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\ShortArticleName{Upper Bounds for Mutations of Potentials}

\ArticleName{Upper Bounds for Mutations of Potentials\footnote{This
paper is a contribution to the Special Issue ``Mirror Symmetry and Related Topics''. The full collection is available at \href{http://www.emis.de/journals/SIGMA/mirror_symmetry.html}{http://www.emis.de/journals/SIGMA/mirror\_{}symmetry.html}}}

\Author{John Alexander CRUZ MORALES~$^{\dag^1}$ and Sergey GALKIN~$^{\dag^2\dag^3\dag^4\dag^5}$}

\AuthorNameForHeading{J.A.~Cruz Morales and S.~Galkin}

\Address{$^{\dag^1}$ Department of Mathematics and Information Sciences, Tokyo Metropolitan University, \\
\hphantom{$^{\dag^1}$}~Minami-Ohsawa 1-1, Hachioji, Tokyo 192-037, Japan}
\EmailDD{\href{mailto:cruzmorales-johnalexander@ed.tmu.ac.jp}{cruzmorales-johnalexander@ed.tmu.ac.jp}, \href{mailto:alekosandro@gmail.com}{alekosandro@gmail.com}}

\Address{$^{\dag^2}$~Kavli Institute for the Physics and Mathematics of the Universe, The University of Tokyo,\\
\hphantom{$^{\dag^2}$}~5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan}

\Address{$^{\dag^3}$~Independent University of Moscow, 11 Bolshoy Vlasyevskiy per., 119002, Moscow, Russia}

\Address{$^{\dag^4}$~Moscow Institute of Physics and Technology, 9 Institutskii per.,\\
\hphantom{$^{\dag^4}$}~Dolgoprudny, 141700, Moscow Region, Russia}
\EmailDD{\href{mailto:Sergey.Galkin@phystech.edu}{Sergey.Galkin@phystech.edu}}

\Address{$^{\dag^5}$ Universit\"at Wien, Fakult\"at f\"ur Mathematik, Garnisongasse 3/14, A-1090 Wien, Austria}

\ArticleDates{Received May 31, 2012, in f\/inal form January 16, 2013; Published online January 19, 2013}

\Abstract{In this note we provide a new, algebraic proof of the excessive Laurent phenomenon for mutations of potentials (in the sense of [Galkin~S., Usnich~A., {P}reprint IPMU 10-0100, 2010]) by introducing to this theory the analogue of the upper bounds from~[Berenstein~A., Fomin~S., Zelevinsky~A., \textit{Duke Math.~J.} \textbf{126} (2005), 1--52].}

\Keywords{cluster algebras; Laurent phenomenon; mutation of potentials; mirror symmetry}

\Classification{13F60; 14J33; 53D37}

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\section{Introduction}

The idea of mutations of potentials was introduced in \cite{GU}
and the Laurent phenomenon was established in the two dimensional case
by means of birational geometry of surfaces.
More precisely, in op.\ cit.\ the authors considered
a toric surface $X$ with a rational function $W$ (a~potential),
and using certain special birational transformations (mutations),
they established the (excessive) Laurent phenomenon which roughly says that
if $W$ is a Laurent polynomial whose mutations are Laurent polynomials, then
all subsequent mutations of these polynomials are also Laurent polynomials
(see Theorem~\ref{ulemma} in Appendix~\ref{appendix2} for a precise statement of the excessive Laurent phenomenon
as established in~\cite{GU}).
The motivating examples of such potentials
come from the mirror images of special Lagrangian tori
on del Pezzo surfaces~\cite{FOOO}
and Auroux's wall-crossing formula relating invariants of dif\/ferent tori~\cite{AU}.

The cluster algebras theory of Fomin and Zelevinsky~\cite{FZ} provides
an inductive way to construct \emph{some} birational transformations
of $n$ variables as a consecutive composition of elementary ones (called elementary mutations)
with a choice of $N=n$ directions at each step.

The theory developed in \cite{GU} can be seen as an extension of the theory of cluster algebras~\cite{FZ} when the number of directions of mutations $N$ is allowed to be (much) bigger than the number of variables $n$, but at least one function remains to be a Laurent polynomial after all mutations. So, it is natural to try to extend the machinery of the theory of cluster algebras for this new setup. The main goal of this paper is to give the f\/irst step in such an extension by means of the introduction of the upper bounds (in the sense of~\cite{BFZ}) and establishing the excessive Laurent phenomenon \cite{GU} in terms of them.
It is worth noticing that a further generalization can be done and in a forthcoming work~\cite{CG} we plan to study the quantization of the mutations of potentials and their upper bounds. Naturally, this quantization can be seen as an extension of the theory of quantum cluster algebras developed in~\cite{BZ,BZ1} and the theory of cluster ensembles in~\cite{FG}.

The upper bounds introduced in this paper can be described as a collection of regular functions that remain regular after one elementary mutation in any direction. Thus, we can establish the main result of this paper in the following terms (see Theorem~\ref{ublemma} for the exact formulation).

\begin{theorem*}[Laurent phenomenon in terms of the upper bounds]
The upper bounds are preserved by mutations.
\end{theorem*}

Aside from providing a new proof for the excessive Laurent phenomenon and the already mentioned generalization in the quantized setup, the algebraic approach that we are introducing here is helpful for tackling the following two problems:
\begin{enumerate}\itemsep=0pt
\item
Develop a higher dimensional theory (i.e.\ dimension higher than $2$) for the mutations of potentials. Some work in that direction is carried out in \cite{CGA}.
\item
Present an explicit construction to compactify Landau--Ginzburg models (Problem~44 of~\cite{GU}).
\end{enumerate}

In the present paper we do not deal with the above two problems (only a small comment on~2 will be made at the end of the paper). We plan to give a detailed discussion of them in~\cite{CG} too. We just want to mention that the new algebraic approach has interesting geometrical applications.

Some words about the organization of the text are in order. In Section~\ref{sec-def} we extend the theory developed in~\cite{GU} to lattices of arbitrary rank and general bilinear forms (i.e., we can consider even degenerate and not unimodular forms) and introduce the notion of upper bounds in order to establish our main theorem. In Section~\ref{section3} we actually establish the main theorem and present its proof when the rank of the lattice is two and the form is non-degenerate which is the case of interest for the geometrical setup of~\cite{GU}. In the last section some questions and future developments are proposed. For the sake of completeness of the presentation we include two appendices. In Appendix~\ref{sec-bfz} we review some def\/initions of~\cite{BFZ} and brief\/ly compare their theory with ours. Appendix~\ref{appendix2} is dedicated to presenting the Laurent phenomenon in terms of~\cite{GU}.

\section{Mutations of potentials and upper bounds} \label{sec-def}

Now we present an extension of the theory of mutations of potentials~\cite{GU}
(as formulated by the second author and Alexandr Usnich)
and introduce our modif\/ied def\/initions with the new def\/inition of upper bound.
Notice that a slightly dif\/ferent theory (which f\/its into the framework of this paper, but not \cite{GU})
is used in our software code\footnote{\url{http://member.ipmu.jp/sergey.galkin/degmir.gp}.}.

\subsection{Combinatorial data}

Let $(\cdot,\cdot) : L^* \times L \to \mathbb{Z}$
be the canonical pairing between
a pair of dual lattices
$L \simeq \mathbb{Z}^r$ and $L^* = \Hom(L,\mathbb{Z}) \simeq \mathbb{Z}^r$.

In what follows the lattice $L$ is endowed with a skew-symmetric bilinear integral form $\omega: L \times L \to \mathbb{Z}$
(we use the notation $\ang{v,v'} = \omega(v,v')$).
In the most important (both technically, and from the point of view of applications) case
$r = \rk L = 2$, we have $\Lambda^2 L \simeq \mathbb{Z}$,
so all integer skew-symmetric bilinear forms are integer multiples $\omega_k = k \omega_1$ ($k \in \mathbb{Z}$)
where a~generator~$\omega_1$ is f\/ixed by the choice of orientation on $L \otimes \mathbb{R}$
so that $\omega_1((1,0),(0,1)) = 1$.
We would occasionally use notations
$\ang{\cdot,\cdot}_1 = \omega_1(\cdot,\cdot)$
and
$\ang{\cdot,\cdot}_k = \omega_k(\cdot,\cdot)$.

The bilinear form $\omega$ gives rise to a map $i = i_{\omega} : L \to L^*$ that sends an element $v \in L$ into a linear form
$i_\omega(v) \in L^*$ such that $(i_\omega(v),v') = \omega(v,v')$ for any $v' \in L$.
The map $i_\omega$ is an isomorphism $\iff$ the form $\omega$ is non-degenerate and unimodular,
when~$\omega$ is non-degenerate but not unimodular the map $i$ identif\/ies the lattice $L$
with a full sublattice in $L^*$ of index~$\det \omega$,
f\/inally if $\omega$ is degenerate then both the kernel and the cokernel of the map $i_\omega$ has positive rank.

We would like to have some functoriality,
so we consider a category whose objects are given by pairs $(L,\omega)$ of the lattice $L$
and a skew-symmetric bilinear form $\omega$,
and the morphisms $\Hom((L',\omega'),(L,\omega))$ are linear maps $f : L' \to L$
such that $\omega' = f^* \omega$,
i.e.\ $\omega(v_1,v_2) = \omega'(f(v_1),f(v_2))$ for all $v_1,v_2 \in L'$.
Any linear map $f : L' \to L$ def\/ines an adjoint $f^* : L^* \to L'^*$
and if it respects the bilinear forms,
then $i_{\omega'} = f^* i_{\omega} f$.

For a vector $u \in L$ we def\/ine
a \emph{symplectic reflection} $R_u$
and a \emph{piecewise linear mutation} $\mu_u$
to be the (piecewise)linear automorphisms of the set $L$ given by the formulae
\begin{gather*}
R_{\omega,u} (v) = v + \omega(u,v) u, \\
\mu_{\omega,u} v = v + \max(0,\omega(u,v)) u.
\end{gather*}
For any morphism $f \in \Hom((L',\omega'),(L,\omega))$ and any vector $u \in L'$ we have
$R_{\omega, fu} f = f R_{\omega',u}$ and
$\mu_{\omega, fu} f = f \mu_{\omega',u}$.
Indeed, $f \mu_u v = f (v + \max(0,\omega'(u,v)) u) = fv + \max(0,\omega'(u,v)) (fu) =  fv + \max(0,\omega(fu,fv)) (fu) = \mu_{fu} (fv)$.

Note that
$R_{a \omega, bu} = R_{\omega,u}^{a b^2}$ for all $a,b \in \mathbb{Z}$
and
$\mu_{a \omega, bu} = \mu_{\omega,u}^{a b^2}$ for all $a,b \in \mathbb{Z}_+$.
However
$\mu_{\omega,-u} v = - \mu_u (-v) =  v + \min(0,\omega(u,v)) u$,
hence
$\mu_{\omega,-u} \mu_{\omega,u} = R_{\omega,u}$.
Both $R_{\omega,u}$ and $\mu_{\omega,u}$ are invertible:
$R_{\omega,u}^{-1} v = R_{-\omega,u} v  = v - \omega(u,v) u$,
$\mu_{\omega,u}^{-1} v = \mu_{-\omega,-u} v = R_{\omega,u}^{-1} \mu_{\omega, -u} v = v - \max(0,\omega(u,v)) u$.
Note that
$\mu_{\omega,-u}^{-1}  v = R_{\omega,-u}^{-1} \mu_{\omega,u}  v = R_{\omega,u}^{-1} \mu_{\omega,u}(v) = v - \min(0,\omega(u,v)) u$.
Therefore, changing $\max$ by $\min$ and $+$ by $-$, simultaneously,
corresponds to changing the form $\omega$ to the opposite $-\omega$.
Further we omit $\omega$ from the notations of $R_u$ and $\mu_u$ where the choice of the form is clear.

