$\displaystyle\left(\frac{\mathcal{Q}_{1|1}^{+}}{\mathcal{Q}_{1|1}^{-}}\right)\bigg{\rvert}_{\mathcal{Q}_{1|\emptyset}=0}=1\,,\quad\left(\frac{\mathcal{Q}_{1|1}^{[2]}}{\mathcal{Q}_{1|1}^{[-2]}}\frac{\mathcal{Q}^{-}_{1|\emptyset}\mathcal{Q}^{-}_{1|12}}{\mathcal{Q}^{+}_{1|\emptyset}\mathcal{Q}^{+}_{1|12}}\right)\bigg{\rvert}_{\mathcal{Q}_{1|1}=0}=-1\,,\quad\left(\frac{\mathcal{Q}^{-}_{12|12}\mathcal{Q}^{+}_{1|1}}{\mathcal{Q}^{+}_{12|12}\mathcal{Q}^{-}_{1|1}}\right)\bigg{\rvert}_{\mathcal{Q}_{1|12}=0}=1\,.$
(84)

The first equation written out explicitly is

$\displaystyle\frac{\mathcal{Q}^{+}_{1|1}}{\mathcal{Q}^{-}_{1|1}}\bigg{\rvert}_{\mathcal{Q}_{1|\emptyset}=0}=1\implies\frac{\mathtt{x}_{1}}{\mathtt{y}_{1}}\prod_{k=1}^{M_{\theta}}\sqrt{\frac{y^{-}_{k}}{y^{+}_{k}}}\frac{y^{+}_{k}-\frac{1}{x^{(i)}_{1|\emptyset}}}{y^{-}_{k}-\frac{1}{x^{(i)}_{1|\emptyset}}}\prod_{k=1}^{M_{1|1}}\frac{u^{(i)}_{1|\emptyset}-u^{(k)}_{1|1}+\frac{\mathtt{i}}{2}}{u^{(i)}_{1|\emptyset}-u^{(k)}_{1|1}-\frac{\mathtt{i}}{2}}=1\,.$
(85)

For the middle node Bethe equations, all factors of $\mathtt{f}$ cancel and
only polynomials in $u$ remain, the Bethe equations become

$\frac{1}{\mathtt{y}^{2}_{1}}\prod_{k=1}^{M_{1|1}}\frac{u^{(i)}_{1|1}-u^{(k)}_{1|1}+\mathtt{i}}{u^{(i)}_{1|1}-u^{(k)}_{1|1}-\mathtt{i}}\prod_{k=1}^{M_{1|\emptyset}}\frac{u^{(i)}_{1|1}-u^{(k)}_{1|\emptyset}-\frac{\mathtt{i}}{2}}{u^{(i)}_{1|1}-u^{(k)}_{1|\emptyset}+\frac{\mathtt{i}}{2}}\prod^{M_{1|12}}_{k=1}\frac{u^{(i)}_{1|1}-u^{(k)}_{1|12}-\frac{\mathtt{i}}{2}}{u^{(i)}_{1|1}-u^{(k)}_{1|12}+\frac{\mathtt{i}}{2}}=-1\,.$
(86)

The last equation is almost identical to (85):

$\displaystyle\frac{\mathcal{Q}_{12|12}^{-}}{\mathcal{Q}_{12|12}^{+}}\frac{\mathcal{Q}^{+}_{1|1}}{\mathcal{Q}^{-}_{1|1}}\bigg{\rvert}_{\mathcal{Q}_{1|12}=0}=1\implies\frac{\mathtt{x}_{1}}{\mathtt{y}_{1}}\prod_{k=1}^{M_{\theta}}\sqrt{\frac{y^{-}_{k}}{y^{+}_{k}}}\frac{x^{(i)}_{1|12}-y^{+}_{k}}{x^{(i)}_{1|12}-y^{-}_{k}}\prod_{k=1}^{M_{1|1}}\frac{u^{(i)}_{1|12}-u^{(k)}_{1|1}+\frac{\mathtt{i}}{2}}{u^{(i)}_{1|12}-u^{(k)}_{1|1}-\frac{\mathtt{i}}{2}}=1\,.$
(87)

These equations are well known as part of the asymptotic Bethe equations for
AdS5/CFT4 Beisert:2005fw . In their own right, they describe a spin chain with
centrally extended $\mathfrak{su}(2|2)$ symmetry Beisert:2006qh  (a.k.a.
inhomogeneous Hubbard model).

###  6.3 Lieb-Wu equations

To study the original ‘homogeneous’ Hubbard model we can take
$F=\frac{1}{x^{M_{\theta}}}=F_{\text{pol}}(\theta_{i}\to\infty)$. The above
derivation goes through with minor modifications. Instead of repeating it, we
use the procedure from Beisert:2006qh : Introduce the parameterisation

$x^{(i)}_{1|\emptyset}=\mathtt{i}e^{-\mathtt{i}k_{i}}\,,\quad
x^{(i)}_{1|12}=\frac{e^{\mathtt{i}k_{i}}}{\mathtt{i}}\,,\quad
u^{(i)}_{1|\emptyset}={\mbox{h}}\sin{k_{i}}\,,\quad
u^{(i)}_{1|12}={\mbox{h}}\sin
k_{i}\,,\quad\lambda_{i}=\frac{1}{{\mbox{h}}}u^{(i)}_{1|1}\,;$ (88)

Fix the twist
$\mathtt{y}_{1}=\mathtt{y}_{2}=1,\mathtt{x}_{1}=\frac{1}{\mathtt{x}_{2}}=\prod_{k=1}^{M_{\theta}}\mathtt{i}$
and take the limit
$y^{+}_{k}\sim\frac{1}{\epsilon},y^{-}_{k}\sim\epsilon,\epsilon\rightarrow 0$
to obtain

$\displaystyle\prod_{j=1}^{M}\frac{\lambda_{i}-\lambda_{j}+\frac{\mathtt{i}}{{\mbox{h}}}}{\lambda_{i}-\lambda_{j}-\frac{\mathtt{i}}{{\mbox{h}}}}\prod_{j=1}^{N}\frac{\lambda_{i}-\sin{k_{j}}-\frac{\mathtt{i}}{2{\mbox{h}}}}{\lambda_{i}-\sin{k_{j}}+\frac{\mathtt{i}}{2{\mbox{h}}}}=-1\,,\quad
i=1,\dots,M\,,$ (89a)
$\displaystyle\prod_{j=1}^{M}\frac{\sin{k_{i}}-\lambda_{j}+\frac{\mathtt{i}}{2{\mbox{h}}}}{\sin{k_{i}}-\lambda_{j}-\frac{\mathtt{i}}{2{\mbox{h}}}}=e^{\mathtt{i}M_{\theta}k_{i}}\,.\quad
i=1,\dots,N\,,$ (89b)

where $M=M_{1|1}$ and $N=M_{1|\emptyset}+M_{1|12}$. These are exactly Lieb-Wu
equations LiebWu  describing the spectrum of homogeneous Hubbard model.
Comparing with equations $(3.95)-(3.96)$ from HubbardBook  shows that
$\mathbf{u}=\frac{1}{2\,{\mbox{h}}}$ is the coupling constant appearing in the
Hamiltonian

$H_{\text{Hubbard}}=-\sum_{j=1}^{M_{\theta}}\sum_{a=\uparrow,\downarrow}(c^{\dagger}_{j,a}c_{j+1,a}+c^{\dagger}_{j+1,a}c_{j,a})+\mathbf{u}\sum_{j=1}^{M_{\theta}}(1-2n_{j,\uparrow})(1-2n_{j,\downarrow})\,.$
(90)

We refer to HubbardBook  for an in-depth treatment of the model and
explanation of the notation used in (90).

The limit $\epsilon\to 0$ gives the source term
$F=\sqrt{\frac{\mathbb{Q}^{-}}{\mathbb{Q}^{+}}}\frac{B_{(+)}}{B_{(-)}}=\sqrt{\frac{R_{-}}{B_{-}}\frac{B_{+}}{R_{+}}}\rightarrow\frac{1}{x^{M_{\theta}}}$.
Of course the choice to send $y^{+},\frac{1}{y^{-}}$ to infinity was
arbitrary, we could equally well consider
$\frac{1}{y^{+}_{k}},y^{-}_{k}\rightarrow\infty$ which would lead to
$F=x^{M_{\theta}}$. We remark that the limit is non-trivial. For one thing, it
violates $|y^{-}_{k}|>1$ requiring continuing the inhomogeneities under the
cut and changing the reality property of ${F_{\text{ext}}}$ to
$\overline{{F_{\text{ext}}}}={F_{\text{ext}}}$. For another, the centrally
extended symmetry undergoes a type of contraction, only the bosonic subalgebra
survives manifestly in the limit deLeeuw:2015ula .

There exist other interesting choices of $y^{+}$ and $y^{-}$ apart from the
homogeneous Hubbard limit. In particular, Frolov:2011wg  studied the cases
$y^{+}y^{-}=-1$ and $\frac{y^{+}}{y^{-}}=-1$ which are Hermitian parity-
invariant models. It would be interesting to investigate the thermodynamic
limit of these models using QSC.

###  6.4 T- and Y-systems

Among suggested source terms (77), we still have to consider
${F_{\text{exp}}}(\theta)$. It exhibits non-polynomial behaviour natural to
the case of thermodynamic Bethe Ansatz (TBA).

TBA for Hubbard model was first developed in the work of Takahashi Takahashi .
Later on, this approach became a part of technology in derivations of mirror
TBA equations and Y-/T-systems of AdS5/CFT4 integrability Gromov:2009tv
,Bombardelli:2009ns ; Gromov:2009bc ; Arutyunov:2009ur . These equations, by a
meticulous analysis of the discontinuity properties Cavaglia:2010nm
,Gromov:2011cx  of Y- and T-functions superposed on the Wronskian solution
Gromov:2010km  of T-functions in terms of Q-functions, were eventually reduced
to AdS5/CFT4 QSC Gromov:2013pga ; Gromov:2014caa .

One of the original motivations that launched our work was to circumvent this
laboured approach to derive QSC’s. We presumably succeeded for the example of
Hubbard model, yet the question remains whether the derived QSC by the
monodromy bootstrap can be also derived via the TBA route. Fortunately, we do
not need to repeat the full TBA computation as most of the work had been
carried out in Cavaglia:2015nta  by Cavaglià, Cornagliotto, Mattelliano, and
Tateo. In fact, the discontinuity relations equivalent to (71) are already
present in that paper, we just need to make a proper decoding of functions
(which is done at the end of this section). What was not done in
Cavaglia:2015nta  is connecting analytic properties of the full
${\mathfrak{gl}}_{2|2}$ Q-system and of the corresponding T-system, and we
shall focus mainly on this task.

Relation between Q- and T-functions is purely algebraic and is valid
independently of whether we discuss TBA or not. Hence, our general discussion
will be done without any assumption on function $F$. At the end of the
section, we return to the concrete choices of $F$, and in particular
explicitly relate ${F_{\text{exp}}}(\theta)$ to Hubbard model at finite
temperature.

$\mathsf{SU}(2|2)$ T-functions are functions $T_{a,s}$ defined on the
$\mathsf{SU}(2|2)$ $\mathbb{L}$-hook, $\\{(a,s)|\,s\in[0,2],a\geq
0\\}\cup\\{(a,s)|\,a\in[0,2],s\geq 0\\}$. They satisfy the T-system:

$\displaystyle
T^{+}_{a,s}T^{-}_{a,s}=T_{a,s+1}T_{a,s-1}+T_{a+1,s}T_{a-1,s}\,.$ (91)

Like Q-functions, T-functions shall also have branch points, and then one
needs to specify whether (91) is valid in the physical or the mirror
kinematics. To distinguish the two cases we use the notation $\check{T}_{a,s}$
for T-functions of the mirror T-system, and $\hat{T}_{a,s}$ for T-functions of
the physical T-system. No check or hat over $T_{a,s}$ shall refer to general
discussion. There is no guarantee that $\check{T}$-system and $\hat{T}$-system
are related by a direct analytic continuation precisely due to the recurring
stumbling block of the paper: equation (91) is non-local. Nevertheless, we
shall see that certain $\check{T}$\- and $\hat{T}$-functions are indeed
related by the continuation, somewhat surprisingly and due to the specific
nature of Riemann-Hilbert problems (71).

Given a solution to the T-system one can construct another one using gauge
transformations

$\displaystyle T_{a,s}\rightarrow
g_{(++)}^{[a+s]}g_{(+-)}^{[a-s]}g_{(-+)}^{[-a+s]}g_{(--)}^{[-a-s]}T_{a,s}\,,$
(92)

implying that not $T_{a,s}$ but rather their invariant combinations

$Y_{a,s}=\frac{T_{a,s-1}T_{a,s+1}}{T_{a-1,s}T_{a+1,s}}$ (93)

encode physical properties. Just as we distinguish physical and mirror
T-functions, we should distinguish $\hat{Y}_{a,s}$ and $\check{Y}_{a,s}$.

There exist a well-established procedure to generate a solution of the
T-system using Q-functions Tsuboi:2009ud . In the specific case of
$\mathsf{SU}(2|2)$ it reads

$\displaystyle T_{0,s\geq
0}=(-1)^{s}Q^{[s]}_{\emptyset|\emptyset}Q^{[-s]}_{12|12}\,,$ $\displaystyle
T_{1,s\geq 1}=(-1)^{s+1}\epsilon^{ab}Q^{[s]}_{a|\emptyset}Q^{[-s]}_{b|12}\,,$
$\displaystyle T_{2,s\geq
2}=(-1)^{s}Q^{[s]}_{12|\emptyset}Q_{\emptyset|12}^{[-s]}\,,$ (94a)
$\displaystyle T_{a\geq
0,0}=(-1)^{a}Q_{12|12}^{[a]}Q_{\emptyset|\emptyset}^{[-a]}\,,$ $\displaystyle
T_{a\geq 1,1}=-\epsilon^{ij}Q^{[a]}_{12|i}Q^{[-a]}_{\emptyset|j}\,,$
$\displaystyle T_{a\geq
2,2}=(-1)^{a}Q^{[a]}_{12|\emptyset}Q^{[-a]}_{\emptyset|12}\,.$ (94b)

The signs are picked following Kazakov:2015efa . As a default, we evaluate
T-functions on the real axis. Hence, assuming UHPA Q-functions, it is
important to specify kinematics when evaluating $Q^{[-s]}_{A|I}$: evaluating
them in the physical kinematics yields $\hat{T}_{a,s}$ while the mirror
kinematics evaluation yields $\check{T}_{a,s}$.

Gauge transformations of the Q-system generate two gauge transformations of
the T-system:

$\displaystyle Q_{A|I}\rightarrow
g_{(+)}^{[|A|-|I|]}g_{(-)}^{[-|A|+|I|]}Q_{A|I}\,,$ $\displaystyle
T_{a,s}\rightarrow
g_{(+)}^{[a+s]}g_{(+)}^{[-a-s]}g_{(-)}^{[s-a]}g_{(-)}^{[-s+a]}T_{a,s}\,.$ (95)

Wronskian solution (94) is not invariant under Hodge transformation of the
underlying Q-system but the transformation rule is not difficult: switching
the Q-system with its Hodge dual is equivalent, up to overall signs, to
sending all shifts in (94) to their negative values. We will denote the
T-functions obtained using Hodge duality by $T^{*}_{a,s}$

Using Wronskian solution (94), we can deduce the analytic properties of the T-
and Y-system from Q-functions. There does not exist a global gauge in which
all T-functions have the simplest possible analytic properties. It is then
advantageous to focus either on the right ($s\geq a$) or the upper ($a\geq s$)
band of the $\mathbb{L}$-hook.

Consider first the right band. From the Q-system we know that
$\mathcal{Q}_{a|\emptyset}\propto\hat{\mathbf{p}}_{a},\mathcal{Q}_{a|12}\propto\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\hat{\mathbf{p}}}\vphantom{\hat{\mathbf{p}}}_{a}$,
where all omitted factors are independent of the index of the Q-functions. It
follows that it is possible to do a gauge transformation such that

$\displaystyle\hat{T}_{0,s\geq 0}=1\,,$ $\displaystyle\hat{T}_{1,s\geq
1}=\epsilon^{ab}\hat{\mathbf{p}}^{[s]}_{a}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\hat{\mathbf{p}}}\vphantom{\hat{\mathbf{p}}}_{b}^{[-s]}\,,$
$\displaystyle\hat{T}_{2,s\geq
2}=(\epsilon^{ab}\hat{\mathbf{p}}^{+}_{a}\hat{\mathbf{p}}^{-}_{b})^{[s]}(\epsilon^{cd}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\hat{\mathbf{p}}}\vphantom{\hat{\mathbf{p}}}^{+}_{c}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\hat{\mathbf{p}}}\vphantom{\hat{\mathbf{p}}}^{-}_{d})^{[-s]}\,,$
(96a) $\displaystyle\check{T}_{0,s\geq 0}=1\,,$
$\displaystyle\check{T}_{1,s\geq
1}=\epsilon^{ab}\hat{\mathbf{p}}^{[s]}_{a}\hat{\mathbf{p}}_{b}^{[-s]}\,,$
$\displaystyle\check{T}_{2,s\geq
2}=\check{T}_{1,1}^{[s]}\check{T}_{1,1}^{[-s]}\,.$ (96b)

This gauge transformation is not of Wronskian type (95) because no such
transformation can set $T_{0,s\geq 0}=1$. Since $\hat{\mathbf{p}}_{a}$ is a
single-valued function of Zhukovsky variable, $T_{1,s\geq 1}$ has only two
cuts. It’s interesting to notice that the four different sheets of $T_{1,s\geq
1}$ can be interpreted as a choice of the kinematics and Hodge duality:

$\displaystyle\hat{T}_{1,s}=\epsilon^{ab}\hat{\mathbf{p}}^{[s]}_{a}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\hat{\mathbf{p}}}\vphantom{\hat{\mathbf{p}}}^{[-s]}_{b}\,,$
$\displaystyle\check{T}_{1,s}=\epsilon^{ab}\hat{\mathbf{p}}^{[s]}_{a}\hat{\mathbf{p}}^{[-s]}_{b}\,,$
$\displaystyle\hat{T}^{*}_{1,s}=\epsilon^{ab}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\hat{\mathbf{p}}}\vphantom{\hat{\mathbf{p}}}^{[s]}_{a}\hat{\mathbf{p}}^{[-s]}_{b}\,,$
$\displaystyle\check{T}^{*}_{1,s}=\epsilon^{ab}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\hat{\mathbf{p}}}\vphantom{\hat{\mathbf{p}}}^{[s]}_{a}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\hat{\mathbf{p}}}\vphantom{\hat{\mathbf{p}}}^{[-s]}_{b}\,.$
(97a)

Hence deciding what is the right kinematics is to a large extent artificial
(although it is of course necessary to make the discussion concrete).

