We found that a sufficient _quantisation condition_ is the following
constraint on this matrix:

${\Omega}_{1}^{2}(u)={\Omega}_{2}^{1}(u)={\Omega}_{3}^{4}(u)={\Omega}_{4}^{3}(u)=0\
.$ (6.19)

This condition was also obtained independently in UpcomingFishnet , and can be
justified in various ways. In particular, one can argue that the same
condition is valid for the Quantum Spectral Curve of $\mathcal{N}=4$ SYM, see
Appendix F, and therefore it should be inherited by the fishnet model. For the
states with one scalar, we also found, together with F. Levkovich-Maslyuk, a
direct proof383838This proof is similar to one found for the cusp operators in
the ladders limit in Cavaglia:2018lxi . that (6.19) is equivalent to the
quantisation conditions for the Schrödinger equation (4.13), see section 6.5.
The same quantisation condition was also used to study different operators
with length three in Gromov:2019jfh . It is expected to be valid for arbitrary
operators.

####  6.2.2 Introducing the coupling constant

For states with $J_{1}>J_{2}$ it can be shown (see, Gromov:2019jfh ;
UpcomingFishnet ) that the coupling constant appears as a coefficient of a
pole in the Baxter equation:

$\xi^{2J_{1}}=\lim_{u\rightarrow 0}u^{J_{1}-J_{2}}a_{2}(u).$ (6.20)

The case with $|J_{1}|=|J_{2}|$ is more complicated. The method to take into
account the coupling constant for generic states was found in UpcomingFishnet
and we borrow this unpublished result here.393939We thank the authors of
UpcomingFishnet  for sharing this result with us. We construct a function
$Q_{+}(u)$, defined as

$\begin{split}Q_{+}(u)&=A\left(q_{1}^{\downarrow}\left(u-\frac{i}{2}\right)q^{\uparrow}_{2}\left(u+\frac{i}{2}\right)-q_{2}^{\downarrow}\left(u-\frac{i}{2}\right)q^{\uparrow}_{1}\left(u+\frac{i}{2}\right)\right)\\\
&+q_{3}^{\downarrow}\left(u-\frac{i}{2}\right)q^{\uparrow}_{4}\left(u+\frac{i}{2}\right)-q_{4}^{\downarrow}\left(u-\frac{i}{2}\right)q^{\uparrow}_{3}\left(u+\frac{i}{2}\right)\
.\end{split}$ (6.21)

The coefficient $A$ should be chosen in such a way that this combination of
Q-functions does not have a pole at $u\sim\pm i/2$. It can be shown that this
can be done only when the quantisation condition (6.19) is satisfied, see
Appendix F. Once this condition is enforced, we can relate the function
$Q_{+}(u)$ to the coupling constant. For the states studied in this paper this
relation is

$\xi^{2}=\lim_{\epsilon\to
0}\epsilon\frac{Q_{+}\left(\frac{3i}{2}-i\epsilon\right)}{Q_{+}\left(\frac{i}{2}\right)}\
.$ (6.22)

####  6.2.3 Baxter equations at length one

For the families of length-one operators described in this paper, the form of
the Baxter equation is almost entirely fixed by the structure described above.

##### Baxter equation for $J_{1}=1$, $J_{2}=0$.

Requiring compatibility of the asymptotics (6.15) with the polynomial-type
ansatz for the coefficients $a_{n}(u)$ fixes

$\displaystyle a_{0}(u)$ $\displaystyle=$ $\displaystyle a_{4}(u)=u\ ,\quad
a_{1}(u)=a_{+}(u)\ ,\,\,a_{3}(u)=a_{-}(u)\ ,$ $\displaystyle a_{2}(u)$
$\displaystyle=$
$\displaystyle\frac{\kappa}{u}+2u\,\left(1+\cos\theta_{1}+\cos\theta_{2}\right)+2\,S_{1}\sin\theta_{1}+2\,S_{2}\sin\theta_{2}\
,$ (6.23) $\displaystyle a_{\pm}(u)$ $\displaystyle=$
$\displaystyle-2\left(\cos\tfrac{\theta_{1}}{2},\sin\tfrac{\theta_{1}}{2}\right).\left(\begin{array}[]{cc}2u&S_{2}\\\
S_{1}&\mp
i\Delta\end{array}\right).\left(\cos\tfrac{\theta_{2}}{2},\sin\tfrac{\theta_{2}}{2}\right)^{\intercal}\
,$ (6.26)

for the family of operators with $J_{1}=1$, $J_{2}=0$. Notice that only one
coefficient, $\kappa$, is left unfixed. Using (6.20), it can be directly
related to the coupling constant, $\kappa=\xi^{2}$. Therefore, the Baxter
equation contains both $\Delta$ and $\xi^{2}$ as parameters. Their mutual
dependence for physical states is fixed by the quantisation condition (6.19).

##### Baxter equation for $J_{1}=1$, $J_{2}=1$

With similar considerations one can see that, for the states built with two
scalars and charges $J_{1}=1$, $J_{2}=1$, the Baxter equation takes the form

$\displaystyle a_{0}(u)$ $\displaystyle=$ $\displaystyle a_{4}(u)=u(u-i),\quad
a_{1}(u)=b_{+}(u)(u-i/2),\,\,a_{3}(u)=b_{-}(u)(u-i/2),$ $\displaystyle
a_{2}(u)$ $\displaystyle=$
$\displaystyle\rho+2\left(u^{2}-iu\right)(1+\cos\theta_{1}+\cos\theta_{2})+(2\,u-i)\,\left(S_{1}\sin\theta_{1}+S_{2}\sin\theta_{2}\right)\
,$ (6.27) $\displaystyle b_{\pm}(u)$ $\displaystyle=$
$\displaystyle-{2}\left(\cos\tfrac{\theta_{1}}{2},\sin\tfrac{\theta_{1}}{2}\right).\left(\begin{array}[]{cc}2u-i&S_{2}\\\
S_{1}&\mp
i\Delta\end{array}\right).\left(\cos\tfrac{\theta_{2}}{2},\sin\tfrac{\theta_{2}}{2}\right)^{\intercal}\
.$ (6.30)

Notice that also in this case there is precisely one unfixed coefficient,
denoted as $\rho$. In this case the relation between this coefficient and the
coupling constant is much less straightforward. It is, however, determined
implicitly by equation (6.22). The physical states are again selected by the
quantisation condition (6.19).

###  6.3 Exact solutions for $\theta_{1}=\theta_{2}$

Before discussing numerical results at generic values of the twists, let us
consider the case $\theta_{1}=\theta_{2}\equiv\theta$ which is solvable
analytically. In this case, there are no unfixed parameters in the Baxter
equation. The quantisation condition (6.19) is automatically satisfied and the
spectrum can be found explicitly. For $\theta_{1}=\theta_{2}$, we have an
enhanced symmetry $U(1)_{L}\times SU(2)_{R}$. In terms of the related spins,
$S_{R}=\frac{S_{1}-S_{2}}{2}$, $S_{L}=\frac{S_{1}+S_{2}}{2}$, the asymptotics
of the Q-functions are

$q_{i}\sim\left(e^{\theta u}u^{\frac{\Delta-2S_{L}-D_{0}}{2}},e^{-\theta
u}\,u^{\frac{\Delta+2S_{L}-D_{0}}{2}},u^{\frac{-\Delta-2S_{R}-D_{0}}{2}},u^{\frac{-\Delta+2S_{R}-D_{0}+2}{2}}\right)\
.$ (6.31)

Notice that, as compared to (6.15), the asymptotics of the last Q-function is
modified by a power of $u$. This is a typical effect of removing one twist
Kazakov:2015efa .

##### States with $J_{1}=1$, $J_{2}=0$.

The compatibility of the asymptotics (6.31) with the Baxter equation (6.2.3)
at $\theta_{1}=\theta_{2}$ fixes

$\kappa=-\sin^{2}\frac{\theta}{2}\,(\Delta^{2}-(1+2S_{R})^{2})\ .$ (6.32)

This new constraint is a special feature of the equal angles limit. Together
with the condition $\kappa=\xi^{2}$, it gives precisely the exact spectrum
(4.29) determined earlier from Feynman diagrams.

##### States with $J_{1}=1$, $J_{2}=1$.

In this case the asymptotics (6.31) for equal angles imposes the condition

$\rho=\frac{-4S_{R}^{2}(t-1)^{2}-4S_{R}(t-1)^{2}+\Delta^{2}(t-1)^{2}-4(t^{2}+1)}{4t},\,\,\,\,t=e^{i\theta}\
.$ (6.33)

To compute the spectrum, we must map this parameter to the coupling via
(6.22), and to do this we need to solve explicitly the Baxter equations. The
solution can be found after transforming the finite-difference equation to a
differential equation through a Mellin transform. Imposing analyticity in the
upper half plane and the asymptotics (6.31), we find the solutions404040We
have written the solutions with a different normalisation for $q_{3}$, as
compared to (6.15).