The underlying combinatorial gadget of our story
is a collection of $n$ vectors in $L$:
\begin{definition
An \emph{exchange collection} $V$ is an element of $L^n$, i.e.\ an $n$-tuple $(v_1,\dots,v_n)$ of
vectors~$v_i \in L$.
Some~$v_i$ may coincide.
For a~vector $v$ its \emph{multiplicity} $m_V(v)$ in the exchange collection $V$ equals the number of vectors in $V$ that coincide with $v$:
$m_V (v) = \# \{ 1 \leqslant i \leqslant n : v_i = v\}$. We say that an exchange collection $V'$ is a \emph{subcollection} of exchange collection $V$ if
$m_{V'} \leqslant m_V$.
Equivalently, one may def\/ine an exchange collection $V$ by its (non-negative integer) multiplicity function
$m_V : L \to \mathbb{Z}_{\geqslant 0}$. In this case $n = \sum\limits_{v \in L} m_V(v)$.
\end{definition}

The exchange collections could be pushed forward by morphisms $f \in \Hom((L',\omega'),(L,\omega))$:
$v_1',\dots,v_n' \in L'^n$ will go to $f v_1,\dots, f v_n \in L^n$.
This gives rise to a natural diagonal action of $\mathrm{Aut} (L,\omega) = \operatorname{Sp}(L,\omega)$
on $L^n$. This action commutes with the permuting action of~$S_n$.

A vector $n \in L$ is called \emph{primitive}
if it is nonzero and its coordinates are coprime, i.e.~$n$ does not belong to the sublattice $k L$ for any $k>1$,
in other words $n$ is not a multiple of other vector in~$L$.
We denote the set of all primitive vectors in~$L$ as~$L_1$.
Similarly one can def\/ine primitive vectors in the dual lattice~$L^*$.
Note that if $\det \omega \neq \pm{1}$
then $i_\omega(n)$ may be a non-primitive element of~$L^*$ even for primitive elements~$n \in L_1$.

\subsection{Birational transformations}
Consider the group ring $\mathbb{Z}[L^*]$~-- ring of Laurent polynomials of $r$ variables.
Its spectrum $T = {\text{Spec }} \mathbb{Z}[L^*] \simeq \mathrm{G}_m^r(\mathbb{Z})$ is the $r$-dimensional torus over the integers,
in particular $T(\mathbb{C}) = \Hom(L^*,\mathbb{C}^*)$,
$L^* = \Hom(T,\mathrm{G}_m)$ is the lattice of characters of $T$
and
$L = \Hom(\mathrm{G}_m,T)$ is the lattice of $1$-parameter subgroups in~$T$.
Def\/ine \emph{the ambient field} $\mathbb{K} = \mathbb{K}_L = \mathcal{Q}(L^*)$ as the fraction f\/ield of $\mathbb{Z}[L^*]$
extended by all roots of unity ($\mathcal{Q} = \mathbb{Q}(\exp(2 \pi i \mathbb{Q})$).

A vector $u \in L$ def\/ines a birational transformation of $\mathbb{K}_L$ (and its various subf\/ields and subrings) as follows
\begin{equation*}
\mu_{u,\omega} : \ X^m \to X^m \big(1+X^{i_\omega(u)}\big)^{(u,m)}.
\end{equation*}

If $f : T_1 \to T_2$ is a rational map between two tori, and $u : \mathrm{G}_m \to T_1$ is a one-parameter subgroup of $T$
then its image $f u : \mathrm{G}_m \to T_2$ is not necessarily a one-parameter subgroup, but asymptotically behaves like one,
this def\/ines a \emph{tropicalization} map $T(F) : \Hom(\mathrm{G}_m,T_1) \to \Hom(\mathrm{G}_m, T_2)$. The tropicalization of the birational
map $\mu_{u,\omega} : T_1 \to T_2$ is the piecewise-linear map
$\mu_{u,\omega} : L_1 \to L_2$ def\/ined in the previous subsection.

One can easily see most of the relations of the previous subsection on the birational level.
For example, $\mu_{-u} \mu_u = \mu_u \mu_{-u} = R_u$ and $R_v \mu_u R_v^{-1} = \mu_{R_v u}$,
where $R_u$ is the homomorphism of the torus $T$ given by
$R_{u,\omega} : X^m \to X^{m+(u,m) i_\omega u}$. Also $R_{a u, b \omega} = R_{u,\omega}^{a^2 b}$ for any $a,b \in \mathbb{Z}$,
and $\mu_{a u,\omega} = (\mu_{u, a \omega})^a$ for any $a \in \mathbb{Z}$, however neither of them is a power of $\mu_{u,\omega}$.\footnote{Since $(1+x^a)$ is not a power of $(1+x)$.} In particular, $(\mu_{u,\omega})^{-1} = \mu_{-u,-\omega}$.

Note that if $M \subset L^*$ is some sublattice of $L^*$ that contains $i_\omega(u)$ then $\mu_u$ preserves the fraction
f\/ield of $\mathbb{Z}[M] \subset \mathbb{Z}[L^*]$.
For any morphism $f \in \Hom((L',\omega'), (L,\omega))$ and a vector $u \in L'$ we have a homomorphism
$f^* : \mathbb{Z}[L^*] \to \mathbb{Z}[L'^*]$ and two birational transformations
$\mu_u \in \mathrm{Aut} \mathbb{K}_{L'}, \mu_{fu} \in \mathrm{Aut} \mathbb{K}_L$ that commute:
$\mu_u f^* = f^* \mu_{fu}$.

\begin{remark} \label{fun1}
We have the following functoriality of the mutations with respect to the lattice~$L$:
let $L' \subset L$ be a sublattice of index $k$ in the lattice~$L$,
so $L^* = \Hom(L,\mathbb{Z})$ is a sublattice of index~$k$ in~$L'^* = \Hom(L',\mathbb{Z})$,
and assume that the vector $u$
lies in the sublattice $L'$.
Then the Abelian group $G = (L/L')$ of order $k$ acts on $\mathcal{Q}[L'^*]$,\footnote{An element $n$ in $L$ multiplies monomial $X^{m'}$ by the root of unity $\exp((2 \pi i) (n,m'))$, here $(n,m')$ is bilinear pairing between
$L$ and $L'^*$ with values in $\mathbb{Q}$ extended by linearity from the pairing $L' \otimes L'^* \to \mathbb{Z}$.}
and its invariants is the subring $\mathcal{Q}[L^*]$,
so~$G$ acts on the torus $T' = {\text{Spec }} \mathcal{Q}[L'^*]$ and the torus $T = {\text{Spec }} \mathcal{Q}[L^*]$ is the quotient-torus $T = T' / G$,
let $\pi : T' \to T$ be the projection to the quotient.
The vector $u$ def\/ines the birational transformation~$\mu_{u,T}$ of the torus $T$ and the birational transformation~$\mu_{u,T'}$ of the torus~$T'$.
Then the mutation~$\mu_u$ commutes with the action of the group~$G$ and with the projections:
$\pi \mu_{u,T'} = \mu_{u,T} \pi$
and $g \mu_{u,T'} = \mu_{u,T'} g$
for any $g \in G$.
\end{remark}

\subsubsection{Rank two case}
Let us see the mutations explicitly in case $\rk L = 2$.
Let $e_1$, $e_2$ be a base of $L$ and $f_1$, $f_2$ be the dual base of~$L^*$,
so $(e_i,f_j) = \delta_{i,j}$. Also let $x_i = X^{f_i}$ be the respective monomials in $\mathbb{Z}[L^*]$.
For
the skew-symmetric bilinear form $\omega_k$ def\/ined by $\omega_k(e_1,e_2) = k$
and a vector $u = u_1 e_1 + u_2 e_2 \in L$
we have
$i_{\omega_k} (u_1 e_1 + u_2 e_2) = (-k u_2) f_1 + (k u_1) f_2$ and so
\begin{equation*}
\mu_{u,\omega_k} : \ (x_1,x_2) \to \big(x_1 \cdot \big(1+x_1^{-k u_2} x_2^{k u_1}\big)^{u_1}, x_2 \cdot \big(1+x_1^{-k u_2} x_2^{k u_1}\big)^{u_2}\big),
\end{equation*}
in particular the inverse map to $\mu_{u,\omega_1}$ is given by
$\mu_{-u,-\omega_1} : (x_1,x_2) \to (x_1 \cdot (1+x_1^{u_2} x_2^{-u_1})^{-u_1}, x_2 \cdot (1+x_1^{u_2} x_2^{-u_1})^{-u_2})$.
In particular, $\mu_{(0,1)}^* f = f\big(x_1,\frac{x_2}{1+x_1}\big)$.

For any matrix $A = \left(\begin{matrix} a & b  \\ c & d\end{matrix} \right) \in \operatorname{SL}(2,\mathbb{Z}) = \operatorname{Sp}(L,\omega)$
there is a regular automorphism of the torus $t_A^* (x_1,x_2) = (x_1^a x_2^b, x_1^c x_2^d)$.
Conjugation by this automorphism acts on the set of mutations:
$\mu_{A u}^* = (t_A \mu_u t_A^{-1})^*$.
So any mutation commutes with an inf\/inite cyclic group given by the stabilizer of $u$ in $\operatorname{Sp}(L,\omega)$,
explicitly if $u=(0,1)$ then in coordinates $(x_1,x_2)$ and $(x_1,x_2' = x_1 x_2)$ the mutation $\mu_{(0,1)}$ is given by the same formula.
Also every mutation commutes with $1$-dimensional subtorus of~$T$,
in case of $u=(0,1)$ the action of the subtorus is given by $(x_1,x_2) \to (x_1, \alpha x_2)$.

\subsection{Mutations of exchange collections and seeds}
Let $L$ be a lattice equipped with a bilinear skew-symmetric form $\omega$.
A~\emph{cluster} $\mathbf{y} \in \mathbb{K}_L^m$ is a collection
$\mathbf{y} = (y_1,\dots,y_m)$
of $m$ rational functions
$y_i \in \mathbb{K}_L$.
We call $\mathbf{y}$ \emph{a base cluster} if $\mathbf{y} = (y_1,,\dots,y_r)$
is a base of the ambient f\/ield $\mathbb{K}_L$.
A~$C$-seed (supported on  $(L,\omega)$)
is a pair $(\mathbf{y},V)$ of a cluster $\mathbf{y} \in \mathbb{K}_L^m$ and an exchange collection
$V = (v_1,\dots,v_n) \in L^n$.
A~$V$-seed (supported on  $(L,\omega)$)
is a pair $(W,V)$ of a rational function $W \in \mathbb{K}_L$ and
an exchange collection $V = (v_1,\dots,v_n) \in L^n$.

Given two exchange collections
 $V'=(v_1',\dots,v_n') \in L'^n$
and
 $V = (v_1,\dots,v_n) \in L^n$
we say that $V'$ is \emph{a mutation} of $V$
in the direction $1\leqslant j\leqslant n$ and denote it by $V' = \mu_j V$ if
under the given identif\/ication $s_j : L \simeq L'$
we have $v'_j = s_j(-v_j)$
and
$v'_i = s_j(\mu_{v_j} v_i)$ for $i\neq k$.

