+ \tfrac12 \Tr\Big[ \big({\rm k} + k + (1+\ka^{-1})\big) J_+ M^{-1} K_-
- \big({\rm k}-k - 3 (1+\ka^{-1})\big) J_- M^{-1} K_+ \Big]
\\
&\quad \qquad - \tfrac{3+\ka}{2(1-\ka)} (\del_+ \log \det M)(\del_- \log \det M) - \tfrac{2(1+\ka)}{1-\ka} {\PCM}(M) + {\WZ}(M) \ \\
&M(g) = \big( \tfrac{1-\ka}{1+\ka}\big)^2 - \Ad_g \ .
\end{aligned}\end{equation}

We note that coefficients of the WZ terms in originlag are RG invariants, $\kk-2$ (see eq. knr) and $1$. This is consistent with usual expectations and suggests that the representation originlag is a natural one.

In the WZW limit $\ka \to 1$ when the RG invariant $\kk$ in b2 reduces to the usual shifted WZ level

\begin{equation}
\kk\big|_{\ka = 1} = k+2 \ ,
\end{equation}

the corrections to the metric and $\BB$-field in b2 vanish, so that
the expression in originlag  reduces to the classical WZW Lagrangian
$k \big[\L_{\PCM}(g) + \L_{\WZ}(g) \big]$. Indeed, even though the coefficients $ \tfrac{3+\ka}{2(1-\ka)}$ and $\tfrac{2(1+\ka)}{1-\ka}$ in b2 blow up in the limit $\ka \to 1$, the corresponding $M$-dependent expressions vanish faster and all $M$-dependence  disappears in the $\ka=1$ limit.

§.§.§ Eta-model limit

Like the $SU(2) \ov U(1)$ example discussed above, the $SU(2)$  b1 admits a formal “boost” scaling limit,

\begin{equation}
\alpha \to \sin(\a + i \zeta) \ , \qquad \qquad \zeta \to \infty \ , \la{fl}
\end{equation}

when it becomes (dropping a closed 2-form contribution to the $\BB$-field)

\begin{equation}\begin{split}\la{class}
\widetilde{ds^2}  & = k \Big[ \frac{1}{\kappa} d\alpha^2 + \frac{ \kappa}{1-\kappa^2} ( d\beta^2 + \sin^2\beta d\gamma^2)\Big] \ ,
\qquad
\widetilde{\BB} = k\cos\beta  \, d \alpha\wedge d\gamma \ .
\end{split}\end{equation}

This geometry is related by abelian T-duality (in the $\a$ direction) to the squashed 3-sphere <cit.>,

\begin{equation}\la{323}
{ ds^2} = k\Big[ \kappa (d\tilde\alpha - \cos\beta d\gamma)^2 + \frac{ \ka}{1-\ka^2}(d\beta^2 + \sin^2\beta d\gamma^2)\Big] \ , \qquad  \BB=0 \ ,
\end{equation}

which in turn has the interpretation as the $\eta$-deformation (<ref>) of the $SU(2)$ PCM (cf. <cit.>). The squashed 3-sphere geometry 323 is an anomalous case of an $\eta$-model that
is automatically renormalizable at 2-loops without needing quantum corrections (this statement was checked in App. B of <cit.> — and more generally for “squashed” models interpolating between a PCM and a symmetric space). As we shall now demonstrate, the quantum corrections derived above for the  $SU(2)$ $\l$-model are consistent with this fact.