$\displaystyle q_{1}^{\downarrow}(u)$ $\displaystyle=$
$\displaystyle\mathcal{F}(S_{R},S_{L},\Delta,t,u),\,\,\,\,\,q_{2}^{\downarrow}(u)=\mathcal{F}(S_{R},-S_{L},\Delta,1/t,u),$
(6.34) $\displaystyle q_{4}^{\downarrow}(u)$ $\displaystyle=$
$\displaystyle\mathcal{G}(S_{R},S_{L},\Delta,t,u),\,\,\,\,\,q_{3}^{\downarrow}(u)\propto\mathcal{G}(S_{R},-S_{L},\Delta,1/t,u)-\mathcal{G}(S_{R},S_{L},\Delta,t,u)\
,$

with

$\displaystyle\frac{\mathcal{F}(S_{R},S_{L},\Delta,t,u)}{\Gamma(-iu)t^{-iu-1/2}\,(it-i)^{-\frac{\Delta}{2}+S_{L}+1}}$
(6.35)
$\displaystyle\equiv\sum_{l=0}^{S_{R}-S_{L}}(-1)^{l}\frac{\Gamma(S_{R}-S_{L}+1)\Gamma(S_{R}+S_{L}+1-l)}{\Gamma(l+1)\Gamma(S_{R}+S_{L}+1)\Gamma(S_{R}-S_{L}-l+1)}\,(1/t+1)^{l}\,$
$\displaystyle\hskip
85.35826pt\times\;{}_{2}\tilde{F}_{1}\left(-iu,S_{L}-\frac{\Delta}{2}+1-l,-iu+S_{L}-\frac{\Delta}{2}+1-l,t\right)\
,$

and

$\displaystyle\frac{\mathcal{G}(S_{R},S_{L},\Delta,t,u)}{\Gamma(-iu)\,(it-i)^{\frac{\Delta}{2}-S_{R}}}=$
$\displaystyle=$
$\displaystyle\sum_{l=0}^{S_{R}+S_{L}}\frac{\Gamma(S_{R}+S_{L}+1)\Gamma(2S_{R}-l+1)\Gamma^{2}(S_{R}-\frac{\Delta}{2}+1)}{\Gamma(l+1)\Gamma(2S_{R}+1)\Gamma(S_{R}+S_{L}-l+1)\Gamma^{2}(S_{R}-\frac{\Delta}{2}-l+1)}\,(t+1)^{l}\,$
$\displaystyle\hskip
85.35826pt\times\;{}_{2}\tilde{F}_{1}\left(-iu,-S_{R}+\frac{\Delta}{2}+l,-iu-
S_{R}+\frac{\Delta}{2}+l,t\right)\ ,$

where ${}_{2}\tilde{F}_{1}(a,b,c,z)\equiv{}_{2}F_{1}(a,b,c,z)/\Gamma(c)$. One
can verify explicitly that the quantisation condition (6.19) is automatically
satisfied. All we need to do is compute $Q_{+}(u)$ and use equation (6.22) to
obtain the coupling dependence of the spectrum.

For instance, in the case $S_{R}=S_{L}=0$, we find that the constant $A$
ensuring the absence of poles in $Q_{+}(u)$ at $u\sim i/2$ is

$A=\frac{\Gamma^{2}(1-\frac{\Delta}{2})}{\Gamma^{2}(\frac{\Delta}{2})}\,\left(4\sin^{2}\frac{\theta}{2}\right)^{\Delta-1}\
,$ (6.37)

and we find, with the definition (6.22) and using
${}_{2}F_{1}(1,n,n+1,z)=n\Phi(z,1,n)$,

$\displaystyle\lim_{u\rightarrow
0}u\,Q_{+}(u+3/2i)\equiv-2\Delta\sin\theta\,\sin\tfrac{\theta}{2}\,A^{\frac{1}{2}},$
(6.38) $\displaystyle
Q_{+}(i/2)=-2\sin\frac{\theta}{2}\,A^{\frac{1}{2}}\,\left(\Phi(e^{i\theta},1,-\tfrac{\Delta}{2})-\Phi(e^{-i\theta},1,-\tfrac{\Delta}{2})-\Phi(e^{i\theta},1,\tfrac{\Delta}{2})+\Phi(e^{-i\theta},1,\tfrac{\Delta}{2})\right)\
,$

from where we see that the result (5.23) earlier computed from Feynman
diagrams is exactly reproduced. We can similarly compute the result for any
values of the spins. We did not manage to find a general closed form
expression for generic spins, but we list several results in Appendix G.

Figure 19: The states with $\Delta(0)=2,4,6,8$ for $J=1,\;M=1$ with
$\theta_{1}=5/6$ and $\theta_{2}=1/6$, $S_{1}=S_{2}=0$. The number of states
and weak coupling match precisely with the predictions obtained in section 5.
Among the states with a given $\Delta(0)$, one level reaches a square-root
branch point where $\Delta=0$, and the scaling dimension splits into two
imaginary levels after this point, with a classical scaling at strong
coupling, $\Delta(\xi^{2})\sim\pm i\xi$, $\xi\rightarrow\infty$. The states
that do not reach a branch point appear to stay real for all values of the
coupling, and tend to constant values at strong coupling.

###  6.4 Numerics for generic angles

##### Method.

In this section, we briefly discuss how to find the spectrum and Q-functions
numerically, at a finite value of the coupling constant. The numerical method
is essentially the same as the one introduced in Gromov:2015wca  in the
context of $\mathcal{N}$=4 SYM. In this approach, one regards the coefficients
entering the polynomials $a_{n}(u)$ in the Baxter equation as variational
parameters. These coefficients are tuned using a generalisation of Newton’s
method in order to find a solution satisfying the quantisation condition
(6.19). This quantisation condition fixes all the coefficients in the Baxter
equation except for one, which is a continuous independent parameter of the
solution, and can be related to the coupling constant using the equations
(6.20) or (6.22).

To implement the method, we need a routine to compute the Q-functions
$q_{i}^{\downarrow}$ and $q_{i}^{\uparrow}$ for any given choice of
parameters. Consider for instance the solution analytic in the upper half
plane. It satisfies a large-$u$ expansion:

$q_{i}^{\downarrow}(u)\sim\mathbf{y}_{i}^{-iu}u^{\hat{M}_{i}}\left(1+\sum_{n=1}^{\infty}\frac{B_{i,n}}{u^{n}}\right)\
,\;\;\text{Im}(u)\rightarrow\infty,$ (6.39)

where the coefficients $B_{i,n}$ are completely determined by the coefficients
in the Baxter equation. With a suitable choice of truncation for the sum,
(6.39) gives an arbitrarily good approximation to the solution for large
enough values of $\text{Im}(u)$ in the upper half plane. We can then use the
Baxter equation to translate down the Q-function from this region to generic
points in a strip around the real axis.

From the Q-functions, one can then construct the matrix $\Omega_{i}^{j}(u)$ as
a ratio of determinants. It is given by414141This relation can be obtained
using the $i$-periodicity of $\Omega$ and the Baxter equation.

$\Omega_{a}^{b}(u)=u^{J_{1}}(u+i)^{J_{1}+J_{2}}(u+2i)^{J_{2}}\,\frac{\epsilon^{bb_{1}b_{2}b_{3}}\,\underset{n=0,1,2,3}{\text{det}}\left\\{q^{\uparrow}_{a}(u-in)q^{\downarrow}_{b_{1}}(u-in)q^{\downarrow}_{b_{2}}(u-in)q^{\downarrow}_{b_{3}}(u-in)\right\\}}{3!\,\prod_{i<j}(\mathbf{y}_{i}-\mathbf{y}_{j})}.$
(6.40)

To turn the quantisation condition (6.19) into a numerical condition, it is
convenient to expand (6.40) around $u=0$. For the operators studied in this
paper, every matrix element of $\Omega$ has at most a single pole at the
origin. We impose the vanishing of the residue of the pole for
$\Omega_{1}^{2}$, $\Omega_{2}^{1}$, $\Omega_{4}^{3}$, $\Omega_{3}^{4}$ (it is
sufficient to consider only one of these matrix elements for the convergence
of the algorithm). This procedure fixes the trajectory $\Delta(\kappa)$ for
$J_{2}=0$, or $\Delta(\rho)$ for $J_{2}=1$, where $\kappa$ and $\rho$ are the
parameters in the Baxter equations (6.2.3),(6.2.3). We can then compute the
associated value of the coupling constant using the on-shell condition
(6.22).424242Alternatively, one can fix the desired value of the coupling
constant and use the quantisation condition, together with (6.22), to fix all
parameters in the Baxter equation as functions of the coupling.

Figure 20: Some states with $\Delta(0)=4,6,8,10$ for $J=1,\;M=1$ with
$\theta_{1}=11/4$ and $\theta_{2}=9/4$, $S_{1}=S_{2}=0$. The number of states
and weak coupling expansions again match the predictions in section 5,
however, for this choice of angles the spectrum is qualitatively different. As
for the case of figure 19, exactly one level, for any given $\Delta(0)=2n$,
reaches a branch point with $\Delta_{c}=0$, and then splits into a pair of
imaginary levels which show classical behaviour at strong coupling. All other
states merge in pairs at branch points with $\Delta_{c}\neq 0$. After these
branch points, they split into two complex conjugate levels with a nonzero
real part, which appear to tend to constant values at strong coupling.

