aainstitutetext:  Mathematics Department, King’s College London, The Strand,
London WC2R 2LS, UK bbinstitutetext: St.Petersburg INP, Gatchina, 188 300,
St.Petersburg, Russiaccinstitutetext:  School of Physics and Astronomy, Tel
Aviv University, Ramat Aviv 69978, Israelddinstitutetext:  CERN, Theoretical
Physics Department, 1211 Geneva 23, Switzerland

# The Holographic Dual of Strongly $\gamma$-deformed N=4 SYM Theory:
Derivation, Generalization, Integrability and Discrete Reparametrization
Symmetry

Nikolay Gromov c,d Amit Sever<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Recently, we constructed the first-principle derivation of the holographic
dual of ${\cal N}=4$ SYM in the double-scaled $\gamma$-deformed limit directly
from the CFT side. The dual fishchain model is a novel integrable chain of
particles in $AdS_{5}$. It can be viewed as a discretized string and revives
earlier string-bit approaches. The original derivation was restricted to the
operators built out of one of two types of scalar fields. In this paper, we
extend our results to the general operators having any number of scalars of
both types, except for a very special case when their numbers are equal.
Interestingly, the extended model reveals a new discrete reparametrization
symmetry of the “world-sheet", preserving all integrals of motion. We use
integrability to formulate a closed system of equations, which allows us to
solve for the spectrum of the model in full generality, and present non-
perturbative numerical results. We show that our results are in agreement with
the Asymptotic Bethe Ansatz of the fishnet model up to the wrapping order at
weak coupling.

CERN-TH-2019-144

##  1 Introduction

The strongly $\gamma$-deformed limit of ${\cal N}=4$ SYM, also known as the
fishnet model Gurdogan:2015csr ; Nielsen:1970bc ; Zamolodchikov:1980mb , is an
ideal playground for developing our exact computational methods and
understanding holography. It is an interacting four-dimensional QFT with
$SU(N)$ symmetry that one could hope is solvable exactly in the planar limit.
Just like the planar limit of ${\cal N}=4$ SYM theory, this model is conformal
for any value of the ’t Hooft coupling $\xi$. It consists of two complex
$N\times N$ matrix scalars $\phi_{1}$ and $\phi_{2}$ that are weakly coupled
when $\xi\to 0$. At strong coupling $\xi\to\infty$, a new classical dual
description emerges in terms of a chain of scalars in $AdS_{5}$ Gromov:2019aku
. At finite $\xi$ the theory becomes rather complicated. At the same time, it
is much simpler than ${\cal N}=4$ SYM. For example, at each order of the
perturbation theory, there are only very few Feynman diagrams that contribute.

There are several properties that make the fishnet theory particularly
interesting and tractable. First, the model can be proven to be integrable at
the quantum level Gurdogan:2015csr ; Nielsen:1970bc ; Zamolodchikov:1980mb .
Second, its Holographic dual fishchain theory can be derived rigorously and
quantized exactly Gromov:2019aku ; Gromov:2019bsj . This contrasts with the
current status of ${\cal N}=4$ theory, where the integrability is conjectured
and the dual super-string in $AdS_{5}\times S^{5}$ can only be quantized semi-
classically.

So far, our derivation and quantization of the fishchain model was restricted
to a set of states/operators that belong to the so-called ${\mathfrak{u}}(1)$
sector. These are operators of the type ${\cal O}={\rm
tr~{}}[\phi_{1}^{J}\partial^{m}(\phi_{2}\phi_{2}^{\dagger})^{n}\dots]$, which
can carry arbitrary spin and are only charged under the
${\mathfrak{u}}(1)_{1}$ subgroup of the total
${\mathfrak{u}}(1)_{1}\times{\mathfrak{u}}(1)_{2}$ symmetry that rotates the
two types of scalars. The main objective of this paper is to extend the
derivation to general states, carrying both quantum numbers. This involves
introduction of the magnons i.e. $\phi_{2}$ fields in the background of $J$
$\phi_{1}$’s. While most of the steps go through very similar to our previous
work, we find that the inclusion of the magnons leads naturally to a new
notion of a discrete reparametrization gauge symmetry. It is the string-bit
counterpart of the smooth string reparametrization freedom, which relates
different ways of parametrizing the same state.111This symmetry is reminiscent
of the Yangian symmetry of planar scattering amplitudes studied in
Chicherin:2017frs . Furthermore, extending the integrability construction to
all operators allows us, in particular, to solve the spectrum with magnons
included, which is a new result by itself. The solution takes the form of a
Baxter TQ-relations, subjected to the quantization conditions of
Gromov:2019cja  and generalizing results of Gromov:2017cja . It allows us to
compute the conformal dimension of almost all single trace operators222There
are some additional technical difficulties in deriving the spectrum for a
special case when both $U(1)$ charges are equal.. Moreover, this construction
gives the general framework for development of the integraiblity based
separation of variables method (SoV) for more general observables such as
correlation functions.

The paper is organized as follows. In section 2 we introduce our notations and
review the main previous results, which are important for this paper. In
particular, we recall the main steps from Gromov:2019bsj  for the quantization
procedure of the fishchain. In section 3, we generalize the notion of the CFT
wave function, initially introduced in Gromov:2019bsj , to operators with
magnons. We then construct the corresponding graph building operator and
derive its holographic counterpart. In section 4, we establish the
integrability of the fishchain model with magnons. In section 5, we identify
redundancy in the space of CFT wave functions that we associate to a discrete
reparametrization symmetry of the fishchain model. In section 6, we solve the
spectrum of the model with magnons and prove its reparametrization symmetry.
We conclude in the section 7.

Figure 1: a) Feynman diagrams that contribute to the correlation function of
an operator of the form ${\rm
tr~{}}\\!(\partial^{m}\phi_{1}^{J}(\phi_{2}\phi_{2}^{\dagger})^{n}\dots)$ are
all of fishnet type – made of the iterative wheel as shown on the diagram
Nielsen:1970bc . This structure can be resummed and leads to integrability
Zamolodchikov:1980mb . b) For operators with also net $\phi_{2}$ charge,
(2.3), the diagrams are of iterative spider type Caetano:2016ydc .

##  2 Notations and set-up

Here, we introduce the notations and give a short introduction to our results
from Gromov:2019aku ; Gromov:2019bsj .

###  2.1 Fishnet model

The fishnet model Gurdogan:2015csr  is defined by the following action333We
have suppressed double trace interactions which are not relevant non-
perturbatively Fokken:2013aea ; Sieg:2016vap ; Gromov:2018hut .

${\cal L}_{4d}=N\,{\rm
tr}\left(|\partial\phi_{1}|^{2}+|\partial\phi_{2}|^{2}+(4\pi)^{2}\xi^{2}\phi_{1}^{\dagger}\phi_{2}^{\dagger}\phi_{1}\phi_{2}\right)\,.$
(2.1)

This model was obtained in Gurdogan:2015csr  by taking a double scaling limit
of $\gamma$-deformed ${\cal N}=4$ SYM theory. Hence, we will still impose the
Gauss constraint and will only consider operators that are neutral under the
$SU(N)$ part of the global symmetries.

Our previous consideration was restricted to a subset of all operators, known
as the ${\mathfrak{u}}(1)$ sector

${\cal O}_{J}={\rm
tr~{}}[\partial^{m}\phi_{1}^{J}(\phi_{2}\phi_{2}^{\dagger})^{n}\dots]+\dots$
(2.2)

containing any number of derivatives, $J$-scalar fields $\phi_{1}$ and any
neutral combination of $\phi_{2}$ and $\phi_{2}^{\dagger}$. Correlation
functions of ${\mathfrak{u}}(1)$ operators are given by the sum of fishnet
type diagrams. These are diagrams that are made of the iterative wheel
structure as is demonstrated in figure 1.a.

General single trace operators in the model can also carry
${\mathfrak{u}}(1)_{2}$ charge. They take the schematic form

${\cal O}_{J_{1}}={\rm
tr~{}}[\partial^{m_{1}}\phi_{1}^{J_{1}}\partial^{m_{2}}\phi_{2}^{J_{2}}(\phi_{2}\phi_{2}^{\dagger})^{n_{1}}(\phi_{1}^{\dagger}\phi_{1})^{n_{2}}\dots]+\dots\;.$
(2.3)

In this paper we allow for arbitrary values of $J_{1}$ and $J_{2}$ as long as
$|J_{1}|\neq|J_{2}|$. Surprisingly, this case seems to be very different as
one can see already in the explicit results of Gromov:2018hut .

Correlation functions of these type of operators consist of a very specific
type of planar Feynman diagrams that have the shape of a spider web. In figure
1.b we draw an example of such a diagram for the case where $J_{2}=1$. In
general, the diagrams can be constructed by first contracting all the
$\phi_{1}$ fields in the trace, (the black lines in figure 1). Then, the
$\phi_{2}$’s have no other option but to spiral around the trace, crossing the
$\phi_{1}$ lines from the right to the left, (the red line in figure 1.b).
Each time a $\phi_{2}$ line crosses a $\phi_{1}$ line we have the four scalars
interaction of the fishnet model (2.1). The absence of the complex conjugate
interaction vertex forbids any other diagram to contribute in the planar
limit.

###  2.2 Fishchain model

Figure 2: Dynamics of the fishchain consisting of $J=4$ particles (denoted by
different colours). The initial conditions are restricted to a $2D$ plane
after projection on the boundary. The plane is parametrized in the radial
parameters by the angle $\phi$ and the radius $r$. The third axis corresponds
to the time. The drift in $\phi$ is due to a non-zero total angular momentum
$S$ and the exponential expansion in $r$ is due to $\Delta>0$.

The holographic fishchain dual of the ${\mathfrak{u}}(1)$ operators of charge
$J_{1}$ consists of a chain of $J_{1}$ point-like scalar particles in
$AdS_{5}$. In the classical/strong coupling limit, $\xi\to\infty$, the theory
is defined by the action

$S_{\rm dual}=\xi\int
dt\sum_{i=1}^{J_{1}}\left[\frac{\dot{X}_{i}^{2}}{2}+\prod_{k}\left(-X_{k}.X_{k+1}\right)^{-\frac{1}{J_{1}}}\right]\;,$
(2.4)

where $X_{i}\in{\mathbb{R}}^{1,5}$. While at the quantum level the particles
live on $AdS_{5}$ space, at the classical limit its radius collapses to zero
and the dynamics take place on the light cone, $X_{i}^{2}=0$. One can think
about (2.4) as a discretized version of the gauge fixed Polyakov action for a
string, which requires a Virasoro condition. An analog of the Virasoro
condition in the case of the fishchain is $\dot{X}_{i}^{2}=1$. In addition to
the constraints

$X_{i}^{2}=0\ ,\qquad\dot{X}_{i}^{2}=1\ ,\qquad i=1,2,\dots,J_{1}\;$ (2.5)

the Hamiltonian of the system should vanish to ensure the time
reparametrization symmetry

$H_{q}=0\ ,\qquad H_{q}={\rm
tr~{}}\left(\frac{q_{J}^{2}}{2}\dots\frac{q_{1}^{2}}{2}\right)-1\;,$ (2.6)

where $q_{i}$ is the local charge density
$q_{i}^{MN}\equiv\dot{X}_{i}^{M}X_{i}^{N}-\dot{X}_{i}^{N}X_{i}^{M}$. The
classical Hamiltonian $H_{q}$ follows from the action (2.4) together with the
constraints (2.5), see Gromov:2019bsj .

As can be seen from figure 2, the action (2.4) produces rather non-trivial
dynamics. Even though this dynamic may look chaotic, the model is integrable.

##### Quantization.

We now briefly go through the main steps of the canonical quantization
procedure. We will follow the Dirac procedure for the quantization of a system
with constraints. To begin with, we notice that in addition to the primary
constraints (2.5) we also have to include the secondary constraint
$\dot{X}_{i}.X_{i}=0$. Using that the canonically conjugate to $X_{i}$
variable is $P_{i}=\dot{X}_{i}$ we have the following constraints

$\varphi_{i,X}=X_{i}^{2}\ ,\qquad\varphi_{i,P}=P_{i}^{2}-1\
,\qquad\varphi_{i}=X_{i}.P_{i}\qquad\text{and}\qquad H_{q}\ .$ (2.7)

Among these, $\varphi_{j}$ and $\varphi_{j,P}$ are second class constraints as
follows from their Poisson bracket. Hence, in the quantization of the model we
first introduce corresponding Dirac brackets and only then promote them into
quantum commutation relations. For example, the commutator between the
operators $\hat{X}$ and $\hat{P}$ takes a non-standard form

$[\hat{X}_{k}^{M},\hat{P}_{j}^{N}]=\frac{i\delta_{kj}}{\xi}\left(\eta^{MN}-\hat{P}_{k}^{M}\hat{P}_{K}^{N}\right)\;.$
(2.8)

Similarly, the commutator between different components of $\hat{X}_{k}^{M}$ is
no longer zero. To resolve these commutation relations we introduce the
operator $\hat{Y}_{i}^{N}=\frac{i}{\xi}\partial_{P_{i,N}}$, which commutes in
the standard way with $\hat{P}_{k}^{N}$. Then we show that we automatically
solve all commutation relations if
$\hat{X}_{i}^{M}=\hat{Y}_{i}^{M}-(\hat{Y}_{i}.\hat{P}_{i})\;\hat{P}_{i}^{M}$.
Next, we have to impose the constraints (2.7) at the quantum level and resolve
possible ordering ambiguities. What we find is that

  * •

The quantum constraint $\hat{\varphi}_{i}=\hat{X}_{i}.\hat{P}_{i}$ is
automatically satisfied. Here, we had to choose an ordering between
$\hat{X}_{i}$ and $\hat{P}_{i}$ in $\hat{\varphi}_{i}$. To satisfy the
constraint, we had to correlate this choice with the one in the definition of
$\hat{X}_{i}$ in terms of $\hat{Y}_{i}$.