Starting  from the 1-loop corrected  background b2
and taking the same scaling limit fl, we get a corrected version of class,

ds^2      = (k/κ + 2/^2) dα^2 + k κ/1-κ^2 ( dβ^2 + sin^2βdγ^2)  ,

= [k + 4+(1+^2)^2/4]  cosβd∧dγ .
The background simpler solves the 2-loop RG equations (<ref>) with the same $\b$-functions  \rf{0} as the $ł$-model and vanishing diffeomorphism vector  and $$-gauge transformation  ($ξ_m=ł_m=0$).
To extend the T-duality relation between \rf{class} and \rf{323} to the quantum level,
we  use the 1-loop corrections to the T-duality transformation laws
of \cite{Kaloper:1997ux}
(see also \cite{Parsons:1999ze}), generalizing the quantum correction in \eqref{5}.\foot{We note that, in order to directly apply the results of \cite{Kaloper:1997ux}, one first has to perform a scheme change, $(\GG+\BB)_{\mu\nu} \to (\GG+\BB)_{\mu\nu} + \frac14 \H^2_{\mu\nu}$ .}
As a result, the T-dual of \rf{simpler} is found to be
\begin{equation}\la{2loopdual} \begin{aligned}
ds^2 {}&{}= \big[k \ka +\tfrac{1}{2} (1+\ka^2)^2\big]\big(d\tilde\alpha - \cos\beta d\gamma\big)^2 + \big[\frac{k \kappa}{1-\kappa^2} + \tfrac{1}{2} (1-\ka^2)\big] \big( d\beta^2 + \sin^2\beta d\gamma^2\big) \ , \\
\BB{}&{}=0 \ ,
\end{aligned}
\end{equation}
where we have rescaled $α̃$.
Indeed this just the squashed 3-sphere geometry \rf{323} with redefined parameters,
\begin{equation}
(k, \ka) \to (\hat k, \hat \ka) \ , \qquad \quad  k = \hat{k} - \frac{3+\hat{\ka}^4}{2\hat{\ka}} \ , \qquad  \ka = \hat{\ka}+\frac{1-\hat{\ka}^2}{\hat{k}} \ .
\end{equation}
Thus the fact that the quantum corrected $ł$-model \rf{b2} is stable under the 2-loop RG flow is perfectly consistent, in the $η$-model or ``boost'' limit, with the fact that the squashed 3-sphere is automatically stable without  quantum deformation.

We see in this example that
the known quantum corrections to the T-duality transformation rules
are naturally consistent with the
quantum deformations of integrable models for higher-loop renormalizability.
As we shall discuss in the next section, similar quantum corrections are expected for non-abelian duality.

\renewcommand{\la}{\label}
\renewcommand{\rf}[1]{(\ref{#1})}
\def \kk {k}

\subsection{Non-abelian dual limit \la{NADqS}}

In a particular limit,
\be \begin{aligned}
&k \to \infty \ , \qquad \l= 1 - \tfrac{\h}{2k}  \ , \qquad g=e^{-\tfrac{h}{2k}v} \ ,  \\
&v \in \Lie(G) \text{ and } \h  \text{ fixed} \la{limr}
\end{aligned} \ee
the \lam\ \rf{40} becomes the non-abelian dual (NAD)  of the PCM or symmetric space (see Section \ref{TS})
\be
\L = -\ha \h \, \Tr[ \del_+ v \, \frac{1}{P + \ad_v} \del_- v ] \ . \la{NADrep}
\ee
It was noted earlier in the literature  \cite{n1,n2,n3,Bonneau:2001za} that the NAD theories \rf{NADrep} appear not to be renormalizable beyond the 1-loop order, prompting doubt about their inequivalence with the PCM and  symmetric space \sms\unskip. In this section, we shall resolve this confusion by observing that the classical NAD models \rf{NADrep} must be supplemented with particular finite counterterms to maintain renormalizability beyond 1-loop order.
We shall produce the necessary counterterms in the $SU(2) U(1)$ and $SU(2)$ examples by taking the NAD limit of the corresponding  $ł$-model corrections  constructed above.