Turning to the upper-band $a\geq s$, it is worth remembering that there is a
gauge where ${\bf Q}_{i}$ become ${\hat{\mathbf{q}}}_{i}$ \- single-valued
functions of Zhukovsky variable. We can then simply repeat all the above
argumentation for the upper band using ${\bf Q}_{i}$ instead of ${\bf P}_{a}$.
It is hence more informative to keep using the same gauge as for the right
band and check what are the upper-band T-functions:

$\displaystyle T_{a\geq
0,0}=(-1)^{a}\left(\frac{U^{[2]}}{\bar{U}^{[-2]}}\frac{1}{F}\right)^{[a]_{D}}\,,$
$\displaystyle T_{a\geq
1,1}=\left(\frac{U^{+}}{\bar{U}^{-}}\right)^{[a]_{D}}\left(\frac{1}{F\bar{U}^{[-2]}U^{[2]}}\right)^{[-a]}\epsilon^{ij}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\hat{\mathbf{q}}}\vphantom{\hat{\mathbf{q}}}^{[a]}_{i}\hat{\mathbf{q}}^{[-a]}_{j}\,.$
(98)

The negative shifts are considered in the physical kinematics to obtain
physical T-functions, and in the mirror kinematics to get $\check{T}_{a,s}$.
We also use the following notation: $f^{[a]_{D}}=f^{[a-1]}f^{[a-3]}\dots
f^{[-a+1]}$.

We discuss now properties of $Y_{1,1},Y_{2,2}$. Recall the explicit
expressions in terms of T-functions

$\displaystyle Y_{1,1}=\frac{T_{1,0}T_{1,2}}{T_{0,1}T_{2,1}}\,,$
$\displaystyle Y_{2,2}=\frac{T_{2,1}T_{2,3}}{T_{1,2}T_{3,2}}\,.$ (99)

Using the gauge of the Wronskian solution (94), we have $T_{2,3}=T_{3,2}$ and
so the product of $Y_{1,1}$ and $Y_{2,2}$ is given by

$Y_{1,1}Y_{2,2}=\frac{\hat{T}_{1,0}}{\hat{T}_{0,1}}=\frac{\check{T}_{1,0}}{\check{T}_{0,1}}=\frac{Q_{12|12}^{+}}{Q_{12|12}^{-}}=\frac{(f^{[2]})^{2}}{f^{2}}=\frac{1}{F^{2}}\,,$
(100)

where all expressions are evaluated slightly above the real axis.

Clearly $Y_{1,1}Y_{2,2}$ have a cut on the real axis and furthermore, using
the Wronskian parameterisation, we can deduce that this cut is present in both
$Y_{1,1}$ and $Y_{2,2}$. To find the analytic continuation around this cut we
use that $Y_{2,2}=\frac{T_{2,1}}{T_{1,2}}$ in the Wronskian gauge and
substitute in the explicit form of Q-functions:
$\mathcal{Q}_{a|12}=-\bar{U}^{[-2]}U\frac{\hat{\mathbf{p}}_{a}}{\sigma}\mathtt{f}\mathtt{f}^{[2]},\mathcal{Q}_{\emptyset|i}=\mathtt{f}^{[2]}\frac{\hat{\mathbf{q}}_{i}}{\sigma}$.
Then the analytic continuation around this cut is computed as

$\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{Y}\vphantom{Y}_{2,2}=F^{2}Y_{2,2}=\frac{1}{Y_{1,1}}\,.$
(101)

Let us now comment on reality. To constrain to real solutions, we shall demand
that complex conjugation of Q-functions is a symmetry of the Q-system and can
then include a combination of a gauge transformation, H-rotation, and possibly
Hodge duality. Since complex conjugation changes the positive shifts to
negative we must also include extra signs to preserve QQ-relations. Explicitly

$\overline{\mathcal{Q}}_{A|I}=(-1)^{\frac{(|A|+|I|)(|A|+|I|-1)}{2}}S\cdot\mathcal{Q}_{A|I}\,,$
(102)

where $S$ is a symmetry transformation.

As usual, working either with or without Hodge duality gives two distinct
options. Moreover, we have to decide in which kinematics we demand the reality
property, which a priory gives us four options in total to consider. We shall
pick one case for now and focus on the physical kinematics and no Hodge
duality:

$\displaystyle\overline{{\bf P}}_{a}=h\,(H_{b})_{a}{}^{b}{\bf P}_{b}\,,$
$\displaystyle\overline{{\bf
Q}}_{i}=\frac{1}{h\,\omega}\,(H_{f}^{\prime})_{i}{}^{j}{\bf
Q}_{j}=-\frac{1}{h\,F}\,(H_{f}^{\prime})_{i}{}^{k}\epsilon_{kj}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
Q}}\vphantom{{\bf Q}}^{j}\,,$ $\displaystyle\overline{F}=\frac{1}{F}\,,$ (103)

where the fermionic H-rotation is performed by the matrix $H_{f}$ with
coefficients $(H_{f})_{i}{}^{j}=\frac{1}{\omega}(H_{f}^{\prime})_{i}{}^{j}$,
we pulled out the factor $1/\omega$ for future convenience. To derive the last
relation use $\mathcal{Q}_{12|12}=\frac{1}{g^{2}}$ to find
$\frac{g^{2}}{\bar{g}^{2}}=\det({H^{\prime}_{f}}^{+})\det(H_{b}^{+})/(\omega^{+})^{2}$.
Periodicity of $H^{\prime}_{f},H_{b}$ and $\omega$ in the physical kinematics
results in $\frac{\overline{g}^{+}}{\overline{g}^{-}}=\frac{g^{+}}{g^{-}}$ so
that $\overline{F}=\frac{1}{F}$.

Let us find the analytic properties of $H_{b}$ and $H_{f}^{\prime}$. Following
the steps detailed in footnote 12 with $\overline{{\bf P}}_{a}$ playing the
role of ${\bf P}^{\downarrow}_{a}$ we find that it is always possible to
reparametrise $h$ and $H_{b}$ so that $h$ has only one short cut on the real
axis and $H_{b}$ has no branch points. Recall then that
$\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
Q}}\vphantom{{\bf Q}}^{i}$ is analytic in the lower half-plane. Because
$\bar{{\bf Q}}_{i}$ is analytic there as well, we use periodicity of
$H_{f}^{\prime}$ to derive that $H_{f}^{\prime}$ cannot have any cuts. In
summary both $H_{b}$ and $H_{f}^{\prime}$ are periodic functions without cuts.

Now we turn to the analytic structure of $h$. It is useful to write down the
relations that follow from compatibility with
$\bar{\bar{\mathcal{Q}}}=\mathcal{Q}$ 242424To derive the relation for
$\bar{H}_{b}H_{b}$ we use that $\bar{H}_{b}H_{b}{\bf
P}^{[2n]}=\frac{1}{(\bar{h}h)^{[2n]}}{\bf P}^{[2n]}$ and assume that ${\bf P}$
and ${\bf P}^{[2]}$ are linearly-independent vectors. Since $\bar{H}_{b}H_{b}$
is a $2\times 2$ matrix it has 2 eigenvalues. If the eigenvalue is degenerate
then $(h\bar{h})^{[2]}=h\bar{h}$ and $h\bar{h}\bar{H}_{b}H_{b}=1$ as claimed.
If there are two distinct eigenvalues then use also
$\bar{H}_{b}H_{b}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
P}}\vphantom{{\bf
P}}=\frac{1}{\overset{\scalebox{0.8}{\raisebox{-1.75pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{h\bar{h}}\vphantom{h\bar{h}}}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
P}}\vphantom{{\bf P}}$. It must be that either
$\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{h\bar{h}}\vphantom{h\bar{h}}=h\bar{h}$
or
$\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{h\bar{h}}\vphantom{h\bar{h}}=(h\bar{h})^{[2]}$.
If
$\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{h\bar{h}}\vphantom{h\bar{h}}=h\bar{h}$
then ${\bf
P}\epsilon\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
P}}\vphantom{{\bf P}}=0$ but this is a contradiction because ${\bf
P}\epsilon\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
P}}\vphantom{{\bf P}}\propto\frac{1}{F}-F\neq 0$; if
$\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{h\bar{h}}\vphantom{h\bar{h}}=(h\bar{h})^{[2]}$
then $h\bar{h}=(h\bar{h})^{[2]}$ because $h,\bar{h}$ only have cuts on the
real axis. We conclude that $h\bar{h}\bar{H}_{b}H=1$, the same argumentation
also gives $\frac{1}{h\bar{h}}\bar{H}_{f}H_{f}=1$.:

$\displaystyle\bar{h}\,h\,\bar{H}_{b}\,H_{b}=1\,,$
$\displaystyle\frac{1}{\bar{h}\,h}\frac{1}{\bar{\omega}\,\omega}\bar{H}^{\prime}_{f}\,H^{\prime}_{f}=1\,.$
(104)

Multiplying these two consistency conditions results in
$\omega\,\bar{\omega}=H_{b}\,\bar{H}_{b}\,H^{\prime}_{f}\,\bar{H}^{\prime}_{f}$
and thus $\omega\,\bar{\omega}$ has no cuts. Analytic continuation of $h$
through its single cut follows from the ${\bf P}\mu$-system:
$\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mathcal{Q}}\vphantom{\mathcal{Q}}_{a|\emptyset}=-\frac{1}{\omega}\mathcal{Q}_{a|12}$.
We take the complex conjugation of this equation and use that the analytic
continuation commutes with complex conjugation. Since
$\overline{\mathcal{Q}}_{a|12}=-\frac{1}{h\omega^{2}}\det
H^{\prime}_{f}\,(H_{b})_{a}{}^{b}\mathcal{Q}_{b|12}$, we find

$\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{h}\vphantom{h}h=-\frac{1}{\omega\bar{\omega}}\det
H^{\prime}_{f}\,.$ (105)

We conclude that $h$ is a single-valued function of Zhukovsky variable and
that
$\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{h}\vphantom{h}h$
is a periodic function.

Perform now a symmetry transformation

$\displaystyle{\bf
P}_{a}\rightarrow\bar{h}_{A}\bar{P}^{A}_{b}(\bar{A}_{b})_{a}{}^{b}{\bf
P}_{b}\,,$ $\displaystyle{\bf
Q}_{i}\rightarrow\frac{1}{\bar{h}_{A}}\bar{P}^{A}_{f}(\bar{A}_{f})_{i}{}^{j}{\bf
Q}_{j}\,,$ (106)

with $A_{b/f}$ $\mathtt{i}$-periodic matrices with unit determinant,
$P^{A}_{b/f}$ periodic functions without cuts and $h_{A}$ a single-valued
function of Zhukovsky variable. Our goal is to simplify
$h\,H_{b},\,\frac{1}{h}\,H^{\prime}_{f}$. It will be convenient to factorise
expressions: $h\,H_{b}=h\,\sqrt{\det H_{b}}\times\frac{H_{b}}{\sqrt{\det
H_{b}}}$, $\frac{1}{h}\,H_{f}^{\prime}=\frac{1}{h}\,\sqrt{\det
H_{f}^{\prime}}\times\frac{H_{f}^{\prime}}{\sqrt{\det H_{f}^{\prime}}}$, and
furthermore we shall define
$M_{b}=\mathtt{i}\,\frac{H_{b}\,\epsilon}{\sqrt{\det H_{b}}}$,
$M_{f}=\mathtt{i}\,\frac{H_{f}^{\prime}\epsilon}{\sqrt{\det H_{f}^{\prime}}}$.

Matrices $M_{b},M_{f}$ are Hermitian due to consistency conditions (104) and
they change under transformation (106) following the rule

$\displaystyle M_{b}\rightarrow A_{b}M_{b}A_{b}^{\dagger}\,,$ $\displaystyle
M_{f}\rightarrow A_{f}M_{f}A_{f}^{\dagger}\,.$ (107)

Use the standard trick
$M_{b/f}=x_{\mu}\sigma^{\mu},\sigma^{\mu}=(\mathds{1}_{2\times
2},\sigma^{i=1,2,3})$ to identify $M_{b/f}$ with the 4-vector $x_{\mu}$.
Recall that the action (107) in the case $A\in\mathsf{SL}(2,\mathbb{C})$ is
equivalent to Lorentz transformations of $x_{\mu}$. Since $\det M_{b/f}=-1$
the 4-vector $x_{\mu}$ is space-like and we can bring it to any Pauli-matrix
using $A_{b/f}$.

We are left with prefactors, we parametrise these as $h\sqrt{\det
H_{b}}=\mathtt{i}\sqrt{\frac{h}{\overset{\scalebox{0.8}{\raisebox{-1.75pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{h}\vphantom{h}}}P_{b},\,\frac{1}{h}\sqrt{\det
H^{\prime}_{f}}=\mathtt{i}\sqrt{\frac{\overset{\scalebox{0.8}{\raisebox{-1.75pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{h}\vphantom{h}}{h}}P_{f}$
with $P_{b/f}$ periodic functions. We consider first the non-periodic factor
$\sqrt{\frac{h}{\overset{\scalebox{0.8}{\raisebox{-1.75pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{h}\vphantom{h}}}$,
under (106) we find
$\sqrt{\frac{h}{\tilde{h}}}\rightarrow\frac{h_{A}}{\bar{h}_{A}}\sqrt{\frac{h}{\tilde{h}}}$.
By taking
$h_{A}=(\frac{\overset{\scalebox{0.8}{\raisebox{-1.75pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{h}\vphantom{h}}{h})^{\frac{1}{4}}$
we set this factor to $1$ since $\bar{h}_{A}h_{A}=1$. Finally, the periodic
factors transform as

$\displaystyle P_{b}\rightarrow\frac{P^{A}_{b}}{\bar{P}^{A}_{b}}\,P_{b}\,,$
$\displaystyle\frac{1}{\omega}P_{f}\rightarrow\frac{P^{A}_{f}\bar{P}^{A}_{f}}{\omega^{\prime}}\,P_{f}\,.$
(108)

Here we have used that
$\omega\rightarrow\omega^{\prime}=(\bar{P}^{A}_{f})^{2}\omega$. By picking
$P_{b}^{A}$ and $P_{f}^{A}$ appropriately we can send $P_{b}\rightarrow
1,P_{f}\rightarrow 1$. This is possible since
$P_{b}\bar{P}_{b}=1,\bar{P}_{f}=P_{f}$ which can be verified using the
explicit expressions $P_{b}=\frac{\sqrt{\det H_{b}}\sqrt{\det
H^{\prime}_{f}}}{\sqrt{\omega\bar{\omega}}},P_{f}=\sqrt{\omega\bar{\omega}}$
as follows from (105) 252525The sign choice when taking the square root of
(105) is irrelevant because it can be reabsorbed by an $A_{b/f}$ rotation..

In conclusion we can pick
$hH_{b}=\frac{1}{h}H^{\prime}_{f}=(\vec{n}\,\cdot\vec{\sigma})\epsilon$ with
$\vec{n}$ a unit vector. It follows that
$\bar{\omega}\omega=1,\bar{\mu}=\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mu}\vphantom{\mu}$
and $\check{T}_{a,s}$, in gauge (96b), are manifestly real, and so
$\check{Y}_{a,s}$ are real. A choice of $\vec{n}$ amounts to picking a
specific basis of Q-functions. There exists a preferred basis in physical
systems singled out by asymptotic conditions, in those cases there also exists
a natural choice of $\vec{n}$.

One of the natural choices is $\vec{n}\vec{\sigma}=\sigma^{3}$ which results
in

$\displaystyle\overline{{\bf P}}_{a}=|\epsilon_{ab}|{\bf P}_{b}\,,$
$\displaystyle\overline{{\bf Q}}_{i}=\frac{|\epsilon_{ij}|}{\omega}{\bf
Q}_{j}\,,$ $\displaystyle\overline{F}=\frac{1}{F}\,,$
$\displaystyle\text{(physical kinematics)}\,,$ (109a)
$\displaystyle\overline{{\bf P}}_{a}=\frac{\mu}{F}(-1)^{a}{\bf P}^{a}\,,$
$\displaystyle\overline{{\bf Q}}_{i}=F(-1)^{i+1}{\bf Q}^{i}\,,$
$\displaystyle\overline{F}=F$ $\displaystyle\text{(mirror kinematics)}\,,$
(109b)

where $|\epsilon_{ab}|=\delta_{a1}\delta_{b2}+\delta_{a2}\delta_{b1}$.