The mutation of a $C$-seed $(\mathbf{y},V)$ in the direction $1\leqslant j\leqslant n$
is a new $C$-seed $(\mathbf{y}_j,V_j)$ where
$V_j = \mu_j V$ is a mutation of the exchange collection,
and $\mathbf{y}_j = \mu_{v_j,\omega} \mathbf{y}$ where each variable is
transformed by the birational transformation $\mu_{v_j,\omega}$.

The identity $\mu_{-u} \mu_u = R_u$ implies that $\mu_j(\mu_j(V))$ and $V$ are
related by the $\operatorname{Sp}(L,\omega)$-trans\-for\-ma\-tion~$R_u$.

\subsection[Upper bounds and property $(V)$]{Upper bounds and property $\boldsymbol{(V)}$}

\begin{definition}[property $(V)$] \label{pv}
We say a $V$-seed $(W,V)$ satisf\/ies property $(V)$ if~$W$ is a Laurent polynomial and for all $v \in L$ the functions ${(\mu_{v}^*)}^{m_V(v)} W$ are also Laurent polynomials.
\end{definition}

In this paper we introduce the upper bound of an exchange collection.
\begin{definition}[upper bounds] \label{ub}
For a $C$-seed $\Sigma = (\mathbf{y},V)$
def\/ine its \emph{upper bound} $\mathcal{U}(\Sigma)$ to be the $\mathcal{Q}$-subalgebra of $\mathbb{K}_L$ given by
\begin{equation*}
\mathcal{U}(\Sigma) = \mathcal{Q}\big[\mathbf{y}^{\pm 1}\big] \cap \big(\cap_{v \in L} \mathcal{Q}\big[{(\mu_{v}^*)}^{m_V(v)} \mathbf{y}^{\pm 1}\big]\big).
\end{equation*}
In case $\mathbf{y}$ is a base cluster (by abuse of notation)
we denote $\mathcal{U}(\Sigma)$ just by~$\mathcal{U}(V)$.
\end{definition}

The upper bounds def\/ined here are a straightforward generalization of the upper bounds in~\cite{BFZ},
but also they can be thought of as the gatherings of all potentials satisfying property~$(V)$.

\begin{proposition}[relation between property $(V)$ and upper bounds] \label{vub1}
The upper bound $\mathcal{U}(V)$ of an exchange collection $V$ consists of all functions $W \in \mathbb{K}_L$
such that the $V$-cluster $(W,V)$ satisfies property~$(V)$.
\end{proposition}

\begin{proposition} \label{prop-isom}
Any morphism $f : (L,\omega) \to (L',\omega')$
induces a dual morphism $f^* : L'^* \to L^*$,
a homomorphism of algebras $f^* : \mathbb{Z}[L'^*] \to \mathbb{Z}[L^*]$.
Assume that this homomorphism has no kernel\footnote{One can bypass this assumption by def\/ining the upper bound $\mathcal{U}(L,\omega;V)$ as a subalgebra in
some localization of $\mathbb{Z}[L^*]$ determined by the exchange collection $V$.}.
Then it induces a homomorphism of upper bounds $f^* : \mathcal{U}(L',\omega';f V) \to \mathcal{U}(L,\omega;V)$.
In particular, if $f$ is an isomorphism, then maps $f^*$ and $(f^{-1})^*$ establish the isomorphisms
between the upper bounds $f^* : \mathcal{U}(L',\omega';f V) \simeq \mathcal{U}(L,\omega;V)$.
\end{proposition}

\begin{proposition} \label{fun2}
Consider a seed $\Sigma = (L,\omega;v_1,\dots,v_n)$.
For a sublattice $L' \subset L$ that contains all vectors $v_i \in L' \subset L$
consider the seed $\Sigma' = (L',\omega_{|_{L'}};v_1,\dots,v_n)$.
By Remark~{\rm \ref{fun1}} there is a~natural action of
$G = L/L'$ on $\mathbb{K}_{L'}$
with $\mathbb{K}_L = \mathbb{K}_{L'}^G$.
Moreover,
the action of $G$ obviously preserves
the property of being a~Laurent polynomial
$($i.e.\ it preserves the subalgebras $\mathcal{Q}[L'^*])$,
and the mutations~$\mu_{v_i}$ commute with the $G$-action.
Thus the upper bound with respect to the overlattice $L$
is the subring of $G$-invariants of the upper bound with respect to the sublattice~$L'$:
$\mathcal{U}(\Sigma) = \mathcal{U}(\Sigma')^G = \mathcal{U}(\Sigma') \cap \mathcal{Q}[L^*]$.
\end{proposition}

\section{Laurent phenomenon}\label{section3}

In what follows we restrict ourselves to the case $\rk L = 2$, $\omega$ is a non-degenerate form and the vectors of exchange collection are primitive,
however none of these conditions is essential.

Next theorem is the analogue of Theorem~1.5 in~\cite{BFZ}, presented here as Theorem~\ref{bfz-theorem}.

\begin{theorem}[Laurent phenomenon in terms of upper bounds] \label{ublemma}
Consider two $C$-seeds: $\Sigma = (L,\omega;v_1,\dots,v_n)$ and
$\Sigma' = (L',\omega';v_1',\dots,v_n')$.
If $\Sigma' = \mu_i \Sigma$ is a mutation of $\Sigma$ in direction $1\leqslant i\leqslant n$
then
the upper bounds for $\Sigma$ and $\Sigma'$ coincide: $\mathcal{U}(\Sigma) = \mu_{v_i}^* \mathcal{U}(\Sigma')$.
As a~corollary, if a seed~$\Sigma'$ is obtained from a seed $\Sigma$
by a sequence of mutations, then the upper bound $\mathcal{U}(\Sigma')$
equals to the upper bound~$\mathcal{U}(\Sigma)$ under identification of the ambient field
by composition of the birational mutations.
\end{theorem}

By Proposition \ref{vub1} Theorem~\ref{ublemma} is equivalent to the next corollary,
which is easier to check in practice
and has almost the same consequences as the main theorem
of~\cite{GU}, presented here as Theorem~\ref{ulemma}.
\begin{corollary}[$V$-lemma] \label{vlemma}
If $V$-seeds $\Sigma$ and $\Sigma'$ are related by a mutation
then the seed $\Sigma$ satisfies property $(V)$ $\iff$
the seed $\Sigma'$ satisfies property~$(V)$.
\end{corollary}

In the rest of this section we prove Theorem~\ref{ublemma}.
Our proof is quite similar to that of~\cite{BFZ}\footnote{See Appendix~\ref{sec-bfz} and Remark~\ref{compare-bfz} for the detailed comparison.}:
The set-theoretic argument
reduces the problem to
exchange collection $V$ with small number of vectors~($1$~or~$2$) without counting of multiplicities.
Actually, when the collection~$V$ has only one vector
the equality of the upper bounds is obvious from the def\/initions.
When the exchange collection consists of two base vectors one can explicitly compute the upper bounds
and compare them. Finally, the case of two non-base non-collinear vectors is thanks to functoriality.

First of all, let us f\/ix the notations.
If the rank two lattice~$L$ is generated by a pair of vectors~$e_1$ and~$e_2$,
then the dual lattice $L^* = \Hom(L,\mathbb{Z})$ has the dual base $f_1$, $f_2$ determined by
$(f_i,e_j) = \delta_{i,j}$. The form $\omega$ is uniquely determined by its value
$k = \omega(e_1,e_2)$, and further we denote this isomorphism class of forms by~$\omega_k$.
We assume that $k\neq 0$, i.e.\ the form $\omega$ is
non-degenerate\footnote{If $k=0$ then $\omega=0$ and all mutations are trivial.},
by swapping~$e_1$ and~$e_2$ one can exchange $k$ to $-k$.
A base~$e_i$ of $L$ corresponds to a base $x_i = X^{f_i}$ of $\mathbb{Z}[L^*]$.

\begin{lemma} \label{l41a}
Let $V$ be an exchange collection in $(L,\omega)$ and $\Sigma = (L,\omega;V)$ be the respective seed.
\begin{enumerate}\itemsep=0pt
\item[$1.$]
If $V$ is empty, then obviously $\mathcal{U}(L,\omega;V) = \mathcal{Q}[L^*]$.
\item[$2.$]
Otherwise, let $V_\alpha$ be a set of exchange collections
such that for any $v\in L$ we have
$m_V(v) = \max_{\alpha} m_{V_\alpha}(v)$.
Then
\[ \mathcal{U}(V) = \cap_{\alpha} \mathcal{U}(V_\alpha) . \]
\item[$3.$]
In particular, if for a vector $v \in L$ we define $V_v = m_V(v) \times v$
to be an exchange collection that consists of a single vector $v$
with multiplicity $m_V(v)$ and $\Sigma_v = (L,\omega;V_v)$ be the respective seed,
then
$ \mathcal{U}(\Sigma) = \cap_{v \in L} (\mathcal{Q}[\mathbf{y}^\pm] \cap \mathcal{Q}[{(\mu_v^*)}^{m_V(v)} \mathbf{y}^\pm])
= \cap_{v \in L} \mathcal{U}(\Sigma_v)$.
In other words, the upper bound of a $C$-seed $\Sigma = (L,\omega;\mathbf{y},V)$ can be expressed as the intersection of
the upper bounds for its $1$-vector subseeds.
\item[$4.$]
Let $V$ consist of a vector $v_1$ with multiplicity $m_+ \geqslant 1$,
a vector $v_2 = - v_1$ with multiplicity \mbox{$m_- \geqslant 0$},
and vectors $v_k$ $(k\geqslant 3)$ that are non-collinear to $v_1$ with some multiplicities \mbox{$m_k \geqslant 0$}.
Consider exchange subcollections $V_0 = \{m_+\times v_1, m_-\times (-v_1)\}$
and $V_k = \{1\times v_1,$ $m_k \times v_k\}$ $(k\geqslant 3)$.
Then
\[ \mathcal{U}(V) = \mathcal{U}(V_0) \cap \mathcal{U}(V_3) \cap \mathcal{U}(V_4) \cap \cdots . \]
\item[$5.$]
Let $V' = \mu_1 V$ be an exchange collection obtained by mutation of~$V$ in~$v_1$;
it consists of vector $-v_1$ with multiplicity $m_- + 1\geqslant 1$,
vector $v_1$ with multiplicity $m_+ - 1 \geqslant 0$
and vectors $v_k' = \mu_{v_1} v_k$ $(k\geqslant 3)$ with multiplicities $m_k$.
Similarly to the previous step define $V_0' = \{(m_- + 1)\times (-v_1), (m_+ - 1)\times v_1\}$
and $V_k' = \{1\times (-v_1), m_k \times v_k'\}$ $(k \geqslant 3)$. Then
\[ \mathcal{U}(V') = \mathcal{U}(V_0') \cap \mathcal{U}(V_3') \cap \mathcal{U}(V_4') \cap \cdots. \]
\item[$6.$]
Hence, to proof Theorem~{\rm \ref{ublemma}} it is necessary and sufficient
to show that
\[ \mathcal{U}(V_0) = \mu_{v_1}^* \mathcal{U}(V_0')
\qquad \text{and} \qquad
\mathcal{U}(V_k) = \mu_{v_1}^* \mathcal{U}(V_k') \quad(\text{for all} \ k\geqslant 3 ). \]
We will prove these equalities in Proposition~{\rm \ref{l43c1}} and Lemma~{\rm \ref{l46}}.
\end{enumerate}
\end{lemma}