##### Results.

We computed the scaling dimension as a function of the coupling for several
states in the length-one sectors, see figures 11, 19, 20. In the $J_{2}=0$
case, these results agree with the spectrum of the Schrödinger problem which
was already discussed.

Some findings for the more complicated one-magnon case are shown in figures 19
and 20. Notice that, at weak coupling, our numerical results confirm very
clearly the pattern of degeneracies due to the mixing between different
combinations of derivatives, described in section 5.2. For this case where
$J_{1}=J_{2}=1$, at strong coupling, we see that most of the states do not
display a classical scaling of the type $|\Delta|\sim\xi$, but instead tend to
constant values. This non-classical behaviour is a generic feature of the
states with $|J_{1}|=|J_{2}|$, and is reminiscent of the behaviour of states
in the SYK model. Presently, these states elude a dual description in terms of
the fishchain model of Gromov:2019aku .

However, we find that, for any group of states with a given $\Delta(0)=2n$,
there is exactly one state which behaves classically at strong coupling. For
these special _classical states_ , the scaling dimension, as a function of the
coupling constant, has qualitatively the same behaviour as for the solutions
of the zero-magnon case: the dimension decreases monotonically with the
coupling, until it reaches a branch point where $\Delta=0$, and then splits
into a pair of purely imaginary levels. At strong coupling, they scale as
$\Delta\sim\pm i\xi$. It would be interesting to determine whether these
special states can be captured by the fishchain model.

All the other levels, instead, reach a constant plateau at strong coupling,
which may be either real or complex, depending on the values of the two
angles. We find that, depending on these parameters, the structure of the
spectrum is qualitatively different, compare figures 19 and 20. In particular,
while in figure 19 all the non-classical states have a real scaling dimension,
in the case studied in figure 20 we see that all of them fuse pairwise at
square-root branch points that occur for $\Delta\neq 0$. After these branch
points, the levels split into complex conjugate pairs, which approach complex
constant values at strong coupling. It might be interesting to study the
boundaries of these different phases, determined by the values of the angles.
We leave these investigations for future studies.

###  6.5 Mapping the Q-functions to the CFT wave functions

So far, we had two different descriptions of the same physics. One is in terms
of the CFT wave function (4.2). Its dynamics are governed by the Schrödinger
equation (4.9) with boundary conditions (4.13). The other is in terms of the
SoV wave function, namely the Q-functions. They are fixed by the Baxter
equation (6.12), (6.2.3) and quantisation condition (6.19). In this section we
close the circle by explicitly mapping one to the other. We will be short, and
only consider the simplest non-trivial case of $J_{1}=1$ and
$J_{2}=S_{1}=S_{2}=0$. In colortwist2  we will extend this map and study it in
greater detail. We thank Fedor Levkovich-Maslyuk for collaboration on this
part.

Our starting point is the Baxter equation (6.12), (6.2.3). After setting
$S_{1}=S_{2}=0$ this equation becomes

$\begin{split}0=&\left(2u(1+\cos\theta_{2}+\cos\theta_{2})+{\xi^{2}\over
u}\right)q(u)\\\
&-2\left((2u+i)\cos\tfrac{\theta_{1}}{2}\cos\tfrac{\theta_{2}}{2}-i\Delta\sin\tfrac{\theta_{1}}{2}\sin\tfrac{\theta_{2}}{2}\right)q(u+i)+(u+i)q(u+2i)\\\
&-2\left((2u-i)\cos\tfrac{\theta_{1}}{2}\cos\tfrac{\theta_{2}}{2}+i\Delta\sin\tfrac{\theta_{1}}{2}\sin\tfrac{\theta_{2}}{2}\right)q(u-i)+(u-i)q(u-2i)\
.\end{split}$ (6.41)

To map this equation to the Schrödinger equation (4.9), we first use a Mellin
transform similar to the one introduced in Cavaglia:2018lxi

$\Theta(\sigma)=\int\limits_{c-i\infty}^{c+i\infty}{du\over 2\pi
iu}q(u)\,w(\sigma)^{iu}\ ,\qquad
w(\sigma)={e^{-\frac{i}{2}\theta_{2}}\sin\frac{\theta_{1}}{2}-e^{2\sigma}\,e^{+\frac{i}{2}\theta_{1}}\sin\frac{\theta_{2}}{2}\over
e^{+i\frac{\theta_{1}}{2}}\sin\frac{\theta_{1}}{2}-e^{2\sigma}\,e^{-i\frac{\theta_{2}}{2}}\sin\frac{\theta_{2}}{2}}\
,$ (6.42)

where $c>0$. We then plug the result into the integral

$\psi(\sigma)=\int\limits^{V(\sigma)}_{V_{-}}{dV^{\prime}\over
V^{\prime}}{V\over V^{\prime}}\,\left({V-V^{\prime}\over
V^{\prime}}\right)^{{\Delta-3\over 2}}\Theta(\sigma(V^{\prime}))\ ,$ (6.43)

where

$V(\sigma)=\tanh\sigma-{\sin^{2}{\theta_{1}\over 2}+\sin^{2}{\theta_{2}\over
2}\over\sin^{2}{\theta_{1}\over 2}-\sin^{2}{\theta_{2}\over 2}}\ ,\qquad
V_{-}={2\sin^{2}{\theta_{1}\over 2}\over\sin^{2}{\theta_{1}\over
2}+\sin^{2}{\theta_{2}\over 2}}\ ,$ (6.44)

and ${\rm Re}\,\Delta>1$. It can be checked the $\psi(\sigma)$ in (6.43)
satisfies the Schrödinger equation (4.9), and obeys the boundary conditions
(4.13) exactly when the quantisation conditions for the Q-functions (6.19) are
satisfied.

Importantly, this relation and its generalisations allows one to explicitly
express the results of correlation functions that are computed using field
theory techniques in terms of the SoV variables colortwist2 .

##  7 Conclusions and Discussion

In this paper, we have introduced a novel type of operators, which can be
constructed in any theory with a ’t Hooft large-$N$ limit by twisting the
colour-trace: _colour-twist operators_. We have studied some simple examples
of these operators perturbatively, at finite coupling, and in the strong
coupling limit in the fishnet model. This was done in two complementary ways,
first by a direct field theory calculation and second, using the integrability
based Quantum Spectral Curve (QSC) technique. A perfect match was obtained for
the spectrum of scaling dimensions computed in the two approaches. Finally, we
explicitly mapped the wave function in the separation of variable basis, known
as the Q-function, to the CFT wave function.

This paper sets the ground for the computation of planar correlation functions
in terms of the same objects that are used for computing the quantum spectrum
of anomalous dimensions, namely in terms of the QSC Q-functions. Further
progress in this direction will be reported in colortwist2 . We end this paper
by enumerating some of the many future directions.

It would be interesting to study colour-twist operators in other integrable
quantum field theories and in ${\cal N}=4$ SYM theory in particular. We expect
that our field theory definition will give results that match with the twisted
Bethe ansatz of Beisert:2005if , and, more generally, with the QSC predictions
Kazakov:2015efa .

In this paper, we have only considered colour-twist operators in the leading
planar limit. It would be interesting to extend our definition beyond the
planar limit and study $1/N$ corrections to the correlation functions between
colour-twist operators. Specifically, the topology of planar Feynman diagrams
is that of a sphere with punctures. At higher orders in the $1/N$ expansion,
the Feynman diagrams have topologies with higher genus. To extend our
definition to these cases, one should allow the twist cuts to wind around the
cycles of the diagrams. For twist transformations of the type (3.1), such a
cutting prescription can be obtained starting from the non-planar Feynman
diagrams of the $\gamma$-deformed theory and cutting them into disks in a
similar way to the cylinder-cut of section 3.1.

As explained in section 2, an operator can only be twisted by a symmetry
transformation that leaves it invariant. The twist-symmetry transformations we
have considered in this paper are continuously connected to the identity.
Namely, they are of the form $R(\theta)=e^{i\theta\hat{J}}$. Correspondingly,
the operators we studied have zero charge under the symmetry generator
$\hat{J}$. Another way of making an operator invariant under $R(\theta)$ is to
have its charge quantised in units of $1/\theta$, but not necessarily equal to
zero. In particular, when the twist transformation is a spacetime rotation or
boost, the corresponding colour-twist operator would have non-integer spin.
Operators with non-integer spin exist in any CFT Kravchuk:2018htv . These so-
called light-ray operators give a continuous interpolation between different
operators that are on the same Regge trajectory. They appear as intermediate
states in correlation functions, but otherwise they annihilate the vacuum;
therefore, their correlators in the un-twisted theory are vanishing. It would
be interesting to see if colour-twist operators with non-integer spin are
somehow related to the light-ray operators in the planar limit. If so, the
twisting procedure may provide a direct way of studying them and their
correlation functions at large $N$.