All reasoning above was performed assuming the physical kinematics and (109a)
is the final outcome. Then we used ${\bf P}\mu$\- and ${\bf
Q}\omega$-relations to derive (109b) which is clearly a reality demand (102)
with Hodge duality and in the mirror kinematics. Moreover, assuming the most
general reality demand (102) with Hodge duality and the mirror kinematics, we
can simplify it to (109b), and then use ${\bf P}\mu$\- and ${\bf
Q}\omega$-equations to get to (109a) as well. Hence changing the kinematics in
(102) is equivalent to the change of the choice of whether to take Hodge dual.
So there are only two different reality demands, not four which was our
original expectation.

The analysis of the Hodge dual case in the physical kinematics (or of the no-
Hodge dual case in the mirror kinematics) is analogous. The reality condition,
in a basis similar to the one in (109), simplifies to

$\displaystyle\overline{{\bf P}}_{a}=(-1)^{a}{\bf P}^{a}\,$
$\displaystyle\overline{{\bf Q}}_{i}=\omega(-1)^{i}{\bf Q}^{i}\,$
$\displaystyle\overline{F}=F\,,$ $\displaystyle\text{(physical kinematics)},$
(110a) $\displaystyle\overline{{\bf P}}_{a}=\frac{F}{\mu}|\epsilon_{ab}|{\bf
P}_{b}\,,$ $\displaystyle\overline{{\bf
Q}}_{i}=-\frac{1}{F}|\epsilon_{ij}|{\bf Q}_{j}\,,$
$\displaystyle\overline{F}=\frac{1}{F}\,,$ $\displaystyle\text{(mirror
kinematics)}.$ (110b)

With these reality conditions, $\hat{T}_{a,s}$ (96a) are real up to the gauge
factor $\frac{\bar{U}^{[-2-s]}\bar{U}^{[-2+s]}}{U^{[2+s]}U^{[2-s]}}$, and
$\hat{Y}_{a,s}$ are real.

Now we comment on concrete examples. The simplest possible solution to a
twisted $\mathsf{SU}(2|2)$ Q-system, and so equivalently to the T-system, is
the character solution. In this case, all Q-functions are simply proportional
to the twist factors,
$\mathcal{Q}_{A|I}\propto\left(\frac{\mathtt{x}_{A}}{\mathtt{y}_{I}}\right)^{-\mathtt{i}u}$.
Using the unimodularity condition
$\mathtt{x}_{1}\mathtt{x}_{2}/\mathtt{y}_{1}\mathtt{y}_{2}=1$, it follows that
all T- and Y-functions do not depend on $u$. Furthermore Gromov:2010vb
,Kazakov:2015efa

$T_{a,s}\propto\Delta\,\chi_{a,s}\,,$ (111)

where $\chi_{a,s}(G)$ is the character and
$\Delta=\det\frac{1}{\mathtt{x}_{a}-\mathtt{y}_{i}}$ is the Cauchy double
alternant. Also, using additionally
$\mathtt{x}_{1}\mathtt{x}_{2}=\mathtt{y}_{1}\mathtt{y}_{2}=1$ imposed by ${\bf
P}\mu$-equations, we observe that Hodge duality acts trivially on the
character solution: $T^{*}_{a,s}=T_{a,s}$ 262626If no constraints on
$\mathtt{x}_{1},\mathtt{x}_{2},\mathtt{y}_{1},\mathtt{y}_{2}$ are imposed,
T-functions of the character solution are of the form
$T_{a,s}=(\frac{\mathtt{x}_{1}\mathtt{x}_{2}}{\mathtt{y}_{1}\mathtt{y}_{2}})^{-\mathtt{i}u}T^{\text{const}}_{a,s}$
with $T^{\text{const}}_{a,s}$ being independent of $u$. Hodge duality acts
non-trivially by sending
$T^{\text{const}}_{a,s}(\mathtt{x}_{a},\mathtt{y}_{i})\rightarrow
T^{\text{const}}_{a,s}(\frac{1}{\mathtt{x}_{a}},\frac{1}{\mathtt{y}_{i}})$
which corresponds to the map from a covariant to the corresponding contra-
variant representation.. There exists a natural basis for the Q-functions of
the character solution:
$\mathcal{Q}_{a|\emptyset}=\mathtt{x}_{a}^{-\mathtt{i}u},\,\mathcal{Q}_{\emptyset|i}=\mathtt{y}_{i}^{\mathtt{i}u}$.
If $\mathtt{x}_{1},\mathtt{y}_{1}$ are real, the basis (109) is defined by
$\mathcal{Q}^{\prime}_{a|\emptyset}=\mathcal{Q}_{a|\emptyset},\mathcal{Q}^{\prime}_{\emptyset|i}=\frac{1}{\sqrt{\omega}}\mathcal{Q}_{\emptyset|i}$.
If instead $\mathtt{x}_{1},\mathtt{y}_{1}$ are phases,
$\mathcal{Q}^{\prime}_{\pm|\emptyset}=\frac{1}{\sqrt{2}}(\mathcal{Q}_{1|\emptyset}\pm\mathtt{i}\mathcal{Q}_{2|\emptyset}),\,\mathcal{Q}^{\prime}_{\emptyset|\pm}=\frac{1}{\sqrt{2\omega}}(\mathcal{Q}_{\emptyset|1}\pm\mathtt{i}\mathcal{Q}_{\emptyset|2})$
gives the basis (109).

Analytic properties of T-system (and of some of Q-functions) emerging from TBA
were derived in Cavaglia:2015nta . Our results perfectly match and further
complement those findings thus demonstrating that the monodromy bootstrap
works also for derivation of QSC’s describing TBA equations. Comparing the
expressions for $Y_{1,1}Y_{2,2}$, equation $(3.1)$ in Cavaglia:2015nta  and
(100), we see that the relevant source term is
$F={F_{\text{exp}}}(-\frac{\mathtt{i}}{T})$ for the finite-temperature Hubbard
model. To match other expressions in Cavaglia:2015nta  with ours, we should
identify

$\displaystyle(-1)^{-\mathtt{i}u}{\bf
P}^{\mathtt{H}}_{+}\propto\hat{\mathbf{p}}_{1}\,,$
$\displaystyle(-1)^{\mathtt{i}u}{\bf
P}^{\mathtt{H}}_{-}\propto\hat{\mathbf{p}}_{2}\,,$ $\displaystyle{\bf
P}^{\mathtt{V}}_{+}\propto\sqrt{F}\,\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\hat{\mathbf{q}}}\vphantom{\hat{\mathbf{q}}}_{2}\,,$
$\displaystyle{\bf
P}^{\mathtt{V}}_{-}\propto\sqrt{F}\,\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\hat{\mathbf{q}}}\vphantom{\hat{\mathbf{q}}}_{1}\,$
(112)

as single-valued functions of Zhukovsky variable. The proportionality
$\propto$ used in this matching is simply proportionality up to a complex
number. Clearly $\bar{F}=\frac{1}{F}$ and furthermore the Q-functions obey
conjugation properties (109) 272727This reality property as stated has only
been verified for the ground state, see the resolvent parametrisation in
section 4.3 of Cavaglia:2015nta , and with purely real twist..

T-system is known to emerge not only as a result of TBA but also in algebraic
Bethe Ansatz for spin chains. Indeed, T-system exists for any $F$ and we
already saw that $F=x^{M_{\theta}}$ describes the finite-size Hubbard model;
an equivalent statement was made in Cavaglia:2015nta . The choice
$F={F_{\text{ext}}}$ also allows building T-systems, now for inhomogeneous
Hubbard models discussed in previous sections.

In the spin chain case, T-functions are expected to have interpretation of
transfer matrices. For a centrally extended ${\mathsf{su}}(2|2)$ spin,
eigenvalues for the transfer matrix $T_{1,1}$ were proposed in Beisert:2006qh
and later derived from algebraic Bethe Ansatz in MartinMello . Our expressions
for $\hat{T}_{1,1}$ match these eigenvalues up to redefinitions and an overall
gauge prefactor.

The basis which realise (109) is in general not appropriate for studying spin
chains, the reason being that there exist a distinguished basis with ${\bf
P}_{a}$ having different asymptotic. To match this basis we should pick
$\vec{n}\cdot\vec{\sigma}=\sigma^{1}$. This gives
$\overline{\hat{\mathbf{p}}_{a}}=(-1)^{a}\hat{\mathbf{p}}_{a}$ and
$\overline{\hat{\mathbf{q}}_{i}}=(-1)^{i}\hat{\mathbf{q}}_{i}\frac{1}{FU^{[2]}\bar{U}^{[-2]}}$.
The extra factors in the conjugation property of $\hat{\mathbf{q}}_{i}$
suggest redefining $\hat{\mathbf{q}}_{i}$. Indeed, for an $\mathsf{SU}(2|2)$
spin chain this is exactly what we did in (81), this definition leads to
purely imaginary/real $\mathbf{q}_{i}(x)$. Notice that to have a sensible
definition of reality we need to restrict to all twists being phases and pick
$y^{\pm}_{k}$ appropriately so that $\bar{F}=\frac{1}{F}$. $\hat{T}_{a,s}$,
including $\hat{T}_{1,1}$, are not real, this is the typical case for e.g.
rational spin chains if the symmetry is larger than $\mathsf{SU}(2)$.

##  7 Asymptotic Q-systems for the massive sector of AdS3/CFT2 integrability

The relevant to us study of integrability in AdS3/CFT2 correspondence was
initiated in Babichenko:2009dk  where a set of Bethe equations was proposed.
An important feature in AdS3/CFT2 is that the isometry group of space-time
splits as a product and one obtains two coupled sets of Bethe equations. Using
the S-matrices found in SMatricesAdS3  it was realised in
AdS${}_{3}\times$S${}^{3}\times$S${}^{3}\times$S1 Borsato:2012ss  that the
relative grading between the two sets of Bethe equations should be different.
This analysis was then extended Borsato:2013qpa  to
AdS${}_{3}\times$S${}^{3}\times$T4 where the following set of Bethe equations
were found

$\displaystyle
1=\prod^{M_{2}}_{j=1}\frac{y_{i,k}-x^{-}_{j}}{y_{i,k}-x^{+}_{j}}\prod^{M_{\bar{2}}}_{j=1}\frac{1-\frac{1}{y_{i,k}\overline{x}^{+}_{j}}}{1-\frac{1}{y_{i,k}\overline{x}^{-}_{j}}}\,,$
(113a) $\displaystyle
1=\prod^{M_{\bar{2}}}_{j=1}\frac{y_{\bar{i},k}-\overline{x}^{+}_{j}}{y_{\bar{i},k}-\overline{x}^{-}_{j}}\prod^{M_{2}}_{j=1}\frac{1-\frac{1}{y_{\bar{i},k}x^{-}_{j}}}{1-\frac{1}{y_{\bar{i},k}x^{+}_{j}}}\,,$
(113b)
$\displaystyle\left(\frac{x^{+}_{k}}{x^{-}_{k}}\right)^{L}=\prod_{\begin{subarray}{c}j=1\\\
j\neq
k\end{subarray}}^{M_{2}}\frac{u_{k}-u_{j}+i}{u_{k}-u_{j}-i}\left(\sigma^{\bullet\bullet}(x_{k},x_{j})\right)^{2}\prod_{j=1}^{M_{\bar{2}}}\frac{1-\frac{1}{x^{+}_{k}\overline{x}^{+}_{j}}}{1-\frac{1}{x^{-}_{k}\overline{x}^{-}_{j}}}\frac{1-\frac{1}{x^{+}_{k}\overline{x}^{-}_{j}}}{1-\frac{1}{x^{-}_{k}\overline{x}^{+}_{j}}}\left(\hat{\sigma}^{\bullet\bullet}(x_{k},\overline{x}_{j})\right)^{2}\times$
(113c)
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\prod_{j=1}^{M_{1}}\frac{x^{-}_{k}-y_{1,j}}{x^{+}_{k}-y_{1,j}}\prod_{j=1}^{M_{3}}\frac{x^{-}_{k}-y_{3,j}}{x^{+}_{k}-y_{3,j}}\prod_{j=1}^{M_{\bar{1}}}\frac{1-\frac{1}{x^{-}_{k}y_{\bar{1},j}}}{1-\frac{1}{x^{+}_{k}y_{\bar{1},j}}}\prod_{j=1}^{M_{\bar{3}}}\frac{1-\frac{1}{x^{-}_{k}y_{\bar{3},j}}}{1-\frac{1}{x^{+}_{k}y_{\bar{3},j}}}\,,$
$\displaystyle\left(\frac{\overline{x}^{+}_{k}}{\overline{x}^{-}_{k}}\right)^{L}=\prod_{\begin{subarray}{c}j=1\\\
j\neq
k\end{subarray}}^{M_{\bar{2}}}\frac{\overline{x}^{-}_{k}-\overline{x}^{+}_{j}}{\overline{x}^{+}_{k}-\overline{x}^{-}_{j}}\frac{1-\frac{1}{\overline{x}^{+}_{k}\overline{x}^{-}_{j}}}{1-\frac{1}{\overline{x}^{-}_{k}\overline{x}^{+}_{j}}}\left(\sigma^{\bullet\bullet}(\overline{x}_{k},\overline{x}_{j})\right)^{2}\prod_{j=1}^{M_{2}}\frac{1-\frac{1}{\overline{x}^{-}_{k}x^{-}_{j}}}{1-\frac{1}{\overline{x}^{-}_{k}x^{+}_{j}}}\frac{1-\frac{1}{\overline{x}^{+}_{k}x^{-}_{j}}}{1-\frac{1}{\overline{x}^{+}_{k}x^{+}_{j}}}\left(\hat{\sigma}^{\bullet\bullet}(\overline{x}_{k},x_{j})\right)^{2}\times$
(113d)
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\prod_{j=1}^{M_{\bar{1}}}\frac{\overline{x}^{+}_{k}-y_{\bar{1},j}}{\overline{x}^{-}_{k}-y_{\bar{1},j}}\prod_{j=1}^{M_{\bar{3}}}\frac{\overline{x}^{+}_{k}-y_{\bar{3},j}}{\overline{x}^{-}_{k}-y_{\bar{3},j}}\prod_{j=1}^{M_{1}}\frac{1-\frac{1}{\overline{x}^{+}_{k}y_{1,j}}}{1-\frac{1}{\overline{x}^{-}_{k}y_{1,j}}}\prod_{j=1}^{M_{3}}\frac{1-\frac{1}{\overline{x}^{+}_{k}y_{3,j}}}{1-\frac{1}{\overline{x}^{+}_{k}y_{3,j}}}\,.$

Here $x_{j}$ and $\overline{x}_{j}$ are distinct momentum-carrying roots;
$y_{i,k},y_{\bar{i},k},i=1,3$ are auxiliary roots; and
$\sigma^{\bullet\bullet}$ and $\hat{\sigma}^{\bullet\bullet}$ are dressing
phases. These phases are constrained by crossing equations considered and
solved in Borsato:2013hoa .

An important ingredient in AdS3/CFT2 correspondence are massless modes, there
also exist a full set of Bethe equations incorporating them Borsato:2016xns .

One should expect that Bethe equations can be obtained from QQ-relations and
that there thus should exist a Q-system of type $\mathsf{PSU}(1,1|2)^{2}$ to
reflect the symmetry of the full theory. In the language of Q-systems the
$\mathsf{P}$ of $\mathsf{PSU}$ is the zero central charge condition and the
specified non-compact real form means that the monodromy properties of
fermionic Q-functions, $\mathcal{Q}_{\emptyset|i}$, should be compatible with
non-integer asymptotics.

Bethe equations describe the spectrum only asymptotically and hence the
Q-system that reproduces them should do the same, we shall refer to it as the
asymptotic Q-system. Given the success in AdS5/CFT4 Gromov:2014caa  and
AdS4/CFT3 Bombardelli:2017vhk  integrability, one could expect that the
asymptotic Q-system might be improved to become a quantum spectral curve
describing the spectrum of the theory at finite value of the coupling constant
and charges. In Section 5, we suggested two non-trivial
$\mathsf{SU}(2|2)\times\mathsf{SU}(2|2)$ quantum spectral curves which would
be natural candidates for this role. Because we are interested only in a
system with zero central charge, the two quantum spectral curves cannot be
distinguished, see Section 4.1. We therefore have the unique QSC coming from
the monodromy bootstrap. It is the natural candidate for exact QSC of
AdS3/CFT2 integrability.

AdS3/CFT2 QSC Conjecture:  The QSC of AdS3 $\times$ S3 $\times$ T4 is given by
model C in the special case of
$F=\bar{F}=\det\mu_{{\dot{a}}a}=\det\bar{\mu}_{a{\dot{a}}}=\det\omega^{i\dot{i}}=\det\bar{\omega}^{\dot{i}i}=1$,
${{\rm p}_{\omega}}={{\rm p}_{\mu}}={{\rm p}_{\bar{\omega}}}={{\rm
p}_{\bar{\mu}}}=0$ and a power-like asymptotic behaviour of Q-functions at
infinity.