\begin{proposition} \label{isom2d}
Let $v_1,v_2 \in L$ be a pair of vectors $v_1 = a e_1 + b e_2$, $v_2 = c e_1 + d e_2$
such that  $ad-bc = 1$.
Consider the lattice $L'$ with the base $e_1'$, $e_2'$
and the form $\omega'(e_1',e_2') = \omega(v_1,v_2)$;
let~$f_1'$,~$f_2'$ be the dual base of~$L'^*$.
Consider a map $m : L \to L'$ given by
$m(e_1) = d e_1' - b e_2', m(e_2) = -c e_1' + a e_2'$;
note that
$m(v_1) = m(a e_1 + b e_2) = e_1'$ and  $m(v_2) = m(c e_1 + d e_2) = e_2'$.
The dual isomorphism $m^*: L'^* \to L^*$ is given by the transposed map
$m^* (f_1') =  d f_1 - c f_2$ and $m^*(f_2') = -b f_1 + a f_2$.
Let $z_1 = X^{f_1'} = x_1^d x_2^{-c}$ and $z_2 = X^{f_2'} = x_1^{-b} x_2^a$.
Since map $m^*$ is invertible by Proposition~{\rm \ref{prop-isom}} it gives the equality
\[ \mathcal{U}(L,\omega;m_1\times v_1,m_2\times v_2) = \mathcal{U}\big(L',\omega';m_1\times e_1',m_2\times e_2'\big)\big|_{{z_1 = x_1^d x_2^{-c},\,z_2 = x_1^{-b} x_2^a}}. \]
\end{proposition}

\begin{lemma} \label{l42}
Assume a seed $\Sigma = (L,\omega;m_1\times v_1)$ consists of a unique vector $v_1$ with multiplicity $m_1 \geqslant 1$.
\begin{enumerate}\itemsep=0pt
\item[$1.$]
If $v_1 = e_2 = (0,1)$
then the upper bound $\mathcal{U}(\Sigma)$ consists of all Laurent polynomials $W$ of the form
$W = \sum_l c_l(x_1) x_2^l$
where $c_l \in \mathcal{Q}[x_1^\pm]$ and for $l \leqslant 0$ we have that $c_l$ is divisible by
$(1+x_1^k)^{-m_1 l}$.
Moreover, $\mathcal{U}(\Sigma) = \mathcal{Q}\big[x_1^\pm, x_2, \frac{1}{x_2^{\prime \prime}} = \frac{(1+x_1^k)^{m_1}}{x_2}\big]$.
\item[$2.$]
If $v_1 = a e_1 + b e_2 = (a,b)$ is an arbitrary primitive vector
then
$\mathcal{U}(\Sigma) = \mathcal{Q}\big[z^{\pm}, z_1, \frac{(1 + z^k)^{m_1}}{z_1}\big]$
 where
$z =  \frac{x_1^a}{x_2^b}$,
$z_1 = x_1^r x_2^s$
and $(r,s)\in\mathbb{Z}^2$ satisfies $rb+sa=1$.
\end{enumerate}
\end{lemma}

\begin{proof}
Recall that mutation in the direction $e_2$ is given by
$x_1' = x$ and $x_2' = \frac{x_2}{1+x_1^k}$.
Assume we have a Laurent polynomial $W = \sum_{l \in \mathbb{Z}} c_l(x_1) x_2^l$.
Then $W$ can be expressed in terms of $x_1$ and $x_2'$ as
$W = \sum_l c_l(x_1) (1+x_1^k)^l (x_2')^l$.
This function is a Laurent polynomial in terms of $(x_1,x_2')$ $\iff$
$c_l(x_1) (1+x_1^k)^l$ is a Laurent polynomial of $x_1$ for all $l$. This is equivalent to
$c_l$ being divisible by $(1+x_1^k)^{-l}$ for $l \leqslant 0$.
Similarly if we do $m_1$ mutations then $x_2^{\prime \prime} = \frac{x_2}{(1+x_1^k)^a}$
and $W = \sum c_l (1+x_1^k)^{m_1 l} (x_2^{\prime \prime})^l$
so for $l\leqslant 0$ we have that $c_l$ is divisible by $(1+x_1^k)^{-m_1 l}$.
Let $c_l(x_1) = (1+x_1^k)^{-m_1l} c'_{-l}(x_1)$ for $l < 0$, $c'_l$ are also Laurent polynomials.
Denote $W_+ = \sum_{l \geqslant 0} c_l(x_1) x_2^l$ and $W_- = \sum_{l <0} c_l(x_1) x_2^l = \sum_{l > 0} c'_l(x_1) (x_2^{\prime \prime})^{-l}$.
Then obviously both $W_+$ and $W_-$ belong to $\mathcal{Q}\big[x_1^\pm,x_2,\frac{1}{x_2^{\prime \prime}}\big]$.
The reverse inclusion is straightforward.

Part (2) follows from Proposition~\ref{isom2d}.
\end{proof}

\begin{proposition} \label{lhz2}
Let exchange collection $V$ consists of a vector $v_1 = e_2 = (0,1)$ with multiplicity $m_1\geqslant 0$
and its inverse $v_2 = -v_1 = -e_2 = (0,-1)$ with multiplicity $m_2 \geqslant 0$,
\begin{enumerate}\itemsep=0pt
\item[$1.$]
The upper bound $\mathcal{U}(L,\omega;m_1\times e_2,m_2\times (-e_2))$
consists of all Laurent polynomials $W$ of the form
$W = \sum_l c_l(x_1) x_2^l$ where $c_l \in \mathcal{Q}[x_1^\pm]$ and for $l \leqslant 0$ we have that $c_l$ is divisible by $(1+x_1^k)^{-m_1l}$
and for $l \geqslant 0$ we have that $c_l$ is divisible by $(1+x_1^k)^{m_2l}$.
\item[$2.$]
$\mathcal{U}(L,\omega;m_1\times e_2,m_2\times(-e_2)) = \mathcal{Q}\big[x_1^\pm, x_2(1+x_1^k)^{m_2},\frac{(1+x_1^k)^{m_1}}{x_2}\big].
$
\end{enumerate}
\end{proposition}

\begin{proof}
The f\/irst statement is a straightforward corollary of Lemmas~\ref{l41a} and~\ref{l42}(1).
The proof of the second statement is similar to the end of the proof of Lemma~\ref{l42}(2):
separate the Laurent polynomial $W$ into positive and negative parts~$W_+$ and~$W_-$;
then both parts lie in the ring $\mathcal{Q}\big[x_1^\pm,x_2 (1+x_1^k)^{m_2},\frac{(1+x_1^k)^{m_1}}{x_2}\big]$.
\end{proof}

\begin{proposition} \label{l43c1}
Assume a seed $\Sigma$ consists of a vector $v_1 = (0,1)$ with multiplicity $m_1$
and its inverse $-v_1 = (0,-1)$ with multiplicity $m_2$.
Then its mutation $\Sigma' = \mu_1(\Sigma)$ consists of~$v_1$ and~$-v_1$ with respective multiplicities $m_1-1$ and $m_2+1$.
Then $\mathcal{U}(\Sigma) = \mathcal{U}(\Sigma')$.
\end{proposition}
\begin{proof}
By Proposition~\ref{lhz2} the upper bounds are expressed as:
$\mathcal{U}(\Sigma) = \mathcal{Q}\big[x_1^\pm,x_2 (1+x_1^k)^{m_2}$, $\frac{(1+x_1^k)^{m_1}}{x_2}\big]$,
$\mathcal{U}(\Sigma') = \mathcal{Q}\big[x_1'^\pm,\frac{(1+x_1'^k)^{m_1-1}}{x_2'},x_2' (1+x_1'^k)^{m_2+1}\big]$.
Since $x_1' = x_1$ and $x_2' = \frac{x_2}{1+x_1^k}$ we have the desired equality of the upper bounds.
\end{proof}

\begin{proposition} \label{lhz3}
Assume that the seed $\Sigma = (L,\omega;m_1\times v_1,m_2\times v_2)$
consists of vectors $v_1$ with multiplicity $m_1 \geqslant 0$
and $v_2$ with multiplicity $m_2 \geqslant 0$.
\begin{enumerate}\itemsep=0pt
\item[$1.$]
If $v_1 = e_1$ and $v_2 = e_2$,
then the upper bound $\mathcal{U}(\Sigma)$ equals
$\mathcal{Q}\big[x_1, x_2, \frac{(1 + x_2^k)^{m_1}}{x_1}, \frac{(1 + x_1^k)^{m_2}}{x_2}\big]$.
\item[$2.$]
If $v_1 = a e_1 + b e_2$ and $v_2 = c e_1 + d e_2$ with $ad-bc = 1$
then the upper bound $\mathcal{U}(\Sigma)$ equals
$\mathcal{Q}\big[z_1, z_2, \frac{(1 + z_2^k)^{m_1}}{z_1}, \frac{(1 + z_1^k)^{m_2}}{z_2}\big]$
with
$z_1 = x_1^dx_2^{-c}$
and
$z_2 = x_1^{-b}x_2^a$.
\end{enumerate}
\end{proposition}

\begin{proof}
For the f\/irst case,
by Lemmas \ref{l41a} and \ref{l42}
we have
$\mathcal{U}(\Sigma) = \mathcal{Q}\big[x_1^\pm,x_2,\frac{(1+x_1^k)^{m_2}}{x_2}\big] \cap \mathcal{Q}\big[x_2^\pm,x_1,\frac{(1+x_2^k)^{m_1}}{x_1}\big]$. If $m_1 = m_2 = 1$,
by Proposition~4.3 of~\cite{BFZ} (with $|b_{12}| = |b_{21}| = b = c = k$ and $q_1=q_2=r_1=r_2=1$)
this intersection equals
$\mathcal{Q}\big[x_1,x_2,\frac{1+x_2^k}{x_1},\frac{1+x_1^k}{x_2}\big]$.
Lemma~\ref{l42} covers cases with $m_1 = 0$ or $m_2 = 0$.
If $m_1$ and $m_2$ are greater than $1$,
the proof of Proposition~4.3 in~\cite{BFZ} can be easily modif\/ied to include the case we need since $x_2 \frac{1 + x_1^k}{x_2}(1 + x_1^k)^{m_{2}-1} = (1+ x_1^k)^{m_2}$ and $x_1 \frac{1 + x_2^k}{x_1}(1 + x_2^k)^{m_{1}-1} = (1+ x_2^k)^{m_1}$ , then the intersection equals $\mathcal{Q}\big[x_1,x_2,\frac{(1+x_2^k)^{m_1}}{x_1},\frac{(1+x_1^k)^{m_2}}{x_2}\big]$.