  * •

The constraint $\hat{\varphi}_{i,X}$ is proportional to ${\rm
tr~{}}q_{i}^{2}$, which is nothing but the quadratic Casimir, $\hat{\bf
C}_{i,2}/\xi^{2}$. Instead of setting it to zero we identify it with the
dimension $1$ of the scalar field $\phi_{1}$, corresponding to $\hat{\bf
C}_{i,2}=-3$.

  * •

The constraint $\hat{\varphi}_{i,P}$ fixed the dependence of the wave function
on $R_{i}^{2}\equiv-Y_{i}.Y_{i}$, as a result all the dynamical information is
contained in the reduced wave-function $\Psi(Z_{1},\dots,Z_{J})$ depending on
$J$ coordinates $Z_{i}^{M}$ on $AdS_{5}$ that are related to the $Y_{i}$’s via
$Y_{i}^{M}=R_{i}Z_{i}^{M}$.

  * •

Finally, we have to resolve the ordering ambiguity in the quantum version of
$H_{q}$. Since the matrix elements of $\hat{q}_{i}^{NM}$ do not commute with
each other, the quantum version of $q_{i}^{2}$ suffers from ambiguities. As we
explain below, the relevant definition is the symmetric and traceless
combination

$:\hat{q}_{i}^{2}:\,=\hat{q}_{i}^{2}-\frac{2i}{\xi}\hat{q}_{i}-\frac{1}{\xi^{2}}\;,$
(2.9)

where the charge density in the $Z$-coordinates is

$\hat{q}^{MN}_{j}=-{\frac{i}{\xi}}(Z_{j}^{N}\partial_{Z_{j\,M}}-Z_{j}^{M}{\frac{\partial}{\partial
Z_{j\,N}}})\ .$ (2.10)

The corresponding quantum fishchain Hamiltonian takes the form

$\hat{H}_{q}={\rm
tr~{}}\left(\frac{:\hat{q}_{J}^{2}:}{2}\dots\frac{:\hat{q}_{1}^{2}:}{2}\right)-1\;.$
(2.11)

At the end of the day we end with a quantum theory in $AdS_{5}$ that is
defined by two relations that the fishchain wave function
$\Psi(Z_{1},\dots,Z_{J})$ has to satisfy

$\hat{H}_{q}\circ\Psi=0\ ,\qquad\text{and}\qquad{\rm
tr~{}}\hat{q}_{i}^{2}\circ\Psi=\frac{2\Delta_{i}(4-\Delta_{i})}{\xi^{2}}\Psi$
(2.12)

where $\Delta_{i}=1$ is the dimension of the scalar $\phi_{1}$.

##### The Holographic map.

The key object which allows to establish the Holographic duality between the
fishchain model and the fishnet CFT is the CFT wave function introduced in
Gromov:2019bsj . This is a $J+1$ point correlator of a local operator ${\cal
O}$ and $J$ scalar fields

$\varphi_{\cal O}(x_{1},\dots,x_{J})=\langle{\cal O}(x_{0})\,{\rm
tr~{}}[\phi^{\dagger}_{1}(x_{1})\dots\phi^{\dagger}_{J}(x_{J})]\rangle\ .$
(2.13)

The knowledge of this function is equivalent to specifing the local operator
${\cal O}$. In order to compute $\varphi_{\cal O}$ in perturbation theory one
has to sum an infinite number of wheel diagrams like in figure 1(a). To sum
these diagrams one can use the so-called graph building operator $\hat{\cal
B}$, which is an operator acting on the wave function $\varphi_{\cal O}$ by
adding one more wheel to the diagrams

$\hat{\cal B}\circ
f(\vec{x}_{1},\dots,\vec{x}_{J})=\int\prod_{i=1}^{J}d^{4}y_{i}\prod_{j=1}^{J}\frac{\xi^{2}/\pi^{2}}{(\vec{y}_{i}-\vec{y}_{i+1})^{2}(\vec{y}_{j}-\vec{x}_{j})^{2}}f(\vec{y}_{1},\dots,\vec{y}_{J})\;.$
(2.14)

Physical non-protected operators in the CFT spectrum are in one to one
correspondence with stationary wave functions

$\hat{\cal B}\circ\varphi_{\cal O}(x_{1},\dots,x_{J})=\varphi_{\cal
O}(x_{1},\dots,x_{J})\;.$ (2.15)

The map between the CFT and the fishchain wave functions reads Gromov:2019bsj

$\psi_{\cal O}^{\bf
I}(Z_{1},\dots,Z_{J})=\int\prod_{i=1}^{J}\frac{D^{4}X_{i}}{-4\pi^{2}(Z_{i}.X_{i})^{3}}\Phi_{\cal
O}(X_{1},\dots,X_{J})\;,$ (2.16)

where $X^{M=-1,\dots,4}$ are six dimensional embedding coordinates with
$(-,+,+,+,+,+)$ signature Dirac:1936fq . They realize flat four dimensional
space as the protective lightcone of ${\mathbb{R}}^{1,5}$ as

$X={\frac{1}{2}}X^{+}\left(1+x^{2},1-x^{2},2\vec{x}\right)\;.$ (2.17)

In this parameterization we have for example $1/(x-y)^{2}=X^{+}Y^{+}/(-2X.Y)$.
When rewriting the correlation functions in embedding space we are following
the general prescription of SimmonsDuffin:2012uy . According to it we have to
introduce extra factors of $X^{+}$ in agreement with the scaling dimensions of
the corresponding operators. In particular, the embedding space wave function
in (2.16) is defined as

$\Phi_{\cal O}(X_{1},\dots,X_{J})\equiv{\frac{\varphi_{\cal
O}(\vec{X}_{1}/X_{1}^{+},\dots,\vec{X}_{J}/X_{J}^{+})}{X_{1}^{+}\,\dots\,X_{J}^{+}\,(X_{0}^{+})^{\Delta_{\cal
O}}}}\;.$ (2.18)

The introduction of these extra factors ensures that the r.h.s. stays
invariant under all conformal transformations.

It was shown in Gromov:2019bsj  that if $\phi_{\cal O}$ obeys the condition
(2.15), then the l.h.s. of (2.16) will be automatically annihilated by
$\hat{H}_{q}$. At the same time the second condition in (2.12) is also
satisfied just because the kernel $\frac{1}{(Z.X)^{3}}$ solves this equation
for any null vector $X$. In the next section we will show how this Holographic
map naturally generalizes to the case where both quantum numbers $J_{1}=J$ and
$|J_{2}|<J$ are non-zero.

##  3 CFT wave function and the graph building operator

In this section we generalize the fishchain model to include magnons i.e.
insertions of the $\phi_{2}$ fields into the “vacuum" of $\phi_{1}$’s. We
assume that the number of $\phi_{2}$ fields is less the number of
$\phi_{1}$’s. The starting point for the construction is the CFT wave
function, which we introduced in (2.13) for the $\mathfrak{u}(1)$ sector. Like
before it is given by the $(J+1)$-point correlation function between a single
trace operator of the type (2.3) and the trace of $J$ local operators, see
figure 1.b

$\varphi^{\bf I}_{\cal O}(x_{1},\dots,x_{J})=\langle{\cal O}(x_{0})\,{\rm
tr~{}}[\chi_{I_{1}}(x_{1})\dots\chi_{I_{J}}(x_{J})]\rangle\ .$ (3.1)

The new ingredient is that the local operators $\chi_{a}$ are not just single
scalars $\phi_{1}^{\dagger}$, but come in two types

${\includegraphics[scale={0.5}]{magnon_types1}}\ .$ (3.2)

We use the multi-index notation ${\bf I}=(I_{1},\dots,I_{J})$ where the index
$I_{j}\in\\{0,1\\}$. For example, when $I_{j}=1$ it indicates the position of
the $\chi_{1}$ insertion, which is a composite of two scalars
$\phi_{1}^{\dagger}$ and $\phi_{2}^{\dagger}$. Like before, the knowledge of
this correlator is equivalent to the knowledge of the initial operator. Any
other planar correlators, such as those with $\phi_{2}\phi_{2}^{\dagger}$ and
$\phi_{1}^{\dagger}\phi_{1}$, can be expressed in terms of this one. In
section 5 we show that the order of the magnons inside the correlator can be
interchanged. Hence, in order to reconstruct the local operator it is
sufficient to know the CFT wave function for only one of all possible magnon
orderings. In this section we consider all magnon orderings on equal footing.

As for the case with no magnons, the Feynman diagrams which contribute to the
correlator $\varphi^{\bf I}_{\cal O}$ in (3.1) are of iterative structure and
can be represented as a simple geometric series built from the corresponding
graph building operator $1/(1-\hat{\cal B})$. The only difference in
comparison to the previous section is that now the graph building operator
$\hat{\cal B}$ is slightly more complicated and is given by

$\hat{\cal B}\circ
f(x_{1},\dots,x_{J})\equiv\int\prod_{i=1}^{J}d^{4}y_{i}\left(\prod_{j=1}^{J}b_{I_{j}}(x_{j},y_{j},y_{j-1})\right)\,f(\vec{y}_{1},\dots,\vec{y}_{J})\
,$ (3.3)

PDF-picture replacement

Figure 3: The graph-building operator $\hat{{\cal B}}$ for the state with two
magnons. The locations of the magnons are indicated by the blue circles.

where $f$ is a probe function. In (3) the integration kernel is built out of
elementary building blocks $b_{a}$, which are different for the sites with
magnons and without

$b_{I}(x,y,\tilde{y})={\frac{\xi^{2}}{\pi^{2}(x-y)^{2}}}\times\left\\{\begin{split}{1\over(y-\tilde{y})^{2}}&\qquad
I=0\;\;\text{(no magnon)}\\\ {1\over(x-\tilde{y})^{2}}&\qquad
I=1\;\;\text{(magnon)}\end{split}\right.\;.$ (3.4)

We see that at the sites with a magnon, the $\phi_{2}$ propagator connects to
the external point $x_{j}$ instead of the integrated $y_{j}$ point as in a
site without a magnon. This is illustrated in figure 3, where the sites with
magnons are highlighted by the blue circles.

As we explained in the previous section the key equation which the CFT wave
function should satisfy is (2.15). The wave function is the correlator
computed at some finite value of the coupling $\xi$. It is given by an
infinite sum of planar Feynman diagrams. As such it should stay invariant
under addition of an extra layer i.e. the wave function should be an
eigenstate of the graph building operator with a unit eigenvalue, (2.15). For
explicit examples we refer to Gromov:2018hut . As was demonstrated in
Gromov:2018hut  the condition (2.15) singles out the physical states in the
theory and leads to the discrete spectrum of the scaling dimensions $\Delta$
corresponding to the properly regularized and renormalized single trace
operators of the type (2.3). Like in our previous paper Gromov:2019bsj , we
will now rewrite (2.15) in a form suitable for uplifting to $AdS_{5}$ space
and show that the dynamics induced by (2.15) is exactly this of the quantum
fishchain (QFC).

###  3.1 Inverting the graph building operator

Following Gromov:2019bsj  in order to relate the graph building operator $\cal
B$ to the dual fishchain model in $AdS_{5}$, we have to invert it and rewrite
(2.15) as a zero energy Hamiltonian constraint

$\hat{H}\,|\varphi_{\cal O}\rangle\equiv\left(\hat{\cal
B}^{-1}-1\right)|\varphi_{\cal O}\rangle=0\;.$ (3.5)

This constraint is then interpreted as the result of a time reparametrization
symmetry of the fishchain. We show below that, defined in this way, $\hat{\cal
B}^{-1}$ and $\hat{H}$ are differential operators.

In the case without magnons the invertion of $\hat{\cal B}$, given by (2.14),
was straightforward. We simply act with $\Box_{j}$ on all the external $x_{k}$
points in (2.14). This has the effect of eliminating the corresponding $y_{k}$
integrals and the $1/(4\pi^{2}(x_{i}-y_{i})^{2})$ factors.444Recall that
$\Box_{i}\frac{1}{4\pi^{2}(x_{i}-y_{i})^{2}}=-\delta^{4}(x_{i}-y_{i})$. In the
case at hand we have an extra complication – there are two propagators that
end at a site with a magnon. Acting with $\Box_{j}$ on the magnon site does
not remove the integration anymore, see figure 3.

Let us show that the inversion of ${\cal B}$ is still possible in the
situation when there is at least one site without a magnon, i.e. for
$J_{2}<J_{1}$. Without loss of generality, we assume that there is a magnon on
the $k$’th site, but there is no magnon at the neighboring site to the right
i.e. $I_{k-1}=0,\;I_{k}=1$, see figure 4. In this case we can hit $x_{k-1}$
with $\Box_{k-1}$, removing the integration in $y_{k-1}$. After that the
propagator connecting $y_{k-1}$ (which is now equal to $x_{k}$) with $x_{k}$
can be removed by multiplication with $x_{k,k-1}^{2}$.555We use the standard
notation $x_{ab}=x_{a}-x_{b}$. After this we are left with a single propagator
connecting $x_{k}$ with an integration point $y_{k}$, which now can be treated
with $\Box_{k}$.