In component form, ${\bf P}\mu$-system (68) is

$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\bar{\mu}}\vphantom{\bar{\mu}}_{a{\dot{a}}}-\bar{\mu}_{a{\dot{a}}}=\frac{1}{r}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mathbf{P}}\vphantom{\mathbf{P}}_{a}\bar{\mathbf{P}}_{{\dot{a}}}-\frac{1}{\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{r}\vphantom{r}}\mathbf{P}_{a}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\bar{\mathbf{P}}}\vphantom{\bar{\mathbf{P}}}_{{\dot{a}}}\,,$
$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mu}\vphantom{\mu}_{{\dot{a}}a}-\mu_{{\dot{a}}a}=\frac{1}{\bar{r}}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\bar{\mathbf{P}}}\vphantom{\bar{\mathbf{P}}}_{{\dot{a}}}\mathbf{P}_{a}-\frac{1}{\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\bar{r}}\vphantom{\bar{r}}}\bar{\mathbf{P}}_{{\dot{a}}}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mathbf{P}}\vphantom{\mathbf{P}}_{a}\,,$
(114a)
$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
P}}\vphantom{{\bf
P}}_{a}=r\bar{\mu}_{a{\dot{a}}}\bar{\mathbf{P}}^{{\dot{a}}}\,,$
$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
P}}\vphantom{{\bf
P}}^{a}=-\frac{1}{r}\bar{\mathbf{P}}_{{\dot{a}}}\bar{\mu}^{{\dot{a}}a}\,,$
(114b)
$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\bar{\mathbf{P}}}\vphantom{\bar{\mathbf{P}}}_{{\dot{a}}}=\bar{r}\mu_{{\dot{a}}a}{\bf
P}^{a}\,,$
$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\bar{\mathbf{P}}}\vphantom{\bar{\mathbf{P}}}^{{\dot{a}}}=-\frac{1}{\bar{r}}{\bf
P}_{a}\mu^{a{\dot{a}}}\,.$ (114c)

${\bf Q}\omega$-system reads

$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\omega}\vphantom{\omega}^{i\dot{j}}-\omega^{i\dot{j}}=\frac{1}{r}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
Q}}\vphantom{{\bf
Q}}^{i}\bar{\mathbf{Q}}^{\dot{j}}-\frac{1}{\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{r}\vphantom{r}}{\bf
Q}^{i}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\bar{\mathbf{Q}}}\vphantom{\bar{\mathbf{Q}}}^{\dot{j}}\,,$
$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\bar{\omega}}\vphantom{\bar{\omega}}^{\dot{i}j}-\bar{\omega}^{\dot{i}j}=\frac{1}{\bar{r}}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\bar{\mathbf{Q}}}\vphantom{\bar{\mathbf{Q}}}^{\dot{i}}\mathbf{Q}^{j}-\frac{1}{\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\bar{r}}\vphantom{\bar{r}}}\bar{\mathbf{Q}}^{\dot{i}}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mathbf{Q}}\vphantom{\mathbf{Q}}^{j}\,,$
(115a)
$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
Q}}\vphantom{{\bf
Q}}_{i}=-\frac{1}{\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\bar{r}}\vphantom{\bar{r}}}\bar{\omega}_{i\dot{j}}\bar{\mathbf{Q}}^{\dot{j}}\,,$
$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
Q}}\vphantom{{\bf
Q}}^{i}=\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\bar{r}}\vphantom{\bar{r}}\bar{\mathbf{Q}}_{\dot{j}}\bar{\omega}^{\dot{j}i}\,,$
(115b)
$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\bar{\mathbf{Q}}}\vphantom{\bar{\mathbf{Q}}}_{\dot{i}}=-\frac{1}{\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{r}\vphantom{r}}\omega_{\dot{i}j}{\bf
Q}^{j}\,,$
$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\bar{\mathbf{Q}}}\vphantom{\bar{\mathbf{Q}}}^{\dot{i}}=\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{r}\vphantom{r}{\bf
Q}_{j}\omega^{j\dot{i}}\,.$ (115c)

For systems of $\mathsf{PSU}$-type, $r$ and $\bar{r}$ can be eliminated using
a gauge transformation. Indeed, from $\mathbf{P}_{a}\mathbf{P}^{a}=0$ it
follows that $\mathbf{P}_{a}=\alpha\epsilon_{ab}\mathbf{P}^{b}$. We can
compute then

$r\bar{\mu}_{a{\dot{a}}}\bar{\mathbf{P}}^{{\dot{a}}}=\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
P}}\vphantom{{\bf
P}}_{a}=\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\alpha}\vphantom{\alpha}\epsilon_{ab}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
P}}\vphantom{{\bf
P}}^{b}=-\frac{\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\alpha}\vphantom{\alpha}}{r}\epsilon_{ab}\bar{\mathbf{P}}_{{\dot{a}}}\bar{\mu}^{{\dot{a}}b}=-\frac{\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\alpha}\vphantom{\alpha}}{r}\epsilon_{ab}\epsilon^{{\dot{a}}\dot{b}}\epsilon^{bc}\bar{\mu}_{c\dot{b}}\bar{\mathbf{P}}_{{\dot{a}}}=\frac{\bar{\alpha}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\alpha}\vphantom{\alpha}}{r}\bar{\mu}_{a{\dot{a}}}\bar{\mathbf{P}}^{{\dot{a}}}\,,$
(116)

and so

$\displaystyle
r^{2}=\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\alpha}\vphantom{\alpha}\bar{\alpha}\,,$
$\displaystyle\bar{r}^{2}=\alpha\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\bar{\alpha}}\vphantom{\bar{\alpha}}\,.$
(117)

Analyticity of $\alpha$ was studied in Section 4.1. It is possible to pick a
gauge such that $r=\bar{r}=\pm 1$, and we can enforce the plus sign for
$r,\bar{r}$ by changing the sign of $\mu,\bar{\mu}$ and $\omega,\bar{\omega}$.

Setting $r=1$ is basically the same as enforcing re-definitions like ${\bf
P}_{1}\to\sqrt{{\bf P}_{1}{\bf P}^{2}}$. This may introduce extra square root
branch points (that are typically not correlated to Zhukovsky ones). To avoid
this potentially unwanted drawback in explicit computations, we shall keep $r$
unfixed.

In this section we shall demonstrate that the proposed exact system is
compatible, in the asymptotic limit, with the above Bethe equations and
furthermore we will attempt to gain better understanding of how the square
root property may fail. Thus we will assume the existence of two exact
Q-systems, $\mathcal{Q},\overline{\mathcal{Q}}$ with a well-defined asymptotic
limit reproducing (113). We will use the font $\mathtt{Q}$ to distinguish
asymptotic Q-functions from Q-functions of a full exact theory. In terms of
$\mathtt{Q}$, (113) becomes

$\displaystyle\frac{\overline{\mathtt{Q}}^{-}_{\dot{1}|\dot{1}}}{\overline{\mathtt{Q}}^{+}_{\dot{1}|\dot{1}}}\bigg{\rvert}_{\overline{\mathtt{Q}}_{\dot{1}|\emptyset}=0}=\frac{\mathtt{Q}^{+}_{1|1}}{\mathtt{Q}^{-}_{1|1}}\bigg{\rvert}_{\mathtt{Q}_{\emptyset|1}=0}=1\,,$
$\displaystyle\frac{\overline{\mathtt{Q}}^{-}_{\dot{1}|\dot{1}}}{\overline{\mathtt{Q}}^{+}_{\dot{1}|\dot{1}}}\bigg{\rvert}_{\overline{\mathtt{Q}}_{\dot{1}|\dot{1}\dot{2}}=0}=\frac{\mathtt{Q}^{+}_{1|1}}{\mathtt{Q}^{-}_{1|1}}\bigg{\rvert}_{\mathtt{Q}_{12|1}=0}=1\,,$
$\displaystyle\frac{\mathtt{Q}^{[2]}_{1|1}}{\mathtt{Q}^{[-2]}_{1|1}}\frac{\mathtt{Q}^{-}_{\emptyset|1}\mathtt{Q}^{-}_{12|1}}{\mathtt{Q}^{+}_{\emptyset|1}\mathtt{Q}^{+}_{12|1}}\bigg{\rvert}_{\mathtt{Q}_{1|1}=0}=-1\,,$
$\displaystyle\frac{\overline{\mathtt{Q}}^{[-2]}_{\dot{1}|\dot{1}}}{\overline{\mathtt{Q}}^{[2]}_{\dot{1}|\dot{1}}}\frac{\overline{\mathtt{Q}}^{+}_{\dot{1}|\emptyset}\overline{\mathtt{Q}}^{+}_{\dot{1}|\dot{1}\dot{2}}}{\overline{\mathtt{Q}}^{-}_{\dot{1}|\emptyset}\overline{\mathtt{Q}}^{-}_{\dot{1}|\dot{1}\dot{2}}}\bigg{\rvert}_{\overline{\mathtt{Q}}_{\dot{1}|\dot{1}}=0}=-1\,.$

There also exist different versions of Bethe equations corresponding to other
gradings of $\mathsf{PSU}(1,1|2)$ Kac-Dynkin diagram, see DifferentGradingPSU
. Using also these additional equations, or equivalently fermionic QQ-
relations, we find the asymptotic Q-functions

$\displaystyle\mathtt{Q}_{1|1}\propto\mathbb{Q}\,\mathtt{f}^{+}\bar{\mathtt{f}}^{+}\,,$
$\displaystyle\mathtt{Q}_{12|12}=1\,,$ (118)
$\displaystyle\mathtt{Q}_{1|\emptyset}\propto
x^{-L/2}B_{-}R_{\overset{\scalebox{0.8}{\raisebox{-1.75pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{y}\vphantom{y}_{1}}\sigma\hat{\bar{\sigma}}\,,$
$\displaystyle\mathtt{Q}_{1|12}\propto
x^{-L/2}B_{+}R_{\overset{\scalebox{0.8}{\raisebox{-1.75pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{y}\vphantom{y}_{3}}\sigma\hat{\bar{\sigma}}\,,$
$\displaystyle\mathtt{Q}_{\emptyset|1}\propto
x^{L/2}\frac{\bar{\mathtt{f}}}{\bar{B}_{+}}\mathtt{f}R_{y_{1}}\frac{1}{\sigma\hat{\bar{\sigma}}}\,,$
$\displaystyle\mathtt{Q}_{12|1}\propto
x^{L/2}\frac{\bar{\mathtt{f}}}{\bar{B}_{+}}\mathtt{f}^{[2]}R_{y_{3}}\frac{1}{\sigma\hat{\bar{\sigma}}}\,,$
$\displaystyle\overline{\mathtt{Q}}_{\dot{1}|\dot{1}}\propto\bar{\mathbb{Q}}\,\bar{\mathtt{f}}^{+}\mathtt{f}^{+}\,,$
$\displaystyle\overline{\mathtt{Q}}_{\dot{1}\dot{2}|\dot{1}\dot{2}}=1\,,$
$\displaystyle\overline{\mathtt{Q}}_{\dot{1}|\emptyset}\propto
x^{-L/2}\bar{B}_{-}B_{y_{1}}\hat{\sigma}\bar{\sigma}\,,$
$\displaystyle\overline{\mathtt{Q}}_{\dot{1}|\dot{1}\dot{2}}\propto
x^{-L/2}\bar{B}_{+}B_{y_{3}}\hat{\sigma}\bar{\sigma}\,,$
$\displaystyle\overline{\mathtt{Q}}_{\emptyset|\dot{1}}\propto
x^{L/2}\frac{\mathtt{f}}{B_{+}}\bar{\mathtt{f}}B_{\overset{\scalebox{0.8}{\raisebox{-1.75pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{y}\vphantom{y}_{1}}\frac{1}{\hat{\sigma}\bar{\sigma}}\,,$
$\displaystyle\overline{\mathtt{Q}}_{\dot{1}\dot{2}|\dot{1}}\propto
x^{L/2}\frac{\mathtt{f}}{B_{+}}\bar{\mathtt{f}}^{[2]}B_{\overset{\scalebox{0.8}{\raisebox{-1.75pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{y}\vphantom{y}_{3}}\frac{1}{\hat{\sigma}\bar{\sigma}}\,.$

The notation, mirroring the standard notation from AdS5/CFT4, is as follows
for objects related to momentum-carrying nodes

$\displaystyle
B_{(\pm)}=\prod_{j=1}^{M_{2}}\sqrt{\frac{{\mbox{h}}}{2x^{\mp}_{j}}}(\frac{1}{x}-x^{\mp}_{j})\,,$
$\displaystyle\bar{B}_{(\pm)}=\prod_{j=1}^{M_{\bar{2}}}\sqrt{\frac{{\mbox{h}}}{2\overline{x}^{\mp}_{j}}}(\frac{1}{x}-\overline{x}^{\mp}_{j})\,,$
(119a) $\displaystyle
R_{(\pm)}=\prod_{j=1}^{M_{2}}\sqrt{\frac{{\mbox{h}}}{2x^{\mp}_{j}}}(x-x^{\mp}_{j})\,,$
$\displaystyle\bar{R}_{\pm}=\prod^{M_{\bar{2}}}_{j=1}\sqrt{\frac{{\mbox{h}}}{2\overline{x}^{\mp}_{j}}}(x-\overline{x}^{\mp}_{j})\,,$
(119b)
$\displaystyle\mathtt{f}=\prod_{n=0}^{\infty}\frac{B^{[2n]}_{(+)}}{B^{[2n]}_{(-)}}\,,$
$\displaystyle\bar{\mathtt{f}}=\prod^{\infty}_{n=0}\frac{\bar{B}^{[2n]}_{(+)}}{\bar{B}^{[2n]}_{(-)}}\,.$
(119c)

For the auxiliary roots $y_{1,k},y_{3,k},y_{\bar{1},k},y_{\bar{3},k}$, we have
used

$\displaystyle
R_{y_{i}}=\prod_{k=1}^{M_{i}}(x-y_{i,k})\prod_{k=1}^{M_{\bar{i}}}(\frac{1}{x}-y_{\bar{i},k})\,,$
$\displaystyle
B_{y_{i}}=\prod_{k=1}^{M_{\bar{i}}}(x-y_{\bar{i},k})\prod_{k=1}^{M_{i}}(\frac{1}{x}-y_{i,k})\,,$
(120)

so that in particular
$\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{R}\vphantom{R}_{y_{i}}=B_{y_{i}}$.
The auxiliary roots are not independent, for example QQ-relation
$\mathtt{Q}_{1|\emptyset}\mathtt{Q}_{\emptyset|1}=\mathtt{Q}_{1|1}^{+}-\mathtt{Q}_{1|1}^{-}=-\mathtt{Q}_{1|12}\mathtt{Q}_{12|1}$
enforces the relation
$R_{\overset{\scalebox{0.8}{\raisebox{-1.75pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{y}\vphantom{y}_{1}}R_{y_{1}}\propto
R_{y_{3}}R_{\overset{\scalebox{0.8}{\raisebox{-1.75pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{y}\vphantom{y}_{3}}$.
Also note that we have assumed the zero-momentum condition

$\displaystyle\prod_{k=1}^{M_{2}}\frac{x^{+}_{k}}{x^{-}_{k}}\prod_{k=1}^{M_{\bar{2}}}\frac{\overline{x}^{+}_{k}}{\overline{x}^{-}_{k}}=1\,$
(121)

in order to fully reproduce (113).

For convenience we will also use
$\mathtt{f}_{\text{tot}}\equiv\mathtt{f}\bar{\mathtt{f}}$. Finally we note
that the dressing phases can be computed as

$\displaystyle\frac{\sigma^{+}(x)}{\sigma^{-}(x)}\equiv\prod_{j=1}^{M_{2}}\sigma^{\bullet\bullet}(x,x_{k})\,,$
$\displaystyle\frac{\hat{\sigma}^{+}(x)}{\hat{\sigma}^{-}(x)}\equiv\prod_{j=1}^{M_{2}}\hat{\sigma}^{\bullet\bullet}(x,x_{k})\,,$
(122)

with the natural generalisation for the right-moving quantities. We did not
impose any constraints on $\sigma$ and $\hat{\sigma}$ so far, deriving
crossing equations for them from comparison with the exact QSC will be an
important check.