Finally, our main motivation for studying the colour-twist comes from the
separation of variables (SoV) approach. The twist removes the degeneracy in
the spectrum, allowing for a one-to-one correspondence between the
integrability description (say in terms of the Q-functions) and the actual
states in the theory. In the case of integrable spin chains, the twisting
procedure is essential for the SoV to be well defined. Equally, in the fishnet
theory, or ${\cal N}=4$ SYM theory, we expect the twist to play an essential
role for a possible SoV construction for the correlation functions. An example
of SoV structures was recently found in Cavaglia:2018lxi ; McGovern:2019sdd ;
Giombi:2018qox ; Giombi:2018hsx  for correlation functions of cusp operators,
and following this paper for single trace operators colortwist2 .

## Acknowledgements

We thank Fedor Levkovich-Maslyuk for collaboration on section 6.5. We thank V.
Kazakov and A. Zhiboedov for invaluable discussions. A.C. was supported by the
STFC grant (ST/P000258/1) Fundamental Physics from the Planck Scale to the
LHC. D.G. was supported by the EPSRC Research Studentship (EP/N509498/1). 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).

##  Appendix A $SO(4)$ representations

In the main body of the paper, we compute the spectrum of operators in the
fishnet model that are twisted by rotation. For that aim, it is useful to
explicitly realise the symmetries that are preserved by (3.14) and to
decompose the $SO(4)$ irreducible representations into representations of the
residual symmetry (3.15) or (3.16).

We work in the canonical frame where the twist transformation is given by
$R_{\vec{\theta}}$ in (3.13). In this conformal frame, the two fixed points of
the twist are $x_{0}=0$ and $x_{\bar{0}}=\infty$. Any point
$x\in{\mathbb{R}}^{4}$ can be represented as a $2\times 2$ matrix

$x=\left(\\!\\!\begin{array}[]{cc}\ \ z_{1}&z_{2}\\\
-\bar{z}_{2}&\bar{z}_{1}\end{array}\\!\right),$ (A.1)

where, as before, $z_{1}=x_{1}+ix_{2}$ and $z_{2}=x_{3}+ix_{4}$. In this
convention, rotations around the origin act as
$SO(4)\simeq\left(SU(2)_{L}\times SU(2)_{R}\right)/{\mathbb{Z}}_{2}$ right and
left multiplications

$x\rightarrow g_{L}\cdot x\cdot g_{R}\ ,\qquad g_{L/R}\in SU(2)_{L/R}\ .$
(A.2)

These $SU(2)_{L/R}$ transformations are generated by the differential
operators

$\begin{array}[]{ll}J_{L}^{+}=z_{2}\partial_{\bar{z}_{1}}-z_{1}\partial_{\bar{z}_{2}}\
,&J_{R}^{+}=z_{1}\partial_{z_{2}}-\bar{z}_{2}\partial_{\bar{z}_{1}}\ ,\\\
J_{L}^{-}=\bar{z}_{1}\partial_{z_{2}}-\bar{z}_{2}\partial_{z_{1}}\
,&J_{R}^{-}=z_{2}\partial_{z_{1}}-\bar{z}_{1}\partial_{\bar{z}_{2}}\ ,\\\
J_{L}^{3}=(z_{1}\partial_{z_{1}}-\bar{z}_{1}\partial_{\bar{z}_{1}}+z_{2}\partial_{z_{2}}-\bar{z}_{2}\partial_{\bar{z}_{2}})/2\
,&J_{R}^{3}=(z_{1}\partial_{z_{1}}-\bar{z}_{1}\partial_{\bar{z}_{1}}-z_{2}\partial_{z_{2}}+\bar{z}_{2}\partial_{\bar{z}_{2}})/2\
.\end{array}$ (A.3)

Irreducible representations of $SO(4)$ are characterised by two irreducible
representations of these two commuting $SU(2)$-factors, and are labelled by
the corresponding spins $(j_{L},j_{R})$. The allowed representations for
fields built with scalars and their derivatives (such as in the fishnet model
that we consider in this paper) are the ones with
$j_{L}+j_{R}\in{\mathbb{Z}}$. Moreover, operators with a single scalar can
only have $j_{L}=j_{R}\equiv S/2$, which correspond to traceless symmetric
tensors with $S$ indices. Operators in such a
$(2j_{L}+1)(2j_{R}+1)$-dimensional representation are labelled by the
eigenvalues of $(J^{3}_{L},J^{3}_{R})$ in (A.3), and $(m_{L},m_{R})$, that
take values in the range $-j_{L/R}\leq m_{L/R}\leq j_{L/R}$. The rest of the
generators act on these operators in the standard way

$\begin{split}J^{\pm}_{L}\circ\mathcal{O}_{\Delta,j_{L},j_{R},m_{L},m_{R}}(x)&=\sqrt{j_{L}(j_{L}+1)-m_{L}(m_{L}\pm
1)}\,\mathcal{O}_{\Delta,j_{L},j_{R},m_{L}\pm 1,m_{R}}(x)\ ,\\\
J^{\pm}_{R}\circ\mathcal{O}_{\Delta,j_{L},j_{R},m_{L},m_{R}}(x)&=\sqrt{j_{R}(j_{R}+1)-m_{R}(m_{R}\pm
1)}\,\mathcal{O}_{\Delta,j_{L},j_{R},m_{L},m_{R}\pm 1}(x)\ .\end{split}$ (A.4)

In this representation, the rotation symmetry (3.13) that we will use to twist
the operators takes the form

$R_{\vec{\theta}}=e^{i\theta_{1}(J^{3}_{L}+J^{3}_{R})+i\theta_{2}(J^{3}_{L}-J^{3}_{R})}.$
(A.5)

In particular, for $\theta_{1}=\theta_{2}=\theta$, (A.5) reduces to
$\exp(2i\theta J_{L}^{3})$, which makes it clear why the remaining symmetry is
enhanced from (3.15) to (3.16).

##  Appendix B Wave function in a generic frame

In this section, we compute the wave function in a generic frame, using two
distinct but equivalent points of view.

##### From covariance of correlation functions

The wave function is defined as a particular correlator (4.2):

$\Psi_{\tilde{R}}(x)\equiv\langle\mathcal{O}_{\tilde{R}}(x_{0})\,\text{tr}\left(\bar{Z}(x)\,\tilde{R}^{-1}\right)\rangle.$
(B.1)

In this general definition, $R$ is a conformal transformation with fixed
points $x_{0}$, $x_{\bar{0}}$, which we assume to take the form
$\tilde{R}=K\circ R_{\vec{\theta}}\circ K^{-1}$, for a conformal map $K$
satisfying $K(0)=x_{0}$, $K(\infty)=x_{\bar{0}}$. As a result of a generic
conformal change of coordinates, correlation functions transform as explained
in section 2.2. This allows us to relate the wave function (B.1) to the one in
the frame where the fixed points are $0$ and $\infty$. In fact, the
transformation rule (2.12) gives

$\Psi_{\tilde{R}}(x)=\Psi_{R_{\vec{\theta}}}(K^{-1}\circ
x)\,\left|\frac{\partial K^{-1}}{\partial
y}\right|_{y=x_{0}}^{\frac{\Delta}{4}}\,\left|\frac{\partial K^{-1}}{\partial
y}\right|_{y=x}^{\frac{1}{4}}.$ (B.2)

To evaluate the Jacobian determinants on the right hand side, we can use the
explicit form of the change of frame given in (4.14).434343Here we do not lose
generality. The most general change of frame satisfying $K(0)=x_{0}$,
$K(\infty)=x_{\bar{0}}$ is related to (4.14) by a rotation $r\in SO(4)$ and
dilatation $d$, $K\rightarrow d\circ r\circ K$. The effect of this
modification on the map $\tilde{R}$ only affects the final result (4.15) by
redefining the null vectors $n_{i}$ as $n_{i}\rightarrow r\circ n_{i}$.
Alternatively, we can say that the geometry of the map is fully specified by
the data $\left\\{x_{0},x_{\bar{0}},n_{i}\right\\}$, and this fixes the form
of the wave function (4.15).

For the special conformal transformation (4.14), we have, for any $x$, $y$,

$\left|\frac{\partial K^{-1}}{\partial
y}\right|^{\frac{1}{4}}=\left|\frac{\partial(K^{\prime})^{-1}}{\partial
y}\right|^{\frac{1}{4}}=\frac{1}{(y-x_{\bar{0}})^{2}},\quad\quad(K^{-1}\circ
x)^{2}=\frac{(x_{0}-x)^{2}}{(x_{0}-x_{\bar{0}})^{2}\,(x-x_{\bar{0}})^{2}}.$
(B.3)

Using these identities, (B.2) leads to the result (4.15).