\subsubsection{NAD of \texorpdfstring{$S^2$}{S^2} sigma model}

For the $SU(2) U(1)$ $ł$-model in the coordinates \rf{43}, the NAD limit amounts to
\be \la{455}
p= \ka \barp\ , \qquad q= 1 - \ka^2 \barq \ , \qquad
\kk \to \infty\ , \quad \ka\to 0
\ , \qquad \h\equiv 4 k\ka ={\rm fixed}
\ . \ee
The resulting theory \rf{NADrep} is the NAD of the \sm\ on $S^2$, with metric
\be \la{46}
ds^2 =\frac{\h}{4}\, {dx^2 + d y^2 \ov  2 y-x^2}\ , \ee
and trivial $$-field since the target space is 2d. In the NAD limit \rf{455}, the leading quantum correction to the $ł$-model in \rf{48} suggests a particular correction to the NAD metric \rf{46},
\be \la{57}
ds^2 =\frac{\h}{4}\, {dx^2 + d y^2 \ov  2 y-x^2} - \frac{1}{2} \Big[\frac{d ( 2 y-x^2 )}{2 y-x^2}\Big]^2 + \OO(h^{-1})
\ ,
\ee
which indeed solves the 2-loop RG equation \rf{27} with only $$ running,
\be \la{58}
\dt \h =  4 + \frac{16}{\h} + \mathcal{O}(h^{-2}) \ , \ee
and matching the 2-loop $$̱-function the dual undeformed $S^2={SU(2) \ov U(1)}$ $\s$-model (eq. (<ref>) with $c_{_{SU(2)}} =2$, $c_{_{U(1)}}=0$). Note that, since the coordinate redefinition in the NAD limit 455 depends on $\ka=\ka(t)$,
the diffeomorphism vector gets an additional contribution compared to 50 and
components are given by

ξ_x =  4x 2 y-x^2 - 1 8 x(1 + 3 x^2 - 4 y)( 2 y-x^2)^2 + (^-2)  ,

ξ_y =   2 (  2 y-1) 2 y-x^2 + 1  8(1 + 2 x^2 - 2 y)( 2 y-x^2)^2 + (^-2)  .

If one takes a further scaling limit,

\begin{equation}\la{addsca} \begin{aligned}
&x \to \gamma \cos\chi \ , \qquad y \to \tfrac12  \gamma^2 + \gamma y \ , \qquad \gamma \to \infty \ ,
% \\
%&ds^2 \to \frac{h}{1 - x^2 }(dx^2 + dy^2) \  .
\end{aligned}
\end{equation}

the NAD metric  46 degenerates to an abelian T-dual of $S^2$ (in the $y$ direction),The T-dual metric h2dual can also be found by directly taking the limit $\kk\to \infty$, $\ka \to 0$ with
$p = \cos \chi$, $q = \ka y$ and $\h\equiv 4 k\ka$ fixed in the $\l$-model metric 43.
ds^2 = 14  ( d χ^2 +1/sin^2χ d y^2)  .
Taking a particular 1-loop modification of the limit addsca (cf. 317),
x →γ(1+ 4^-1) cosχ  ,     y →12  γ^2 + γ(1+4 ^-1) y  ,
γ→∞ ,
and shifting the coupling $\h \to \h -8$ in  the quantum-corrected NAD metric 57, we obtain a quantum-corrected T-dual metric h2dual,
ds^2 =14   [dχ^2 +1/sin^2 χ (1+8 ^-1^2χ)dy^2] + (^-1)  ,
which coincides with the  known corrections  5 to the T-duality transformation rule <cit.>
(with ${\cal M} = \sin^2{\chi}$ and $\h \to \tfrac{1}{4}\h$ so that 5 gives $\tilde {\cal M} =\frac{1}{\sin^2 \chi} \big(1+2 (\h/4)^{-1}\cot^2\chi\big)$ in this case).

§.§.§ NAD of SU(2) principal chiral model

In the case of the $SU(2)$ $\lambda$-model in the coordinates b1, the NAD limit (<ref>) amounts to

\begin{equation}\la{nadlimit2}
\a \to \ka \, \alpha\ , \ \ \ \qquad
\kk \to \infty\ , \quad \ka\to 0
\ , \qquad \h\equiv 4 k\ka ={\rm fixed}
\end{equation}

where $\hh$ and the new coordinate $\a$ are fixed.
Taking this limit in the 1-loop-corrected $\l$-model background b2 suggests a particular quantum correction to the $SU(2)$ NAD model,