Part (2) follows from Proposition~\ref{isom2d}.
\end{proof}

\begin{lemma} \label{l46}
Let $\Sigma = (L,\omega;1 \times v_1, m_2 \times v_2)$ be a seed of two non-collinear vectors $v_1$ and $v_2$
with $m(v_1) = 1$ and $m(v_2) = m_2 \geqslant 0$
and
$\Sigma' = \Sigma_1 = (L'=L,\omega'=\omega; v_1' = -v_1, m_2 \times (v_2' = \mu_{v_1} v_2) )$
be the mutation of the seed $\Sigma$ in~$v_1$.
Then $\mathcal{U}(\Sigma) = \mu_{v_1}^* \mathcal{U}(\Sigma')$.\footnote{Denote $\Sigma_2 = \{(\mu_{v_2})^{m_2} v_1, -v_2 \}$~-- $m_2$-multiple mutation of $\Sigma$ in $v_2$,
and $\Sigma'_2 = \{ (\mu_{v_2'})^{m_2} v_1',  -v_2' \}$~-- $m_2$-multiple mutation of $\Sigma'$ in $v_2'$.
We are going to prove that
 $\mathcal{U}(\Sigma) = \mathcal{Q}[\mathbf{y}] \cap \mathcal{Q}[\mathbf{y}' = \mathbf{y_1}] \cap \mathcal{Q}[\mathbf{y_2}]$
equals to
$\mathcal{U}(\Sigma') = \mathcal{Q}[\mathbf{y}] \cap \mathcal{Q}[{\mathbf{y}'} = \mathbf{y_1}] \cap \mathcal{Q}[\mathbf{y_2'}]$.}
\end{lemma}
\begin{proof}
First of all note that, $\omega'(v_1',v_2') = -\omega(v_1,v_2)$ and since
$\mu_{-v_1} \mu_{v_1} = R_{v_1}$ it is suf\/f\/icient to consider only the case
$\omega(v_1,v_2) > 0$.
We f\/irst consider the case when $v_1 = e_1 = (1,0)$ and $v_2 = e_2 = (0,1)$;
denote $k = \omega(e_1,e_2) > 0$.
Let $e_1'$, $e_2'$ be the base of $L'$ that corresponds to~$e_1$,~$e_2$ under the natural
identif\/ication of $L' \simeq L$;
f\/inally consider a base $e_1''$, $e_2''$ of $L'$
given by $e_1'' = v_2' = k e_1' + e_2'$, $e_2'' = v_1' = -e_1'$.
Let $f_1'$, $f_2'$ and $f_1''$, $f_2''$ be the respective dual bases of~$L'^*$.
Thus we have one natural regular system of coordinates
$x_1 = X^{f_1}$, $x_2 = X^{f_2}$
on the torus $T = {\text{Spec }} \mathbb{Z}[L^*]$,
and two regular systems of coordinates
$x_1' = X^{f_1'}$, $x_2' = X^{f_2'}$; $x_1'' = X^{f_1''}$, $x_2'' = X^{f_2''}$
on the torus $T' = {\text{Spec }} \mathbb{Z}[L'^*]$.
Taking $a=k$, $b=1$, $c=-1$ and $d=0$
in Proposition~\ref{isom2d}
we have that $x_1'' = x_2'$ and $x_2'' = \frac{x_2'^k}{x_1'}$, since $x_2' = x_2$ and $x_1' = \frac{x_1x_2^k}{1+x_2^k}$ (they are the mutations of~$x_1$ and~$x_2$ with respect to $v_1$), thus what we need to show is that the rings
$\mathcal{Q}\big[x_1,x_2,\frac{1+x_2^k}{x_1}, \frac{1+x_1^k}{x_2}\big]$
and $\mathcal{Q}\big[x_1,x_2, \frac{1+x_2^k}{x_1}, \frac{x_1^{k} + (1 + x_2^k)^k}{x_1^kx_2}\big]$ are equal.
We will f\/irst show that
$\frac{x_1^{k} + (1 + x_2^k)^k}{x_1^kx_2} \in \mathcal{Q}\big[x_1,x_2, \frac{1+x_2^k}{x_1}, \frac{1+x_1^k}{x_2}\big]$.
We have that $ \frac{x_1^{k} + (1 + x_2^k)^k}{x_1^kx_2} = \big(\frac{1 + x_1^k}{x_2}\big)\big(\frac{(1+x_2^k)^k}{x_1^k}\big) - \sum\limits_{j=1}^k \frac{k!}{j!(k-j)!} x_2^{kj-1}$. Clearly the expression in the right side belongs to $\mathcal{Q}\big[x_1,x_2, \frac{1+x_2^k}{x_1}, \frac{1+x_1^k}{x_2}\big]$.
Now, we will show that $\frac{1+x_1^k}{x_2} \in \mathcal{Q}\big[x_1,x_2, \frac{1+x_2^k}{x_1}, \frac{x_1^k + (1 + x_2^k)^k}{x_1^kx_2}\big]$. We have that $\frac{1+x_1^k}{x_2} = x_1^k  \frac{x_1^{k} + (1 + x_2^k)^k}{x_1^kx_2} - \sum\limits_{j=1}^k \frac{k!}{j(k-j)!} x_2^{kj-1}$. Again, clearly the expression in the right side belongs to $\mathcal{Q}\big[x_1,x_2, \frac{1+x_2^k}{x_1}, \frac{x_1^{k} + (1 + x_2^k)^k}{x_1^kx_2}\big]$. Thus, we have the equality between the rings. Similarly, if the multiplicity of $v_2$ is $m_2 > 1$, we have that $\mathcal{Q}\big[x_1,x_2,\frac{(1+x_2^k)}{x_1}, \frac{(1+x_1^k)^{m_2}}{x_2}\big] = \mathcal{Q}\big[x_1,x_2, \frac{1+x_2^k}{x_1}, \frac{(x_1^{k} + (1 + x_2^k)^k)^{m_2}}{x_1^{m_2k}x_2}\big]$.\footnote{The argument for showing the equality of these two rings is the same of that when $m_2 = 1$, but the computations are slightly longer, so we omit them.}
If $v_1$ and $v_2$ are another basis of $\mathbb{Z}^2$ the result follows from
Proposition~\ref{isom2d}.
In case~$v_1$ and~$v_2$ is a pair of non-collinear vectors which are not a basis for~$\mathbb{Z}^2$,
consider the sublattice $L' \subset L$ generated by $e_1' = v_1$ and $e_2' = v_2$
with the form $\omega' = \omega|_{{L'}}$.
As we just saw upper bounds with respect to the sublattice coincide:
$\mathcal{U}(L',\omega';v_1, m_2\times v_2) = \mu_{v_1}^* \mathcal{U}(L',\omega';-v_1,m_2\times \mu_{v_1} v_2)$.
Now the statement follows from the Proposition~\ref{fun2}.
\end{proof}

\begin{remark}
If a mutation of a Laurent polynomial with integer coef\/f\/icients
happened to be a Laurent polynomial, then its coef\/f\/icients are also integer.
Let $u \in L$ be a primitive vector
and $W, W' \in \mathcal{Q}[L^*]$ be a pair of Laurent polynomials with arbitrary coef\/f\/icients
such that $W = \mu_u^* W'$.
Then $W \in \mathbb{Z}[L^*] \iff W' \in \mathbb{Z}[L^*]$.
\end{remark}

\begin{proof}
Choose coordinates on $L$ so that $u = e_2$.
Assume $W$ has integer coef\/f\/icients.
By Lemma~\ref{l42}(1) $W = \sum c_l(x_1) x_2^l$
and $W' = \sum_{l\in\mathbb{Z}} c_l'(x_1) x_2'^l$
with $c_l' = c_l (1+x_1^k)^{-l}$ for all $l\in \mathbb{Z}$.
Clearly $W \in \mathbb{Z}[L^*] \iff$ all coef\/f\/icients of $W$ are integer $\iff$
for all $l\in \mathbb{Z}$ all coef\/f\/icients of $c_l$ are integer.
Since $(1+x_1^k) \in \mathbb{Q}[x_1^{\pm}]$ it is clear that $c_l' \in \mathbb{Q}[x_1^{\pm}]$.
Recall that for a Laurent polynomial $P \in \mathbb{Q}[x]$ its Gauss's content
$C(P) \in \mathbb{Q}$ is def\/ined as the greatest common divisor of all its coef\/f\/icients:
if $P = \sum a_i x^i$ then $C = \gcd(a_i)$.
Clearly $C(P) \in \mathbb{Z} \iff P \in \mathbb{Z}[x^{\pm}]$.
Gauss's lemma says that $C(P \cdot P') = C(P) \cdot C(P')$.
Since $C(1+x_1^k) = \gcd(1,1) = 1$
we see that $C(c_l') = C(c_l) \cdot 1^{-l} = C(c_l)$,
hence $c_l' \in \mathbb{Z}[x_1^\pm] \iff c_l \in \mathbb{Z}[x_1^\pm]$.
\end{proof}

\section{Questions and future developments}

In the introduction was pointed out that our def\/inition of upper bounds makes plausible to consider a quantum version of mutations of potentials and the corresponding quantum Laurent phenomenon. On the other hand, in \cite{K} a non-commutative version of the Laurent phenomenon is discussed. Thus, we would like to ask:

\begin{question}
Is it possible to consider a non-commutative version of the Laurent phenomenon for mutation of potentials and develop a theory of upper bounds in this context?
\end{question}

In \cite{GU} the following problem (Problem 44) was proposed

\begin{question} \label{ques2}
Construct a fiberwise-compact canonical mirror of a Fano variety as a~gluing of open charts given by $($all$)$ different toric degenerations.
\end{question}

Conjecture \ref{conj} (which will be proved in \cite{CG}) gives a partial answer for the above question.

\begin{conjecture} \label{conj}
For the $10$ potentials $W$ $($i.e., $(W_1,W_2,\dots,W_9,W_Q))$ listed in {\rm \cite{GU}} $($or rather the exchange collections $V$ $($resp.\ $V_1,\dots,V_9,V_Q))$ the upper bound $\mathcal{U}(V)$ is the algebra of polynomials in one variable. Moreover, this variable is $W$.
\end{conjecture}

Conjecture~\ref{conj} is useful for symplectic geometry as long as one knows two (non-trivial) properties of the FOOO's potentials~$m_0$~\cite{FOOO} (here $W = m_0$):
\begin{enumerate}\itemsep=0pt
\item
$W$ is a Laurent polynomial (this is some kind of convergence/f\/initeness property).

\item
$W$ is transformed according to Auroux's wall-crossing formula~\cite{AU},
and more specif\/ically by the mutations described in Section~\ref{sec-def}. The directions of the mutations/walls are encoded by an exchange collection~$V$.
\end{enumerate}

What we believe is that once one knows these assumptions, one should be able to prove that some disc-counting potential equals some particularly written $W$ (formally) without any actual disc counting. Needless to say this is a speculative idea.