##### From the Dyson-type equation in a generic frame

Alternatively, the wave function can be characterised as a solution of the
Dyson-type equation

$\tilde{\mathcal{B}}\circ\Psi_{\tilde{R}}(x)=\Psi_{\tilde{R}}(x),$ (B.4)

where $\tilde{\mathcal{B}}$ is the graph-building operator. According to the
perturbative rules explained in this paper, for a generic twist map
$\tilde{R}$ the graph-building operator is given by

$\tilde{\mathcal{B}}\circ\tilde{F}(y)\equiv\frac{\xi^{2}}{\pi^{2}}\,\int\frac{d^{4}x\,\left|{\partial\tilde{R}(x)\over\partial
x}\right|^{\frac{1}{4}}}{(y-x)^{2}(x-{\tilde{R}}(x))^{2}}\tilde{F}(x).$ (B.5)

To relate this expression to the Dyson-type equation in the original frame, we
use (2.5) with $x_{\circlearrowleft}\equiv K^{-1}(x)$, which leads to

$\frac{d^{4}x\,\left|{\partial\tilde{R}(x)\over\partial
x}\right|^{\frac{1}{4}}}{(x-{\tilde{R}}(x))^{2}}=\frac{d^{4}x^{\prime}}{(x^{\prime}-R_{\vec{\theta}}(x^{\prime}))^{2}}\,\left|{\partial
K^{-1}\over\partial x}\right|^{-\frac{1}{2}},$ (B.6)

with $x^{\prime}=K^{-1}\circ x$. Changing integration variables $x\rightarrow
x^{\prime}$ in (B.5), with the help of (B.6) we find

$\left(\tilde{\mathcal{B}}\circ\tilde{F}\right)(y)=\,\left|{\partial
K^{-1}\over\partial y}\right|^{\frac{1}{4}}\,\left(\mathcal{B}\circ
F\right)(K^{-1}\circ y),$ (B.7)

where $\mathcal{B}$ is the graph-building operator defined in the frame with
fixed points at $0$ and $\infty$, and the relation between $F$ and $\tilde{F}$
is

$\tilde{F}(x)=\left|{\partial K^{-1}\over\partial
x}\right|^{\frac{1}{4}}\,F(K^{-1}\circ x).$ (B.8)

Thanks to this property, the solution of the Dyson-type equation (B.4) can be
constructed as $\Psi_{\tilde{R}}(x)\propto\Psi_{R_{\vec{\theta}}}(K^{-1}\circ
x)\,\left|{\partial K^{-1}\over\partial x}\right|^{\frac{1}{4}},$ where
$\Psi_{R_{\vec{\theta}}}$ is the wave function stationary under the graph-
building operator $\mathcal{B}$. Therefore, we have rederived the
transformation rule (B.2), apart from a proportionality factor (independent of
$x$). This factor is simply $1/|x_{0\bar{0}}|^{2\Delta}$, and can be recovered
by demanding that the wave function reduces to $\Psi_{R_{\vec{\theta}}}$ for
$x_{0\bar{0}}\rightarrow\infty$.

##  Appendix C One-magnon eigenvalue

In this appendix, we describe the details of the calculation of the integral
(5.22) for the eigenvalue of the graph-building operator at $J_{1}=J_{2}=1$
and $\theta_{2}=\theta_{1}\equiv\theta$. States in this case are parametrised
by their scaling dimension $\Delta$ as well as their $U(1)_{L}\times
SU(2)_{R}$ quantum numbers, $(m_{L},j_{R},m_{R})$. The method we use is
applicable for any values of these spins. For simplicity, we only consider the
case where $m_{L}=j_{R}=m_{R}=0$. In this case, the wave function (5.21)
simplifies to $\Psi_{\Delta,0,0,0}(x)=|x|^{-2\Delta-2}$. The action of the
graph building operator on this function is given by the integral (5.22)

$\displaystyle
E_{1,1}={\mathcal{B}\circ\Psi_{\Delta,0,0,0}(x)\over\xi^{2}\,\Psi_{\Delta,0,0,0}(x)}$
$\displaystyle=$
$\displaystyle\frac{x^{2\Delta+2}}{\pi^{2}}\int\frac{d^{4}w}{(x-w_{\circlearrowleft})^{2}(x-w)^{2}w^{2\Delta+2}}$
$\displaystyle=$
$\displaystyle\int\frac{(r_{1}dr_{1}d\phi_{1})\,(r_{2}dr_{2}d\phi_{2})\,(r_{1}^{2}+r_{2}^{2})^{-\Delta/2-1}}{\pi^{2}\left(r_{1}^{2}+r_{2}^{2}+1-2r_{1}\cos\phi_{1}\right)\left(r_{1}^{2}+r_{2}^{2}+1-2r_{1}\cos\left(\theta-\phi_{1}\right)\right)}\
,$

where, without loss of generality, we have fixed $x=(1,0,0,0)$ and
parametrised the integration point as
$w=(r_{1}\cos\phi_{1},r_{1}\sin\phi_{1},r_{2}\cos\phi_{2},r_{2}\cos\phi_{2})$.
The integral over $\phi_{2}$ factorises trivially. The integral over
$\phi_{1}$ can be done explicitly by changing variables to $z=e^{i\phi_{1}}$,
blowing up the contour of integration, and picking up the poles. This results
in

$E_{1,1}=\int\limits_{0}^{\infty}\\!d\rho\,\int\limits_{0}^{\pi/2}\\!d\sigma\,\frac{2\left(\rho^{2}+1\right)\rho^{1-\Delta}\sin(2\sigma)}{\sqrt{\rho^{4}-2\rho^{2}\cos(2\sigma)+1}\left(\rho^{4}-\rho^{2}\left(2\cos^{2}\frac{\theta}{2}\cos(2\sigma)+\cos\theta-1\right)+1\right)}\
,$ (C.2)

where we have used the parametrisation $r_{1}+i\,r_{2}=\rho\,e^{i\sigma}$. The
integration in $\sigma$ can be performed explicitly and gives

$E_{1,1}={i\over\sin\theta}\int\limits_{0}^{\infty}\\!{d\rho\over\rho^{\Delta+1}}\log{\rho^{2}-e^{i\theta}\over\rho^{2}-e^{-i\theta}}\
.$ (C.3)

Finally, by integrating over $\rho$, we arrive at (5.23).

##  Appendix D One-loop one-magnon data

In the table below we present the one-loop scaling dimensions of operators
with $J_{1}=J_{2}=1$ and $\theta_{1}=\theta_{2}\equiv\theta$. Operators in
that group can only have an anomalous dimension if they are the twist of a
conformal descendant of $\mathcal{O}_{0}=\text{tr}(XZ)$. Such operators are
characterised by the number of boxes in $\Box^{n}\mathcal{O}_{0}$ and the
$U(1)_{L}\times SU(2)_{R}$ representation, $(m_{L},j_{R})$.

$\begin{array}[]{|c|c|c|l|}\hline\cr j_{R}&m_{L}&n&\Delta\\\ \hline\cr
0&0&0&2-2\xi^{2}\\\ 0&0&1&4-2\xi^{2}\cos(\theta)\\\
\frac{1}{2}&\frac{1}{2}&0&3-2e^{\frac{i\theta}{2}}\xi^{2}\cos\left(\frac{\theta}{2}\right)\\\
\frac{1}{2}&\frac{1}{2}&1&5-\frac{2}{3}e^{\frac{i\theta}{2}}\xi^{2}\left(\cos\left(\frac{\theta}{2}\right)+2\cos\left(\frac{3\theta}{2}\right)\right)\\\
1&0&0&4-\frac{2}{3}\xi^{2}(\cos(\theta)+2)\\\
1&0&1&6-\frac{2}{3}\xi^{2}(2\cos(\theta)+\cos(2\theta))\\\
1&1&0&4-\frac{2}{3}e^{i\theta}\xi^{2}(2\cos(\theta)+1)\\\
1&1&1&6+\frac{1}{3}e^{i\theta}\xi^{2}(-2\cos(\theta)-3\cos(2\theta)-1)\\\
\frac{3}{2}&\frac{1}{2}&0&5+\frac{1}{3}e^{\frac{i\theta}{2}}\xi^{2}\left(-5\cos\left(\frac{\theta}{2}\right)-\cos\left(\frac{3\theta}{2}\right)\right)\\\
\frac{3}{2}&\frac{1}{2}&1&7+\frac{1}{15}e^{\frac{i\theta}{2}}\xi^{2}\left(-7\cos\left(\frac{\theta}{2}\right)-17\cos\left(\frac{3\theta}{2}\right)-6\cos\left(\frac{5\theta}{2}\right)\right)\\\
\frac{3}{2}&\frac{3}{2}&0&5-2e^{\frac{3i\theta}{2}}\xi^{2}\cos\left(\frac{\theta}{2}\right)\cos(\theta)\\\
\frac{3}{2}&\frac{3}{2}&1&7+e^{\frac{3i\theta}{2}}\xi^{2}\left(-\frac{3}{5}\left(\cos\left(\frac{\theta}{2}\right)+\cos\left(\frac{3\theta}{2}\right)\right)-\frac{4}{5}\cos\left(\frac{5\theta}{2}\right)\right)\\\
2&0&0&6-\frac{2}{15}\xi^{2}(6\cos(\theta)+\cos(2\theta)+8)\\\
2&0&1&\frac{1}{5}\xi^{2}(-5\cos(\theta)-4\cos(2\theta)-\cos(3\theta))+8\\\
2&1&0&6+\frac{1}{5}e^{i\theta}\xi^{2}(-6\cos(\theta)-\cos(2\theta)-3)\\\
2&1&1&8-\frac{2}{15}e^{i\theta}\xi^{2}(4\cos(\theta)+7\cos(2\theta)+2\cos(3\theta)+2)\\\
2&2&0&6-\frac{2}{5}e^{2i\theta}\xi^{2}(2\cos(\theta)+2\cos(2\theta)+1)\\\
2&2&1&8-\frac{2}{15}e^{2i\theta}\xi^{2}(4\cos(\theta)+4\cos(2\theta)+5\cos(3\theta)+2)\\\
\frac{5}{2}&\frac{3}{2}&0&7-\frac{2}{15}e^{\frac{3i\theta}{2}}\xi^{2}\left(7\left(\cos\left(\frac{\theta}{2}\right)+\cos\left(\frac{3\theta}{2}\right)\right)+\cos\left(\frac{5\theta}{2}\right)\right)\\\
\frac{5}{2}&\frac{3}{2}&1&9-\frac{2}{105}e^{\frac{3i\theta}{2}}\xi^{2}\left(27\cos\left(\frac{\theta}{2}\right)+27\cos\left(\frac{3\theta}{2}\right)+41\cos\left(\frac{5\theta}{2}\right)+10\cos\left(\frac{7\theta}{2}\right)\right)\\\
\frac{5}{2}&\frac{5}{2}&0&7-\frac{2}{3}e^{\frac{5i\theta}{2}}\xi^{2}\cos\left(\frac{\theta}{2}\right)(2\cos(2\theta)+1)\\\
\frac{5}{2}&\frac{5}{2}&1&9+\frac{2}{21}e^{\frac{5i\theta}{2}}\xi^{2}\left(-5\left(\cos\left(\frac{\theta}{2}\right)+\cos\left(\frac{3\theta}{2}\right)+\cos\left(\frac{5\theta}{2}\right)\right)-6\cos\left(\frac{7\theta}{2}\right)\right)\\\
3&2&0&8-\frac{2}{21}e^{2i\theta}\xi^{2}(8\cos(\theta)+8\cos(2\theta)+\cos(3\theta)+4)\\\
3&2&1&10+\frac{1}{21}e^{2i\theta}\xi^{2}(-10\cos(\theta)-10\cos(2\theta)-14\cos(3\theta)-3\cos(4\theta)-5)\\\
3&3&0&8-\frac{2}{7}e^{3i\theta}\xi^{2}(2\cos(\theta)+2\cos(2\theta)+2\cos(3\theta)+1)\\\
3&3&1&10+\frac{1}{14}e^{3i\theta}\xi^{2}(-6\cos(\theta)-6\cos(2\theta)-6\cos(3\theta)-7\cos(4\theta)-3)\\\
\hline\cr\end{array}$ (D.1)