\begin{align}
ds^2 & = \Big[\frac{\hh}{4} + \frac{2\a^4}{(1+\a^2)^2} \Big] d\alpha^2 + \frac{h}{4}\frac{\a^2}{(1+\a^2)} (d\beta^2 + \sin^2\beta d\gamma^2) \ , \no \\
\BB & = \Big[\frac{\hh+5}{4}\frac{\a^3 }{(1+\a^2)} + \frac{2\a}{1+\a^2} - 2\arctan{\a}\Big] \sin \b d\b\wedge d\g \ , \la{f2zero} \\
\H & = \Big[\frac{\hh+5}{4}\frac{\a^2(3+\a^2) }{4(1+\a^2)^2} - \frac{4\a^2}{(1+\a^2)^2}\Big] \sin \b d\a\wedge d\b\wedge d\g \ .  \no
\end{align}

Indeed it solves the 2-loop RG equations (<ref>) with
\begin{align}
\frac{d}{dt} \hh = 2 + {2}{\hh}^{-1} \ , \la{333}
\end{align}
matching that of the $SU(2)$ PCM (i.e. \eqref{Prg} with $c__SU(2)=2$) and diffeomorphism vector,
\begin{align}
\la{xcorr2}
\xi^\a = \frac{2\a}{\hh(1+\a^2)}\Big[(1 -\a^2) - \frac{4(5+5\a^2+3\a^4+\a^6)}{h(1+\a^2)^2}\Big] \ , \qquad \xi^{\b,\g} = \l_{\a,\b,\g} = 0 \ ,
\end{align}
equalling the $ł$-model diffeomorphism vector \rf{diffv}  in the NAD limit \rf{nadlimit2} plus an extra contribution due to the RG-dependent rescaling of $$.

We can also take the NAD limit \eqref{limr} in the corrected Lagrangian written in the form \rf{originlag},
\begin{equation}\begin{aligned}\la{originlagnad}
&\L  = - (\tfrac{1}{2} \hh + \tfrac{9}{4}) \Tr\big[\partial_+ v\, M^{-1} \partial_- v \big]
- \tfrac{7}{4} \Tr\big[\partial_- v\, M^{-1} \partial_+ v \big] \qquad \qquad
\\
&\qquad \quad - \tfrac{3}{2} (\del_+ \log \det M)(\del_- \log \det M) - 2 \, \L_{\rm PCM}(M) + \L_{\rm WZ}(M) \ ,\\
&M \equiv 1 +\ad_v \ .
\end{aligned}\end{equation}
This indeed matches with \rf{f2zero} upon choosing the coordinates,
\begin{equation}
v = -\tfrac{i}{2} \a \big[\cos{\b} \, \s_2 + \sin{\b}(\cos{\g} \, \s_3 - \sin{\g}\, \s_1)\big] \ .
\end{equation}

In a further limit of  infinite shift of $$,
\begin{equation}\la{limit}
\alpha \to \alpha + \ell \ , \qquad \ \ \ \ell \to \infty \ ,
\end{equation}
the NAD  background  \rf{f2zero} degenerates to an abelian T-dual of the PCM on $SU(2)$ (in the $$ direction). The corrected NAD background then leads to a corrected T-dual background in this limit (dropping a trivial closed 2-form contribution to the $$-field),
\begin{align}
\wt{ds^2} = \tfrac{\hh+8}{4}\,d\alpha^2 + \tfrac{\hh}{4}\big(d\beta^2 + \sin^2\beta d\gamma^2\big) \ ,\qquad \ \ \
\BB = \tfrac{\hh+5}{4} \, \cos \b\, d\a \wedge d\g \la{f2zerolim1}
\ .
\end{align}
This background ($ℝ ×S^2$ supported by a $$-field) is indeed correctly related to the  $SU(2)$ PCM,
\begin{equation}\begin{split}
{ds^2}%& = \tfrac{\hh+2}{4}\big(d\b^2 + d\tilde{\a}^2 + d\g^2 - 2\cos\b\, d\tilde\a d \g\big)
= \hh \big( d \theta ^2+ \sin^2 \theta\, d \psi^2 + \cos^2 \theta \, d \chi^2 \big) \ ,
\end{split}\end{equation}
by (i) the 1-loop corrected T-duality transformation $→$ of \cite{Kaloper:1997ux}, (ii) shifting the coupling $→-2$, and (iii) changing coordinates  $= ψ+ χ$, $= ψ- χ$ and $=̱ 2θ$.