\section{Introduction \label{Introduction}}

From the string worldsheet perspective, two-dimensional non-linear $\sigma$-models with boundaries provide a rich area  to describe curved background geometries with  D-brane configurations in string theory. These  non-perturbative degrees of freedom are essential higher-dimensional objects on which open strings can end, and of which the geometry is completely determined by the allowed worldsheet boundary conditions. The answer of what boundary conditions are \textit{allowed} is decided by symmetry. In the case of string theory, e.g., they should preserve worldsheet conformal invariance. For a $\sigma$-model describing  strings in curved backgrounds, the answer is usually challenging and tractable only when the precise (boundary) CFT description is available.\\
\indent A simple but non-trivial example where one can make progress is provided by the Wess-Zumino-Witten model \cite{Witten:1983ar} describing strings in group manifolds supported by an NS-flux. The exact conformal invariance of this model is  covered by the existence of two holomorphic currents underlying two copies of an affine Kac-Moody current algebra and two copies of a Virasoro algebra. The inclusion of boundaries in the WZW model has been studied in a number of works \cite{Kato:1996nu,Alekseev:1998mc,Felder:1999ka,Stanciu:1999id} by identifying maximally symmetric gluing conditions on the holomorphic currents at the boundary preserving one copy of both the Kac-Moody and Virasoro algebra. Although the former is not necessary for conformal invariance it leads to  an elegant geometrical picture of the allowed D-brane configurations: they should wrap \textit{twisted conjugacy classes} of the group manifold. For example in the $SU(2)_k$ WZW model one finds two  D0-branes and a further  $k-1$ D2-branes  that are blown up to wrap the conjugacy classes described by $S^2 \subset S^3$ \cite{Alekseev:1998mc}.\\
\indent When the precise CFT formulation is not available, however, we will see in this note an elegant D-brane picture can arise also in  the context of $\sigma$-models with worldsheet integrability. Integrable stringy $\sigma$-models attracted considerable attention since the observation of worldsheet integrability in the AdS$_5\times$S$^5$ superstring \cite{Bena:2003wd}. Classically,  they are characterised by the existence of an infinite number of local or non-local conserved charges in involution leading, in principle, to a dramatic simplicity and exact solvability. Including boundaries in the theory typically destroys  conserved charges such as, e.g., the loss of translational invariance through the boundary. In this note, we will focus on \textit{allowed}  boundary conditions that preserve  the classical integrable structure by demanding the conservation of a tower of non-local charges generated by a monodromy matrix. This method has been introduced in \cite{Cherednik:1985vs,Sklyanin:1988yz} and further developed from a classical string point of view in \cite{Dekel:2011ja}.\\
\indent A suitable integrable $\sigma$-model that makes contact between the above methods is the integrable $\lambda$-deformed WZW model introduced by Sfetsos in \cite{Sfetsos:2013wia}. The deformation parameter $\lambda \in [0,1]$ interpolates between the WZW model at $\lambda = 0$ and the non-Abelian T-dual of the Principal Chiral Model (PCM) in  a scaling limit $\lambda \rightarrow 1$. On ordinary Lie group manifolds, accommodating only bosonic field content, the deformation is marginally relevant \cite{Itsios:2014lca,Appadu:2015nfa}. However, significant evidence from both a worldsheet \cite{Hollowood:2014qma,Appadu:2015nfa} and target space \cite{Borsato:2016zcf,Chervonyi:2016ajp,Borsato:2016ose} perspective implies that, when applied to super-coset geometries, the $\lambda$-model is a truly marginal deformation introducing no Weyl anomaly. Hence, the deformation of the WZW group manifold can be thought of as a bosonic trunctation of a truly superstring theory. The question of establishing D-branes in this deformed geometry is therefore natural and has been pursued in  the article  \cite{Driezen:2018glg} on which this proceedings is based. We will see, by demanding integrability, that the geometrical picture of twisted conjugacy classes of the WZW point persists and naturally fits in the deformed geometry. The semi-classical flux quantisation will turn out to consistently be independent of the continuous $\lambda$-deformation parameter.   Additionally the $\lambda$-deformation allows to track the behaviour of D-branes under generalised dualities \cite{Driezen:2018glg} --again  a challenging question in general curved backgrounds-- by the non-Abelian T-dual scaling limit and Poisson-Lie T-duality to the integrable $\eta$-deformation of the PCM \cite{Vicedo:2015pna,Hoare:2015gda,Klimcik:2015gba}. Illustrated for the $G=SU(2)$ manifold one will find under both dualities D2-branes transforming  to space-filling D3-branes that can be shown to preserve the classical integrable structure of the dual theories.\\

We lay out in section \ref{s:bmm} the general procedure to construct \textit{integrable} boundary conditions of two-dimensional $\sigma$-models. We apply this method in section \ref{s:lambda} to the integrable $\lambda$-deformation where we first review the model's construction, then interpret the allowed integrable boundary conditions as twisted conjugacy classes (illustrated in the $G=SU(2)$ manifold) and discuss their behaviour under generalised T-dualities. We end with some conclusions and outlook directions in section \ref{s:concl}.

\section{The boundary monodromy method for integrable systems}\label{s:bmm}

The boundary monodromy method, introduced by Cherednik and Sklyanin in \cite{Cherednik:1985vs,Sklyanin:1988yz}, is a powerful tool to derive boundary conditions preserving the integrability property of two-dimensional integrable field theories. The method consists of demanding that a monodromy matrix constructed from a Lax connection generates an infinite tower of conserved \textit{non-local} charges when a  boundary is present\footnote{To have a truly classically integrable (boundary) theory one should moreover show these charges to Poisson commute. We will not discuss this here, but see e.g. \cite{Mann:2006rh}.}. We will briefly review it here, following \cite{Dekel:2011ja,Driezen:2018glg}, as well as the  case without boundaries  to introduce notations.

Let us first consider the no-boundary case in a general two-dimensional field theory on a periodic or infinite line. We denote the coordinates by $(\tau,\sigma)$  by analogy with the closed string worldsheet theory. It is known that an infinite tower of conserved charges can be generated when the equations of motion of the theory can be represented by a zero-curvature condition of a so-called $\mathfrak{g}^{\mathbb{C}}$-valued Lax connection ${\cal L}(z)$ that depends on a generic \textit{spectral} parameter $z \in \mathbb{C}$ \cite{Zakharov:1973pp},
\begin{equation} \label{eq:zerocurvatureLax}
\begin{aligned}
d {\cal L}(z) + {\cal L}(z) \wedge {\cal L}(z)  = 0 , \qquad \forall\, z \in \mathbb{C}.
\end{aligned}
\end{equation}
In this case the transport matrix  defined by,
\begin{equation}\label{eq:transport}
\begin{aligned}
T^\Omega(b,a ; z) =  \overleftarrow{P \exp} \left( - \int^b_a d\sigma\, \Omega [{\cal L}_\sigma (\tau,\sigma ;z ) ] \right) \in G^{\mathbb{C}},
\end{aligned}
\end{equation}
(with $\Omega : \mathfrak{g} \rightarrow \mathfrak{g}$  a constant Lie algebra automorphism included for generality) satisfies,
\begin{equation}\label{eq:TransportToTime}
\begin{aligned}
\partial_\tau T^\Omega (b,a ; z ) = T^\Omega (b,a ; z ) \Omega [{\cal L}_\tau (\tau , a ; z)] -  \Omega [{\cal L}_\tau (\tau , b ; z)] T^\Omega (b,a ; z ).
\end{aligned}
\end{equation}
Indeed, under periodic boundary conditions $\sigma \sim \sigma + 2\pi$ or asymptotic fall-off boundary conditions, one can then show that the monodromy matrix $T(2\pi,0 ; z)$ (for $\Omega = \mathbf{1}$) generates conserved charges by,
\begin{equation}
\begin{aligned}
\partial_\tau \mathrm{Tr} T(2\pi, 0 ; z )^n , \qquad \forall\ n\in \mathbb{N} \;\; \mathrm{ and } \;\; \forall\ z\in \mathbb{C} .
\end{aligned}
\end{equation}
Hence, every value of $n$ or every term in the  expansion of $\mathrm{Tr} T(2\pi, 0 ; z )$ in $z$, corresponds to a conserved charge.

When the two-dimensional theory is defined on a finite line $\sigma \in [0,\pi]$, describing by analogy an open string worldsheet theory, one can determine \textit{integrable} boundary conditions on the endpoints by demanding the production of conserved charges along similar lines as above. Reminiscent to the method of image charges, one can derive these by taking a  copy of the finite-line theory and act on it with a reflection $R: \sigma \rightarrow 2\pi - \sigma$. The boundary monodromy matrix $T_b (z)$ is then constructed by gluing the usual transport matrix $T(\pi, 0 ;z )$ in the original region to the transport matrix $T_R^\Omega (2\pi, \pi ; z)$ in the reflected region,
\begin{equation}
T_b (\lambda) = T_R^\Omega (2\pi, \pi ;z) T(\pi, 0 ;z).
\end{equation}
Notice that in the reflected region we have included the possibility of a non-trivial automorphism $\Omega$ acting on the Lax connection in the path-ordered exponential as in \eqref{eq:transport}. By demanding that  the time derivative of the boundary monodromy matrix is given by a commutator,
\begin{equation}\label{eq:BoundMonToTime}
\partial_\tau T_b (z) = \left[ T_b (z) , N(z) \right] \, ,
\end{equation}
for some matrix $N(z)$ one will indeed find that  $\partial_\tau \mathrm{Tr} T_b(z)^{n} =0$ for any $n\in \mathbb{N}$ and $z\in\mathbb{C}$. Assuming\footnote{In general  this strongly depends on the specific form of the Lax connection $\mathcal{L}(z)$
but the  procedure described here can be easily adapted to other cases. },
 \begin{equation}\label{eq:ReflectedTransport}
 T^\Omega_R (2\pi , \pi ;z) = T^\Omega(0,\pi ; z_R)\, ,
 \end{equation}
we find  explicitly using \eqref{eq:TransportToTime} that the time derivative of the boundary monodromy matrix satisfies,
\begin{equation}
\begin{aligned}
\partial_\tau T_b (z) =\; & \left[ T^\Omega(0,\pi; z_R) \Omega[\mathcal{L}_\tau ( \tau, \pi; z_R)] - \Omega[\mathcal{L}_\tau ( \tau , 0 ;z_R)] T^\Omega(0,\pi ; z_R) \right]  T(\pi , 0 ; z)  \\
& + T^\Omega(0,\pi ; z_R)  \left[ T(\pi ,0 ; z)\mathcal{L}_\tau ( \tau , 0 ;z) - \mathcal{L}_\tau ( \tau, \pi ; z) T(\pi,0;z) \right] \, .
\end{aligned}
\end{equation}
The  condition \eqref{eq:BoundMonToTime}  sufficiently holds when $N(z) =\mathcal{L}_\tau (\tau , 0;z)$ and when we require the following boundary conditions on both the endpoints\footnote{In principle the boundary conditions can be different on each endpoint (see e.g.\ \cite{Driezen:2018glg}) which in the string theory application allows the open string to connect distinct D-brane configurations.},
\begin{equation}\label{eq:BoundCondLax1}
\mathcal{L}_\tau (\tau, 0 ;z ) = \Omega[ \mathcal{L}_\tau (\tau ,0 ; z_R) ] \, ,
\end{equation}
and similarly on $\sigma = \pi$. When studying a specific two-dimensional integrable model with a known Lax connection, and knowing its behaviour under the reflection $R$, one can now easily derive the \textit{integrable} boundary conditions on the field variables by eq.~\eqref{eq:BoundCondLax1}.  Typically this will involve additional conditions on the automorphism $\Omega$ as we will see in the coming section.

\section{Applied to $\lambda$-deformations} \label{s:lambda}

We will now apply the boundary monodromy method to the (standard) $\lambda$-deformation introduced by Sfetsos in \cite{Sfetsos:2013wia}. The interest in this particular model is that it is a two-dimensional integrable field theory deforming the exactly conformal Wess-Zumino-Witten (WZW) model on group manifolds.  We will therefore be able to relate the integrable boundary conditions of the $\lambda$-model to known results of stable D-brane configurations  wrapping twisted conjugacy classes in the group manifold \cite{Alekseev:1998mc,Felder:1999ka,Stanciu:2000fz,Figueroa-OFarrill:2000lcd,Bachas:2000fr}.