It is easy now to figure out the general procedure. Assuming again that at the
site $k-1$ there is no magnon we begin by acting with $\Box_{k-1}$, removing
integration in $y_{k}$. Then, if at the site $k$ there is a magnon, we act
with the combination $\Box_{k}x_{k,k-1}^{2}$, see figure 4. Otherwise, on the
sites with no magnon we act with the operator $x_{k,k-1}^{2}\Box_{k}$. Acting
in this way we can invert $\hat{\cal B}$ completely.

PDF-picture replacement

Figure 4: In the presence of a magnons, the inverse of the graph building
operator is more involved. Here, we demonstrate how the operator is inverted
for the case where there is a magnon at the $k$’th site but no magnons at the
neighboring sites.

For example for $J=3,M=1$ with $I_{3}=0$, $I_{2}=1$ and $I_{1}=0$ we find

$\hat{\cal
B}^{-1}=\frac{1}{(-4\xi^{2})^{3}}x_{1,3}^{2}\,(x_{2,3}^{2}\Box_{3})\,(\Box_{2}x_{2,1}^{2})\,\Box_{1}\;.$
(3.6)

We conclude that it is always possible to write $\hat{\cal B}^{-1}$ as a
differential operator. Below we rewrite $\hat{\cal B}^{-1}$ in a simple and
manifestly conformally invariant way, applicable for all cases. For that we
will have to first change to embedding coordinates, introduced in section 2.2.

##### Embedding coordinates.

When manipulating conformal integrals such as the ones that are generated by
the graph building operator, it is always advantageous to work in the
embedding coordinates. In the case of the correlator (3.1), we have the
$\chi_{I}$ operators in (3.2) at positions $X_{1},\dots,X_{J}$ and ${\cal
O}_{\Delta}$ at $X_{0}$. Hence, the corresponding embedding space wave
function is defined as

$\Phi^{\bf I}_{\cal O}(X_{1},\dots,X_{J})\equiv{\varphi^{\bf I}_{\cal
O}(\vec{X}_{1}/X_{1}^{+},\dots,\vec{X}_{J}/X_{J}^{+})\over(X_{1}^{+})^{1+I_{1}}\dots(X_{J}^{+})^{1+I_{J}}\,(X_{0}^{+})^{\Delta_{\cal
O}}}\;.$ (3.7)

The advantage of the covariant formalism is that the inverted $\hat{\cal B}$
can be written in a very concise form

$\hat{\cal B}^{-1}={\rm
tr~{}}\left(\beta_{J,I_{J}}.\dots.\beta_{1,I_{1}}\right)$ (3.8)

where $\beta_{k,I_{k}}$ is a local operator that only acts on the $X_{k}$’th
coordinate. It is given by

$\left(\beta_{k,I_{k}}\right)^{M\,N}=-{1\over
2}\times\left\\{\begin{split}X_{k}^{M}X_{k}^{N}\,K_{k}^{2}&\qquad I_{k}=0\\\
X_{k}^{M}\,K_{k}^{2}\,X_{k}^{N}&\qquad I_{k}=1\end{split}\right.\
,\qquad\text{where}\qquad K^{M}=-{i\over\xi}{\partial\over\partial X_{M}}\;.$
(3.9)

To understand the mechanism behind (3.8) we look at the same example as in
(3.6). There we took $I_{3}=0$, $I_{2}=1$ and $I_{1}=0$. For this case (3.8)
gives

${\cal
B}^{-1}=\frac{1}{(-2)^{3}}(X_{3}.X_{1})(X_{3}.X_{2})K_{3}^{2}K_{2}^{2}(X_{2}.X_{1})K_{1}^{2}\;.$
(3.10)

By noticing that
$-\xi^{2}K_{i}^{2}=\Box_{i}^{4D}-4\partial_{X_{i}^{+}}\partial_{X_{i}^{-}}$
and due to the fact that $\Phi^{\bf I}_{\cal O}$ from (3.7) does not depend on
$X_{-}$, we can simply replace $K_{i}^{2}$ by $-\frac{1}{\xi^{2}}\Box_{i}$
when acting on the CFT wave function. In addition we use
$x_{ab}^{2}=\frac{-2X_{a}.X_{b}}{X^{+}_{a}X^{+}_{b}}$ to reproduce precisely
(3.6). It is equally easy to see that the equation (3.8) works in the general
situation, provided that $J_{2}<J_{1}$.

Finally, let us rewrite the local site operators, $\beta_{k,I_{k}}$, in terms
of the local generator of the conformal symmetry, namely in terms of
${\mathfrak{q}}_{k}$ defined as

$\hat{\mathfrak{q}}_{i}^{M\,N}\equiv X_{i}^{N}K_{i}^{M}-X_{i}^{M}K_{i}^{N}\;.$
(3.11)

We can use that the CFT wave function is a homogeneous function of $X_{k}$
with degree of homogeneity $-(1+I_{k})$, as follows directly from its
definition (3.7). This property allows us to make the replacements
$\hat{X}_{k}.\hat{K}_{k}\to{i\over\xi}(1+I_{k})$. Using that in analogy with
(2.9), we find

$\beta_{j,I_{j}}^{MN}\simeq\left(\hat{\mathfrak{q}}^{2}_{k}\right)^{M\,N}-{i\over\xi}(2+I_{k})\,{\mathfrak{q}}_{k}^{M\,N}-{1\over\xi^{2}}(1+I_{k})\eta^{MN}\;.$
(3.12)

where the "$\simeq$" means equality when acting on the wave function (3.7).

##### The strong coupling classical limit.

Before we proceed to the holographic interpretation of equation (3.8) at the
quantum level, we can already make a nontrivial conclusion about the strong
coupling limit of the theory, where the dual description becomes classical
Gromov:2019aku .

In order to understand the classical limit we recall that the operator $({\cal
B}^{-1}-1)$ can be interpreted as a worldline Hamiltonian. The classical limit
is obtained by simply replacing $-\frac{i}{\xi}\partial_{x_{i}}$ by the
classical momentum $p_{i}$ Gromov:2019aku . As one can see from e.g. (3.9) the
only difference between the situation with magnon and without magnon is the
ordering of the operators $X_{i}$ and $K_{i}$, which becomes irrelevant in the
classical limit. From this simple observation we conclude that the classical
theory is not affected by the presence of magnons. But there will be a
difference visible already at the first quasi-classical correction.

One way to think about this is that in the strong coupling limit the
difference between a dimension one scalar and a dimension two composite
operator made of two scalars (which is the magnon), becomes negligible.
Namely, in units of $1/\xi$, both of them are indistinguishable from zero.

###  3.2 Quantum fishchain with magnons

In the previous section we understood that the classical limit of the theory
with or without magnons should be the same. This means that essentially all
the quantization procedure developed in Gromov:2019bsj  and described shortly
in section 2.2 still applies here. There are only two steps in our
quantization procedure which allow for a modification. First, in (2.12) we
assumed that the local Casimir operator ${\rm tr~{}}\hat{q}^{2}_{i}$, which is
a c-number, should be associated with the dimension of the scalar, or more
precisely its $AdS_{5}$ mass $m^{2}=-3$. This assumption does not look natural
anymore as the magnons are the bound-states of two scalars
$\phi_{1}^{\dagger}$ and $\phi_{2}^{\dagger}$ of the total dimension
$\Delta_{i}=2$, meaning that the $AdS_{5}$ mass becomes $-4$ or equivalently
${\rm tr~{}}\hat{q}^{2}_{i}=\frac{8}{\xi^{2}}$.

Another subtle point in the quantization procedure, which needs to be
revisited, is the ordering ambiguity in the quantum Hamiltonian $\hat{H}_{q}$
(see (2.11) and (2.9)). We have a freedom to add to the naive term
$(\hat{q}_{i}^{2})^{NM}$ the extra terms $(\hat{q}_{i})^{NM}$ and
$\eta^{NM}{\rm tr~{}}q_{i}^{2}\sim\eta^{NM}/\xi^{2}$ like in (2.9). The
general ansatz is

$(q_{k}^{2})^{NM}\to:(q_{k}^{2})^{NM}:_{I_{k}}=(\hat{q}^{2}_{k})^{NM}-i\frac{c_{I_{k}}}{\xi}\hat{q}_{k}^{NM}-\frac{d_{I_{k}}}{\xi^{2}}\eta^{NM}\;,$
(3.13)

where $c_{{I_{k}}}$ and $d_{I_{k}}$ are some numbers, which for the case with
no magnons were set to $2$ and $1$ correspondingly i.e. $c_{0}=2,\;d_{0}=1$.
In this section we fix this quantization ambiguity by requiring that this
quantum fishchain model is exactly dual to the Fishnet model for the states
with magnons.

##### Map between the wave functions.

In Gromov:2019bsj  we have built an explicit map between the CFT wave function
in the ${\mathfrak{u}}(1)$-sector and the wave function of the quantum
fishchain in $AdS_{5}$. This construction was reviewed in section 2.2, see
(2.16). We now extend this map to the case with magnons.

The map (2.16) takes the 4D flat space CFT wave function and returns a
function of $J$ points $Z_{1},\dots,Z_{J}$, parametrizing a unit radius
$AdS_{5}$ space, $Z_{i}^{2}=-1$. These are the coordinates of the particles
constituting the fishchain.

The factors $1/(Z_{i}.X_{i})^{3}$ in (2.16) are bulk to boundary propagators
for $J$ scalars in $AdS_{5}$ of mass $m^{2}=-3$. It was designed to solve the
constraint ${\rm tr~{}}\hat{q}^{2}_{i}=\frac{6}{\xi^{2}}$. As for the sites
with magnons we should modify this constraint to ${\rm
tr~{}}\hat{q}^{2}_{i}=\frac{8}{\xi^{2}}$, so we can take
$1/(Z_{i}.X_{i})^{2}$. Hence, the generalization of the map (2.16) to
wavefunctions with both $\chi_{0}$ and $\chi_{1}$ operators is

$\psi_{\cal O}^{\bf
I}(Z_{1},\dots,Z_{J})=\int\prod_{i=1}^{J}\frac{D^{4}X_{i}}{-4\pi^{2}(Z_{i}.X_{i})^{3-I_{i}}}\Phi_{\cal
O}^{\bf I}(X_{1},\dots,X_{J})\;.$ (3.14)

As before, $1/(Z_{i}.X_{i})^{2}$ is the bulk to boundary propagator for an
$AdS_{5}$ scalar of $m^{2}=-4$. It is the fastest decaying solution to the
bulk equation of motion with a source at the boundary point $X_{i}$.666The
other solution is $\log(Z_{i}.X_{i})/(Z_{i}.X_{i})^{2}$.

It is not too hard to invert the map (3.14). One way is to go to the boundary
of $AdS_{5}$. While with no magnon excitations the bulk wave function exactly
reduces to the CFT wave function Gromov:2019bsj , in the case with magnons
there is an addition logarithmic factor

$\lim_{Z_{i}^{+}\to\infty}\frac{\psi_{\cal O}^{\bf
I}(Z_{1},\dots,Z_{J})}{\prod_{i}(2\log Z_{i}^{+})^{I_{i}}}=\Phi_{\cal O}^{\bf
I}(Z_{1},\dots,Z_{J})\ .$ (3.15)

##### Exact duality.

Having the map between the wave functions extended to the general case (3.14),
we can prove the duality and also fix the remaining quantization ambiguity
manifested by the existence of two arbitrary constants in the QFC Hamiltonian
(3.13). Following Gromov:2019bsj  we can establish the map between the QFC
Hamiltonian $\hat{H}_{q}$ and the CFT Hamiltonian $H$ by requiring

$\hat{H}_{q}\circ\psi_{\cal O}^{\bf
I}(Z_{1},\dots,Z_{J})=\int\prod_{i=1}^{J}\frac{D^{4}X_{i}}{-4\pi^{2}(Z_{i}.X_{i})^{3-I_{i}}}H\circ\Phi_{\cal
O}^{\bf I}(X_{1},\dots,X_{J})=0\;,$ (3.16)

where $H=(\hat{\cal B}^{-1}-1)$ as defined in (3.5). Exactly like in
Gromov:2019bsj  we simply use the identity

$\int\prod_{i=1}^{J}\frac{D^{4}X_{i}}{-4\pi^{2}(Z_{i}.X_{i})^{3-I_{i}}}\,\hat{\mathfrak{q}}_{i}\,F(X_{1},\dots,X_{J})=\hat{q}_{i}\,\int\prod_{i=1}^{J}\frac{D^{4}X_{i}}{-4\pi^{2}(Z_{i}.X_{i})^{3-I_{i}}}\,F(X_{1},\dots,X_{J})\;,$
(3.17)

where the $q$’s are the $SO(1,5)$ isometry generators of $AdS_{5}$, (2.10).
From this we immediately establish the values of the missing constants in
(3.13) by reading them off (3.12): $c_{I_{k}}=2+I_{k}$ and
$d_{I_{k}}=1+I_{k}$. Thus we arrive at the fishchain quantum Hamiltonian
constraint

${\hat{H}}_{q}={\rm tr~{}}\left[{:\hat{q}_{J}:_{I_{J}}\over
2}\dots:\hat{q}_{1}:_{I_{1}}\over 2\right]-1\ .$ (3.18)

Here, $:{q}_{k}:_{I_{k}}$ is given by

$\
\left(:q^{2}_{k}\\!:\\!_{I_{k}}\right)^{M\,N}\equiv\left(q^{2}_{k}\right)^{M\,N}-{i\over\xi}(2+I_{k})\,q_{k}^{M\,N}-{1\over\xi^{2}}(1+I_{k})\eta^{MN}\;.$
(3.19)

This derivation extends the proof Gromov:2019bsj  of the exact duality between
the QFC and the Fishnet CFT models beyond the ${\mathfrak{u}}(1)$ sector.