\renewcommand{\la}{\label}
\renewcommand{\rf}[1]{(\ref{#1})}
\def \kk {k}

\subsubsection{Summary}

These findings extend the previous conclusions \cite{Fridling:1983ha,Fradkin:1984ai} about the 1-loop quantum
equivalence of the models related by the non-abelian duality to the 2-loop level. As for abelian T-duality, NAD (properly modified by
quantum $'$ corrections) should also be a symmetry of the \sm\  $β$-functions or the string effective action to all orders in $'$.

Resolving earlier problems checking NAD at 2-loop level \cite{n1,n2,n3,Bonneau:2001za},
we found that the preservation of quantum equivalence requires a non-trivial 1-loop correction to the
classical NAD model metric.
In the $SU(2) U(1)$ and $SU(2)$ cases, we explicitly constructed the corrections restoring
2-loop renormalizability, with the 2-loop $$̱-functions matching the associated symmetric space and group models. This is a non-trivial result since these 2-loop $\b$-functions are  invariant under scheme changes (redefinitions of $h$) since the PCM/symmetric space and their NADs are \textit{single-coupling} theories.

In further abelian T-dual limits, we obtained corrected versions of T-dual models matching the known quantum corrections to the T-duality rules (the simple case of \rf{5} and the more general case of \cite{Kaloper:1997ux}). In these special cases, this approach  serves as an alternate derivation of the leading correction to the T-duality transformation.

\section{Discussion}

We have demonstrated that, just as for gauged WZW models,
the invariance of integrable $$-models under the
two-loop (and higher) RG flow requires a specific quantum
deformation of the classical Lagrangian (i.e. of the target space metric and $B$-field).

In particular, we proposed an exact metric for the $η$-deformation of the $S^2$ (and $H^2$) $$-model
that solves the 3-loop RG flow equations and is consistent with the gWZW limit.
We also found the leading-order deformation of the \lam\
for $SU(2) U(1)$, which solves the
2-loop RG equations and is consistent with the gWZW and $η$-model limits.
For this example we identified the origin of the deformation as a finite counterterm
resulting from the determinant of integrating over the auxiliary 2d gauge field
(the same determinant that leads to the correction of the dilaton term on a curved 2d background).

We also found the leading-order deformation of the \lam\ for $SU(2)$ that solves the 2-loop RG equations,
but in that case the necessary counterterm does not take the same simple and identifiable form.
We checked the consistency of the corrections with the WZW model limit and the limit T-dual to the squashed 3-sphere ($η$-model).

In the non-abelian dual limit, the corrections for the $ł$-models generated predictions
for leading corrections to the $SU(2) U(1)$ and $SU(2)$ NAD models.
These corrections were indeed found to restore the NAD models' 2-loop renormalizability,
resolving  the earlier problems  \cite{n1,n2,n3,Bonneau:2001za} in checking the consistency of non-abelian duality with the 2-loop RG flow.

\

The classical $ł$-model's $ℤ_2$ transformation  \rf{z2sym}
is respected by the leading corrections found in the above examples.
In explicit coordinates, this is seen as invariance of the counterterm in \rf{48} (in the $SU(2) U(1)$ case) under $k →-k$, $→-$, $p →-p$; and invariance of the counterterms in \rf{1910b2} (in the $SU(2)$ case) under $k→-k$, $→-$, $→-$.
The resulting 2-loop $$̱-functions 49 and 19100 are thus invariant under $k \to -k$, $\ka \to -\ka$. As we shall see later in Chapter <ref> (see eq. 19102777), the preservation of this $\mathbb{Z}_2$ invariance requires a particular formulation of the quantum theory, and  in generic subtraction schemes it would require a certain coupling redefinition to become manifest.