\subsection{Construction of the $\lambda$-action}

Let us first briefly introduce the construction of  $\lambda$-deformations  of \cite{Sfetsos:2013wia}. One starts by doubling the degrees of freedom on a Lie group manifold $G$, by combining the WZW model on $G$ at level $k$ with the Principal Chiral Model (PCM)  on $G$ with a coupling constant $\kappa^2$, i.e.,
 \begin{equation}\label{eq:doubledaction}
 \begin{aligned}
 S_{k,\kappa^2}(g,\widetilde{g}) &= S_{\text{WZW},k}(g) + S_{\text{PCM},\kappa^2}(\widetilde{g}) ,\\
   S_{\text{WZW,k}}(g) &= -\frac{k}{2\pi}\int_\Sigma  d  \sigma d\tau    \langle g^{-1} \partial_+ g , g^{-1} \partial_- g \rangle - \frac{  k}{24\pi }       \int_{M_3}   \langle   \bar g^{-1} d\bar g, [\bar g^{-1} d\bar g,\bar g^{-1} d\bar g]  \rangle , \\
 S_{\text{PCM},\kappa^2}(\widetilde{g}) &= -  \frac{\kappa^2}{\pi} \int d\sigma d\tau \, \langle \widetilde{g}^{-1}\partial_+ \widetilde{g} , \widetilde{g}^{-1}\partial_- \widetilde{g} \rangle \, ,
 \end{aligned}
 \end{equation}
which are both realised through distinct group elements $g\in G$ and $\widetilde{g}\in G$ respectively\footnote{We have taken    conventions on the worldsheet $\Sigma$  in which the two-dimensional metric is  fixed  as $\text{diag}(+1,-1)$, $\epsilon_{\tau\sigma} = 1$ and $\partial_\pm = \frac{1}{2}(\partial_\tau \pm \partial_\sigma)$. Moreover, we consider compact semi-simple Lie groups $G$ of which the generators $T_A$, $A \in \{1, \cdots, \text{dim}(G) \}$ of the Lie algebra $\mathfrak{g}$ are Hermitean and normalised with respect to the  Cartan-Killing billinear form $\langle \cdot , \cdot \rangle$ as $\langle T_A, T_B \rangle = \frac{1}{x_R} \mathrm{Tr} (T_A T_B) = \delta_{AB}$ (with $x_R$ the index of the representation $R$). It is known that for compact groups the level $k$ should be integer quantised while for non-compact groups it can be free \cite{Witten:1983ar}.}. The fields $\bar{g}$ are an extension of $g$ into $M_3 \subset G$ such that $\partial M_3 = g(\Sigma)$. Altogether the doubled model  \eqref{eq:doubledaction} has a global $G_L \times G_R$ symmetry. Next, one reduces back to $\text{dim}(G)$ degrees of freedom by gauging a subgroup acting as,
\begin{equation}
\begin{aligned}
G_L : \widetilde{g} \rightarrow h^{-1} \widetilde{g}, \qquad G_{\text{diag}}: g \rightarrow h^{-1} g h, \qquad \text{with} \; h \in G,
\end{aligned}
\end{equation}
using a common gauge field $A \rightarrow h^{-1} A h - h^{-1} d h$. Doing a minimal substitution on the PCM, by replacing $\partial_\pm \widetilde{g} \rightarrow D_\pm \widetilde{g} = \partial_\pm \widetilde{g} - A_\pm \widetilde{g}$ and replacing the WZW model by the $G/G$ gauged WZW model,
  \begin{equation}
  S_{\text{gWZW,k}}(g,A) = S_{\text{WZW},k}(g) + \frac{k}{\pi} \int d\sigma d\tau \, \langle A_- , \partial_+ g g^{-1} - A_+ ,  g^{-1}\partial_- g  + A_+ , g^{-1} A_- g  -  A_+ , A_- \rangle \ .
  \end{equation}
one finds the $\lambda$-deformation after fixing the gauge by $\widetilde{g} = \mathbf{1}$,
\begin{equation}
  \begin{aligned}\label{eq:LambdaAction1}
  S_{k,\lambda}(g, A) =  S_{\text{WZW},k} (g)
  - \frac{k}{ \pi} \int d\sigma d\tau \langle A_+ ,  (\lambda^{-1} - D_{g^{-1}}) A_- \rangle
 - \langle A_- , \partial_+ g g^{-1} \rangle + \langle A_+ , g^{-1}\partial_- g   \rangle   \ .
  \end{aligned}
  \end{equation}
Here we have introduced the adjoint operator $D_g : \mathfrak{g} \rightarrow \mathfrak{g}$, $D_g (T_A) = g T_A g^{-1} = T_B (D_g){}^B{}_A$ with $g\in G$ and the parameter $\lambda$,
\begin{equation}
\lambda = \frac{k}{k+\kappa^2} .
\end{equation}
The gauge fields are now auxiliary and can be integrated out. Varying  the action $S_{k,\lambda}(g,A) $ with respect to $A_{\pm}$ we find the constraints,
\begin{equation}\label{eq:GaugeConstraints}
A_+ = \left( \lambda^{-1} -  D_g \right)^{-1} \partial_+ g g^{-1}\, , \qquad
A_-  = -\left( \lambda^{-1} - D_{g^{-1}} \right)^{-1} g^{-1} \partial_- g\, .
\end{equation}
Substituting these  into eq.~\eqref{eq:LambdaAction1} gives the large $k$ effective action,
\begin{equation}\label{eq:LambdaAction2}
\begin{aligned}
S_{k,\lambda}(g) = S_{\text{WZW},k}(g) - \frac{k }{\pi}\int \mathrm{d}\sigma \mathrm{d}\tau \, \partial_+ g g^{-1} \left( \lambda^{-1} -  D_{g^{-1}} \right)^{-1} g^{-1}\partial_- g \ ,
\end{aligned}
\end{equation}
which is an all-loop in $\lambda$ deformation of the WZW theory with a global $g \rightarrow g_0 g g_0^{-1}$, $g_0\in G$ symmetry left.  Effectively, the $\lambda$-theory thus  deforms the target space metric and Kalb-Ramond field  of the WZW $\sigma$-model. In addition, the Gaussian elimination of the gauge fields in the path integral results in a non-constant dilaton profile,
\begin{equation}
\Phi = \Phi_0 - \frac{1}{2} \ln \det \left( \mathbf{1} - \lambda D_{g^{-1}} \right) ,
\end{equation}
with $\Phi_0$ constant.
While the integrability of the $\lambda$-model (with periodic boundary conditions) has been first shown in \cite{Sfetsos:2013wia} starting from the effective $\sigma$-model action \eqref{eq:LambdaAction2}, one can straightforwardly show it starting from \eqref{eq:LambdaAction1} as  done in  \cite{Hollowood:2014rla}. The Lax connection ${\cal L}(z)$ representing the equations of motion of the fields $g$ satisfying the zero-curvature condition \eqref{eq:zerocurvatureLax} $\forall\ z \in \mathbb{C}$ is,
\begin{equation} \label{eq:LambdaLax}
{\cal L}_\pm (z) = - \frac{2}{1 +\lambda} \frac{A_\pm}{1 \mp z} ,
\end{equation}
upon the constraints \eqref{eq:GaugeConstraints}.

\subsection{Interpretation as (twisted) conjugacy classes}
To apply the boundary monodromy method to the $\lambda$-model  we first need to consider the behaviour of the transport matrix $T_R^\Omega (2\pi, \pi ; z)$ under the reflection $R: \sigma \rightarrow 2\pi - \sigma$. For the $\lambda$-Lax \eqref{eq:LambdaLax} one will find that eq.\  \eqref{eq:ReflectedTransport} holds for $z_R = -z$. The resulting integrable boundary conditions \eqref{eq:BoundCondLax1} of the $\lambda$-model are then, after expanding order by order in the spectral parameter $z$,
\begin{equation}
\left. A_+ \right\vert_{\partial\Sigma} =\left.  \Omega \left[ A_- \right] \right\vert_{\partial\Sigma},
\end{equation}
together with the requirement that $\Omega$ is an \textit{involutive} automorphism of the Lie algebra,
\begin{equation}
\Omega^2 = 1 .
\end{equation}
Moreover, to interpret the above boundary conditions as Dirichlet and (generalised) Neumann conditions, the automorphism $\Omega$ should be such that no energy-momentum is flowing through the boundary, i.e. the energy-momentum tensor must satisfy $T_{01}| = 0$, which  turns out to require that it is metric-preserving in the sense of $\langle \Omega (T_A) , \Omega (T_B) \rangle = \langle T_A , T_B \rangle$.

Upon the constraint \eqref{eq:GaugeConstraints} the integrable boundary conditions are now given by,
\begin{equation} \label{eq:intbc}
(\mathbf{1} - \lambda D_g)^{-1} \partial_+ g g^{-1} = - \Omega (\mathbf{1} - \lambda D_{g^{-1}})^{-1} g^{-1}\partial_- g .
\end{equation}
At the WZW conformal point ($\lambda = 0$) one  consistently finds the (twisted) gluing conditions of \cite{Alekseev:1998mc,Felder:1999ka,Stanciu:2000fz} of the holomorphic Kac-Moody currents $J_+ = -k\partial_+ g g^{-1},\, J_- = kg^{-1} \partial_- g$  on the boundary,
\begin{equation} \label{eq:wzwgluing}
\lambda \rightarrow 0: \qquad    J_+ = \Omega (J_- ) ,
\end{equation}
preserving precisely one copy of both the Kac-Moody current algebra (iff.\ $\Omega$ is a Lie algebra automorphism, which is the case here) and the Virasoro algebra. Because only the latter property is a necessity to preserve conformal invariance, the former property led to the description of  the corresponding  D-brane configurations as being `maximally symmetric'.  In \cite{Alekseev:1998mc,Felder:1999ka,Stanciu:2000fz} (see also \cite{Figueroa-OFarrill:1999cmq}) it was  shown, starting from the corresponding Dirichlet conditions of eq.~\eqref{eq:wzwgluing},  that the D-brane worldvolumes  wrap (twisted) conjugacy classes of the group $G$,
\begin{equation}
C_\omega (g) = \{ h g \omega (h^{-1} ) \, | \, \forall h\in G \} , \qquad \omega (e^{tX}) \equiv e^{t \Omega (X)} \in G , \;\; X\in \mathfrak{g},
\end{equation}
classified by the quotient of metric-preserving outer automorphisms $\omega \in \text{Out}_0 (G) = \text{Aut}_0(G)/\text{Inn}_0(G)$ \cite{Figueroa-OFarrill:1999cmq}. When $\omega \in \text{Inn}_0(G)$ is inner, i.e. $\omega(h) = \text{ad}_w (h) = w h w^{-1}$ for some $w\in G$, the twisted conjugacy class $C_\omega (g)$ is related to the ordinary conjugacy class $C_{\text{Id}}(g)$ by a (right) group translation,
\begin{equation}
C_{\text{ad}_w} (g) = C_{\text{Id}} (gw) w^{-1} ,
\end{equation}
which is a symmetry of the WZW model. The automorphisms $\omega$ are in principle not constrained any further here.

