in an
indirect way.



Chapter 2. Hexagons and the classical limit 40

2.4.2 SL(2) sector

Here we derive the classical limit of the hexagon in the non-compact SL(2)
sector. As in the previous section we first find a new representation for the hexagon
based on the recursion relations of Section 2.2 and with them we compute the
hexagon’s classical limit in some specific cases.

A new representation for the SL(2) hexagon

For the compact sector, we found a new expression for the hexagon at tree
level from tailoring in (2.63). The question now is that if we can do the same for
the more complicated case of the SL(2) sector. We remember that the recursion
relations (2.42) - (2.47) uniquely define for us the hexagon partition function since
it’s a rational function, therefore if we find a solution for these we have solved the
problem. Based on the SU(2) case we propose for the non-compact scenario

H(u1, u2, u3) = ∑
α1∪ᾱ1=u1
α2∪ᾱ2=u2
α3∪ᾱ3=u3

h|α1|
1 (1− h1)

|ᾱ1|h|α2|
2 (1− h2)

|ᾱ2|h|α3|
3 (1− h3)

|ᾱ3|

× f−(ᾱ1, α1) f−(ᾱ2, α2) f−(ᾱ3, α3) f−(α2, ᾱ1) f−(α3, ᾱ2) f−(α1, ᾱ3), (2.93)

with the new variables h1, h2 and h3 being defined as

h2(1− h1) = t12, h3(1− h2) = t23, and h1(1− h3) = t31. (2.94)

At first sight it is not clear that this new hexagon representation is a polynomial in
the tij variables, however we can do some simple tests to check this. Consider for
example an hexagon with 2, 2 and 1 excitations on each side, then we have

H(u, v, w) = 1+
it23 (v1 + v2 − 2w1 + i)
(v1 − w1) (v2 − w1)

+
it31 (u1 + u2 − 2w1 − i)
(u1 − w1) (w1 − u2)

+ · · · . (2.95)

Note that if indeed (2.93) satisfies the recursion relations then it must be a polyno-
mial of tij. The novelty of this expression is that it reduces the complicated sum
over configurations in the partition function of [67] to a single sum over partitions.
This simplifies the computation at tree level and maybe it can give a hint to an
all loop expansion of it. Below we check that this expression indeed satisfy the
recursion relations.

Let us first compute the residue in expression (2.93), we note that the only



Chapter 2. Hexagons and the classical limit 41

nonzero contribution for the residue happens when x ∈ ᾱ1 and y ∈ α2. The proof
works exactly as in the SU(2) case. The y→ x residue is

Resy→x H(u1 ∪ {x}, u2 ∪ {y}, u3) = ∑
α1∪ᾱ1=u1∪{x}
α2∪ᾱ2=u2∪{x}

α3∪ᾱ3=u3

h|α1|
1 (1− h1)

|ᾱ1|h|α2|
2 (1− h2)

|ᾱ2|

h|α3|
3 (1− h3)

|ᾱ3| f−(ᾱ1, α1) f−(ᾱ2, α2) f−(ᾱ3, α3) f−(α3, ᾱ2) f−(α1, ᾱ3)

f−(α2/{x}, ᾱ1/{x}) f−(α2/{x}, {x}) f−({x}, ᾱ1/{x})Resy→x f−(y, x), (2.96)

redefining the partitions to be

α1 = γ1

ᾱ1 = γ̄1 ∪ {x}
,

α2 = γ2 ∪ {x}
ᾱ2 = γ̄2

, and
α3 = γ3

ᾱ3 = γ̄3
,

we find that

Resy→x H(u1 ∪ {x}, u2 ∪ {y}, u3) = −ih2(1− h1) f−(u2, x) f−(x, u1)

∑
γ1∪γ̄1=u1
γ2∪γ̄2=u2
γ3∪γ̄3=u3

h|γ1|
1 (1− h1)

|γ̄1|h|γ2|
2 (1− h2)

|γ̄2|h|γ3|
3 (1− h3)

|γ̄3| f−(γ̄1, γ1) f−(γ̄2, γ2)

f−(γ̄3, γ3) f−(γ2, γ̄1) f−(γ3, γ̄2) f−(γ1, γ̄3). (2.97)

Which yields the recursion relation (2.42) if we use the definition of hi variables
and the new hexagon representation:

Resy→x H(u1 ∪ {x}, u2 ∪ {y}, u3) = −it12 f−(x, u1) f−(u2, x)H(u1, u2, u3).
(2.98)

With similar manipulations we can prove the remaining recursion relations. Note
that the proof of (2.45) - (2.47) is trivial due to the form of f−(u, v). Therefore the
expression (2.93) is indeed the hexagon partition function since it solves all the
recursion relations.