Note that so far we only considered the CFT wave functions, which contained
$\phi_{1}^{\dagger}$ and $\phi_{1}^{\dagger}\phi_{2}^{\dagger}$ under the
correlator with a local operator ${\cal O}$. Whereas this does indeed go
beyond the ${\mathfrak{u}}(1)$ sector one could still question if that is the
most general case. In particular one can also introduce “anti-magnons" i.e.
$\phi_{2}\phi_{1}^{\dagger}$, as we do in the next subsection. Later, however,
we show that the CFT wave functions with the same magnon number $J_{2}$
(defined as a difference of the number of magnons and anti-magnons) are all
equivalent to each other. It is yet very fruitful to have both magnons and
anti-magnons at the same time as we will see in the next section 3.3.

###  3.3 Adding anti-magnons to the QFC

We consider the most general situation, where at a site of the chain we can
have either no magnon, one magnon, one anti-magnon or a magnon–anti-magnon
pair. The CFT wave function definition (3.1) should be extended by allowing
for $\chi_{j}$ to take two additional values

${\includegraphics[scale={0.6}]{magnon_types2}}\ .$ (3.20)

for anti-magnon and magnon–anti-magnon pair correspondingly. All the steps
from the previous parts of this section are carried out in a similar way. We
summarize in table 1 the key elements.

type | $I_{k}$ | $\chi_{I_{k}}$ | $-2\beta_{I_{k}}^{MN}$ | $c_{I_{k}}$ | $d_{I_{k}}$ | $\Delta_{k}$ | ${\rm tr~{}}q_{k}^{2}$
---|---|---|---|---|---|---|---
no magnon | $0$ | $\phi_{1}^{\dagger}$ | $X^{M}X^{N}K^{2}$ | 2 | 1 | 1 | $6/\xi^{2}$
magnon | $+1$ | $\phi^{\dagger}_{1}{\color[rgb]{1,0,0}\phi^{\dagger}_{2}}$ | $X^{M}K^{2}X^{N}$ | 3 | 2 | 2 | $8/\xi^{2}$
anti-magnon | $-1$ | ${\color[rgb]{1,0,0}\phi_{2}}\phi^{\dagger}_{1}$ | $X^{N}K^{2}X^{M}$ | 1 | 0 | 2 | $8/\xi^{2}$
magnon + anti-magnon | $\bar{0}$ | ${\color[rgb]{1,0,0}\phi_{2}}\phi^{\dagger}_{1}{\color[rgb]{1,0,0}\phi^{\dagger}_{2}}$ | $K^{2}X^{M}X^{N}$ | 2 | 1 | 3 | $6/\xi^{2}$
Table 1: Four types of local composite operators entering into the definition
of the CFT wave function.

As before we assume that $|M-\bar{M}|<J$ where $M$ is the total number of
magnons and $\bar{M}$ is the total number of anti-magnons. In this case we
again have $\hat{{\cal B}}^{-1}={\rm
tr~{}}(\beta_{J,I_{J}}.\dots.\beta_{1,I_{1}})$ where the $\beta$’s are
summarized in table 1.

Next, one can also map the CFT wave function with the QFC wave function in
$AdS_{5}$ using the generalization of (3.14)

$\Psi^{\bf I}(Z_{1},\dots,Z_{J})=\int\prod{D^{4}X_{i}\over-4\pi^{2}}{\Phi^{\bf
I}(X_{1},\dots,X_{J})\over(X_{1}.Z_{1})^{4-\Delta_{1}}\dots(X_{J}.Z_{J})^{4-\Delta_{J}}}$
(3.21)

where again $\Delta_{k}$ is the conformal dimensions of $\chi_{I_{k}}$, given
in the table 1. Finally, the QFC Hamiltonian $\hat{H}_{q}$ is obtained by
uplifting the CFT Hamiltonian $\hat{H}$, which results in fixing the
quantization ambiguity parameters $c_{I}$ and $d_{I}$ as shown in the table 1.

To summarize, we see that the quantum fishchain naturally generalizes to
incorporate magnons. They are in one to one correspondence with all possible
orderings of $K^{2}_{k}$ and the $X_{k}$’s in the site operator $\beta_{k}$.

##  4 Integrability of the fishchain with magnons

In this section we extend the proof of the quantum integrability of the QFC
Gromov:2019bsj  to the general case with magnons. In order to demonstrate the
integrability of the QFC model we will follow the same steps as in
Gromov:2019bsj . The strategy is to identify the Hamiltonian as a part of a
large family of mutually commuting operators, which can be obtained as an
expansion in powers of $u$ of the T-operators $\hat{\mathbb{T}}(u)$.

There are $3$ non-trivial T-operators, which should contain a complete set of
integrals of motion. The building blocks for these operators are the Lax
matrices in antisymmetric finite-dimensional representations of
$\mathfrak{sl}(4)$. They were computed in Gromov:2019bsj  in terms of
$\hat{q}_{k}$. We will see that this derivation is general enough to
accommodate the current case as well. The main building blocks for the
operators $\hat{\mathbb{T}}(u)$ are the L-matrices

$\displaystyle\hat{\mathbb{L}}_{k}^{{\bf 1}}(u)$ $\displaystyle=$
$\displaystyle 1$ (4.1) $\displaystyle\hat{\mathbb{L}}_{k}^{\bf 4}(u)$
$\displaystyle=$ $\displaystyle u-\tfrac{i}{2}\hat{q}_{k}^{MN}\Sigma_{MN}$
$\displaystyle\hat{\mathbb{L}}_{k}^{\bf 6}(u)$ $\displaystyle=$ $\displaystyle
u^{2}+u\hat{q}_{k}+\frac{\hat{q}_{k}^{2}}{2}-\frac{i\hat{q}_{k}}{\xi}-\frac{{\rm
tr~{}}\hat{q}_{k}^{2}}{8}+\frac{1}{4\xi^{2}}$
$\displaystyle\hat{\mathbb{L}}_{k}^{\bar{\bf 4}}(u)$ $\displaystyle=$
$\displaystyle\left(u^{2}-\frac{1}{8}{\rm
tr~{}}\hat{q}_{k}^{2}+\frac{1}{\xi^{2}}\right)\left[-\hat{\mathbb{L}}_{k}^{\bf
4}(-u)\right]^{T}$ $\displaystyle\hat{\mathbb{L}}_{k}^{\bar{\bf 1}}(u)$
$\displaystyle=$ $\displaystyle\left(u^{2}-\frac{1}{8}{\rm
tr~{}}\hat{q}_{k}^{2}+\frac{5}{4\xi^{2}}\right)^{2}+\frac{1}{8\xi^{2}}{\rm
tr~{}}\hat{q}_{k}^{2}-\frac{1}{\xi^{4}}\;.$

Where $\Sigma_{MN}$ are $6$-dimensional $\sigma$-matrices. Their explicit
representation can be found in Gromov:2019bsj . The mutually commuting
families of operators are constructed from (4.1) via

$\displaystyle\hat{\mathbb{T}}(u)={\rm
tr~{}}\\!\\!\left[\hat{\mathbb{L}}_{J}(u-\theta_{J})\dots\hat{\mathbb{L}}_{2}(u-\theta_{2})\hat{\mathbb{L}}_{1}(u-\theta_{1})G\right]\;.$
(4.2)

Where $G$ is a constant twist matrix and $\\{\theta_{j}\\}$ are the so-called
inhomogeneities which can be taken as arbitrary. For the original fishnet
model the twist matrix is trivial, $G=1$. However, here we consider the most
general deformed case, dual to the twisted fishnet of twistingpaper , with
$G={\rm diag}(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4})$ and we assume
$\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}=1$ for unimodularity of $G$.
Next, we have to choose the values of $\\{\theta_{j}\\}$ so that the QFC
Hamiltonian appears in (4.2) for a special value of $u$. More precisely we
require that in the same manner as in the case without magnons we have
$\hat{\mathbb{T}}^{\bf 6}(0)=\hat{H}_{q}+1$.

We now consider all $4$ cases of insertions one by one – the site without
magnon, with magnon, with anti-magnon, and with magnon anti-magnon pair using
the data from table 1.

##### No-magnon case.

The case without magnon corresponds to a single scalar $\phi_{1}$, which
allows for the value of the quandatic Casimir to be $-3$, implying ${\rm
tr~{}}\hat{q}_{k}^{2}=\frac{6}{\xi^{2}}$. As a result (4.1) reduces to

$\displaystyle\hat{\mathbb{L}}_{k}^{\bf 6}(u)$ $\displaystyle=$ $\displaystyle
u^{2}+u\hat{q}_{k}+\frac{\hat{q}_{k}^{2}}{2}-\frac{i\hat{q}_{k}}{\xi}-\frac{1}{2\xi^{2}}$
(4.3) $\displaystyle\hat{\mathbb{L}}_{k}^{\bar{\bf 4}}(u)$ $\displaystyle=$
$\displaystyle\left(u^{2}+\frac{1}{4\xi^{2}}\right)\left[-\hat{\mathbb{L}}_{k}^{\bf
4}(-u)\right]^{T}$ $\displaystyle\hat{\mathbb{L}}_{k}^{\bar{\bf 1}}(u)$
$\displaystyle=$ $\displaystyle u^{2}\left(u^{2}+\frac{1}{\xi^{2}}\right)\;,$

we see that $\hat{\mathbb{L}}_{k}^{\bf
6}(0)=\frac{\hat{q}_{k}^{2}}{2}-\frac{i\hat{q}_{k}}{\xi}-\frac{1}{2\xi^{2}}=\frac{:\hat{q}_{k}^{2}:_{0}}{2}$,
which is the correct factor in $\hat{H}_{q}$ for sites without a magnon,
meaning that for these sites we have to set $\theta_{k}=0$.

##### Magnon case.

For the site with magnon the dimension $\Delta_{k}=2$ corresponds to the
quadratic Casimir $-4$, implying that ${\rm
tr~{}}\hat{q}_{k}^{2}=\frac{8}{\xi^{2}}$. Consequently from (4.1) we get

$\displaystyle\hat{\mathbb{L}}_{k}^{\bf 6}(u)$ $\displaystyle=$ $\displaystyle
u^{2}+u\hat{q}_{k}+\frac{\hat{q}_{k}^{2}}{2}-\frac{i\hat{q}_{k}}{\xi}-\frac{3}{4\xi^{2}}$
(4.4) $\displaystyle\hat{\mathbb{L}}_{k}^{\bar{\bf 4}}(u)$ $\displaystyle=$
$\displaystyle u^{2}\left[-\hat{\mathbb{L}}_{k}^{\bf 4}(-u)\right]^{T}$
$\displaystyle\hat{\mathbb{L}}_{k}^{\bar{\bf 1}}(u)$ $\displaystyle=$
$\displaystyle\left(u+\frac{i}{2\xi}\right)^{2}\left(u-\frac{i}{2\xi}\right)^{2}\;.$

This time the ambiguity constants are $c_{I}=3$ and $d_{I}=2$ i.e. we need to
tune $\theta_{k}$ to obtain $\hat{\mathbb{L}}_{k}^{\bf
6}(-\theta_{k})=\frac{\hat{q}_{k}^{2}}{2}-\frac{3i\hat{q}_{k}}{2\xi}-\frac{1}{\xi^{2}}$.
Note that with only one parameter $\theta_{k}$ it is not guaranteed that we
manage to fit the two constants correctly. Fortunately, by setting
$\theta_{k}=\frac{i}{2\xi}$777N.G. is grateful to V.Kazakov and Z.Tsuboi for
discussing the possibility of inclusion of inhomogeneities in relation to the
magnons. we indeed obtain $\hat{\mathbb{L}}_{k}^{\bf
6}(0)=\frac{1}{2}:\hat{q}_{k}^{2}:_{1}$.

##### Anti-magnon case.

The building block for the anti-magnon is $\hat{\mathbb{L}}_{k}^{\bf
6}(-\theta_{k})=\frac{\hat{q}_{k}^{2}}{2}-\frac{i\hat{q}_{k}}{2\xi}-\frac{1}{\xi^{2}}$,
which corresponds to $\theta_{k}=-\frac{i}{2\xi}$.

##### Magnon–anti-magnon case.

The Lax matrix, in this case, is identical to the no-magnon case considered
above. The reason is that $\Delta(\chi_{\bar{0}})=4-\Delta(\chi_{0})=3$, so
the quadratic Casimir, given by $c=\Delta(\Delta-4)=-3$ is exactly the same as
that for $\Delta=1$. This implies that in both cases we have ${\rm
tr~{}}q_{k}^{2}=\frac{6}{\xi^{2}}$. Moreover, the quantization ambiguity
constants $c_{I}$ and $d_{I}$ are also exactly the same (see table 1) and
therefore $\theta_{k}=0$ in this case as well. As is explained in the next
section, at the fishnet side both the magnon–anti-magnon case and no-magnon
case describe different ways of cutting the same propagation line of
$\phi_{1}$ along the fishnet diagrams. Hence, the corresponding transfer
matrix that encodes all physical charges should be the same.