For generic $\lambda$ it was shown in \cite{Driezen:2018glg} that the geometrical picture of the integrable boundary conditions \eqref{eq:intbc}  as D-branes wrapping twisted conjugacy classes persists by pleasing cancellations of the $\lambda$-dependence. This is indeed expected, since the deformation affects only  target space data  as the metric, while the worldvolumes are defined through the orthogonal decomposition of the tangent space with respect to the Dirichlet conditions, independently of the target space metric.  However, integrability picks out only the automorphisms $\omega (e^{tX}) = e^{t\Omega(X)}$  that satisfy $\Omega^2 = 1$.  Generic inner automorphisms, or group translations of the conjugacy classes, are thus excluded. Indeed, independent right group translations are not a symmetry of the $\lambda$-model which remarkably follows from demanding integrability structures.

\subsection{$G = SU(2)$ illustration}

To illustrate the above, we focus in this section on the case of the  $G= SU(2)$ group manifold for which $\text{Out}_0(SU(2)) = \text{Id}$ is trivial and one will describe ordinary conjugacy classes\footnote{ For an analysis and explicit example of \textit{twisted} conjugacy classes, possible in $\lambda-SL(2,R)$, we refer to \cite{Driezen:2018glg}. Interestingly, in $G=SL(2,R)$ only the twisted conjugacy classes turn out to correspond to \textit{stable} D-brane configurations \cite{Bachas:2000fr}, telling us it is indeed worth including the possibility of twisted gluing conditions.}. We parametrise the group element $g\in SU(2)$ in Cartesian coordinates,
\begin{equation}
g = \begin{pmatrix}
X_0 + i X_3 & - X_1 + i X_2 \\ X_1 +  i X_2 & X_0 - i X_3
 \end{pmatrix} ,
\end{equation}
constrained to $\det g = X_0^2 + X_1^2 + X_2^2 + X_3^2 = 1$,  making the embedding of $SU(2)$ as an $S^3$ in $\mathbb{R}^4$ apparent. The set of group elements in an ordinary conjugacy class $C(g)$ clearly have a constant trace and here  fix the $X_0$ parameter  to some constant value. The conjugacy classes are thus $S^2$-spheres of varying radius inside the $S^3$ as illustrated  in figure \ref{fig:su2illustration}. When the $\lambda$-parameter is turned on, and the target space metric $G$ gets deformed, what is changed will be the size (or radius) of the $S^2$-spheres by the induced deformed metric $\left. G\right\vert_{S^2}$  on these branes. In figure \ref{fig:su2illustration} it is moreover clear that  the rotational symmetry of the WZW gets lost in the deformation, which is a reflection of the $\Omega^2 = 1$ constraint coming from integrability and preventing the existence of  rotated branes. Indeed, in \cite{Driezen:2018glg} a semi-classical analysis of the quadratic scalar fluctuations of the branes moreover shows that at $\lambda=0$ a massless  p-wave triplet exists which becomes massive for $\lambda \neq 0$.
\begin{figure}[H]
\centering
\includegraphics[scale=0.15]{LambdaSquashin2.pdf}
\caption{For illustrative purposes we portray here the $S^3 \simeq SU(2)$ group manifold in one dimension less. The green lines represent the $S^2$-branes or conjugacy classes in the $S^3$ that change size under the \textit{squashing} of the $S^3$ when the $\lambda$-deformation is turned on.}\label{fig:su2illustration}
\end{figure}
In both cases, there is a total of 2 static D0-branes (corresponding to the north- and southpole) and $l$ static D2-branes (corresponding to the $S^2$'s). The number $l$ is integer quantised and equal to $l=k-1$, following from topological obstructions in formulating the WZ term in \eqref{eq:doubledaction} in the presence of a boundary. In the boundary case  the WZ term should be altered as \cite{Klimcik:1996hp},
\begin{equation}\label{eq:wzbdy}
\int_{M_3} H \rightarrow \int_{M_3} H + \int_{D_2}
 \omega ,
 \end{equation}
 with $\partial M_3 = g(\Sigma) + D_2$, $D_2 \subset g(\partial\Sigma)$ and $\omega$ a two-form on $D_2$ such that $H |_{D_2} = \mathrm{d}\omega$. To cancel  in the path integral  ambiguities in the choice of $M_3$ recall that the closed string WZW theory on compact groups requires the level $k$ to be integer quantised. On the other hand, the open string WZW theory with \eqref{eq:wzbdy} will require the D-branes to be localised on a discrete number of positions. In the case of $G=SU(2)$ the number of branes are then indeed fixed as $l = k-1 \in \mathbb{Z}$, where we refer to \cite{Klimcik:1996hp,Stanciu:2000fz,Figueroa-OFarrill:2000lcd} for more details. Interestingly, this can be seen as a semi-classical stabilisation of the D2-branes, since their localised positions forbids them to shrink smoothly to zero size. When the deformation is turned on, \cite{Driezen:2018glg} shows the \textit{continuous} $\lambda$-dependence precisely cancels in the topological conditions\footnote{Both the $H$-form and $\omega$-form receive a $\lambda$-contribution but these precisely cancel.} and so consistently also in the $U(1)$ flux quantisation. Again, indeed, a semi-classical analysis of the scalar fluctuations gives a massive s-wave with a mass independent of $\lambda$.

\subsection{Interplay with generalised T-dualities}
Another motivation to look at $\lambda$-deformations is the close connection to generalised T-dualities. The $\lambda \rightarrow 1$ scaling limit (obtained by taking $k \rightarrow \infty$) produces e.g.\ the non-Abelian T-dual of the Principal Chiral Model (PCM) \cite{Sfetsos:2013wia}. On the other hand, for generic values of $\lambda \in [0,1]$ the model is Poisson-Lie T-dual \cite{Klimcik:1995ux,Klimcik:1995dy}, up to an additional analytical continuation, to an integrable deformation of the PCM  \cite{Vicedo:2015pna,Hoare:2015gda,Klimcik:2015gba}  known as Yang-Baxter or $\eta$-deformations \cite{Klimcik:2008eq,Delduc:2013fga} which have an action,
\begin{equation}
S_{t, \eta} (\widehat{g}) = \frac{1}{t} \int_\Sigma d\sigma d\tau\; \partial_+ \widehat{g} \widehat{g}^{-1} \left( \mathbf{1} - \eta {\cal R} \right)^{-1} \partial_- \widehat{g} \widehat{g}^{-1} ,
\end{equation}
where ${\cal R} : \mathfrak{g} \rightarrow \mathfrak{g}$ is an operator solving the modified classical  Yang-Baxter equation.

 Tracking the behaviour of D-brane configurations under these generalised T-dualities is, in general, a challenging procedure due to the lack of well-defined boundary conditions in the curved background geometry. In the $\lambda$-deformation, however, integrability dictates us precise boundary conditions (in eq.~\eqref{eq:intbc}) given   in terms of worldsheet derivatives of the phase-space variables. Together with the known canonical transformations of non-Abelian T-duality \cite{Lozano:1995jx,Sfetsos:1996pm} (NATD) and Poisson-Lie (PL) T-duality \cite{Sfetsos:1997pi} this allows us to find the dual D-branes in the e.g.  $G= SU(2)$ case. Schematically we find  \cite{Driezen:2018glg},
 \begin{equation*}
\begin{aligned}
&\text{D2-brane in the NATD of the PCM }  \quad &&\xrightarrow{\text{can.\ transf.\ }} \quad  &&\text{D3-brane in the original PCM} \\
 &\text{D2-brane in the $\lambda$-deformed WZW} \quad &&\xrightarrow{\text{can.\ transf.\ }} \quad &&\text{D3-brane in the $\eta$-deformed PCM}
 \end{aligned}
 \end{equation*}
 so that in both cases the $S^2$-branes pop open to space-filling D3-branes. Remarkably, the boundary conditions obtained in this way in \cite{Driezen:2018glg} match precisely with the boundary conditions that follow from the boundary monodromy method in section \ref{s:bmm} when plugging in the Lax pair of the Principal Chiral Model \cite{Zakharov:1973pp},
 \begin{equation}
 {\cal L }_\pm (z) = \frac{1}{1\mp z} g^{-1} \partial_\pm g,
 \end{equation}
 or of the $\eta$-deformed PCM \cite{Klimcik:2008eq,Delduc:2013fga},
 \begin{equation}
 {\cal L }_\pm (\eta ; z) = \frac{1+\eta^2}{1\pm z} D_g \cdot \frac{1}{1\pm \eta {\cal R}} \cdot \partial_\pm g g^{-1} ,
 \end{equation}
in \eqref{eq:BoundCondLax1} respectively.

\section{Conclusions and outlook}\label{s:concl}
In this overview note we have seen an efficient method to derive classical integrable boundary conditions in $\sigma$-models by demanding that the monodromy matrix of the Lax connection generates conserved charges even in the presence of boundaries. As emphasized in the introduction, this is generically challenging for stringy $\sigma$-models without precise CFT formulations. The boundary monodromy method, however, essentially  only requires  the knowledge of the $\sigma$-model Lax connection. \\
\indent In the context of  $\lambda$-deformations the boundary monodromy method dictates us integrable boundary conditions that are described elegantly by twisted conjugacy classes in the deformed target space. This geometrical picture is independent of the deformation parameter and, indeed, corresponds smoothly to the D-brane configurations dictated by CFT methods of the undeformed WZW point. Illustrated in the $SU(2)$ manifold we have seen that the deformation changes the size of the D-branes and destroys their rotational symmetry in the deformed geometry.  This latter (natural) observation ties nicely together with the constraining features of integrability. Additionally, we have seen that the flux quantisation consistently remains independent of the $\lambda$-parameter and enforces the branes to sit stabilised at localised positions. These conclusions are supported in \cite{Driezen:2018glg} by an analysis of  scalar fluctuations of the D-branes. Finally, armed with precise integrable boundary conditions, one can track them under generalised dualities present in  $\lambda$-deformations. For $G=SU(2)$ we have seen a Dirichlet condition to transform into a generalised Neumann which, to close the circle, turns out to follow from demanding integrability of the dual models as well.\\

Let us stress the analysis so far has been purely classical. It remains an interesting question to understand the quantum description of the integrable boundary conditions in these $\lambda$-models. Here, the bulk $S$-matrix of \cite{Appadu:2017fff} should be supplemented by a boundary $K$-matrix describing particle reflections off the boundary and satisfying the boundary Yang-Baxter equation \cite{Cherednik:1985vs,Sklyanin:1988yz}. Since the $S$-matrix of \cite{Appadu:2017fff} was derived by mapping the quantum $\lambda$-model to  a spin $k$ XXX Heisenberg spin chain, it would be appealing to interpret the boundary $K$-matrix in the corresponding open spin chain as well.\\
\indent Another appealing line of study, returning to the string point of view, is to consider integrable D-branes of $\lambda$-models in supercoset geometries \cite{Hollowood:2014qma} as here the deformation is expected to be truly marginal to all loops.

\section*{Acknowledgements}
I would like to thank the organisers of the ``Dualities and Generalised Geometries" session part of the Corfu Summer Institute 2018 schools and workshops for giving me the opportunity to speak and for a stimulating and interesting workshop. Additionally, I would like to thank my collaborators and advisers Daniel Thompson and Alexander Sevrin  for  the fruitful discussions we have had in the process of \cite{Driezen:2018glg} and Saskia Demulder for a careful read of the manuscript. Finally, I acknowledge also the  support by the ``FWO-Vlaanderen'' through an aspirant fellowship and the project  G006119N.

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