Another simple test we can do is to check the case of abelian hexagon. As
explained in [67], one of the manifestations of this case is given by u2 = u3 = ∅
which yields

H(u, ∅, ∅) = 1. (2.99)



Chapter 2. Hexagons and the classical limit 42

We can prove this directly from (2.93). Here we have

H(u, ∅, ∅) = ∑
α∪ᾱ=u

h|α|1 (1− h1)
|ᾱ| f−(ᾱ, α). (2.100)

This is a rational function with poles for coincident roots, however it is easy to
see that the residue vanishes. Indeed, let x, y ∈ u and consider the residue y→ x.
There are two contributions to it: x ∈ α and y ∈ ᾱ or x ∈ ᾱ and y ∈ α. Since
f−(u, v) is antisymmetric at the pole these two cancel each other. ThenH(u, ∅, ∅)

is a polynomial function and due to the asymptotic conditions (2.45) - (2.47) it
must be unity. Therefore we can properly match our new representation with this
abelian case. Note that this abelian case can also be obtained if (t12, t23, t31) =

(0, 0, 0) which is equivalent to (h1, h2, h3) = (0, 0, 0). Another less trivial abelian
case we can match corresponds to (t12, t23, t31) = (1, 0, 0), which is equivalent to
(h1, h2, h3) = (0, 1, 0) or (h1, h2, h3) = (0, 1, 1). These choices truncate the sum over
partitions (2.93) to

H(u1, u2, u3) = f−(u2, u1), (2.101)

which matches the abelian case (at tree level) in [67]. These tests further validate
our expression (2.93) for the hexagon partition function in the SL(2) sector for
general polarized operators.

For these abelian cases we can even write simple expressions for the full
structure constant G. For this we introduce the so-called A-functional by

A±u [g] = ∑
α∪ᾱ=u

(−1)|α|g(α) f±(α, ᾱ), (2.102)

which has a determinant expression as proved in [94]

A±u [g] =
detab

(
ub−1

a − g(ua)(ua ± i)b−1)
detab(ub−1

a )
. (2.103)

Note that if choose g(z) = aℓ(z) we have the pDWPF of (2.19) minus the polyno-
mial part4. If we pick (t12, t23, t31) = (0, 0, 0) for the polarizations, it can be seen
that

G = A+
u1
[aℓ31 ]A

+
u2
[aℓ12 ]A

+
u3
[aℓ23 ]. (2.104)

4Here we are committing a slight abuse of notation, since we introduce A also in Appendix B,
however for a specific choice of g(z).



Chapter 2. Hexagons and the classical limit 43

Now for the other abelian choice (t12, t23, t31) = (1, 0, 0) we find

G = A+
u3
[aℓ23 ] ∑

α1∪ᾱ1=u1
α2∪ᾱ2=u2

(−1)|α1|+|α2|aℓ31(α1)aℓ12(α2) f−(ᾱ1, α1) f−(ᾱ2, α2)

f−(α2, α1) f−(ᾱ2, ᾱ1). (2.105)

Which is like the type I-I-II correlator in the SU(2) sector. In [67] there are Pfaffian
representations of this correlator.

A case that has no analog in the SU(2) sector and we can analyze is the
one with (t12, t23, t31) = (t, 0, 0) and 0 < t < 1. This is equivalent to choosing
(h1, h2, h3) = (0, t, 0) which then yields the following hexagon:

H(δ1, δ2, δ3) = ∑
α2∪ᾱ2=δ2

t|α2|(1− t)|ᾱ2| f (ᾱ2, α2) f (α2, δ1). (2.106)

Clearly the limits t→ 0 and t→ 1 gives the same abelian cases discussed before.
Using the A-functional we can write this in closed form

H(δ1, δ2, δ3) = (1− t)|δ2|A+
δ2

[
t

t− 1
f−(z, δ1)

]
. (2.107)

Therefore the correlator becomes simply

G = (1− t)M2A+
u3
[aℓ23 ] ∑

α1∪ᾱ1=u1
α2∪ᾱ2=u2

(−1)|α1|+|α2|aℓ31(α1)aℓ12(α2) f−(ᾱ1, α1) f−(ᾱ2, α2)

A+
δ2

[
t

t− 1
f−(z, δ1)

]
A+

δ̄2

[
t

t− 1
f−(z, δ̄1)

]
. (2.108)

Which is a generalization of the previous formula for the abelian cases. The
reader can worry that this expression is not polynomial in t, however the prefactor
(1− t)M2 guarantees this property.

Anothe