To summarize, we see that after fixing the impurities to
$\theta_{k}=\frac{iI_{k}}{2\xi}$ we get indeed $\hat{\mathbb{T}}^{\bf
6}(0)=\hat{H}_{q}+1$. This property implies the quantum integrability of our
model, as we identified the Hamiltonian of the system within a large family of
mutually commuting operators.888Of course one should verify that there are
sufficiently many independent integrals of motion. This is usually rather hard
to verify rigorously, but we strongly believe this is the case. The rough
counting of the integrals of motions indeed gives $\sim 4J$, which coincides
with the number of degrees of freedom of the theory.

In the current formulation the ordering of the magnons is quite important.
Different distributions of magnons within the chain will produce different
Hamiltonian and T-operators. We will see in section 6 that the eigenvalues of
all T-operators are in fact independent of the order of magnons. This
invariance has a very nice physical interpretation – it can be associated with
the discrete reparametrization symmetry of the fishchain as we discuss in the
next section 5. Based on the results of this section, in section 6 we present
the general integrability formalism for the spectrum of the theory.

##  5 Discrete reparametrization symmetry

When introducing the magnons we have to face the following paradox. On the CFT
side, the primary operators with several magnons will be given by a
complicated linear combination of all possible magnon permutations. At the
same time on the fishnet side, it looks like we have the freedom to choose any
ordering for the magnons. This choice is made explicitly in our definition of
the CFT wave function by the correlator (2.3). Moreover, the fishchain
Hamiltonian $\hat{H}_{q}$ itself depends on a particular fixed order. The
danger is that the spectrum of the Hamiltonian is also order dependent, which
would result in an inconsistency of our approach. The physical quantities such
as dimensions of local operators should not depend on an arbitrary choice of
order of insertions of $\phi_{2}$ fields in the CFT wave function, which
probes this local operator.

Figure 5: Two possible cuttings of the two point function of two local
operators ${\cal O}$. Each cutting represents the same two point function as a
contraction of two CFT wave functions. The two cuts produce different ordering
of the magnons inside the chain. The cut $\gamma_{1}$ leads to the wave
function (3.1) with
$\\{I_{k+3},I_{k+2},I_{k+1},I_{k},I_{k-1},I_{k-2}\\}=\\{0,0,1,1,0,0\\}$, while
the cut $\gamma_{2}$ leads to the equivalent wave function with
$\\{I_{k+3},I_{k+2},I_{k+1},I_{k},I_{k-1},I_{k-2}\\}=\\{0,1,0,1,0,0\\}$.

In this section we demonstrate that, luckily, the spectrum does not depend on
the ordering of the magnons. This is due to a novel symmetry of the fishchain
which we interpret as the discrete reparametrization symmetry. We will show
that redistribution of the magnons can be understood as a map between Hilbert
spaces of the two chains, which differ by the magnon positions. We will then
prove that this map preserves the eigenvalues of $\hat{H}_{q}$ and of all
other conserved charges encoded into $\hat{\mathbb{T}}(u)$, introduced in the
previous section.

###  5.1 Cutting and leveling

Consider a correlation function between two local operators of the type (2.3)
that consists of $J_{1}=J$ fields $\phi_{1}$ and $J_{2}=M$ insertions of
$\phi_{2}$’s fields in the trace (as before we assume $M<J$). The planar
Feynman diagrams that contribute to this correlator have cylindrical topology
and are of an iterative spider-net-like structure as discussed in the
introduction, see figure 5. The fishchain state, as given by the correlator in
(3.1), corresponds to a specific choice of cut around this cylinder. For
example the blue cut $\color[rgb]{0,0,1}\gamma_{1}$ in figure 5 corresponds to
the correlator (3.1) with $M=2$, where one magnon ($\chi_{1}$) is at the
$k$’th site of the chain and the second magnon is at the neighboring $k+1$
site. On the other hand, when the same diagram is cut as is indicated by the
$\color[rgb]{0,0,1}\gamma_{2}$ cut in figure 5, the second magnon is at the
$k+2$ site.

In general, a cut is a choice of a decomposition of the fishnet diagrams. It
is specified by a line ${\color[rgb]{0,0,1}\gamma}$ that goes around the
cylinder and crosses a set of propagators on the way. Here, we choose to
restrict our consideration to space-like cuts. These are cuts that cross every
$\phi_{1}$ line and every $\phi_{2}$ line exactly once. Moreover, every
$\phi_{2}$ cut must be followed by a $\phi_{1}$ cut. These are the cuts that
look space-like on the fishnet lattice. They are the ones that allow us to
obtain all the higher loop fishnet diagrams by iterative action of the
corresponding graph building operator. Such cuts are analogues to Cauchy
surfaces on the string worldsheet. The graph building operator generates the
“worldsheet” time propagation on such surfaces.

We notice that a choice of space-like cut corresponds to a choice of a basis
of CFT wave functions of the type (2.3) as follows. A cut of a
$\phi_{1}$-propagator is mapped to a $\phi_{1}^{\dagger}$ field in the
corresponding correlator (2.3) and a cut of a $\phi_{2}$-propagator is mapped
to a $\phi_{2}^{\dagger}$ field (see figure 5). We then merge
$\phi_{1}^{\dagger}$ and $\phi_{2}^{\dagger}$ fields to be at the same point
as a $\chi_{1}$ or $\chi_{-1}$ in accordance with the vertex above the cut. We
will give more explicit examples below.

Let us point out that the situation in which two different cuts result in a
different order of magnons in the chain like in figure 5 is not the most
general. In fact one can also generate an arbitrary number of magnon–anti-
magnon pairs. For example, we can start with a chain with no magnons at all
and deform the cut. After doing so, we end up with a magnon and an anti-
magnon, see figure 6.

PDF-picture replacement

Figure 6:  Another example of a cut which, instead of moving magnons around,
creates a magnon–anti-magnon pair.

One way to think about the difference between different cuts is as a map
between Hilbert spaces. We will now construct explicit operators that
implement this map between the Hilbert spaces of two different deformations of
the cut.

###  5.2 The cut deformation operators

Any deformation of the cut, such as the one in figure 4, can be realized as
the action of a cut deformation operator on the corresponding CFT wave
function. The aim of this section is to construct the CFT cut deformation
operators and then lift them into an operator acting on fishchain
wavefunctions, i.e. acting on the $AdS_{5}$ variables $Z_{i}$. We will use
these operators in section 6 and will prove that they generate a discrete
symmetry of the quantum fishchain, leaving the spectrum invariant.

Any cut deformation of the CFT wave function can be decomposed into a
combination of two elementary operations

PDF-picture replacement

Figure 7: Graphical representation of the action of the operators $v_{k}^{-1}$
and $v_{k}$ on a CFT wave function. The operator $v_{k}^{-1}$ removes a
vertical $\phi_{1}$ propagator, which is equivalent to the creation of a
magnon–anti-magnon pair on one site. Correspondingly, $v_{k}$ annihilates the
magnon–anti-magnon pair.

$\displaystyle v_{k}\circ\Psi(\dots,X_{k},\dots)={\xi^{2}\over\pi^{2}}\int
D^{4}Y_{k}\,\frac{\Psi(\dots,Y_{k},\dots)}{-2X_{k}.Y_{k}}\;\;\;\;$ (add
vertical propagator) (5.1) $\displaystyle
h_{k+1,k}\circ\Psi(\dots,X_{k+1},X_{k},\dots)=\frac{\Psi(\dots,X_{k+1},X_{k},\dots)}{-2X_{k+1}.X_{j}}\;\;\;\;$
$\displaystyle\text{(add horizontal propagator)}\ .$ (5.2)

As one can see from the equation the $v_{k}$ operator adds a vertical
$\phi_{1}$ propagator, and $h_{k}$ adds a horizontal $\phi_{2}$ propagator.
The corresponding inverse of these elementary operations are

PDF-picture replacement

Figure 8: Graphical representation of the action of the operator
$h_{k+1,k}^{-1}$, removing the $\phi_{2}$ propagator stretched horizontally
between the $k+1$ and $k$ sites. It can be interpreted as an operator
annihilating a magnon and anti-magnon situated at two neighbouring sites.

$\displaystyle v_{k}^{-1}=-{1\over 4\xi^{2}}\,\Box_{k}\;\;\;\;$ (remove
vertical propagator) (5.3) $\displaystyle
h_{k+1,k}^{-1}=-2X_{k+1}.X_{k}\;\;\;\;$ $\displaystyle\text{(remove horizontal
propagator)}\ .$ (5.4)

We can build out of $v$ and $h$ the operators which move a magnon/anti-magnon
to the right/left

$\displaystyle r_{k+1,k}\equiv h_{k+1,k}^{-1}\circ v_{k}^{-1}\;\;\;\;$ (moves
magnon right) (5.5) $\displaystyle\bar{r}_{k+1,k}\equiv h_{k+1,k}^{-1}\circ
v_{k+1}^{-1}\;\;\;\;$ $\displaystyle\text{(moves anti-magnon left)}\;.$ (5.6)

Let us show that the operators $v_{k},\;r_{k+1,k},\;\bar{r}_{k+1,k}$ and
$h^{-1}_{k+1,k}$ constitute the $3$ elementary moves which allow us to map any
CFT wave function to a standard form. When acting on a state in a fishchain
Hilbert space, characterized by the indexes ${\bf I}$, they transform it to a
state in a different Hilbert space, characterized by a different set of
indexes $\tilde{\bf I}$ as follows

$\displaystyle v_{k}\;\;:\;\;{\bf I}=\\{\dots
I_{k+1},\bar{0},I_{k-1}\dots\\}\;\;$ $\displaystyle\to$
$\displaystyle\;\;\tilde{\bf I}=\\{\dots I_{k+1},0,I_{k-1}\dots\\}$ (5.7)
$\displaystyle r_{k+1,k}\;\;:\;\;{\bf I}=\\{\dots
I_{k+2},1,0,I_{k-1}\dots\\}\;\;$ $\displaystyle\to$
$\displaystyle\;\;\tilde{\bf I}=\\{\dots I_{k+2},0,1,I_{k-1}\dots\\}$ (5.8)
$\displaystyle\bar{r}_{k+1,k}\;\;:\;\;{\bf I}=\\{\dots
I_{k+2},0,-1,I_{k-1}\dots\\}\;\;$ $\displaystyle\to$
$\displaystyle\;\;\tilde{\bf I}=\\{\dots I_{k+2},-1,0,I_{k-1}\dots\\}$ (5.9)
$\displaystyle h^{-1}_{k+1,k}\;\;:\;\;{\bf I}=\\{\dots
I_{k+2},1,-1,I_{k-1}\dots\\}\;\;$ $\displaystyle\to$
$\displaystyle\;\;\tilde{\bf I}=\\{\dots I_{k+2},0,0,I_{k-1}\dots\\}\;,$
(5.10)

see figures 8 and 9. We are going to show that a state with
${\mathfrak{u}}(1)$ charges $|J_{2}|<J_{1}$ can be mapped to the state in the
Hilbert space, characterized by ${\bf I}$ in the standard configuration ${\bf
I}=\\{1,1,\dots,1,0,0,\dots,0\\}$, for $J_{2}\leq 0$, and ${\bf
I}=\\{-1,-1,\dots,-1,0,0,\dots,0\\}$ otherwise. Indeed, we can use $v_{k}$ to
replace all $\bar{0}$ by $0$’s. Next we can move magnons and anti-magnons
towards each other using $r_{k+1,k}$ and $\bar{r}_{k+1,k}$ and then annihilate
them in pairs using $h^{-1}_{k+1,k}$. After that, depending on the sign of
$J_{2}$ there will be only magnons or anti-magnons left, which we can group
together with $r_{k+1,k}$ or $\bar{r}_{k+1,k}$.

PDF-picture replacement

Figure 9: The operator $r_{k+1,k}$ moves a magnon at the $(k+1)$ site to the
empty site on its right, ($k$). Similarly, the reflection of this figure
describes how the operator $\bar{r}_{k+1,k}$ moves an anti-magnon at the
$k$’th site to the empty site on its left ($k+1$).

In the next section we derive what these operators correspond to in the dual
description.

###  5.3 The fishchain reparametrization operators

Using the map between the CFT and the fishchain wavefunctions (3.21) we now
map $v_{k}^{-1}$ and $h_{k}^{-1}$ into fishchain operators. Consider first the
operator $v_{k}^{-1}$. This operator acts on a site with no magnon and maps it
to a site with a magnon anti-magnon pair. Plugging it into the map (3.21) and
taking into account that the conformal dimension at that site changes from
$\Delta_{k}=1$ to $\Delta_{k}=3$, we get the equation for the corresponding
operator ${\mathbb{V}}^{-1}$ in $AdS_{5}$

$v^{-1}(x)\circ\frac{1}{(Z.{\cal
X})^{1}}={\mathbb{V}}^{-1}(Z)\circ\frac{1}{(Z.{\cal X})^{3}}\;.$ (5.11)

Using that $v^{-1}(x)=-{1\over 4\xi^{2}}\Box_{\vec{x}}$ we conclude that
${\mathbb{V}}^{-1}(Z)={1\over 2\xi^{2}}$, i.e. up to a multiplication by a
constant this operator acts trivially in the dual picture. This is because on
the fishchain side there is no real distinction between no-magnon and a
magnon–anti-magnon pair at the same site, as one can see from the table 1.