We now turn to the proposed exact quantum spectral curve. From symmetry
principles Gromov:2014caa , Q-functions in the large-$u$ limit should scale as

$\displaystyle\mathbf{P}_{a}\sim u^{-\hat{\lambda}_{a}^{L}}\,,$
$\displaystyle\mathbf{P}^{a}\sim u^{\hat{\lambda}^{L}_{a}-1}\,,$
$\displaystyle\mathbf{Q}_{i}\sim u^{-\hat{\nu}_{i}^{L}-1}\,,$
$\displaystyle\mathbf{Q}^{i}\sim u^{\hat{\nu}^{L}_{i}}\,,$
$\displaystyle\bar{\mathbf{P}}_{{\dot{a}}}\sim u^{-\hat{\lambda}^{R}_{a}}\,,$
$\displaystyle\bar{\mathbf{P}}^{\dot{a}}\sim u^{\hat{\lambda}^{R}_{a}-1}\,,$
$\displaystyle\bar{\mathbf{Q}}_{{\dot{i}}}\sim u^{-\hat{\nu}^{R}_{i}-1}\,,$
$\displaystyle\bar{\mathbf{Q}}^{{\dot{i}}}\sim u^{\hat{\nu}^{R}_{i}}\,,$ (123)

where $\hat{\lambda},\hat{\nu}$ are shifted weights of the left and the right
symmetry algebra. In the current $\mathsf{SU}(2)\times\mathsf{SL}(2)$ grading,
we find
$\hat{\lambda}^{a}_{1}=\lambda^{a}_{1}+\delta_{a,L},\hat{\lambda}^{a}_{2}=\lambda^{a}_{2}+\delta_{a,R},\hat{\nu}^{a}_{1}=\nu^{a}_{1}-\delta_{a,L},\hat{\nu}^{a}_{2}=\nu^{2}_{a}-\delta_{a,R}$,
see Appendix C in Gromov:2014caa  for more details. These expressions also
follow from quasi-momentum that defines the classical spectral curve of
AdS3/CFT2 Babichenko:2014yaa . We can express the weights in terms of the
energy and the spin, $\Delta,S$ of AdS3 as well as the two angular momenta of
S3, $J,K$:

$\displaystyle\lambda^{L}_{1}=\frac{1}{2}(J+K)+\Lambda^{L}\,,$
$\displaystyle\lambda^{L}_{2}=-\frac{1}{2}(J+K)+\Lambda^{L}\,,$
$\displaystyle\nu_{1}^{L}=-\frac{1}{2}(\Delta+S)-\Lambda^{L}\,,$
$\displaystyle\nu^{L}_{2}=\frac{1}{2}(\Delta+S)-\Lambda^{L}\,,$
$\displaystyle\lambda^{R}_{1}=\frac{1}{2}(J-K)+\Lambda^{R}\,,$
$\displaystyle\lambda^{R}_{2}=-\frac{1}{2}(J-K)+\Lambda^{R}\,,$
$\displaystyle\nu^{R}_{1}=-\frac{1}{2}(\Delta-S)-\Lambda^{R}\,,$
$\displaystyle\nu^{R}_{2}=\frac{1}{2}(\Delta-S)-\Lambda^{R}\,.$

Explicit expressions for charges in terms of Bethe roots Borsato:2013qpa  set
$\Lambda^{L}=\frac{1}{2}(M_{1}-M_{3}),\,\Lambda^{R}=-\frac{1}{2}(M_{\bar{1}}-M_{\bar{3}})$.

We assume as in Gromov:2014caa  that the large-$u$ scaling dictates, at least
on the defining sheet, the large volume scaling. We define the large volume
regime as the limit $\Delta\simeq J\rightarrow\infty$. To keep track of the
scaling, we introduce the parameter $\epsilon=u^{-\frac{\Delta}{2}}$. Using
the explicit scalings given in (7), we find

$\displaystyle\mathbf{P}_{1}\sim\mathbf{P}^{2}\sim\epsilon\,,$
$\displaystyle\mathbf{P}_{2}\sim\mathbf{P}^{1}\sim\epsilon^{-1}\,,$
$\displaystyle\mathbf{Q}_{1}\sim\mathbf{Q}^{2}\sim\epsilon^{-1}\,,$
$\displaystyle\mathbf{Q}_{2}\sim\mathbf{Q}^{1}\sim\epsilon\,,$ (124a)
$\displaystyle\bar{\mathbf{P}}_{\dot{1}}\sim\bar{\mathbf{P}}^{\dot{2}}\sim\epsilon\,,$
$\displaystyle\bar{\mathbf{P}}_{\dot{2}}\sim\bar{\mathbf{P}}^{\dot{1}}\sim\epsilon^{-1}\,,$
$\displaystyle\bar{\mathbf{Q}}_{\dot{1}}\sim\bar{\mathbf{Q}}^{\dot{2}}\sim\epsilon^{-1}\,,$
$\displaystyle\bar{\mathbf{Q}}_{\dot{2}}\sim\bar{\mathbf{Q}}^{\dot{1}}\sim\epsilon\,.$
(124b)

Use the relations

$\displaystyle\bar{\mu}_{a{\dot{a}}}=-\omega^{i{\dot{i}}}\,\mathcal{Q}^{-}_{a|i}\,\overline{\mathcal{Q}}^{-}_{{\dot{a}}|{\dot{i}}}\,,$
$\displaystyle\mu_{{\dot{a}}a}=-\bar{\omega}^{{\dot{i}}i}\,\overline{\mathcal{Q}}^{-}_{{\dot{a}}|{\dot{i}}}\,\mathcal{Q}^{-}_{a|i}\,.$
(125)

Now, since $\omega^{i{\dot{i}}},\bar{\omega}^{{\dot{i}}i}$ are periodic, they
should scale as constants at large $u$ or decrease exponentially. At the same
time, from the scaling of Q-functions and the QQ-relation
$\mathbf{P}_{a}\mathbf{Q}_{i}=\mathcal{Q}_{a|i}^{+}-\mathcal{Q}^{-}_{a|i}$ we
find that $\mathcal{Q}_{a|i}\sim
u^{-\hat{\lambda}^{L}_{a}-\hat{\nu}^{L}_{i}}$. In particular

$\displaystyle\mathcal{Q}_{1|1}\sim 1\,,$
$\displaystyle\mathcal{Q}_{1|2}\sim\epsilon^{2}\,,$
$\displaystyle\mathcal{Q}_{2|1}\sim\epsilon^{-2}\,,$
$\displaystyle\mathcal{Q}_{2|2}\sim 1\,,$ (126)

with the same scaling for $\overline{\mathcal{Q}}_{{\dot{a}}|{\dot{i}}}$.
Plugging this into (125), we deduce

$\displaystyle(\bar{\mu}_{\text{as}})_{a\dot{a}}=-(\omega_{\text{as}})^{1\dot{1}}\mathtt{Q}^{-}_{a|1}\overline{\mathtt{Q}}^{-}_{\dot{a}|\dot{1}}\,,$
$\displaystyle{(\mu_{\text{as}})}_{{\dot{a}}a}=-(\bar{\omega}_{\text{as}})^{\dot{1}1}\overline{\mathtt{Q}}^{-}_{{\dot{a}}|\dot{1}}\mathtt{Q}^{-}_{a|1}\,,$
(127)

where we write $(\bar{\mu}_{\text{as}})_{a\dot{a}}$ for the asymptotic
versions of $\bar{\mu}_{a\dot{a}}$ and so on. It is reassuring to see that
from the scaling of $\mathcal{Q}_{a|i}$ it follows that

$(\bar{\mu}_{\text{as}})_{1\dot{1}}\sim
u^{-\hat{\lambda}_{1}^{L}-\hat{\nu}_{1}^{L}-\hat{\lambda}_{1}^{R}-\hat{\nu}^{R}_{1}}\sim
u^{\Delta-J}\,,$ (128)

and, using the definition of $\mathtt{f}_{\text{tot}}$, one reproduces the
standard expression for the energy in terms of Bethe roots Gromov:2011cx . We
focus now on
${(\mu_{\text{as}})}_{\dot{1}1},(\bar{\mu}_{\text{as}})_{1\dot{1}}$ as they
depend on $\mathtt{Q}$-functions that appear directly in asymptotic Bethe
equations. From periodicity of $\omega$ we find

$\frac{{(\bar{\mu}_{\text{as}})}^{[2]}_{1\dot{1}}}{(\bar{\mu}_{\text{as}})_{1\dot{1}}}=\frac{\mathtt{Q}^{+}_{1|1}\overline{\mathtt{Q}}^{+}_{\dot{1}|\dot{1}}}{\mathtt{Q}^{-}_{1|1}\overline{\mathtt{Q}}^{-}_{\dot{1}|\dot{1}}}=\frac{\mathbb{Q}^{+}\bar{\mathbb{Q}}^{+}}{\mathbb{Q}^{-}\bar{\mathbb{Q}}^{-}}\left(\frac{\mathtt{f}_{\text{tot}}^{[2]}}{\mathtt{f}_{\text{tot}}}\right)^{2}\,,$
(129)

where we have used the explicit expressions for the asymptotic Q-functions in
the second expression. It follows that $(\bar{\mu}_{\text{as}})_{1\dot{1}}$
and ${(\mu_{\text{as}})}_{\dot{1}1}$ must have square-root branch cuts to
match the proposed asymptotic Q-system.

We also have

$\displaystyle(\bar{\mu}_{\text{as}})_{1\dot{1}}\propto{(\mu_{\text{as}})}_{\dot{1}1}\propto\mathbb{Q}^{-}\bar{\mathbb{Q}}^{-}(\mathtt{f}_{\text{tot}})(\mathtt{f}_{\text{tot}}^{*})^{[-2]}\,,$
$\displaystyle(\bar{\omega}_{\text{as}})^{\dot{1}1}\propto(\omega_{\text{as}})^{1\dot{1}}\propto\frac{\mathtt{f}_{\text{tot}}^{*[-2]}}{\mathtt{f}_{\text{tot}}}\,,$
(130)

where
$\mathtt{f}_{\text{tot}}^{*}=\prod\limits_{n=0}^{\infty}\frac{B^{[-2n]}_{(-)}}{B^{[-2n]}_{(+)}}\frac{\bar{B}^{[-2n]}_{(-)}}{\bar{B}^{[-2n]}_{(+)}}$.
These expressions are clearly similar to those in (36) but
$\mathtt{f}_{\text{tot}}$ is not related to any gauge transformation.

We next compute the asymptotic approximation of exact ${\bf P}$-functions on
their next-to-the defining Riemann sheets (in the physical kinematics). It
will be instructive to calculate both clock-wise and counter-clock-wise:

$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
P}}\vphantom{{\bf
P}}_{a}=r\bar{\mu}_{a{\dot{a}}}\bar{\mathbf{P}}^{{\dot{a}}}=-r\omega^{i{\dot{i}}}\mathcal{Q}^{-}_{a|i}\overline{\mathcal{Q}}^{-}_{{\dot{a}}|{\dot{i}}}\bar{\mathbf{P}}^{{\dot{a}}}=-r\omega^{i{\dot{i}}}\mathcal{Q}^{-}_{a|i}\bar{\mathbf{Q}}_{{\dot{i}}}\sim
r(\omega_{\text{as}})^{1\dot{1}}\mathtt{Q}^{-}_{a|1}\overline{\mathtt{Q}}_{\emptyset|\dot{1}}\,,$
(131a)
$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
P}}\vphantom{{\bf
P}}_{a}=-\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\bar{r}}\vphantom{\bar{r}}\bar{\mathbf{P}}^{{\dot{a}}}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mu}\vphantom{\mu}_{{\dot{a}}a}=\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\bar{r}}\vphantom{\bar{r}}\bar{\omega}^{{\dot{i}}i}\overline{\mathcal{Q}}^{+}_{{\dot{a}}|{\dot{i}}}\mathcal{Q}^{+}_{a|i}\bar{\mathbf{P}}^{{\dot{a}}}=\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\bar{r}}\vphantom{\bar{r}}\bar{\omega}^{{\dot{i}}i}\bar{\mathbf{Q}}_{{\dot{i}}}\mathcal{Q}^{+}_{a|i}\sim\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\bar{r}}\vphantom{\bar{r}}(\bar{\omega}_{\text{as}})^{\dot{1}1}\mathtt{Q}^{+}_{a|1}\overline{\mathtt{Q}}_{\emptyset|\dot{1}}\,.$
(131b)

Can we assume
$\big{(}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
P}}\vphantom{{\bf P}}\big{)}_{\rm
as}=\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{({\bf
P}_{\rm as})}\vphantom{({\bf P}_{\rm as})}$? In general, taking the asymptotic
limit does not commute with the analytic continuation across the cut. However
we expect this to be the case for ${\bf P}_{1}=\mathcal{Q}_{1|\emptyset}$ and
${\bf P}^{2}\propto\mathcal{Q}_{1|12}$ because these Q-functions are small in
the asymptotic limit. In Stokes-type phenomena (and crossing the branch have
similar features), analytic continuation of the small solution across the
Stokes line can be trusted, while analytic continuation of a large solution
cannot be.

We summarise

$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mathtt{Q}}\vphantom{\mathtt{Q}}_{1|\emptyset}\propto
r(\omega_{\text{as}})^{1\dot{1}}\mathtt{Q}^{-}_{1|1}\overline{\mathtt{Q}}_{\emptyset|\dot{1}}\,,$
$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mathtt{Q}}\vphantom{\mathtt{Q}}_{1|12}\propto\frac{1}{r}(\omega_{\text{as}})^{1\dot{1}}\mathtt{Q}^{-}_{1|1}\overline{\mathtt{Q}}_{\dot{1}\dot{2}|\dot{1}}\,,$
(132a)
$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mathtt{Q}}\vphantom{\mathtt{Q}}_{1|\emptyset}\propto\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\bar{r}}\vphantom{\bar{r}}(\bar{\omega}_{\text{as}})^{\dot{1}1}\mathtt{Q}^{+}_{1|1}\overline{\mathtt{Q}}_{\emptyset|1}\,,$
$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mathtt{Q}}\vphantom{\mathtt{Q}}_{1|12}\propto\frac{1}{\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\bar{r}}\vphantom{\bar{r}}}(\bar{\omega}_{\text{as}})^{\dot{1}1}\mathtt{Q}^{+}_{1|1}\overline{\mathtt{Q}}_{\dot{1}\dot{2}|\dot{1}}\,,$
(132b)

and with the clear generalisation for
$\overline{\mathtt{Q}}_{\dot{1}|\emptyset},\overline{\mathtt{Q}}_{\dot{1}|\dot{1}\dot{2}}$.

We use these expressions to compute the analytic continuation of
$\sigma\hat{\bar{\sigma}}$. It is useful to combine
$\mathtt{Q}_{1|0}\mathtt{Q}_{1|12}$ and
$\mathtt{Q}_{1|\emptyset}\mathtt{Q}_{1|12}$ because factors of $r,\bar{r}$
will cancel then. This gives

$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mathtt{Q}}\vphantom{\mathtt{Q}}_{1|\emptyset}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mathtt{Q}}\vphantom{\mathtt{Q}}_{1|12}\propto\left((\omega_{\text{as}})^{1\dot{1}}\mathtt{Q}_{1|1}^{-}\right)^{2}\overline{\mathtt{Q}}_{\emptyset|\dot{1}}\overline{\mathtt{Q}}_{\dot{1}\dot{2}|\dot{1}}=\left((\mathtt{f}_{\text{tot}}^{*})^{[-2]}\right)^{2}(\mathbb{Q}^{-})^{2}\overline{\mathtt{Q}}_{\emptyset|\dot{1}}\overline{\mathtt{Q}}_{\dot{1}\dot{2}|\dot{1}}\,,$
$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mathtt{Q}}\vphantom{\mathtt{Q}}_{1|\emptyset}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mathtt{Q}}\vphantom{\mathtt{Q}}_{1|12}\propto\left((\omega_{\text{as}})^{1\dot{1}}\mathtt{Q}_{1|1}^{+}\right)^{2}\overline{\mathtt{Q}}_{\emptyset|\dot{1}}\overline{\mathtt{Q}}_{\dot{1}\dot{2}|\dot{1}}=\left((\mathtt{f}_{\text{tot}}^{*})^{[-2]}\right)^{2}\left(\frac{R_{(+)}}{R_{(-)}}\right)^{2}\left(\frac{\bar{B}_{(-)}}{\bar{B}_{(+)}}\right)^{2}(\mathbb{Q}^{-})^{2}\overline{\mathtt{Q}}_{\emptyset|\dot{1}}\overline{\mathtt{Q}}_{\dot{1}\dot{2}|\dot{1}}\,.$

We see that these equations are compatible with the relation
$\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{R_{\overset{\scalebox{0.8}{\raisebox{-1.75pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{y}\vphantom{y}_{1}}R_{\overset{\scalebox{0.8}{\raisebox{-1.75pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{y}\vphantom{y}_{3}}}\vphantom{R_{\overset{\scalebox{0.8}{\raisebox{-1.75pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{y}\vphantom{y}_{1}}R_{\overset{\scalebox{0.8}{\raisebox{-1.75pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{y}\vphantom{y}_{3}}}=B_{\overset{\scalebox{0.8}{\raisebox{-1.75pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{y}\vphantom{y}_{1}}B_{\overset{\scalebox{0.8}{\raisebox{-1.75pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{y}\vphantom{y}_{3}}$.
We can then plug in our asymptotic Q-functions to find the following crossing
equations for the dressing phases

$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{(\sigma\hat{\bar{\sigma}})}\vphantom{(\sigma\hat{\bar{\sigma}})}^{2}(\bar{\sigma}\hat{\sigma})^{2}=(\mathtt{f}_{\text{tot}}^{[2]}\mathtt{f}_{\text{tot}}^{*[-2]})^{2}\frac{R_{(-)}}{R_{(+)}}\frac{\bar{B}_{(+)}}{\bar{B}_{(-)}}\,,$
$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{(\hat{\sigma}\bar{\sigma})}\vphantom{(\hat{\sigma}\bar{\sigma})}^{2}(\sigma\hat{\bar{\sigma}})^{2}=(\mathtt{f}_{\text{tot}}^{[2]}\mathtt{f}_{\text{tot}}^{*[-2]})^{2}\frac{\bar{R}_{(-)}}{\bar{R}_{(+)}}\frac{B_{(+)}}{B_{(-)}}\,,$
(133)
$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{(\sigma\hat{\bar{\sigma}})}\vphantom{(\sigma\hat{\bar{\sigma}})}^{2}(\bar{\sigma}\hat{\sigma})^{2}=(\mathtt{f}_{\text{tot}}^{[2]}\mathtt{f}_{\text{tot}}^{*[-2]})^{2}\frac{R_{(+)}}{R_{(-)}}\frac{\bar{B}_{(-)}}{\bar{B}_{(+)}}\,,$
$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{(\hat{\sigma}\bar{\sigma})}\vphantom{(\hat{\sigma}\bar{\sigma})}^{2}(\sigma\hat{\bar{\sigma}})^{2}=(\mathtt{f}_{\text{tot}}^{[2]}\mathtt{f}_{\text{tot}}^{*[-2]})^{2}\frac{\bar{R}_{(+)}}{\bar{R}_{(-)}}\frac{B_{(-)}}{B_{(+)}}\,.$
(134)

We notice that the factor of
$\mathtt{f}_{\text{tot}}^{[2]}\mathtt{f}_{\text{tot}}^{*[-2]}$ is a familiar
BES-type of a phase. Absorb this factor by a redefinition:

$\displaystyle(\sigma\hat{\bar{\sigma}})^{2}=({\sigma_{\text{BES}}})^{2}\sigma^{(+)}\,,$
$\displaystyle(\bar{\sigma}\hat{\sigma})^{2}=({\sigma_{\text{BES}}})^{2}\bar{\sigma}^{(+)}$
(135)

so that we can summarise the remaining equations as

$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\sigma^{(+)}}\vphantom{\sigma^{(+)}}\bar{\sigma}^{(+)}=\left(\frac{R_{(-)}}{R_{(+)}}\frac{\bar{B}_{(+)}}{\bar{B}_{(-)}}\right)^{\eta}\,,$
$\displaystyle\sigma^{(+)}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\bar{\sigma}^{(+)}}\vphantom{\bar{\sigma}^{(+)}}=\left(\frac{\bar{R}_{(-)}}{\bar{R}_{(+)}}\frac{B_{(+)}}{B_{(-)}}\right)^{\eta}\,,$
(136)

where $\eta=\pm 1$ for clock-wise/counter-clock-wise crossing. Clearly we have
two different results and we conclude that with our current assumptions the
branch-cut cannot be of square root type. Instead if we compute what happens
when we cross the cut twice

${\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\approx$}}}}{\sigma^{(+)}}\vphantom{\sigma^{(+)}}}=\left(\frac{B_{(-)}}{B_{(+)}}\frac{\bar{R}_{(+)}}{\bar{R}_{(-)}}\right)^{2\eta}\sigma^{(+)}$
(137)

we see that the cut is of a logarithmic type. The crossing equations found
here match exactly those of Borsato:2013hoa , see also sax2019singularities .
This is a non-trivial result that supports our conjecture.