Another example is given by $\\{m_{L},m_{R},S+2n\\}=\\{0,0,4\\}$. In this
case, there are three operators
$\vec{\mathcal{O}}\equiv\left\\{(\partial_{1}\bar{\partial}_{1})^{2},\,\partial_{1}\partial_{2}(\bar{\partial}_{1}\bar{\partial}_{2}),\,(\partial_{2}\bar{\partial}_{2})^{2}\right\\}\,\mathcal{O}_{0}$.
The corresponding mixing matrix is

$\hat{\Gamma}_{\vec{\theta}}\,\vec{\mathcal{O}}=\xi^{2}\left(\begin{array}[]{ccc}-\frac{32s_{1}^{2}}{5}+8s_{1}-2&\frac{16s_{1}}{15}(5-6s_{1})&-\frac{16s_{1}^{2}}{15}\\\
\frac{4s_{2}}{15}(5-6s_{1})&-\frac{64s_{1}s_{2}}{15}+\frac{8}{3}(s_{1}+s_{2})-2&\frac{4s_{1}}{15}(5-6s_{2})\\\
-\frac{16s_{2}^{2}}{15}&\frac{16s_{2}}{15}(5-6s_{2})&-\frac{32s_{2}^{2}}{5}+8s_{2}-2\end{array}\right)\vec{\mathcal{O}}\
,$ (D.2)

where $s_{i}=\sin^{2}\frac{\theta_{i}}{2}$. It is straightforward to construct
the mixing matrix for any $m_{L}$, $m_{R}$ and $M$.

##  Appendix E Single-wrapping anomalous dimension from fishnet diagrams

The leading diagram contributing to the two-point function
$\langle\mathcal{O}_{J}(x_{0})\,\bar{\mathcal{O}}_{J}(x_{\bar{0}})\rangle$,
where
$\mathcal{O}_{J}(x_{0})=\text{tr}\left(Z^{J}(x_{0})R_{\vec{\theta}}\right)$,
is exemplified in figure 18. The goal of this section is to extract the
logarithmic divergence of the diagram, which corresponds to the anomalous
dimension of the operator.

We introduce a point-splitting regularisation by separating the $Z$ scalar
from the fixed point of the twist transformation $x_{0}$

$\mathcal{O}_{J}^{\text{reg}}=\text{tr}\left(Z^{J}(x)R_{\vec{\theta}}\right)\
,\qquad\bar{\mathcal{O}}_{J}^{\text{reg}}=\text{tr}\left(\bar{Z}^{J}(y)R_{-\vec{\theta}}\right)\
.$ (E.1)

We will be working in the frame with $x_{0}=0$, $x_{\bar{0}}=\infty$ and
$x^{2}=1/y^{2}=\epsilon^{2}$.

The anomalous dimension can be read off from

${\langle\mathcal{O}_{J}^{\text{reg}}\bar{\mathcal{O}}_{J}^{\text{reg}}\rangle_{J\text{-loop}}\over\langle\mathcal{O}_{J}\bar{\mathcal{O}}_{J}\rangle_{\text{tree}}}\sim\log(\epsilon)\,\gamma^{\text{1-wheel}}_{J}\
.$ (E.2)

As illustrated in figure 18, we can isolate the integration over one of the
internal vertices. We remain with a ladder subdiagram times two propagators

$\langle\mathcal{O}_{J}^{\text{reg}}\bar{\mathcal{O}}_{J}^{\text{reg}}\rangle_{J\text{-loop}}=\frac{\xi^{2}}{\pi^{2}}\,\int
d^{4}w\,\frac{\mathcal{G}_{J-1}^{\text{ladder}}(x,y,w,w_{\circlearrowleft})}{(x-w)^{2}\,(y-w)^{2}}\
.$ (E.3)

The ladder integral (grey in figure 18) was evaluated in Usyukina:1993ch  and
is given by

$\mathcal{G}_{n}^{\text{ladder}}(x_{1},x_{2},x_{3},x_{4})=\xi^{2n}\,{1\over
x_{12}^{2n}x_{34}^{2}}{(1-z)(1-\bar{z})\over z-\bar{z}}L_{n}(z,\bar{z})\ ,$
(E.4)

where $z$ and $\bar{z}$ are the two conformal cross ratios

$z\bar{z}={x_{14}^{2}x_{23}^{2}\over x_{13}^{2}x_{24}^{2}}\
,\qquad(1-z)(1-\bar{z})={x_{12}^{2}x_{43}^{2}\over x_{13}^{2}x_{24}^{2}}\ ,$
(E.5)

and the ladder function $L_{n}$ is given by

$L_{n}=\sum_{j=n}^{2n}{j![-\log(z\bar{z})]^{2n-j}\over
n!(j-n)!(2n-j)!}\left[{\rm Li}_{j}(z)-{\rm Li}_{j}(\bar{z})\right]\ .$ (E.6)

The form (E.3) is convenient, since we can directly set $x=0$ and
$y\rightarrow\infty$ in the integrand without encountering any singularity.
The UV divergence will be produced only after performing the integration over
$w$. Defining the integral with a cutoff around $0$ and $\infty$, we have

${\langle\mathcal{O}_{J}^{\text{reg}}\bar{\mathcal{O}}_{J}^{\text{reg}}\rangle_{J\text{-loop}}\over\langle\mathcal{O}_{J}\bar{\mathcal{O}}_{J}\rangle_{\text{tree}}}=\frac{\xi^{2J}}{\pi^{2}}\int
d^{4}w\frac{1}{(w-w_{\circlearrowleft})^{2}}\,{(1-z)(1-\bar{z})\over
z-\bar{z}}L_{n}(z,\bar{z})\ ,$ (E.7)

where the explicit form of the cross ratios in terms of the integration
variable is

$z\bar{z}=1\ ,\qquad(1-z)(1-\bar{z})={(w-w_{\circlearrowleft})^{2}\over
w^{2}}=4{r_{1}^{2}\sin^{2}{\theta_{1}\over
2}+r_{2}^{2}\sin^{2}{\theta_{2}\over 2}\over(r_{1}^{2}+r_{2}^{2})}\ ,$ (E.8)

with $r_{1}^{2}=w_{1}^{2}+w_{2}^{2}$, $r_{2}^{2}=w_{3}^{2}+w_{4}^{2}$ being
the square radial coordinates in the two rotation planes.