Similarly, for $h_{k+1,k}^{-1}$ (which decreases $\Delta_{i}$ by one on two
consecutive sites) we find its $AdS_{5}$ counterpart ${\mathbb{H}}$ from the
following identity

${\mathbb{H}}_{\Delta_{1},\Delta_{2}}(Z_{1},Z_{2})\circ\frac{1}{(Z_{1}.{\cal
X}_{1})^{4-\Delta_{1}}(Z_{2}.{\cal X}_{2})^{4-\Delta_{2}}}=\frac{-2{\cal
X}_{1}.{\cal X}_{2}}{(Z_{1}.{\cal X}_{1})^{5-\Delta_{1}}(Z_{2}.{\cal
X}_{2})^{5-\Delta_{2}}}\;,$ (5.12)

where $\Delta_{j}$ can be either $2$ or $3$. We find that

${\mathbb{H}}_{\Delta_{1},\Delta_{2}}(Z_{1},Z_{2})={-2\over\alpha_{1}\alpha_{2}}\left[\nabla_{1}.\nabla_{2}+\alpha_{1}Z_{1}.\nabla_{2}+\alpha_{2}Z_{2}.\nabla_{1}+\alpha_{1}\alpha_{2}Z_{1}.Z_{2}\right]$
(5.13)

where $\alpha_{k}=4-\Delta_{k}$ and999Remember that the embedding coordinates
should satisfy $Z^{2}=-1$, so the $\partial_{Z}$ operation is in principle not
very well defined, however the combination $\nabla$ commutes with $Z^{2}$,
making it well defined.

$\nabla_{j,M}={\partial\over\partial
Z^{M}_{j}}+Z_{j\,M}\left(Z_{j}^{N}{\partial\over\partial{Z_{j}^{N}}}\right)\;.$
(5.14)

As a consequence, the lift of the operator $r_{k+1,k}=h_{k}^{-1}\circ
v_{k}^{-1}$, that moves a magnon to an empty site to the right reads, up to an
irrelevant constant factor

${\mathbb{H}}_{3,2}=\nabla_{1}.\nabla_{2}+2Z_{2}.\nabla_{1}+Z_{1}.\nabla_{2}+2Z_{1}.Z_{2}\;.$
(5.15)

In the next section we will show that the magnon move operator
${\mathbb{H}}_{3,2}$ and magnon annihilation operator ${\mathbb{H}}_{2,2}$
generate a gauge symmetry of the fishchain that we interpret as discrete
reparametrization symmetry.

##  6 Integrability, Exact Spectrum and Reparametrization Invariance

In section 4 we constructed a set of mutually commuting operators which also
include the Hamiltonian $\hat{H}_{q}$. Then, in section 5, we have constructed
the fishnet cut deformation operators and lifted them to fishchain operators.
In this section, we use the set of commuting operators to solve for the
spectrum of dimensions $\Delta$ of local operators. We then show that the cut
deformation operators generate similarity transformations of the transfer
matrices that can be interpreted as gauge transformations of the corresponding
Baxter equation.

This section generalizes the results of Gromov:2017cja , where the non-
perturbative spectrum was obtained for the operators $J=3,\;M=0$101010We
remind that $M$ denotes the number of magnons and $\bar{M}$ number of anti-
magnons. The ${\mathfrak{u}}(1)$ charges in these notations are
$J_{1}=J,\;J_{2}=M-\bar{M}$., and Gromov:2019cja , where the general $J,\;M=0$
case was considered. An important result of Gromov:2017cja ; Gromov:2019cja ,
which we are going to use here is the quantization condition, which is needed
in addition to the Baxter TQ-relations. We will propose the generalization of
this quantization condition and also present the most general form of the TQ-
relations valid for any values of the charges $J_{1},J_{2}$ such that
$J_{1}>|J_{2}|$.

Using these techniques we will be able to solve numerically for several
examples of states, $J=3,M=1$ and $J=3,M=2$. At finite $\xi$ we explore a very
rich analytic structure of the function $\Delta(\xi)$. At weak coupling we are
able to compare our numerical results with the asymptotic Bethe ansatz, valid
only up to the wrapping order $\xi^{2J+2|M|-2}$.

Our Baxter TQ-relation, which we present in this section, together with the
generalization of the quantization condition of Gromov:2017cja ;
Gromov:2019cja  constitute the complete set of equations, analogous to the
Quantum Spectral Curve equations for N=4 SYM Gromov:2013pga  or ABJM
Cavaglia:2014exa  theory.

###  6.1 T-functions and Baxter TQ relation

We start by formulating the TQ-relations. In order to compare with the
previous partial results of Gromov:2017cja ; Gromov:2019cja  it is convenient
to rescale the spectral parameter by $\xi$. To avoid confusion we denote the
rescaled spectral parameter by $v$, which is related to the initial one by
$u=v/\xi$.111111When writing ${\mathbb{T}}(v)$, strictly speaking, what we
mean is ${\mathbb{T}}(v/\xi)$. We also denote by ${\mathbb{T}}(u)$ the
eigenvalue (or T-functions) of the simultaneously diagonalizable operators
$\hat{\mathbb{T}}(u)$.

The T-functions corresponding to the trivial representations can be computed
explicitly

${\mathbb{T}}^{\bf 1}(v)=1\;\;{\rm and}\;\;{\mathbb{T}}^{\bar{\bf
1}}(v)=\frac{v^{2J}(v+i)^{J-M+\bar{M}}(v-i)^{J+M-\bar{M}}}{\xi^{4J}}\;.$ (6.1)

Also we notice from (4.2)-(4.4) that ${\mathbb{T}}^{\bar{\bf 4}}$ and
${\mathbb{T}}^{{\bf 4}}$ are closely related

${\mathbb{T}}^{\bar{\bf
4}}(v)=\frac{(-1)^{J}}{\xi^{2J}}(v+\tfrac{i}{2})^{J-M+\bar{M}}(v-\tfrac{i}{2})^{J+M-\bar{M}}{\mathbb{T}}^{{\bf
4}*}(-v)$ (6.2)

where $*$ indicates that one should inverse the order of the particles and
interchange magnons and anti-magnons. This transformation is an obvious
symmetry of the system, but of course not all states are invariant under this
transformation.

##### Baxter TQ-relations.

One of the key quantities in quantum integrability are the Q-functions
Baxter:1972wg ; Bazhanov:1996dr . They can be defined as T-functions
corresponding to a rather complicated auxiliary space representation. As a
simple alternative to defining them as eigenvalues of these complicated
Q-operators, one can find them from a finite difference TQ-relation, which in
our case reads121212We use the standard shorthand notations
$f^{[+a]}=f(v+\tfrac{ia}{2})$.

${\mathbb{T}}^{{\bf 1}[+2]}\;Q^{[+4]}-{\mathbb{T}}^{{\bf
4}[+1]}\;Q^{[+2]}+{\mathbb{T}}^{{\bf 6}}\;Q-{\mathbb{T}}^{\bar{\bf
4}[-1]}\;Q^{[-2]}+{\mathbb{T}}^{\bar{\bf 1}[-2]}\;Q^{[-4]}=0\;.$ (6.3)

We notice that the coefficients in this equation are polynomials of different
degree. However, as we can see from (6.2) even though ${\mathbb{T}}^{\bar{\bf
4}}(v)$ is a polynomial of degree $3J$, it has a trivial part of degree $2J$,
which factorises. To remove the trivial factors we introduce the following
gauge transformation

$Q(v)=q(v)\;v^{\bar{M}}\left[\xi^{iv}e^{\frac{\pi
v}{2}}\Gamma(-iv)\right]^{J}\;.$ (6.4)

This transformation is designed so that the equation (6.3) becomes more
symmetric

$\displaystyle-\frac{P_{2J}^{\bf 6}(v)}{v^{J-M-\bar{M}}}\;q_{a}$
$\displaystyle=$ $\displaystyle
v^{M}(v+i)^{J}(v+2i)^{\bar{M}}\;q_{a}^{[+4]}-v^{M}(v+i)^{\bar{M}}P_{J}^{\bf
4}(v+\tfrac{i}{2})\;q_{a}^{[+2]}$ $\displaystyle+$ $\displaystyle
v^{\bar{M}}(v-i)^{J}(v-2i)^{M}\;q_{a}^{[-4]}-v^{\bar{M}}(v-i)^{M}P_{J}^{\bar{\bf
4}}(v-\tfrac{i}{2})\;q_{a}^{[-2]}\;,$

where we defined the degree-$L$ polynomials $P_{L}$ in the following way

$P_{J}^{\bf 4}(v)\equiv\xi^{J}{\mathbb{T}}^{\bf 4}(v)\;\;,\;\;P_{2J}^{\bf
6}(v)\equiv\xi^{2J}{\mathbb{T}}^{\bf 6}(v)\;\;,\;\;P_{J}^{\bar{\bf
4}}(v)\equiv(-\xi)^{J}{\mathbb{T}}^{\bf 4*}(-v)\;.$ (6.6)

The index $a=1,\dots,4$ in $q_{a}$ labels $4$ independent solutions of (6.1).
We specify below exactly how these solutions are defined by their large $v$
asymptotic.

One can see that in the form (6.1) the Baxter TQ-equation generalizes the one
derived in the CFT context in Gromov:2017cja ; Gromov:2019cja  for the case
$M=0$. Furthermore, in Gromov:2017cja  it is in addition assumed that the
state is invariant under the inversion of the order of the sites, which we do
not assume here. Our equation is thus a generalization of the results of
Gromov:2017cja ; Gromov:2019cja  to the most general states with an arbitrary
number of magnons, as long as $|J_{2}|<|J_{1}|$.

##### Asymptotics of q.

The global conformal group charges $\Delta,S_{1}$ and $S_{2}$, enter into the
TQ-relations through the large $v$ asymptotic of T-functions and consequently
Q-functions. Indeed, at large $v$ the product (4.2) becomes a sum of
$\hat{q}_{k}$, which is the global $\mathfrak{so}(1,5)$ charge. This
observation is reflected in the following expressions for the highest powers
of the polynomials $P_{L}$:

$\displaystyle P_{J}^{\bf 4}$ $\displaystyle=$ $\displaystyle
v^{J}(\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4})+\frac{v^{J-1}}{2i}\left[(\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4})(M-\bar{M})+\lambda_{4}\left(\Delta-
S_{1}+S_{2}\right)\right]$ $\displaystyle+$
$\displaystyle\frac{v^{J-1}}{2i}\left[\lambda_{3}\left(\Delta+S_{1}-S_{2}\right)+\lambda_{2}\left(-\Delta-
S_{1}-S_{2}\right)+\lambda_{1}\left(-\Delta+S_{1}+S_{2}\right)\right]+\dots$
$\displaystyle P_{2J}^{\bf 6}$ $\displaystyle=$
$\displaystyle\left[v^{2J}+\frac{v^{2J-1}}{i}(M-\bar{M})\right](\lambda_{1}\lambda_{2}+\lambda_{1}\lambda_{3}+\lambda_{1}\lambda_{4}+\lambda_{2}\lambda_{3}+\lambda_{2}\lambda_{4}+\lambda_{3}\lambda_{4})$
$\displaystyle-$
$\displaystyle\frac{v^{2J-1}}{i}\left[\Delta\left(\lambda_{1}\lambda_{2}-\lambda_{3}\lambda_{4}\right)+S_{2}\left(\lambda_{2}\lambda_{3}-\lambda_{1}\lambda_{4}\right)+S_{1}\left(\lambda_{2}\lambda_{4}-\lambda_{1}\lambda_{3}\right)\right]+\dots$
$\displaystyle P_{J}^{\bar{\bf 4}}$ $\displaystyle=$ $\displaystyle
v^{J}(\bar{\lambda}_{1}+\bar{\lambda}_{2}+\bar{\lambda}_{3}+\bar{\lambda}_{4})+\frac{v^{J-1}}{2i}\left[(\bar{\lambda}_{1}+\bar{\lambda}_{2}+\bar{\lambda}_{3}+\bar{\lambda}_{4})(M-\bar{M})-\bar{\lambda}_{4}\left(\Delta-
S_{1}+S_{2}\right)\right]$ $\displaystyle-$
$\displaystyle\frac{v^{J-1}}{2i}\left[\bar{\lambda}_{3}\left(\Delta+S_{1}-S_{2}\right)+\bar{\lambda}_{2}\left(-\Delta-
S_{1}-S_{2}\right)+\bar{\lambda}_{1}\left(-\Delta+S_{1}+S_{2}\right)\right]+\dots$

where $\bar{\lambda}_{a}\equiv\frac{1}{\lambda_{a}}$. We remind that
$\lambda_{i}$’s are the diagonal elements of the twist matrix $G$. Next, we
plug the asymptotics (6.1) into the TQ-relation (6.1) to deduce the following
four possible asymptotics of the four linearly independent solutions of (6.1)

$q_{a}=\lambda_{a}^{-iv}v^{+\mu_{a}-\frac{J+M+\bar{M}}{2}}+\dots\
,\qquad\mu_{a}=\frac{1}{2}\left(\begin{array}[]{c}+\Delta-S_{1}-S_{2}\\\
+\Delta+S_{1}+S_{2}\\\ -\Delta-S_{1}+S_{2}\\\
-\Delta+S_{1}-S_{2}\end{array}\right)_{a}\;.$ (6.8)

Another important asymptotic is the value of $P_{2J}^{\bf 6}(v)$ at the
origin. Due to the relation (6.6) and the observation that ${\mathbb{T}}^{\bf
6}(0)=H_{q}+1$ we have the following relation

$P_{2J}^{\bf 6}(0)=\xi^{2J}\;.$ (6.9)

This is central for how the fishchain condition $H_{q}=0$ injects the
information about the coupling $\xi$ into the TQ-relations. Except for (6.9)
there is no other explicit dependence on $\xi$ appearing in (6.1).