Let us now turn to the question of cut structure for $\mu_{{\dot{a}}a}$ and
$\bar{\mu}_{a{\dot{a}}}$ in the asymptotic limit. We already know that
$(\bar{\mu}_{\text{as}})_{1\dot{1}},{(\mu_{\text{as}})}_{\dot{1}1}$ have
square root cuts. However, consider now $(\bar{\mu}_{\text{as}})_{2\dot{1}}$.
To compute this object, we need $\mathtt{Q}_{2|1}$ which can be obtained from
the QQ-relation
$\mathtt{Q}_{2|\emptyset}\mathtt{Q}_{\emptyset|1}=\mathtt{Q}_{2|1}^{+}-\mathtt{Q}_{2|1}^{-}$.
We find $\mathtt{Q}_{2|\emptyset}$ from

$\displaystyle\mathtt{Q}_{1|\emptyset}^{+}\,\mathtt{Q}_{2|\emptyset}^{-}-\mathtt{Q}_{1|\emptyset}^{-}\,\mathtt{Q}_{2|\emptyset}^{+}=\mathtt{Q}_{12|\emptyset}\,,$
$\displaystyle\mathtt{Q}_{12|1}^{+}\,\mathtt{Q}_{1|\emptyset}^{-}-\mathtt{Q}_{12|1}^{-}\,\mathtt{Q}_{1|\emptyset}^{+}=\mathtt{Q}_{1|1}\mathtt{Q}_{12|\emptyset}\,,$
(138)

Combining these equations and using $f^{+}\,g^{-}-f^{-}\,g^{+}=h\implies
f=-g\sum_{n=0}^{\infty}\left(\frac{h}{g^{+}g^{-}}\right)^{[2n+1]}$ gives

$\mathtt{Q}_{2|\emptyset}\propto-\frac{\mathtt{Q}_{12|1}}{\mathtt{Q}_{1|1}^{+}}+\mathtt{Q}_{1|\emptyset}\times\dots\,,$
(139)

where the suppressed terms do not have cuts on the real axis. We see that the
dressing factors in $\mathtt{Q}_{2|\emptyset}\mathtt{Q}_{\emptyset|1}$ do not
cancel out as happens for $\mathtt{Q}_{1|\emptyset}\mathtt{Q}_{\emptyset|1}$.
This immediately implies
$\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\approx$}}}}{\mathtt{Q}}\vphantom{\mathtt{Q}}^{-}_{2|1}\neq\mathtt{Q}^{-}_{2|1}$
and so from (127) it follows that $(\bar{\mu}_{\text{as}})_{2\dot{1}}$ does
not have a square-root cut. Thus the no-square root property can be seen in
the large volume limit, not only for some Q-functions but also for some
$\bar{\mu}_{a\dot{a}}$.

Before closing this section we comment on the appearance of massless modes.
Turning off all massive excitations, Bethe equations sourced by massless modes
$z_{k}^{\pm}\equiv(z_{k})^{\pm 1}$ are

$\displaystyle\prod_{k=1}^{M_{\theta}}\frac{y_{i}-z^{-}_{k}}{y_{i}-z^{+}_{k}}=1\,.$
(140)

One can notice that these will be auxiliary equations,
$\frac{\mathcal{Q}^{+}_{1|1}}{\mathcal{Q}_{1|1}^{-}}=1$, in the type-B QSC
with $F=F_{\text{pol}}$, under the replacement $\theta_{k}\to z_{k}$, and
assuming momentum conservation $\prod z_{k}=1$. This suggests that redefining
$\mathtt{f}_{\text{tot}}\to\mathtt{f}_{\text{tot}}\prod\limits_{n=0}^{\infty}F_{\text{pol}}^{[2n]}$
might be a way to include massless modes into the above analysis.

##  8 Conclusions and discussion

In this paper we proposed a method to derive quantum spectral curves of
AdS/CFT type. The method consists of the special prescription for analytic
continuation and the monodromy requirement: Q-system, a fused flag encoding
conserved charges, transforms by a symmetry after the continuation. In the
considered examples, and we believe that the statement is general, the
resulting QSC’s are labelled by the equivalence classes
$\mathsf{Out}\simeq\mathsf{Aut}/\mathsf{Inn}$, where $\mathsf{Inn}$ is the
group of continuous symmetries (H-rotations and gauge transformations) and
$\mathsf{Aut}$ are all symmetries (including also the discrete ones: Hodge
duality and the swap of left and right Lie groups). For the $\mathsf{SU}(2|2)$
case, there are two options (cases A and B), and for the
$\mathsf{SU}(2|2)\times\mathsf{SU}(2|2)$ there are eight options out of which
the two are interesting (cases C and D).

The focus of the proposed classification is to distinguish between the
emerging Riemann-Hilbert problems. On top of them, extra analytic requirements
such as asymptotic behaviour at infinity or additional periodicity features
are needed to nail down the concrete physical system, and different systems
are possible with the same type of monodromy behaviour. For instance, we
explained in Section 6 that the type-B system with square root cuts can encode
Lieb-Wu equations, their inhomogeneous generalisation, and also thermodynamic
Bethe Ansatz equations for Hubbard model, depending on which additional
analytic requirements do we choose. These options are unlikely to form the
exhaustive set of possible physics described by the type-B QSC. For one more
possibility, let us mention q-deformation of the inhomogeneous Hubbard model
Beisert:2008tw . We expect that the same QSC should describe it, an explicit
analysis verifying this proposal is yet to be done.

A situation when an automorphism of the system of conserved charges (Q-system
or T-system) is emerging in the result of ‘large’ change of spectral parameter
appears also in other integrable systems. For instance, in the case of
integrable spin chains with boundary described as representations of twisted
Yangians or quantum twisted affine algebras (or related systems, e.g. in the
context of ODE/IM), the automorphism is due when one makes a reflection
$u\to-u$ (rational case) or the half-way rotation $z\to e^{\mathtt{i}\pi}z$
(trigonometric case), see for instance Molev:1994rs ; Masoero:2015rcz ;
2016Guay ; Frenkel:2016gxg . These examples deal with entire functions of the
spectral parameter so the stated large change of $u$ or $z$ is achieved by the
unambiguous analytic continuation. Furthermore, since $u=0$ and $z=0$ are
regular points in these examples, all possible automorphisms should become
‘inner’ (continuous) which forces folding of Dynkin diagram if the
automorphism involves the reflection of this diagram 282828An alternative is
to create two copies of Q-systems and fold them into one by the relation of
style $Q_{\rm copy~{}1}(u)=Q_{\rm copy~{}2}^{*}(-u)$..

By contrast in our case, the large change of $u$ is the encircling of the
branch point $u={\mbox{h}}$, and a naive attempt of the analytic continuation
interferes with the non-locality of Q-system. We circumvent this difficulty by
performing symmetry transformations while doing the continuation. In the
result, we get a very non-trivial representative in $\mathsf{Aut}$ of the
equivalence class in $\mathsf{Out}$. For instance, a representative of the
class ${\rm[Hodge]}$ defining the type-B system is Hodge transformation
supplemented with gauge transformations, with physically relevant information
packaged in function $F$, and with H-rotations encoded through matrices $\mu$
and $\omega$. Note that the mentioned continuous symmetries are even not
global ones but defined either in the mirror or in the physical kinematics.
Typically, the representative of the equivalence class does not square to the
identity operation, taking higher powers does not generically become the
identity either. A similar effect is observed in Janik’s crossing equation
Janik:2006dc , and essentially due to the same reason – non-locality in the
cut structure of the scattering matrix. In fact, the monodromy bootstrap is a
rather natural development of the crossing equation idea.

The non-triviality of the equivalence class representative, although it does
not imply per se, nicely correlates with the observation that Q-functions
almost never have branch points of square root type (i.e. of the second
order). While it was possible in higher-dimensional examples of AdS/CFT
integrability, here we prove the no-go theorems that show that Hubbard model
is the only possible exception in cases A,B,C.

The equivalence class is nearly fully fixed, in terms of the discontinuity
relations on $\mu$ and $\omega$. This allows us to close the system of
Riemann-Hilbert problems that constitute a concrete realisation of QSC to
solve. Function $F$ is only partially constrained by the monodromy
requirements, it plays the role of a source term similar to the role of
Drinfeld polynomial in more conventional integrable systems. It is very
appealing to provide a more solid justification to this so far rather casual
observation, for instance to relate $F$ and the evaluation representations in
Beisert:2006fmy .

From a physicist’s perspective, the developed approach offers a possibility of
deriving QSC’s while circumventing tedious and not always available TBA
computations. We demonstrated that this is clearly the case on the example of
Hubbard model where a plethora of other methods are available allowing us to
make comparisons; And we used the monodromy bootstrap to give the concrete
conjecture for the spectral curve of AdS3/CFT2 integrability with
AdS${}_{3}\times$ S${}^{3}\times$T4 background supported by RR-flux. In the
latter case, a possibility to derive QSC from the currently available rather
incomplete TBA data Bombardelli:2018jkj ; Fontanella:2019ury  292929There are
also some TBA results for the NS-NS case Dei:2018jyj  which one cannot use for
an immediate comparison. is unlikely. Presence of the massless modes
considerably complicates all steps of the TBA analysis, from formulation of
the initial assumptions till the explicit computations of the spectrum. We
believe that the quantum spectral curve derived from the monodromy bootstrap
should capture correctly both massive and massless modes. In the present
paper, we only made checks against the asymptotic Bethe Ansatz in the massive
sector, although the gained experience with Hubbard model offered us a hint
towards incorporating the massless modes, we mention it at the end of Section
7. With no doubts, this QSC conjecture requires further considerable
verification both in massless and massive sectors, its status is much less
established compared to its higher-dimensional cousins.

Applying the same arguments as for AdS3/CFT2, we can conjecture that the
type-B system at zero central charge (i.e. when $F^{2}=1$ in which case it
also coincides with the type-A system) describes spectrum of AdS2/CFT1
integrability. We have performed a similar analysis to the one of Section 7
for this scenario and obtained asymptotic Bethe equations that are not
consistent with either the Bethe equations proposed in Sorokin:2011rr  or with
the finite gap equations in the same paper that were used to conjecture the
Bethe equations. The reason for the discrepancy is not entirely clear and we
decided to not include the AdS2/CFT1 computation in the current work
postponing it for future publications.

From a mathematician’s perspective, our approach seems to offer a possibility
to derive many different QSC’s of AdS/CFT type thus changing a perspective on
AdS/CFT integrability: if previously the known examples were often viewed as
isolated points, now these points start to fit into a landscape of novel-type
integrable systems which is comparable in variability to well-established
rational or trigonometric cases. We note that these AdS/CFT integrable systems
are not of elliptic type either, we should rather expect that they are built
on top of rational, trigonometric, and probably elliptic cases by introducing
a possibility for branch points.

In future research, it would be important to explicitly verify whether the
monodromy bootstrap indeed works for a variety of other groups to support the
above claim. The equations presented in Section 5 do not depend on the group
rank and hence we expect them to be valid for $\mathsf{SU}(N|N)$ and
$\mathsf{SU}(N|N)\times\mathsf{SU}(N|N)$ systems. Whether we can also assume
non-zero Coxeter number and consider $\mathsf{SU}(N|M)$ systems remains
unclear. Departing from A-series and considering orthosymplectic groups is
even more intriguing because less is known there about the fully extended
Q-system needed to properly describe the group of symmetry transformations.
Two constructions are available in the literature that can be used for
guidance in this research: QSC for AdS4/CFT3 integrability Bombardelli:2017vhk
, and the reproduction procedure on Bethe equations Kang:2021bhx  which should
be equivalent to applying Weyl reflections on Q-system, see e.g. explanations
in Ekhammar:2020enr . Almost no information exists about extension of the
$D(2,1;\alpha)$ Q-system, doing this case has a clear physical motivation as
it should describe AdS/CFT integrability with
AdS${}_{3}\times$S${}^{3}\times$S${}^{3}\times$T1 background.

Of course, QSC provides only information about the spectrum. Probably the most
fundamental question to ask what is the underlying quantum algebra 303030For
AdS5/CFT4 there are doubts whether it is even a Hopf algebra, if yes then with
highly non-trivial braiding, hence we avoid using the term ‘quantum group’.
whose representation is the physical model and whose Bethe algebra, a maximal
commutative subalgebra of conserved charges, is encoded by QSC. This quantum
algebra would be an analog of Yangian or quantum affine algebra, or their
twisted versions. The best handle we have to date is inhomogeneous Hubbard
model where the looked-for algebra is presumably known: it is Yangian for
centrally extended ${\mathsf{su}}(2|2)$. Its first Drinfeld and RTT
realisations are known, and some basic examples of fusion were performed
Beisert:2006fmy ; Beisert:2014hya ; Beisert:2015msa  313131See also
Matsumoto:2008ww ; Borsato:2017lpf  and references therein for derivations of
the mentioned Yangian as contraction from $\mathfrak{d}(2,1;\alpha)$ case..
Yet, the universal R-matrix is unknown and neither systematic representation
theory was developed. Also, Q-functions seem to be constructible as
eigenvalues of Q-operators in ‘prefundamental’ representations meneghelli_2020
, but more research should be done to confirm this point. Having at hand
Hubbard QSC derived in this paper can be a valuable organising principle to
guide future study of these questions.

###### Acknowledgements.

A major obstacle in concluding this work was unwillingness of the authors to
accept the possibility of non-square root branch points. We would like to
thank Bogdan Stefański and Olof Ohlsson Sax for numerous conference talks and
private discussions over the years that eventually persuaded us to overcome
this bias. This work was supported by the Knut and Alice Wallenberg Foundation
under grant “Exact Results in Gauge and String Theories” Dnr KAW 2015.0083.

Note for v1: We learned that A. Cavaglià, N. Gromov, A. Torrielli, and B.
Stefański have been working on a related topic. We agreed to synchronise the
publications. At the stage of publishing v1, we are not aware how large the
overlap between the two works will eventually be.

Note for v2: The quantum spectral curve for AdS${}_{3}\times$ S${}^{3}\times$
T4 proposed independently in Cavaglia:2021eqr  agrees with our proposal. There
are substantial additions in v2 of our paper, but not in sections where
overlap with Cavaglia:2021eqr  exists: there we did not add new results but
only corrected typos and improved clarity of explanations.