In the $SU(2) \ov U(1)$ case, the $\l$-model 43 is invariant under an additional $\mathbb{Z}_2$ transformation: $p \leftrightarrow q$, $\ka \to \ka^{-1}$ (i.e. $\l \to -\l$), associated with the subgroup $H=U(1)$ being abelian.I thank B. Hoare for a discussion of this point.  Indeed this $\mathbb{Z}_2$ is also respected by the counterterm in 48, and the resulting 2-loop $\b$-functions \rf{49} are then invariant under $→^-1$.

\

In summary, while some models (e.g., on groups or symmetric spaces) have sufficient \textit{manifest} global symmetry to ensure renormalizability to all orders, general integrable models may not have any manifest symmetries. However, the \textit{hidden} integrable symmetries still seem to constrain the RG flow. This pattern is found to persist at higher-loop orders, and in such cases the hidden symmetries seem to require a particular choice of finite counterterms for their preservation at the quantum level.

A general open question is how to systematically construct the necessary finite counterterms. One natural approach would be to try to derive them by  imposing  ``Ward identities'' for the  hidden integrable symmetries
at the quantum level, i.e. requiring the  choice of finite counterterms  to be consistent with quantum integrability, and  assuming that such counterterms would also be consistent with renormalizability.
Another approach, assuming a UV fixed point and a known underlying quantum S-matrix,
would be to try to systematically reconstruct the corrected \sm\
from a dual massive model, as considered in \cite{fateev96,Litvinov:2018bou,Alfimov:2020jpy}.

\

An interesting open problem would be to construct an
exact generalization of the 1-loop-corrected metric \rf{48} for the $SU(2) U(1)$ \lam\  reducing in the appropriate limits to the exact gWZW
metric \rf{522} and the proposed exact  \etm\ metric \rf{36},\rf{37}. This is non-trivial given the lack of isometries in the \lam\ metric
and that the effective action approach used for the gWZW model \cite{Tseytlin:1993my} does not appear to apply directly to  non-conformal theories.

It would be interesting to see whether the exact metric \rf{24} of the $SU(2)U(1)$ $η$-model,
translated to the Hamiltonian framework, would have a natural interpretation in terms of  a deformation of
the classical integrable structure.

A case that was not yet considered is the higher-loop RG flow of the $η$-deformation of the PCM (rather than the symmetric space). For the simplest $SU(2)$ case, the $η$-model happens to coincide \cite{Klimcik:2002zj} with the $σ$-model on a `squashed 3-sphere' \cite{cherednik} and is automatically stable due to manifest symmetries. This is because the $η$-deformation breaks the PCM's symmetry to $G_L ×U(1)^r$, i.e. $SU(2)_L×U(1)_R$  in the $SU(2)$ case, with $U(1)_R$ happening to be a subgroup of $SU(2)$ such that $SU(2) U(1)$ is a symmetric space. This implies that the $SU(2)$ $η$-model must coincide with the ``squashed'' model related to  $SU(2) U(1)$ --- part of a general family  built on symmetric spaces $G/H$ considered in App. B of \cite{HLT2}. (Note that general examples from that family are renormalizable due to manifest $G_L×H_R$ symmetry but \textit{not} classically integrable: the $SU(2) U(1)$ example is atypically integrable.) For $η$-models based on general groups, this coincidence does not happen, so the theory is not  protected by manifest symmetries and one may expect to require
additional  finite local counterterms to get higher-loop renormalizability.