###  6.2 Discrete reparametrization symmetry and integrability

The general expectation from the discrete reparametrization symmetry,
described in section 5, is that the spectrum and other integrals of motion
should remain invariant. In this section we will prove this is indeed the case
based on the integrability construction described in the previous section.

##### Magnon reordering symmetry.

One can immediately notice that the Baxter equation (6.1) does not contain any
explicit reference to the ordering of the particles. The equation only depends
on the number of magnons $M$ and anti-magnons $\bar{M}$, but not on their
particular ordering along the fishchain. Furthermore, we will see below that
the Baxter TQ-relations depends non-trivially only on the difference
$J_{2}=M-\bar{M}$.

At the level of the eigenvector this symmetry is less obvious. In section 5 we
have shown that wave functions with different magnon positions corresponds to
different ways of cutting the same Feynman diagrams. Using this picture, we
have argued that a shift in the magnon position should be thought of as a
discrete reparametrization symmetry of the fishchain model. This shift is
generated by the fishchain operator ${\mathbb{H}}_{3,2}$ in (5.15)

${\cal
R}_{21}\equiv{\mathbb{H}}_{3,2}(Z_{1},Z_{2})=\nabla_{1}.\nabla_{2}+\nabla_{2}.Z_{1}+2Z_{2}.\nabla_{1}+2Z_{1}.Z_{2}\;.$
(6.10)

We will now prove that indeed, this operator generates a symmetry of the
transfer matrix. The operator ${\cal R}_{21}$ has the following key property

$\displaystyle{\mathbb{L}}^{ab}_{2}(u){\mathbb{L}}^{bc}_{1}\left(u-\tfrac{i}{2\xi}\right){\cal
R}_{21}-{\cal
R}_{21}{\mathbb{L}}^{ab}_{2}\left(u-\tfrac{i}{2\xi}\right){\mathbb{L}}^{bc}_{1}(u)$
(6.11)
$\displaystyle=\frac{i\Sigma^{ac}_{NM}}{2\xi^{2}}\left[(Z_{2}^{N}\nabla_{1}^{M}+Z_{2}^{N}Z_{1}^{M})\left(\frac{\xi^{2}\;{\rm
tr~{}}\\!q_{2}^{2}}{2}-4\right)-(Z_{1}^{N}\nabla_{2}^{M}+2Z_{1}^{N}Z_{2}^{M})\left(\frac{\xi^{2}\;{\rm
tr~{}}\\!q_{1}^{2}}{2}-3\right)\right]\;.$

At first sight it looks quite complicated, however, one can see easily that on
the states with a magnon on site $1$ and no magnon on $2$ the r.h.s. become
zero and the operator ${\cal R}$ become the interchange operator which
exchanges the roles between the magnons and no-magnons. The existence of such
operators guarantees that the spectrum of all integrals of motion does not
depend on the initial order of the magnons. This ensures the self-consistency
of our approach.

More precisely we can see that the magnon move operation results in the
following similarity transformation

$\hat{\mathbb{T}}^{\bf I}(u)\,|\Psi\rangle^{\bf I}=({\cal
R}_{k,k+1})^{-1}\,\hat{\mathbb{T}}^{\overset{\leftrightarrow}{\bf
I}}(u)\,{\cal R}_{k,k+1}\,|\Psi\rangle^{\bf I}=({\cal
R}_{k,k+1})^{-1}\,\hat{\mathbb{T}}^{\overset{\leftrightarrow}{\bf
I}}(u)\,|\Psi\rangle^{\overset{\leftrightarrow}{\bf I}}$ (6.12)

where the set $\bf I$ is such that $I_{k}=1$ and $I_{k+1}=0$, whereas the
transformed set $\overset{\leftrightarrow}{\bf I}$ corresponds to the magnon
at $k+1$ i.e. $I_{k}=0$ and $I_{k+1}=1$. The equation (6.12) ensures that the
spectrum of all conserved charges before and after the magnon move operation
stays unchanged. Similarly, we can shift an anti-magnon to the left and all
conserved charges are unaffected. Since states are uniquely characterize by
the charges, this is a gauge symmetry that is associated with a redundancy in
our description.

##### Magnon–anti-magnon annihilation.

First, let us demonstrate that the spectrum does not change when we
simultaneously remove one magnon and one anti-magnon. We will establish a
simple relation between the solutions of the Baxter equation with different
numbers of magnons. Consider the transformation $M\to
M-n,\;\bar{M}\to\bar{M}-n$. Under this transformation (6.1) becomes

$\displaystyle-\frac{P_{2J}^{\bf 6}(v)}{v^{J-M-\bar{M}+2n}}\;\tilde{q}_{a}$
$\displaystyle=$
$\displaystyle\frac{v^{M}(v+i)^{J}(v+2i)^{\bar{M}}}{v^{n}(v+2i)^{n}}\;\tilde{q}_{a}^{[+4]}-\frac{v^{M}(v+i)^{\bar{M}}}{v^{n}(v+i)^{n}}P_{J}^{\bf
4}(v+\tfrac{i}{2})\;\tilde{q}_{a}^{[+2]}$ $\displaystyle+$
$\displaystyle\frac{v^{\bar{M}}(v-i)^{J}(v-2i)^{M}}{v^{n}(v-2i)^{n}}\;\tilde{q}_{a}^{[-4]}-\frac{v^{\bar{M}}(v-i)^{M}P_{J}^{\bar{\bf
4}}(v-\tfrac{i}{2})}{v^{n}(v-i)^{n}}\;\tilde{q}_{a}^{[-2]}\;.$

First we see that the overall factor $1/v^{n}$ cancels. Furthermore, by
denoting $q(v)=\frac{\tilde{q}(v)}{v^{n}}$ we return back to the initial form
(6.1).

At the level of the fishchain wave function, we can always shift the positions
of the magnon and anti-magnon until the anti-magnon is at the first site and
the magnon is at the second site. To Annihilate them, we act with

${\mathbb{H}}_{2,2}(Z_{1},Z_{2})\propto\tilde{\cal
R}_{12}=\nabla_{1}.\nabla_{2}+2Z_{2}.\nabla_{1}+2Z_{1}.\nabla_{2}+4Z_{1}.Z_{2}\;.$
(6.14)

This operator has the following property

$\displaystyle{\mathbb{L}}^{ab}_{1}(u){\mathbb{L}}^{bc}_{2}\left(u\right)\tilde{\cal
R}_{12}-\tilde{\cal
R}_{12}{\mathbb{L}}^{ab}_{1}(u-\tfrac{i}{2\xi}){\mathbb{L}}^{bc}_{2}(u+\tfrac{i}{2\xi})$
(6.15) $\displaystyle=\frac{i\Sigma^{ac}_{NM}}{2\xi^{2}}\left[(Z_{1}^{N}{\cal
D}_{2}^{M}+2Z_{1}^{N}Z_{2}^{M})\left(\frac{\xi^{2}\;{\rm
tr~{}}\\!q_{1}^{2}}{2}-4\right)-(Z_{2}^{N}{\cal
D}_{1}^{M}+2Z_{2}^{N}Z_{1}^{M})\left(\frac{\xi^{2}\;{\rm
tr~{}}\\!q_{2}^{2}}{2}-4\right)\right]\;.$

Similar to (6.11) the r.h.s. of the above equation vanishes on the states with
magnon and anti-magnon. This relation leads to an analog of (6.12) where the
initial set ${\bf I}$ has $I_{k+1}=1$ and $I_{k}=-1$, whereas the resulting
set ${\overset{\leftrightarrow}{\bf I}}$ has no magnons at these positions
i.e.
${\overset{\leftrightarrow}{I}}_{k}={\overset{\leftrightarrow}{I}}_{k+1}=0$.
We see again that the magnon annihilation operation does not affect the
spectrum or any of the the conserved charges.

###  6.3 Quantization condition and exact spectrum

In order to adapt the quantization condition of Gromov:2019cja  to the general
case with any number of magnons, we introduce two sets of solutions of the TQ-
relations (6.1). One set, denoted by $q_{a}^{\downarrow}(v)$, is regular in
the upper half-plane and satisfy (6.8) at large $v$. The other set,
$q_{a}^{\uparrow}(v)$, is regular in the lower half-plane. It is easy to see
that such solutions exist and that they can be constructed numerically, using
the methods of Gromov:2015wca .

Since there are only $4$ independent solutions of the $4$th order finite
difference equation (6.1), these two sets $q_{a}^{\downarrow}(v)$ and
$q_{a}^{\uparrow}(v)$ should be linearly dependent with i-periodic
coefficients, i.e.

$q_{a}^{\uparrow}(v)={\Omega_{a}}^{b}(v)\;q_{b}^{\downarrow}(v)\
,\qquad{\Omega_{a}}^{b}(v+i)={\Omega_{a}}^{b}(v)\;.$ (6.16)

The matrix ${\Omega_{a}}^{b}(v)$ itself can be constructed from q-functions as
the simple ratio of two determinants

${\Omega_{a}}^{b}(v)=\frac{\epsilon^{b\;b_{1}b_{2}b_{3}}}{3!}\frac{\det\limits_{n=0,\dots,3}\\{q_{a}^{\uparrow}(v-in),q_{b_{1}}^{\downarrow}(v-in),q_{b_{2}}^{\downarrow}(v-in),q_{b_{3}}^{\downarrow}(v-in)\\}}{\det\limits_{n=0,\dots,3}\\{q_{1}^{\downarrow}(v-in),q_{2}^{\downarrow}(v-in),q_{3}^{\downarrow}(v-in),q_{4}^{\downarrow}(v-in)\\}}\;,$
(6.17)

as follows from (6.16).

In Gromov:2019cja  it is shown that this matrix ${\Omega_{a}}^{b}(v)$ should
satisfy the following quantization condition131313We are grateful to the
authors of Gromov:2019cja  for sharing this unpublished result with us

${\Omega_{1}}^{2}={\Omega_{2}}^{1}={\Omega_{3}}^{4}={\Omega_{4}}^{3}=0\;,$
(6.18)

which singles out physical solutions of the TQ-relations. Furthermore, one can
show Gromov:2019cja  that the $i$-periodic function ${\Omega_{a}}^{b}(u)$ has
simple poles at $u=in$ of order $J$. This implies that each matrix element of
${\Omega_{a}}^{b}(u)$ can be parametrized by $J+1$ constants $C^{(n)}$ as

${\Omega_{a}}^{b}(u)=\frac{\sum\limits_{n=0}^{J}{C^{(n)\;\;b}_{\;\;\;\;\;a}}\;e^{2\pi
nu}}{(1-e^{2\pi u})^{J}}\;,$ (6.19)

where in addition for off-diagonal elements only $J$ constants $C^{(n)}$ are
non-zero. This implies that we have $4J$ equations in (6.18). One can show
that only $4J-3$ of these conditions are linearly independent (for large
enough $J$), which is exactly how many free parameters we have in the
polynomial coefficients $P_{L}$ in (6.1). Thus, using the quantization
condition (6.18) we should be able to fix all coefficients in the Baxter TQ-
relations, which also includes $\Delta$.

In the next section we report on the numerical tests of this procedure for the
case with magnons.

###  6.4 Numerical tests

In this section we apply the procedure described in section 6.3 for two cases,
$J=3,\;M=1$ and $J=3,\;M=2$. In order to solve the Baxter TQ-relations (6.1)
we use the method of Gromov:2015wca ; Gromov:2017cja . Then we use the
generalized gluing conditions (6.18) to fix the parameters in (6.1). In both
cases we found perfect agreement with the Asymptotic Bethe Ansatz (ABA) of
Caetano:2016ydc ; Ipsen:2018fmu ; Beisert:2006ez  at weak coupling. In this
section we fix the twist parameters as
$\lambda_{1}=e^{i/3},\;\lambda_{2}=e^{-i/3},\;\lambda_{3}=e^{i/2},\;\lambda_{4}=e^{-i/2}$.

####  6.4.1 Length three, one magnon

Figure 10: Numerical solution (dots) and the Asymptotic Bethe Ansatz
prediction (solid line) match perfectly at small coupling for the state with
one magnon and $J=3$. The two start to deviate from each other at larger
values of $\xi^{2}$.

We generated to a very high precision numerical values for a wide range of
couplings using the method described in 6.3 (see figure 10). In order to check
our results we have to recall that in the ABA approach for one magnon one only
needs to know the dispersion relation $\Delta_{ABA}=J+\sqrt{1-4\xi^{2}}$,
which is expected to be accurate up to the wrapping order
$\xi^{2J+2M}=\xi^{8}$.