The simplest case to consider is when the twist involves equal angles
$\theta_{1}=\theta_{2}=\theta$. In this case, the cross ratios are independent
on the integration variable and given by

$z=e^{i\theta}\ ,\qquad\bar{z}=e^{-i\theta}\ .$ (E.9)

Therefore, the radial integration factors out and we can immediately extract
the anomalous dimension

$\gamma_{J}^{\text{1-wheel}}(\theta,\theta)=-{i\xi^{2J}\over\sin\theta}\,\left(\begin{array}[]{c}2J-2\\\
J-1\end{array}\right)\left[{\rm Li}_{2J-2}(e^{-i\theta})-{\rm
Li}_{2J-2}(e^{i\theta})\right]\ .$ (E.10)

For generic angles $\theta_{1}\neq\theta_{2}$, the cross ratios instead have a
non-trivial dependence on $r_{1}$, $r_{2}$. Introducing $\varphi$ defined as

$z=e^{-i\varphi}\ ,\quad\bar{z}=e^{i\varphi}\
,\quad(1-z)(1-\bar{z})=4\sin^{2}{\varphi\over 2},$ (E.11)

and changing integration variables to $(r_{2}^{2},\varphi)$, we find that the
integral factorises as

$\displaystyle{\langle\mathcal{O}_{J}^{\text{reg}}\bar{\mathcal{O}}_{J}^{\text{reg}}\rangle_{J\text{-loop}}\over\langle\mathcal{O}_{J}\bar{\mathcal{O}}_{J}\rangle_{\text{tree}}}\sim\frac{\xi^{2J}}{4i}\left(\\!\begin{array}[]{c}2J-2\\\
J-1\end{array}\\!\right)\\!\int\limits_{\epsilon^{2}<r_{1}^{2}+r_{2}^{2}<1/\epsilon^{2}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{d(r_{1}^{2})d(r_{2}^{2})\,\left[{\rm
Li}_{2J-2}(e^{i\varphi})-{\rm
Li}_{2J-2}(e^{-i\varphi})\right]\,\tan\frac{\varphi}{2}\over(r_{1}^{2}+r_{2}^{2})(r_{1}^{2}\sin^{2}{\theta_{1}\over
2}+r_{2}^{2}\sin^{2}{\theta_{2}\over 2})}$ (E.14) $\displaystyle=$
$\displaystyle\underbrace{\left(-\frac{1}{2}\int\limits_{\epsilon<r_{2}<1/\epsilon}{d(r_{2}^{2})\over
r_{2}^{2}}\right)}_{-2\log\epsilon}\times\underbrace{\left(-i\xi^{2J}\left(\\!\begin{array}[]{c}2J-2\\\
J-1\end{array}\\!\right)\int\limits_{\theta_{2}}^{\theta_{1}}d\varphi\,{{\rm
Li}_{2J-2}(e^{i\varphi})-{\rm
Li}_{2J-2}(e^{-i\varphi})\over\cos\theta_{1}-\cos\theta_{2}}\right)}_{\gamma_{J}^{\text{1-wheel}}(\theta_{1},\theta_{2})}\
,$ (E.17)

yielding the anomalous dimension in (6.2).

##  Appendix F Justification of the quantisation condition of Baxter equations
from $\mathcal{N}=4$ SYM

As shown in Gromov:2017cja ; UpcomingFishnet , the Baxter equations for the
fishnet model can be derived by applying the double scaling limit to the
Quantum Spectral Curve of $\mathcal{N}=4$ SYM. These results show that the
Q-functions of the fishnet model are directly connected to a quadruplet of
Q-functions of $\mathcal{N}=4$ SYM, denoted as ${\bf Q}_{i}$. For example, for
the states with charges $J_{1}=J$, $J_{2}=0$, we have the relation
UpcomingFishnet :

${\bf Q}_{i}(u)\rightarrow
q^{\downarrow}_{i}(u)/u^{J/2}\quad\quad\text{(double scaling limit)}\ .$ (F.1)

A dual set of Q-functions, denoted as ${\bf Q}^{i}(u)$, also plays an
important role. In the double scaling limit, they are related to $3\times 3$
determinants constructed out of the $q$-functions. Given this relation, it is
not surprising that there is a way to derive the quantisation condition (6.19)
from the properties of the QSC, which we now explain.

The two sets of Q-functions ${\bf Q}_{i}$ and ${\bf Q}^{i}$ each satisfy a
fourth order Baxter equation444444In this case, the coefficients of the Baxter
equation are not simple rational functions of $u$, but are explicitly
constructed out of other Q-functions usually denoted as ${\bf P}_{i}$, ${\bf
P}^{i}$. , which is a consequence of the algebraic $PSU(2,2|4)$ Q-system
relations Gromov:2014caa ; Gromov:2017blm . In the same way that we explained
in section 6.2.1, one can consider two independent solutions of these
difference equations, one solution being analytic in the upper half $u$-plane,
and the other in the lower half $u$-plane. In this way, we introduce ${\bf
Q}_{i}^{\downarrow}(u)$ and ${\bf Q}_{i}^{\uparrow}(u)$, and similarly ${\bf
Q}^{i\downarrow}(u)$, ${\bf Q}^{i\uparrow}(u)$.454545 In the conventional
notations used in the QSC literature, ${\bf Q}_{i}(u)\equiv{\bf
Q}_{i}^{\downarrow}(u)$ and similarly ${\bf Q}^{i}(u)\equiv{\bf
Q}^{i\downarrow}(u)$. For real values of the parameters these two sets of
solutions are related by complex conjugation ${\bf
Q}^{\uparrow}_{i}(u)=\overline{{\bf Q}_{i}^{\downarrow}}(u)$ in
$\mathcal{N}=4$ SYM. However, this direct relation is lost when considering
generic complex points in parameter space, and for this reason we will not use
it in the following.

Since they satisfy the same Baxter equation, the two sets of Q-functions
analytic in the upper/lower half plane are related by an $i$-periodic matrix,

${\bf Q}^{\uparrow}_{i}(u)=\Omega_{i}^{j}(u)\,{\bf Q}_{j}^{\downarrow}(u)\
,\,\,\,\,\Omega_{i}^{j}(u+i)=\Omega_{i}^{j}(u)\ .$ (F.2)

Importantly, this matrix $\Omega$ is directly related to the corresponding
matrix in the fishnet model (see eq. (6.17)) after taking the double scaling
limit.

While the Q-functions in the fishnet model are meromorphic, their “parent
functions” in $\mathcal{N}=4$ SYM have square-root branch cuts rather than
poles. The branch cuts are at $u\in(-2g,2g)-in$, $n\in\mathbb{N}$, where
$g^{2}=\lambda/(16\pi^{2})$. The analytic continuation of a Q-function through
the branch cut on the real axis is denoted as $\widetilde{\bf Q}^{i}$. The
quantisation conditions for $\mathcal{N}$=4 SYM can be stated as
Gromov:2014caa

$\widetilde{\bf Q}^{i\downarrow}(u)=\omega^{ij}(u){\bf Q}_{j}^{\downarrow}(u)\
,$ (F.3)

where $\omega^{ij}(u)$ is an $i$-periodic function of $u$, satisfying the
algebraic conditions

$\omega^{ij}(u)=-\omega^{ji}(u)\
,\,\,\,\,1=\omega^{12}(u)\omega^{34}(u)-\omega^{13}(u)\omega^{24}(u)+\omega^{14}(u)\omega^{23}(u)\
.$ (F.4)

Combining (F.2) and (F.3), we obtain

$\widetilde{{\bf Q}}^{i\downarrow}(u)=G^{ij}{{\bf Q}}_{j}^{\uparrow}(u)\ ,$
(F.5)

where $G^{ij}$ is defined as

$G^{ik}\Omega_{k}^{j}(u)=\omega^{ik}(u)\ ,$ (F.6)

and is called the gluing matrix Gromov:2015vua . The crucial property of the
gluing matrix is that it has no cuts. For the case of integer spins that we
consider, it is a constant matrix. The form of this matrix was determined for
the theory without twists in Gromov:2015vua . The case of the theory with
twists in AdS was never explicitly considered; however, our results are
consistent with the assumption that the gluing matrix takes the form:

$G^{ij}=\begin{pmatrix}0&\alpha&0&0\\\ -\alpha&0&0&0\\\ 0&0&0&\beta\\\
0&0&-\beta&0\end{pmatrix},$ (F.7)

where $\alpha$, $\beta$ are two constants that may depend on the state and are
in general non-vanishing. The antisymmetry of $\omega^{ij}$, combined with
(F.7), implies precisely that the four components $\Omega_{1}^{2}$,
$\Omega_{2}^{1}$, $\Omega_{3}^{4}$, $\Omega_{4}^{3}$ must vanish. This
justifies the quantisation condition (6.19).