##  Appendix A Monodromy without Hodge duality

In this appendix we fill in details for the monodromy bootstrap of type A,
when the properly defined analytic continuation around the branch point leads
at most to a continuous symmetry transformation of the physical system. We
parameterise LHPA Q-system $\mathcal{Q}_{A|I}^{\downarrow}$ simply in the same
way as we parameterise $\mathcal{Q}_{A|I}$:

$\displaystyle{\bf P}_{a}$ $\displaystyle\equiv\mathcal{Q}_{a|\emptyset}\,,$
$\displaystyle{\bf P}^{a}$ $\displaystyle\equiv\mathcal{Q}^{a|\emptyset}\,,$
$\displaystyle{\bf Q}_{i}$ $\displaystyle\equiv\mathcal{Q}_{\emptyset|i}$
$\displaystyle{\bf Q}^{i}$ $\displaystyle\equiv\mathcal{Q}^{\emptyset|i}\,,$
(141a) $\displaystyle{\bf P}_{a}^{\downarrow}$
$\displaystyle\equiv\mathcal{Q}_{a|\emptyset}^{\downarrow}\,,$
$\displaystyle{\bf P}_{\downarrow}^{a}$
$\displaystyle\equiv\mathcal{Q}_{\downarrow}^{a|\emptyset}\,,$
$\displaystyle{\bf Q}_{i}^{\downarrow}$
$\displaystyle\equiv\mathcal{Q}_{\emptyset|i}^{\downarrow}$ $\displaystyle{\bf
Q}^{i}_{\downarrow}$
$\displaystyle\equiv\mathcal{Q}^{\emptyset|i}_{\downarrow}\,.$ (141b)

Hasse diagram is still given by (21), and the LHPA Q-system is described by
exactly the same diagram but with all objects being sub- or super-scripted
with $\downarrow$.

Rotation in the physical kinematics is required to act on bosonic indices
only. Together with gauge transformations, we have ${\bf
P}_{a}^{\downarrow}={\check{\mathsf{h}}}\,\mu_{a}{}^{b}{\bf P}_{b}$. Rotation
in the mirror kinematics is required to act on fermionic indices only, and we
introduce it by ${\bf
Q}_{i}^{\downarrow}={\hat{\mathsf{h}}}^{-1}\omega_{i}{}^{j}{\bf Q}_{j}$.
Denote $\det\limits_{1\leq a,b\leq 2}\mu_{a}{}^{b}=\mu^{2}$,
$\det\limits_{1\leq i,j\leq 2}\omega_{i}{}^{j}=\omega^{2}$. Then it follows

$\displaystyle\frac{\mathcal{Q}_{12|12}^{\downarrow}}{\mathcal{Q}_{12|12}}=\frac{g^{2}}{g_{\downarrow}^{2}}=\left(\omega^{+}\right)^{2}=\left(\mu^{+}\right)^{2}\,,\
\ \ \ -1/2<\Im(u)<1/2\,.$ (142)

Consider $\mu(u)$ and $\omega(u)$ for $u$ slightly above real axis. According
to (142), these two functions are equal. Now, if we shift the argument by
$\mathtt{i}$ up in the mirror kinematics, the value of $\mu^{2}=\omega^{2}$
won’t change due to the periodicity of $\mu^{2}$. And, similarly, it won’t
change if we shift the argument by $\mathtt{i}$ in the physical kinematics
where $\omega^{2}$ is periodic. Now it is not difficult to deduce that
$\mu^{2}=\omega^{2}$ are just $\mathtt{i}$-periodic functions without any
cuts. By an appropriate H-rotation of Q↓-system, we can set
$\mu^{2}=\omega^{2}=1$, and we choose the sign of square root to get
$\mu=\omega=1$. Then we also have $g^{2}=g_{\downarrow}^{2}$ and therefore
$\mathcal{Q}_{12|12}=\mathcal{Q}_{12|12}^{\downarrow}=\frac{1}{g^{2}}$.

Recall the important invariant combination

$\displaystyle{\bf Q}^{i}\,{\bf Q}_{i}={\bf P}^{a}\,{\bf
P}_{a}=\frac{1}{F}-F\,,\quad F=\frac{g^{+}}{g^{-}}\,.$ (143)

We see that, unlike in the type-B scenario, $F$ is simply analytic everywhere.

With the choice to normalise $\mu=\omega=1$, we can write the full set of
relations between $\mathcal{Q}$ and $\mathcal{Q}^{\downarrow}$:

Mirror

$\displaystyle{\bf P}_{a}^{\downarrow}$
$\displaystyle={\check{\mathsf{h}}}\,\mu_{a}{}^{b}\,{\bf P}_{b}$ (144a)
$\displaystyle{\bf Q}_{i}^{\downarrow}$
$\displaystyle={\check{\mathsf{h}}}^{-1}\,{\bf Q}_{i}$ (144b)
$\displaystyle{\bf P}_{\downarrow}^{a}$
$\displaystyle={\check{\mathsf{h}}}^{-1}{\bf P}^{b}\,(\mu^{-1})_{b}{}^{a}$
(144c) $\displaystyle{\bf Q}_{\downarrow}^{i}$
$\displaystyle={\check{\mathsf{h}}}\,{\bf Q}^{i}$ (144d)
$\displaystyle\mathcal{Q}_{a|i}^{\downarrow}$
$\displaystyle=(\mu_{a}{}^{b})^{+}\,\mathcal{Q}_{b|i}$ (144e)

Physical

$\displaystyle{\bf P}_{a}^{\downarrow}$
$\displaystyle={\hat{\mathsf{h}}}\,{\bf P}_{a}$ (145a) $\displaystyle{\bf
Q}_{i}^{\downarrow}$
$\displaystyle={{\hat{\mathsf{h}}}}^{-1}\,\omega_{i}{}^{j}\,{\bf Q}_{j}$
(145b) $\displaystyle{\bf P}_{\downarrow}^{a}$
$\displaystyle={{\hat{\mathsf{h}}}}^{-1}\,{\bf P}^{a}$ (145c)
$\displaystyle{\bf Q}_{\downarrow}^{i}$
$\displaystyle={\hat{\mathsf{h}}}\,{\bf Q}^{j}(\omega^{-1})_{j}{}^{i}$ (145d)
$\displaystyle\mathcal{Q}_{a|i}^{\downarrow}$
$\displaystyle=(\omega_{i}{}^{j})^{+}\,\mathcal{Q}_{a|j}$ (145e)

We derive further consequences in full analogy with the type-B case.

With

$\displaystyle
r\equiv\frac{{\check{\mathsf{h}}}}{\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\hat{\mathsf{h}}}}\vphantom{{\hat{\mathsf{h}}}}}\,,$
(146)

the monodromy properties of ${\bf P}$ and ${\bf Q}$ are

Clock-wise

$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
P}}\vphantom{{\bf P}}_{a}$ $\displaystyle=r\,\mu_{a}{}^{b}\,{\bf P}_{b}\,,$
(147a)
$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
P}}\vphantom{{\bf P}}^{a}$ $\displaystyle=\frac{1}{r}\,{\bf
P}^{b}(\mu^{-1})_{b}{}^{a}\,,\ $ (147b)
$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
Q}}\vphantom{{\bf Q}}_{i}$ $\displaystyle=\frac{1}{r}\,\omega_{i}{}^{j}\,{\bf
Q}_{j}\,,$ (147c)
$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
Q}}\vphantom{{\bf Q}}^{i}$ $\displaystyle={r}\,{\bf
Q}^{j}\,\left(\omega^{-1}\right)_{j}{}^{i}\,,$ (147d)

Counterclock-wise

$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
P}}\vphantom{{\bf P}}_{a}$
$\displaystyle=\frac{1}{\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{r}\vphantom{r}}\left(\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mu}\vphantom{\mu}^{-1}\right)_{a}{}^{b}\,{\bf
P}_{b}\,,$ (148a)
$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
P}}\vphantom{{\bf P}}^{a}$
$\displaystyle=\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{r}\vphantom{r}\,{\bf
P}^{b}\,\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mu}\vphantom{\mu}_{b}{}^{a}\,,$
(148b)
$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
Q}}\vphantom{{\bf Q}}_{i}$
$\displaystyle=\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{r}\vphantom{r}\left(\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\omega}\vphantom{\omega}^{-1}\right)_{i}{}^{j}\,{\bf
Q}_{j}\,,$ (148c)
$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
Q}}\vphantom{{\bf Q}}^{i}$
$\displaystyle=\frac{1}{\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{r}\vphantom{r}}\,{\bf
Q}^{j}\,\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\omega}\vphantom{\omega}_{j}{}^{i}\,.$
(148d)

From comparison of equations (144e) and (145e) for
$\mathcal{Q}_{a|i}^{\downarrow}$ we get

$\displaystyle\omega_{i}{}^{j}$
$\displaystyle=-\mu_{a}{}^{b}\,\left(\mathcal{Q}_{b|i}\mathcal{Q}^{a|j}\right)^{-}\,,$
$\displaystyle\left(\omega^{-1}\right)_{i}{}^{j}$
$\displaystyle=-\left(\mu^{-1}\right)_{a}{}^{b}\,\left(\mathcal{Q}_{b|i}\,\mathcal{Q}^{a|j}\right)^{-}\,,$
$\displaystyle 0<\Im(u)<1\,.$ (149)

Then it follows

$\displaystyle{\bf P}^{a}\mu_{a}{}^{b}{\bf P}_{b}$ $\displaystyle={\bf
P}^{a}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mu}\vphantom{\mu}_{a}{}^{b}{\bf
P}_{b}$ and hence $\displaystyle{\bf
P}^{a}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
P}}\vphantom{{\bf
P}}_{a}=\frac{r}{\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{r}\vphantom{r}}\,\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
P}}\vphantom{{\bf P}}^{a}{\bf P}_{a}\,,$ (150a) $\displaystyle{\bf
P}^{a}(\mu^{-1})_{a}{}^{b}{\bf P}_{b}$ $\displaystyle={\bf
P}^{a}(\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mu}\vphantom{\mu}^{-1})_{a}{}^{b}{\bf
P}_{b}$ and hence $\displaystyle{\bf
P}^{a}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
P}}\vphantom{{\bf
P}}_{a}=\frac{r}{\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{r}\vphantom{r}}\,\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
P}}\vphantom{{\bf P}}^{a}{\bf P}_{a}\,.$ (150b)

From
$0=\Delta(\mathcal{Q}_{a|i}^{\downarrow\,-})=\Delta(\mu_{a}{}^{b}\mathcal{Q}_{b|i}^{-})$
one gets

$\displaystyle\mu_{a}{}^{b}-\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mu}\vphantom{\mu}_{a}{}^{b}$
$\displaystyle=\frac{F}{r}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
P}}\vphantom{{\bf P}}_{a}{\bf
P}^{b}-\frac{F}{\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{r}\vphantom{r}}{\bf
P}_{a}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
P}}\vphantom{{\bf P}}^{b}=F\left(\mu_{a}{}^{c}{\bf P}_{c}{\bf P}^{b}-{\bf
P}_{a}{\bf
P}^{c}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mu}\vphantom{\mu}_{c}{}^{b}\right)\,,$
(151a)
$\displaystyle\mu_{a}{}^{b}-\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mu}\vphantom{\mu}_{a}{}^{b}$
$\displaystyle=\frac{F}{r}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
P}}\vphantom{{\bf P}}_{a}{\bf
P}^{b}-\frac{F}{\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{r}\vphantom{r}}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
P}}\vphantom{{\bf
P}}}\vphantom{\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
P}}\vphantom{{\bf
P}}}_{a}\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
P}}\vphantom{{\bf P}}^{b}=F\left(\mu_{a}{}^{c}{\bf P}_{c}{\bf
P}^{b}-\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mu}\vphantom{\mu}_{a}{}^{c}\mu_{c}{}^{d}{\bf
P}_{d}{\bf P}^{e}(\mu^{-1})_{e}{}^{b}\right)\,.$ (151b)

##  Appendix B Proof of no-go theorems

In this appendix we prove the no-go theorems that restrict considerably or
forbid completely the possibility of branch points of square root type. By
being a branch point of this type we mean the property
$\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{f}\vphantom{f}=\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{f}\vphantom{f}$.
We shall use mostly tensor notation of Section 5. Hence $\mu$ shall denote a
matrix not a square root of its determinant.

We shall not study case D because fourth-order monodromies are more natural to
be asked about for it. From the remaining cases, it is in principle enough to
consider case C. Indeed, A is reproduced for $\bar{Q}^{*}=Q$ and B is
reproduced for $\bar{Q}=Q$. However A and B have simpler proofs that use extra
symmetries, while case C is considerably more technical. We therefore decided
to outline the proofs for the three cases separately.

Two assumptions are being used: algebraic independence of ${\bf P}$-functions
in the sense explained below, and also the following one: if a function $R$ is
free from Zhukovsky branch points then $\sqrt{R}$ is free from them as well,
this shall be referred to as $\sqrt{\phantom{a}}$-assumption. A supportive
observation for it is that appearance of such branch points is only possible
if $R$ has a pole or a zero at this very specific non-dynamical position.
Moreover, the assumption will be only used when $R$ is periodic in which case
it means the presence of infinite ladders of such poles/zeros to violate
$\sqrt{\phantom{a}}$-assumption. In fact, we already used this type of
assumption by claiming that $F$ has only one branch cut: only the ratio
$\left(g^{+}/g^{-}\right)^{2}$ is guaranteed to be UHPA from the postulated
analytic properties of Q-functions, while we relied on analyticity of
$g^{+}/g^{-}=\sqrt{\left(g^{+}/g^{-}\right)^{2}}$ in the arguments.

### Case B

Equation (51) in tensor notations reads

$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mu}\vphantom{\mu}=\mu\,\left(1+\frac{1}{F}{\bf
P}^{*}\otimes{\bf P}\right)\,\left(1+\frac{1}{F}\mu\,{\bf P}^{*}\otimes{\bf
P}\,\mu^{-1}\right)^{\rm T}\,.$ (152)

We remind that $1-F{\bf P}^{*}\otimes{\bf P}$ is the inverse of
$1+\frac{1}{F}{\bf P}^{*}\otimes{\bf P}$, this guarantees that these matrices
are invertible. We also shall use that $\mu$ is invertible as it realises an
H-symmetry.

It is handy to define ${\mathcal{A}}=\mu\left(1+\frac{1}{F}{\bf
P}^{*}\otimes{\bf P}\right)$. Then, using (152) and (66b), equation
$\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mu}\vphantom{\mu}=\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mu}\vphantom{\mu}$
becomes after matrix rearrangements

$\displaystyle{\mathcal{A}}^{\rm
T}\,\mu^{-1}\,{\mathcal{A}}={\mathcal{A}}\,\mu^{-1}\,{\mathcal{A}}^{\rm T}\,.$
(153)

Let us decompose ${\mathcal{A}}$ into symmetric and antisymmetric parts
${\mathcal{A}}={\mathcal{A}}_{+}+{\mathcal{A}}_{-}$. Then one gets
${\mathcal{A}}_{+}\,\mu^{-1}\,{\mathcal{A}}_{-}={\mathcal{A}}_{-}\,\mu^{-1}\,{\mathcal{A}}_{+}$
and, by adding the tautological
${\mathcal{A}}_{-}\,\mu^{-1}\,{\mathcal{A}}_{-}={\mathcal{A}}_{-}\,\mu^{-1}\,{\mathcal{A}}_{-}$,
we conclude

$\displaystyle{\mathcal{A}}\,\mu^{-1}\,{\mathcal{A}}_{-}={\mathcal{A}}_{-}\,\mu^{-1}\,{\mathcal{A}}\,.$
(154)

Now two-dimensionality of the problem is going to be used. Define
$\epsilon=\left(\begin{smallmatrix}{0}&{1}\\\
{-1}&{0}\end{smallmatrix}\right)$. Any $2\times 2$ anti-symmetric matrix is
proportional to $\epsilon$ which implies, if ${\mathcal{A}}_{-}\neq 0$,
${\mathcal{A}}\,\mu^{-1}\,\epsilon=\epsilon\,\mu^{-1}\,{\mathcal{A}}$. Recall
now the definition of ${\mathcal{A}}$ to simplify the equation to the explicit
linear in $\mu$ equality $\mu\,{\bf P}^{*}\otimes{\bf P}=-\epsilon\,{\bf
P}^{*}\otimes{\bf P}\,\epsilon\,\mu$. Contracting from the left with ${\bf
P}^{*}$, the r.h.s. vanishes leaving us with

$\displaystyle{\bf P}^{*}\,\mu\,{\bf P}^{*}=0$ (155)

which is (63) implying that $\mu$ is antisymmetric.

Let us assume now that ${\bf P}^{*}\,\mu\,{\bf P}^{*}\neq 0$ but instead
${\mathcal{A}}_{-}=0$. Explicitly, ${\mathcal{A}}_{-}=0$ reads

$\displaystyle\left(F\epsilon^{ba}+{\bf P}^{b}{\bf
P}_{c}\epsilon^{ca}\right)^{[2n]}\mu_{ab}=0\,,\quad n=0,1,2,\ldots\,,$ (156)

where we added the shifts by $n$ using that $\mu$ is periodic in the mirror
kinematics. Introduce single-index parameterisation
$\mu_{1}\equiv\mu_{11},\mu_{2}\equiv\mu_{12},\mu_{3}\equiv\mu_{21},\mu_{3}\equiv\mu_{22}$.
Then the last equation can be understood as the matrix equation
$M_{n\alpha}\mu_{\alpha}=0$, where $M$ is the matrix with semi-infinite number
of rows.