Indeed, by making a fit of our numerical values with even powers of $\xi$ we
found the following expansion

$\displaystyle\Delta_{TQ}=4.-2.\xi^{2}-2.\xi^{4}-4.\xi^{6}+0.155845\xi^{8}+\dots\;.$
(6.20)

We see that the first $4$ coefficients are integers, and agree precisely with
the magnon dispersion relation. At order $\xi^{8}$ we get some real number,
which is expected to be a combination of two polylogarithms ${\rm Li}_{3}$ and
${\rm Li}_{5}$ with the arguments depending on the twists.

Figure 11: Single magnon operator ${\rm tr~{}}\phi_{1}^{3}\phi_{2}$ has a
dimension $\Delta$, which is a multivalued function of $\xi$. At weak coupling
the bare dimension of the state is $J_{1}+J_{2}=4$. Analytically continuing in
$\xi$ one can discover an infinite tower of states with the bare dimensions
$4+2n,\;n=0,1,\dots$ ($n=0,1$ are indicated by the red dots on the picture).

What is very interesting is the analytic structure of our result in $\xi$. At
small enough $\xi$ the function has a regular (convergent) expansion in powers
of $\xi^{2}$. At $\xi\sim 0.7$ the dimension $\Delta$ tends to $0$ and after
that point become purely imaginary (see figure 10). The value $\xi\sim 0.7$ is
thus a branch point. This is not the only branch cut of the function
$\Delta(\xi)$. The function $\Delta(\xi)$ is in fact a multivalued function
with infinitely many sheets (see figure 11), containing not just one but
infinitely many local operators, which can be identified at weak coupling as
${\rm tr~{}}{\phi_{1}^{3}\phi_{2}(\phi_{2}\phi_{2}^{\dagger})^{n}}$ (up to
permutations).

####  6.4.2 Length three, two magnons

We also found a numerical solution in the two magnon case for the chain of
length $J=3$. Again our results are in complete agreement with the ABA
prediction at weak coupling. Note that this time, the ABA calculation involves
a rather non-trivial dressing phase, which gives rise to the $\zeta$ values in
the result

$\displaystyle\Delta_{ABA}$ $\displaystyle=$ $\displaystyle
5+\left(\sqrt{5}-1\right)\xi^{2}+\left(\frac{3}{\sqrt{5}}-1\right)\xi^{4}-\left(2+\frac{2}{5\sqrt{5}}\right)\xi^{6}$
$\displaystyle+$
$\displaystyle\left(-\frac{4\zeta_{3}}{\sqrt{5}}-4\zeta_{3}+\frac{81}{25\sqrt{5}}+3\right)\xi^{8}+\left(\frac{112\zeta_{3}}{5\sqrt{5}}-\frac{424}{25\sqrt{5}}+8\right)\xi^{10}+{\cal
O}\left(\xi^{12}\right)\;.$

Our numerical fit of the result of solution of the TQ-relations gives

$\displaystyle\Delta_{TQ}=5+1.23607\xi^{2}+0.341641\xi^{4}-2.17889\xi^{6}-2.50956\xi^{8}-0.575805\xi^{10}+\dots$
(6.22)

which agrees perfectly with the first $4$-loops predicted by ABA (6.4.2). At
the wrapping order $\xi^{2J+2M}=\xi^{10}$, as expected, the ABA result is no
longer valid and deviates from our numerical prediction (6.22).

In conclusion, the fact that we managed to reproduce the ABA prediction from
the TQ-relations provides a very non-trivial test of our approach.

##  7 Conclusion

In this paper, we completed the derivation of the Holographic dual of the
fishnet theory. Introducing the magnons, i.e. $\phi_{2}$ and/or
$\phi_{2}^{\dagger}$ fields, into the consideration brought some surprises.
Firstly, we found that the classical limit $\xi\to\infty$ remain unchanged.
Namely, we still find the same classically integrable fishchain model, which
is a classical chain of particles on a light-cone with nearest-neighbor
interaction. The essence of the magnons is hiding in the details of the
quantization procedure, which also modifies the light-cone into the $AdS_{5}$
space.141414$AdS_{5}$ space first appeared in the regime of large R-charge of
the fishnet model in Basso:2018agi . This is, in fact, in complete agreement
with the exact spectrum and 4-point function computation in Gromov:2018hut
for the length-two operators. Secondly, the introduction of the magnons
revealed a new nice feature, namely the discrete reparametrization symmetry –
an invariance of the physics w.r.t. different sectioning of the discretized
world-sheet of the fishchain into space-like slices. The magnons in this
picture indicate bends of the curve cutting the world-sheet. Mathematically
this symmetry manifests as a map between Hilbert spaces, preserving the
integrability and conserved charges that uniquely characterize the state.

Another important output of our analysis is the universal integrability
picture, which includes all possible insertions on equal footing. We found the
quantum Baxter TQ-relations, which together with the quantization condition
Gromov:2017cja ; Gromov:2019cja , give us direct access to the quantum
spectrum of the model in complete generality.151515This spectrum was
previously only available for a rather narrow subset of states Gromov:2017cja
. This also opens ways to the separation of variables (SoV) construction,
along the lines of the recent works Gromov:2016itr ; Ryan:2018fyo ;
Derkachov:2018rot ; Cavaglia:2019pow .

Among open questions remains a particular configuration when the number of
magnons saturates the number of states, i.e. when both $\mathfrak{u}(1)$
charges are equal $|J_{1}|=|J_{2}|$. In this case, surprisingly, the general
strategy outlined in this paper fails. On some particular examples it is known
that this case produces a more complicated result for the spectrum even for
short operators, tractable with the conformal symmetry alone Gromov:2018hut .
Nevertheless, the integrability should be present in this case too
Gromov:2019cja , but the strong coupling behavior is very different,
indicating that it is mysteriously missing in the current picture. This
problem could be related to another large part of states which happened to be
protected, and decoupled from the dynamical part of the spectrum studied in
this paper. The reason why there is a large number of states with protected
dimension is also unclear, there is no manifest symmetry, like supersymmetry,
which can help to explain this phenomenon.

## Acknowledgements

We thank O. Aharony, D. Grabner, V. Kazakov, G. Korchemsky and A. Zhiboedov
for invaluable discussions. We thank J. McGovern for comments on the
manuscript. N.G. was supported by the STFC grant (ST/P000258/1). A.S. was
supported by the I-CORE Program of the Planning and Budgeting Committee, The
Israel Science Foundation (1937/12).

## References

  * (1) Ö. Gürdoğan and V. Kazakov, “New Integrable 4D Quantum Field Theories from Strongly Deformed Planar $\mathcal{N}=$ 4 Supersymmetric Yang-Mills Theory,” Phys. Rev. Lett. 117, no. 20, 201602 (2016) Addendum: [Phys. Rev. Lett. 117, no. 25, 259903 (2016)] doi:10.1103/PhysRevLett.117.201602, 10.1103/PhysRevLett.117.259903 [arXiv:1512.06704 [hep-th]].
  * (2) H. B. Nielsen and P. Olesen, “A Parton view on dual amplitudes,” Phys. Lett. 32B, 203 (1970). doi:10.1016/0370-2693(70)90474-0
  * (3) A. B. Zamolodchikov, “’fishnet’ Diagrams As A Completely Integrable System,” Phys. Lett. 97B, 63 (1980). doi:10.1016/0370-2693(80)90547-X
  * (4) N. Gromov and A. Sever, “The Holographic Fishchain,” Phys. Rev. Lett. 123 (2019) 081602 doi:10.1103/PhysRevLett.123.081602 [arXiv:1903.10508 [hep-th]].
  * (5) N. Gromov and A. Sever, “Quantum Fishchain in $AdS_{5}$,” arXiv:1907.01001 [hep-th].
  * (6) D. Chicherin, V. Kazakov, F. Loebbert, D. Muller and D. l. Zhong, “Yangian Symmetry for Fishnet Feynman Graphs,” Phys. Rev. D 96, no. 12, 121901 (2017) doi:10.1103/PhysRevD.96.121901 [arXiv:1708.00007 [hep-th]].
  * (7) D. Grabner, N. Gromov, V. Kazakov, G. Korchemsky, in preparation.
  * (8) N. Gromov, V. Kazakov, G. Korchemsky, S. Negro and G. Sizov, “Integrability of Conformal Fishnet Theory,” arXiv:1706.04167 [hep-th].
  * (9) J. Caetano, Ö. Gürdoğan and V. Kazakov, “Chiral limit of $\mathcal{N}$ = 4 SYM and ABJM and integrable Feynman graphs,” JHEP 1803, 077 (2018) doi:10.1007/JHEP03(2018)077 [arXiv:1612.05895 [hep-th]].
  * (10) A. C. Ipsen, M. Staudacher and L. Zippelius, “The one-loop spectral problem of strongly twisted $\mathcal{N}$ = 4 Super Yang-Mills theory,” JHEP 1904, 044 (2019) doi:10.1007/JHEP04(2019)044 [arXiv:1812.08794 [hep-th]].
  * (11) N. Beisert, B. Eden and M. Staudacher, “Transcendentality and Crossing,” J. Stat. Mech. 0701, P01021 (2007) doi:10.1088/1742-5468/2007/01/P01021 [hep-th/0610251].
  * (12) J. Fokken, C. Sieg and M. Wilhelm, “Non-conformality of ${{\gamma}_{i}}$-deformed N = 4 SYM theory,” J. Phys. A 47 (2014) 455401 doi:10.1088/1751-8113/47/45/455401 [arXiv:1308.4420 [hep-th]].
  * (13) C. Sieg and M. Wilhelm, “On a CFT limit of planar $\gamma_{i}$-deformed $\mathcal{N}=4$ SYM theory,” Phys. Lett. B 756, 118 (2016) doi:10.1016/j.physletb.2016.03.004 [arXiv:1602.05817 [hep-th]].
  * (14) N. Gromov, V. Kazakov and G. Korchemsky, “Exact Correlation Functions in Conformal Fishnet Theory,” arXiv:1808.02688 [hep-th].
  * (15) P. A. M. Dirac, “Wave equations in conformal space,” Annals Math. 37, 429 (1936). doi:10.2307/1968455
  * (16) D. Simmons-Duffin, “Projectors, Shadows, and Conformal Blocks,” JHEP 1404, 146 (2014) doi:10.1007/JHEP04(2014)146 [arXiv:1204.3894 [hep-th]].
  * (17) A. Cavaglià, D. Grabner, N. Gromov and A. Sever, To appear.
  * (18) N. Gromov, V. Kazakov, S. Leurent and D. Volin, “Quantum Spectral Curve for Planar $\mathcal{N}=4$ Super-Yang-Mills Theory,” Phys. Rev. Lett. 112 (2014) no.1, 011602 doi:10.1103/PhysRevLett.112.011602 [arXiv:1305.1939 [hep-th]].
  * (19) A. Cavaglià, D. Fioravanti, N. Gromov and R. Tateo, “Quantum Spectral Curve of the $\mathcal{N}=$ 6 Supersymmetric Chern-Simons Theory,” Phys. Rev. Lett. 113 (2014) no.2, 021601 doi:10.1103/PhysRevLett.113.021601 [arXiv:1403.1859 [hep-th]].
  * (20) R. J. Baxter, “Eight vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain. 1. Some fundamental eigenvectors,” Annals Phys. 76, 1 (1973). doi:10.1016/0003-4916(73)90439-9
  * (21) V. V. Bazhanov, S. L. Lukyanov and A. B. Zamolodchikov, “Integrable structure of conformal field theory. 2. Q operator and DDV equation,” Commun. Math. Phys. 190 (1997) 247 doi:10.1007/s002200050240 [hep-th/9604044].
  * (22) N. Gromov, F. Levkovich-Maslyuk and G. Sizov, “Quantum Spectral Curve and the Numerical Solution of the Spectral Problem in AdS5/CFT4,” JHEP 1606 (2016) 036 doi:10.1007/JHEP06(2016)036 [arXiv:1504.06640 [hep-th]].
  * (23) B. Basso and D. l. Zhong, “Continuum limit of fishnet graphs and AdS sigma model,” JHEP 1901, 002 (2019) doi:10.1007/JHEP01(2019)002 [arXiv:1806.04105 [hep-th]].
  * (24) N. Gromov, F. Levkovich-Maslyuk and G. Sizov, “New Construction of Eigenstates and Separation of Variables for SU(N) Quantum Spin Chains,” JHEP 1709 (2017) 111 doi:10.1007/JHEP09(2017)111 [arXiv:1610.08032 [hep-th]].
  * (25) P. Ryan and D. Volin, “Separated variables and wave functions for rational gl(N) spin chains in the companion twist frame,” J. Math. Phys. 60 (2019) no.3, 032701 doi:10.1063/1.5085387 [arXiv:1810.10996 [math-ph]].
  * (26) S. Derkachov, V. Kazakov and E. Olivucci, “Basso-Dixon Correlators in Two-Dimensional Fishnet CFT,” JHEP 1904 (2019) 032 doi:10.1007/JHEP04(2019)032 [arXiv:1811.10623 [hep-th]].
  * (27) A. Cavaglià, N. Gromov and F. Levkovich-Maslyuk, “Separation of variables and scalar products at any rank,” arXiv:1907.03788 [hep-th].