Notice that the ansatz (F.7), together with (F.4), imposes several further
constraints on the form of the matrix $\Omega_{i}^{j}(u)$. In particular, it
must take the form

$\Omega=\left(\begin{array}[]{cccc}\Omega_{1}^{1}&0&\Omega_{1}^{3}&\Omega_{1}^{4}\\\
0&\Omega_{1}^{1}&\Omega_{2}^{3}&\Omega_{2}^{4}\\\
A\,\Omega_{2}^{4}&-A\,\Omega_{1}^{4}&\Omega_{3}^{3}&0\\\
-A\,\Omega_{2}^{3}&A\,\Omega_{1}^{3}&0&\Omega_{3}^{3}\end{array}\right),$
(F.8)

where $A=\alpha/\beta$. Indeed, our numerical study of twisted Baxter
equations in the fishnet model confirms that, once we impose even just one of
the conditions (6.19), the matrix $\Omega$ acquires the form (F.8). This
provides a strong cross-check of the correctness of the quantisation condition
(6.19). It can be proved that the constant $A$ appearing in (F.8) is the same
parameter guaranteeing the absence of poles in the bi-linear combination
$Q_{+}(u)$ defined in (6.22), see UpcomingFishnet .

##  Appendix G Results for the one-magnon spectrum at equal angles from
integrability

Starting from the solutions of the Baxter equation (6.34), we can compute the
spectrum using the exact equation (6.22) derived in UpcomingFishnet :

$\xi^{2}=i\lim_{u\rightarrow 0}u\,\frac{Q_{+}(u+3/2i)}{Q_{+}(i/2)}\ .$ (G.1)

Let us discuss some of the formal steps involved. We listed the explicit form
of the Q-functions analytic in the upper half plane in (6.34). The Q-functions
analytic in the lower half plane are

$\displaystyle q_{1}^{\uparrow}(u)$ $\displaystyle=$ $\displaystyle
e^{i\pi(-\Delta/2+S_{L}+1)}\mathcal{F}(S_{R},S_{L},\Delta,1/t,-u+i)\ ,$ (G.2)
$\displaystyle q_{2}^{\uparrow}(u)$ $\displaystyle=$ $\displaystyle
e^{i\pi(-\Delta/2-S_{L}+1)}\mathcal{F}(S_{R},-S_{L},\Delta,t,-u+i)\ ,$ (G.3)
$\displaystyle q_{4}^{\uparrow}(u)$ $\displaystyle=$ $\displaystyle
e^{i\pi(\Delta/2-S_{R})}\mathcal{G}(S_{R},S_{L},\Delta,1/t,-u+i)\ ,$ (G.4)
$\displaystyle q_{3}^{\uparrow}(u)$ $\displaystyle\propto$ $\displaystyle
e^{i\pi(\Delta/2-S_{R})}\left(\mathcal{G}(S_{R},-S_{L},\Delta,t,-u)-\mathcal{G}(S_{R},S_{L},\Delta,1/t,-u+i)\right)\
.$

Next, one needs to fix the form of the bi-linear combination $Q_{+}(u)$ by
fixing the constant $A$ appearing in (6.21). This constant is fixed in such a
way that $Q_{+}(u)$ is regular at $u\sim i/2$ and can easily be found case by
case. We conjecture the general form of this constant for arbitrary spins to
be

$A=(-1)^{S_{L}+S_{R}}\,\frac{\Gamma^{2}(1-\frac{\Delta}{2}+S_{R})\Gamma(S_{R}+1-S_{L})\Gamma(S_{R}+S_{L}+1)}{\Gamma^{2}(\frac{\Delta}{2}-S_{R})\Gamma^{2}(2S_{R}+1)}\,(4\sin^{2}\frac{\theta}{2})^{\Delta-1}\,\left(\frac{\cos^{2}\frac{\theta}{2}}{\sin^{2}\frac{\theta}{2}}\right)^{S_{R}}.$
(G.5)

Using the explicit form of the Q-functions, one can then explicitly evaluate
(G.1). In particular, the evaluation of the limit $\lim_{u\rightarrow
0}Q_{+}(u+3/2i)$ is straightforward, as the pole comes from the $\Gamma(-iu)$
pre-factor included in the functions $\mathcal{F}$ and $\mathcal{G}$
functions, see (6.35). Stripping off the pole, the result takes the generic
form of a linear combination of Lerch functions $\Phi(t^{\pm 1},1,b)\equiv
1/b\;{}_{2}F_{1}(1,b,b+1,t^{\pm 1})$, multiplied by rational functions of
$\Delta$ and $t$, with a prefactor $A^{\frac{1}{2}}$. Using identities between
the Lerch functions, the result can be significantly simplified, and we find
that the residue of $Q_{+}(u)$ at $u\sim 3/2i$ can always be rewritten as
$A^{\frac{1}{2}}$, multiplied by a rational function of $\Delta$ and $t$. To
evaluate the denominator in (G.1), it is convenient to rewrite the function
$Q_{+}(u)$ in the form

$\begin{split}Q_{+}(u)&=-A\left(q_{1}^{\downarrow}\left(u+\frac{i}{2}\right)q^{\uparrow}_{2}\left(u-\frac{3i}{2}\right)-q_{2}^{\downarrow}\left(u+\frac{i}{2}\right)q^{\uparrow}_{1}\left(u-\frac{3i}{2}\right)\right)\\\
&-q_{3}^{\downarrow}\left(u+\frac{i}{2}\right)q^{\uparrow}_{4}\left(u-\frac{3i}{2}\right)+q_{4}^{\downarrow}\left(u+\frac{i}{2}\right)q^{\uparrow}_{3}\left(u-\frac{3i}{2}\right)\
.\end{split}$ (G.6)

This alternative expression is completely equivalent to (6.21) once we choose
the value of $A$ canceling the poles, namely (G.5). This can be proved using
the form of $\Omega$ in (F.8). The expression (G.6) has the advantage that it
manifestly has no pole at $u=i/2$. The direct evaluation of $Q_{+}(i/2)$ then
yields a quadratic combination of Lerch functions. Again, this can be further
simplified and leads in the end to a simple linear combination involving a
basis of four Lerch functions. For instance, for integer spins we can always
bring the result to the form

$\displaystyle\frac{(1+t)^{2S_{R}}\,\prod_{n=1}^{S_{R}}(\Delta^{2}-(2n)^{2})}{t^{S_{R}}\,\xi^{2}}$
$\displaystyle=$ $\displaystyle
t^{S_{L}}\frac{(1+t)^{2}}{t}\,\,R^{(0)}(t,\Delta,S_{R},S_{L})$
$\displaystyle+t^{S_{L}}\,{R^{(1)}(t,\Delta,S_{R},S_{L})}\,\frac{2t\left(\Phi(t,1,-\frac{\Delta}{2})+\Phi(1/t,1,\frac{\Delta}{2})\right)}{\Delta\,(-1+t^{2})}$
$\displaystyle-t^{S_{L}}\,{R^{(1)}(t,-\Delta,S_{R},S_{L})}\,\frac{2t\left(\Phi(t,1,\frac{\Delta}{2})+\Phi(1/t,1,-\frac{\Delta}{2})\right)}{\Delta\,(-1+t^{2})}\
,$

where $R^{(i)}(t,\Delta,S_{R},S_{L})$ are polynomials in $t^{\pm 1}$ and
$\Delta$. A small sample of results is listed in the table below.

$\footnotesize\begin{array}[]{|c|c| l|l|}\hline\cr
S_{R}&S_{L}&R^{(0)}(\theta,\Delta,S_{R},S_{L})&R^{(1)}(\theta,\Delta,S_{R},S_{L})\\\
\hline\cr 0&0&0&1\\\ \hline\cr
1&\left\\{-1,0,1\right\\}&4S_{L}^{2}&\begin{array}[]{l}S_{L}^{2}\left(2(\Delta-2)t-2(\Delta+2)t^{-1}\right)\\\
+((\Delta+2)t^{-\frac{1}{2}}+(\Delta-2)t^{\frac{1}{2}})^{2}\end{array}\\\
\hline\cr
2&0&-384&\left(6\Delta(\frac{1}{t}-t)+\Delta^{2}\frac{(t+1)^{2}}{t}+8(t-4+\frac{1}{t})\right)^{2}\\\
\hline\cr
2&\left\\{-1,1\right\\}&4\left(\Delta^{2}t^{-1}+\Delta^{2}t+2\Delta^{2}-64\right)&\begin{array}[]{l}(\Delta+(\Delta-4)t+4){\Big{(}}\Delta^{3}\frac{(t+1)^{3}}{t^{2}}-96(\frac{1}{t}-1)\\\
+8\Delta(\frac{1}{t}+1)(t-7+\frac{1}{t})-6\Delta^{2}(t-1)(\frac{1}{t}+1)^{2}{\Big{)}}\end{array}\\\
\hline\cr
2&\left\\{-2,2\right\\}&\begin{array}[]{l}16\Delta^{2}t^{-1}+4\Delta^{2}t^{2}-16t^{2}+16\Delta^{2}t-96t\\\
+24\Delta^{2}+4\Delta^{2}t^{-2}-224-{16}{t^{-2}}-96t^{-1}\end{array}&\begin{array}[]{l}384+16\Delta\left(t^{2}+8t-\frac{8}{t}-\frac{1}{t^{2}}\right)+\Delta^{4}\frac{(t+1)^{4}}{t^{2}}\\\
-4\Delta^{3}\frac{(t-1)(t+1)^{3}}{t^{2}}-4\Delta^{2}(1+\frac{1}{t})^{2}(t(t+14)+1)\end{array}\\\
\hline\cr\end{array}$

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