Prior to continue the main analysis, we need to show that the kernel of $M$
can be spanned by periodic in $u$ functions without branch points, this is a
rather general statement not specific to the details of the problem. Indeed,
let $r<4$ is the rank of $M$. Then there exist a $(r+1)\times(r+1)$ submatrix
of $M$ with $n=0,\ldots,r$ and some subset of $\alpha$’s that has precisely
one zero mode. For definiteness, let this subset is $\alpha=1,\ldots,r+1$
323232If not, we can use re-labelling or even make a linear transformation on
$\mu_{\alpha}$, so no loss of generality for this choice.. Denote the zero
mode by $c_{\alpha}^{(1)}$. It is clear that
$\sum\limits_{\alpha=1}^{r+1}M_{n+1,\alpha}\left(c_{\alpha}^{(1)}\right)^{[2]}=0$
and hence
$\sum\limits_{\alpha=1}^{r+1}M_{n\alpha}\left(c_{\alpha}^{(1)}\right)^{[2]}=0$
for any $n$, therefore $\left(c_{\alpha}^{(1)}\right)^{[2]}$ is a zero mode,
but because there is only one zero mode of the $(r+1)\times(r+1)$ submatrix,
it must be $\left(c_{\alpha}^{(1)}\right)^{[2]}=\Lambda(u)\,c_{\alpha}^{(1)}$.
We can normalise at will and set e.g. $c_{r+1}^{(1)}=1$ (if $c_{r+1}^{(1)}=0$,
pick any other non-zero coefficient) getting the periodicity property
$\left(c_{\alpha}^{(1)}\right)^{[2]}=c_{\alpha}^{(1)}$. Now, recall that any
eigenvector of a matrix can be represented, up to a normalisation, as a
rational combination of matrix entries and the eigenvalue. Because
$M_{n\alpha}$ depends only on ${\bf P}$ and $F$ who are UHPA, we conclude that
$c_{\alpha}^{(1)}$ in the chosen normalisation is also UHPA and, by its
periodicity, is free from cuts everywhere.

If $r<3$, pick $(r+2)\times(r+2)$ submatrix of $M$ with two zero modes. One
zero mode shall be $c_{\alpha}^{(1)}$ with $c_{r+2}^{(1)}=0$ and another shall
be denoted as $\bar{c}_{\alpha}$, we normalise it to have $\bar{c}_{r+2}=1$
(this coefficient cannot be zero because $\bar{c}$ is linearly independent
from $c^{(1)}$). The most general periodicity property with this normalisation
is ${\bar{c}}^{[2]}=\bar{c}+\Lambda(u)\,c^{(1)}$. Notice now that
$\Lambda(u)={\bar{c}}_{r+1}^{[2]}-{\bar{c}}_{r+1}$, hence we can introduce
$c^{(2)}:=\bar{c}-{\bar{c}}_{r+1}c^{(1)}$, this vector is a zero mode of $M$
which is linearly independent from $c^{(1)}$ and is periodic in $u$. By the
same argument as above, it is free from cuts.

If needed, we can keep reasoning in the same manner and construct linearly
independent $c^{(i)}$ which span the kernel of $M$ and are free from branch
points. One then has
$\mu_{\alpha}=\sum\limits_{i}\Lambda_{i}(u)c_{\alpha}^{(i)}$.

If $\mu_{ab}$ is a symmetric matrix or if $M$ has more than one zero modes,
there must exist a zero mode $c$ such that $c_{ab}$ is a symmetric matrix
which therefore satisfies

$\displaystyle\left({\bf P}^{b}{\bf
P}_{c}\epsilon^{ca}\right)^{[2n]}c_{ab}=0\,.$ (157)

Introduce $v=({\bf P}_{1}{\bf P}^{1}-{\bf P}_{2}{\bf P}^{2},{\bf P}_{2}{\bf
P}^{1},{\bf P}_{1}{\bf P}^{2})$. The last equation has a non-trivial solution
only if $v^{[2n]}\wedge v^{[2m]}\wedge v^{[2k]}=0$ for any
$n,m,k\in\mathbb{Z}$. Our assumption on algebraic independence on ${\bf P}$ is
that this property cannot be fulfilled. For instance, if Q-functions have
power-like large-$u$ behaviour and ${\bf P}_{a}\sim u^{-\lambda_{a}}$ then
${\bf P}^{a}\sim u^{\lambda_{a}+{\rm const}}$ which, for
$\lambda_{1}\neq\lambda_{2}$, is already enough to verify the assumed
algebraic independence.

We therefore end up with the case when there is exactly one zero mode of $M$
and such that $\mu_{ab}=\Lambda c_{ab}$ is a matrix which has both non-zero
symmetric and antisymmetric parts. The discontinuity properties of $\mu_{ab}$
are now concentrated in the scalar function $\Lambda$. Using (58), we see
$\Lambda/\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\Lambda}\vphantom{\Lambda}=F^{2}$
and then (60) implies $F^{2}=1$. But this immediately tells us that $\Lambda$
and hence $\mu_{ab}$ have no branch points on the real axis. By periodicity,
$\mu_{ab}$ is then free from any branch points.

In conclusion, (156) cannot be satisfied by $\mu_{ab}$ with branch points and
therefore the only non-trivial option to satisfy
$\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mu}\vphantom{\mu}_{ab}=\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mu}\vphantom{\mu}_{ab}$
is $\mu_{ab}\propto\epsilon_{ab}$.

### Case A

First let us comment on why
$\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
P}}\vphantom{{\bf
P}}\otimes\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
P}}\vphantom{{\bf
P}}^{*}=\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
P}}\vphantom{{\bf
P}}\otimes\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bf
P}}\vphantom{{\bf P}}^{*}$ implies branch points of square root type for
$\mu$. The idea is to use relations (150) which, under this assumption on the
cut structure of ${\bf P}\otimes{\bf P}^{*}$, imply
$r/\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{r}\vphantom{r}=\pm
1$ that in turn implies, using (151), equality of clock- and counter-clock-
wise discontinuities $\Delta(\mu_{ab})$. A possible escape from this
conclusion is ${\bf P}^{a}\mu_{a}{}^{b}{\bf P}_{b}=0$, but for this equation
we shall assume algebraic independence of ${\bf P}$ in the sense of the
previous subsection. Then $({\bf P}^{a}{\bf P}_{b})^{[2n]}\mu_{a}{}^{b}=0$ can
have potentially only one zero mode which constraints $\mu$ to the form
$\mu_{a}{}^{b}=\Lambda(u)c_{a}{}^{b}$, where $c_{a}{}^{b}$ is analytic. Then
$\Lambda^{2}=\det{\mu}/\det{c}$ is analytic and we use
$\sqrt{\phantom{a}}$-assumption to state that $\Lambda(u)$ is also analytic.

To prove the no-go theorem for square root cuts of $\mu$, let us introduce
matrix $\Pi=1+\frac{1}{F}{\bf P}\otimes{\bf P}^{*}$. Using it, QSC equations
become even more elegant

$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\Pi}\vphantom{\Pi}=\mu\,\Pi\,\mu^{-1}\,,\quad\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mu}\vphantom{\mu}=\Pi\,\mu\,\Pi^{-1}\,.$
(158)

Relation (151b) can be re-written in this conventions as
$\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mu}\vphantom{\mu}=\mu\,\Pi^{-1}\,\mu\,\Pi\,\mu^{-1}$,
and then the square root cut assumption
$\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mu}\vphantom{\mu}=\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mu}\vphantom{\mu}$
leads to

$\displaystyle(\mu\,\Pi^{-1})^{2}=(\Pi^{-1}\mu)^{2}\,.$ (159)

Now we shall apply Hamilton-Cayley using the two-dimensionality of the
problem: any $2\times 2$ matrix satisfies $M^{2}-M\,\operatorname{\rm
Tr}M+\det M=0$. It simplifies the last equation to

$\displaystyle[\mu,\Pi^{-1}]\operatorname{\rm Tr}(\mu\,\Pi^{-1})=0\,.$ (160)

Using the same assumptions on independence of ${\bf P}_{a},{\bf P}^{a}$ as
before, we get that $\operatorname{\rm Tr}(\mu\,\Pi^{-1})=0$ implies
$\mu=\Lambda\,c$, we already explained why this leads to $\mu$ without branch
points. As for the possibility of $[\mu,\Pi^{-1}]=0$, we can readily use (158)
to get
$\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mu}\vphantom{\mu}=\mu$.

### Case C

Using matrices ${\Pi}=1+\frac{1}{F}{{{{\bf P}}}}{\vphantom{\bar{{{\bf
P}}}}}\otimes{{{{\bf P}}}{\vphantom{\bar{{{\bf
P}}}}}^{*}}{\vphantom{\bar{{{\bf P}}}}}$,
$\bar{{\Pi}}=1+\frac{1}{\bar{F}}{{{\bar{{\bf P}}}}}{\vphantom{\bar{{{\bf
P}}}}}\otimes{{{\bar{{\bf P}}}}{\vphantom{\bar{{{\bf
P}}}}}^{*}}{\vphantom{\bar{{{\bf P}}}}}$, the QSC equations become

$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\Pi}}\vphantom{{\Pi}}$
$\displaystyle={{\bar{\mu}}}{\vphantom{\bar{\mu}}}\,\bar{{\Pi}}^{-{\rm
T}}\,{{\bar{\mu}}}{\vphantom{\bar{\mu}}}^{-1}\,,$
$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\bar{{\Pi}}}\vphantom{\bar{{\Pi}}}$
$\displaystyle={\mu}{\vphantom{\bar{\mu}}}\,{\Pi}^{-{\rm
T}}\,{\mu}{\vphantom{\bar{\mu}}}^{-1}\,$ (161a)
$\displaystyle{\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mu}\vphantom{\mu}}{\vphantom{\bar{\mu}}}$
$\displaystyle=\bar{{\Pi}}\,{\mu}{\vphantom{\bar{\mu}}}\,{\Pi}^{\rm T}\,,$
$\displaystyle{\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bar{\mu}}}\vphantom{{\bar{\mu}}}}{\vphantom{\bar{\mu}}}$
$\displaystyle={\Pi}\,{{\bar{\mu}}}{\vphantom{\bar{\mu}}}\,\bar{{\Pi}}^{\rm
T}\,,$ (161b)

where $M^{-{\rm T}}$ means the inverse transpose of $M$.

##### Equivalence of square root cut assumptions.

An easy consequence of the above rewriting of QSC is

$\displaystyle\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\Pi}}\vphantom{{\Pi}}\,{\Pi}^{-1}={{\bar{\mu}}}{\vphantom{\bar{\mu}}}\,{\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bar{\mu}}}\vphantom{{\bar{\mu}}}}{\vphantom{\bar{\mu}}}^{-1}\,.$
(162)

From here it is clear that conditions
$\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\Pi}}\vphantom{{\Pi}}=\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\Pi}}\vphantom{{\Pi}}$
and
${\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mu}\vphantom{\mu}}{\vphantom{\bar{\mu}}}={\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mu}\vphantom{\mu}}{\vphantom{\bar{\mu}}}$
are equivalent (and the same holds for the pair
$\bar{{\Pi}},{\mu}{\vphantom{\bar{\mu}}}$). Property
${{\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{{\bf
P}}}\vphantom{{{\bf P}}}}}{\vphantom{\bar{{{\bf
P}}}}}={{\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle*$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{{\bf
P}}}\vphantom{{{\bf P}}}}}{\vphantom{\bar{{{\bf P}}}}}$ is potentially a
slightly stronger condition than that for ${\Pi}$. However, in the proof of
this appendix, only relations of ${\bf P}\mu$-system shall be of relevance
while analytic properties of ${\bf Q}$ shall not be appealed to. For this
reason we can pick a gauge where functions ${{{{\bf P}}}}{\vphantom{\bar{{{\bf
P}}}}}_{1},{{{\bar{{\bf P}}}}}{\vphantom{\bar{{{\bf P}}}}}_{1}$ have square
root cuts (the choice ${{{{\bf P}}}}{\vphantom{\bar{{{\bf
P}}}}}_{1\vphantom{\dot{1}}}={{{\bar{{\bf P}}}}}{\vphantom{\bar{{{\bf
P}}}}}_{\dot{1}}=1$ will do), in this gauge branch cut properties of ${\Pi}$
and ${{{{\bf P}}}}{\vphantom{\bar{{{\bf P}}}}},{{{\bar{{\bf
P}}}}}{\vphantom{\bar{{{\bf P}}}}}$ are the same.

##### Non-vanishing of bilinear combinations.

As we already covered cases A,B, we will assume there is no particular
algebraic relation between ${{{{\bf P}}}}{\vphantom{\bar{{{\bf P}}}}}$ and
${{{\bar{{\bf P}}}}}{\vphantom{\bar{{{\bf P}}}}}$ meaning that the
combinations ${{{\bar{{\bf P}}}}{\vphantom{\bar{{{\bf
P}}}}}^{*}}{\vphantom{\bar{{{\bf P}}}}}\mu{{{{\bf P}}}{\vphantom{\bar{{{\bf
P}}}}}^{*}}{\vphantom{\bar{{{\bf P}}}}}={{{\bar{{\bf
P}}}}}{\vphantom{\bar{{{\bf P}}}}}^{\dot{a}}\mu_{\dot{a}a}{{{{\bf
P}}}}{\vphantom{\bar{{{\bf P}}}}}^{a}\propto{{{\bar{{\bf
P}}}}{\vphantom{\bar{{{\bf P}}}}}^{*}}{\vphantom{\bar{{{\bf
P}}}}}{{\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\bar{{\bf
P}}}}\vphantom{{\bar{{\bf P}}}}}}{\vphantom{\bar{{{\bf P}}}}}$ and all other
bilinears ${{{{\bf P}}}}{\vphantom{\bar{{{\bf P}}}}}\mu{{{{\bf
P}}}}{\vphantom{\bar{{{\bf
P}}}}}\propto{{\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{{\bf
P}}}\vphantom{{{\bf P}}}}}{\vphantom{\bar{{{\bf P}}}}}{{{{\bf
P}}}}{\vphantom{\bar{{{\bf P}}}}}$ (with all consistent arrangements of bars
and stars in ${{{{\bf P}}}}{\vphantom{\bar{{{\bf P}}}}}\mu{{{{\bf
P}}}}{\vphantom{\bar{{{\bf P}}}}}$, and including inverses of $\mu$) shall be
non-vanishing. The argument is of the same type as used for (157).

##### Case $F^{2}=1$

This one is special because ${{{{\bf P}}}}{\vphantom{\bar{{{\bf
P}}}}}\otimes{{{{\bf P}}}{\vphantom{\bar{{{\bf
P}}}}}^{*}}{\vphantom{\bar{{{\bf P}}}}}$ becomes nilpotent in the presence of
the zero charge constraint. By using $\sqrt{\phantom{a}}$-assumption and
redefining ${{{{\bf P}}}}{\vphantom{\bar{{{\bf P}}}}},{{{\bar{{\bf
P}}}}}{\vphantom{\bar{{{\bf P}}}}}$ we can reach precisely all the
normalisations stated for AdS3/CFT2 conjecture, and hence we can use relations
(114) 333333While reaching these normalisations, we perform rescaling
${\mu}{\vphantom{\bar{\mu}}}\to\frac{1}{\sqrt{\det{\mu}{\vphantom{\bar{\mu}}}}}{\mu}{\vphantom{\bar{\mu}}}$.
By $\sqrt{\phantom{a}}$-assumption, $\sqrt{\det{\mu}{\vphantom{\bar{\mu}}}}$
is analytic but it is potentially anti-periodic. Hence such a rescaling could
create an anti-periodic ${\mu}{\vphantom{\bar{\mu}}}$. This is not an issue
for the current proof because periodicity of ${\mu}{\vphantom{\bar{\mu}}}$ is
only used for arguments of type (157), these arguments work for
${\mu}{\vphantom{\bar{\mu}}}$ having any period and not only $\mathtt{i}$. The
remark applies for similar rescalings in other parts of the proof.. We also
choose a gauge with $r=\bar{r}=1$. Then we conclude, from (114a) assuming
square root branch cut, that
${\mu}{\vphantom{\bar{\mu}}}+{{\bar{\mu}}}{\vphantom{\bar{\mu}}}^{\rm T}=G$ is
an everywhere analytic function. If $G$ is non-zero, it should satisfy
$G\,{\Pi}^{-{\rm T}}=\bar{{\Pi}}\,G$ which is impossible given algebraic
independence of ${{{{\bf P}}}}{\vphantom{\bar{{{\bf P}}}}}$ and ${{{\bar{{\bf
P}}}}}{\vphantom{\bar{{{\bf P}}}}}$ (note that we check a matrix equation and
so algebraic relations ${{{{\bf P}}}}{\vphantom{\bar{{{\bf P}}}}}_{a}{{{{\bf
P}}}}{\vphantom{\bar{{{\bf P}}}}}^{a}={{{\bar{{\bf P}}}}}{\vphantom{\bar{{{\bf
P}}}}}_{\dot{a}}{{{\bar{{\bf P}}}}}{\vphantom{\bar{{{\bf P}}}}}^{\dot{a}}=0$
are not enough to get non-vanishing $G$).

Then ${\mu}{\vphantom{\bar{\mu}}}=-{{\bar{\mu}}}{\vphantom{\bar{\mu}}}^{\rm
T}$, from here there are many ways to show impossibility of square root branch
points. For instance, we can use (161a) to deduce that
${\mu}{\vphantom{\bar{\mu}}}^{-1}{\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{\mu}\vphantom{\mu}}{\vphantom{\bar{\mu}}}$
commutes with
$\overset{\scalebox{0.8}{\raisebox{-2.5pt}{\raisebox{0.0pt}{\makebox[2.0pt][l]{$\scriptstyle\hphantom{*}$}}\leavevmode\resizebox{0.0pt}{0.0pt}{$\scriptscriptstyle\sim$}}}}{{\Pi}}\vphantom{{\Pi}}^{\rm
T}$ and then use (162) to decide that