the \etm\  and the \lam\  are, in general, related \cite{Vicedo:2015pna,Hoare:2015gda,Sfetsos:2015nya,Klimcik:2015gba,Hoare:2017ukq}
by  Poisson-Lie (PL) duality \cite{Klimcik:1995dy} (and analytic continuation).
Since the scaling limits between the $ł$-models and (T-duals of) $η$-models revealed that our 2-loop results for the two classes of models are consistent,
then the same observations made above for NAD are likely to apply also to PL duality.
Namely, PL duality (with suitable quantum corrections) should be a symmetry not only at 1-loop order \cite{Valent:2009nv,Sfetsos:2009dj}, but also at higher loops.
This expectation
was recently confirmed by several papers \cite{BW}
proposing 1-loop corrections to the classical PL duality rules (including NAD as a special case\foot{See also \cite{Borsato:2019oip} on the  leading quantum corrections to homogeneous Yang-Baxter deformations (equivalent to particular NAD transformations \cite{homog2,homog3}) in order for the deformation to preserve 2-loop conformal invariance.}).
It would be interesting to check those results against the  corrections \rf{57},\rf{1910originlagnad} found here  for particular NAD models.
In a related approach, it has recently been proposed \cite{hassler} that there is a particular 2-loop ``subtraction scheme''\foot{The PL-symmetric subtraction scheme of \cite{hassler} is not a covariant
subtraction scheme  in the usual sense as it is related to the \GB scheme \rf{BMT} by a Lorentz-non-covariant transformation involving the spin connection.}  in which PL symmetry becomes manifest. The $ł$- and $η$-models were claimed to be automatically renormalizable in that PL-scheme, suggesting the possibility to derive the corresponding quantum corrections in the standard scheme by reversing the scheme transformation.

\renewcommand{\la}[1]{\label{2103#1}}
\renewcommand{\rf}[1]{(\ref{2103#1})}
\input{2103.tex}
\renewcommand{\la}{\label}
\renewcommand{\rf}[1]{(\ref{#1})}

\renewcommand{\la}[1]{\label{1910#1}}
\renewcommand{\rf}[1]{(\ref{1910#1})}
\input{1910v2.tex}
\renewcommand{\la}{\label}
\renewcommand{\rf}[1]{(\ref{#1})}

\renewcommand{\la}[1]{\label{2008#1}}
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\chapter{Conclusions and future directions \la{cfd}}

\section{Integrability vs. RG flow}
This thesis is based on the interesting observation that integrable 2d $$-models appear to be stable under the leading 1-loop RG flow, i.e. they are solutions of the  (generalized) Ricci flow equation \rf{BMT} with a few couplings running.
Our purpose was to investigate this property beyond the leading 1-loop order. We found in Chapter \ref{QCS} for integrable $η$- and $ł$-deformed models, and in Chapter \ref{GGS} for integrable $G×G/H$ models,  that generic examples will require quantum deformation of the target space geometry to remain stable under the 2-loop and higher RG flow. Thus, in each case, there is a particular finite counterterm  --- or particular choice of bare action --- that restores renormalizability beyond 1-loop.  Moreover, this is a scheme-independent statement since these counterterms cannot be eliminated by a covariant scheme transformation.

In Section \ref{Seta}, we explicitly constructed these counterterms for the simplest $η$-deformed $S^2$ (or ``sausage'') model. In that case, by analogy with the gauged WZW model (whose corrections are well-known, and which arises as a particular limit of the sausage model) we conjectured the exact form \rf{24} of the quantum deformation to all orders in $'$ (or the $$-model loop expansion). We checked explicitly up to 3 loops that these corrections indeed restore renormalizability  in a particular subtraction scheme.

In Section \ref{Slam}, we constructed the leading corrections restoring 2-loop renormalizability to the $ł$-models based on $SU(2) U(1)$ and $SU(2)$. In the simpler $SU(2) U(1)$ case, the corrections were understood as a familiar $(logM)^2$ determinant \rf{47} from the quadratic integral over the $ł$-model's 2d gauge field in \rf{lam}. However, such 2d integrals are known to be ill-defined \cite{Schwarz:1992te} with a local term ambiguity depending on a choice of prescription or regularization. Indeed we find that this simple formula is not sufficient to restore renormalizability in the $SU(2)$ case, where the leading corrections take the more complicated form \rf{1910originlag}, or \rf{1910b2} in explicit coordinates.

In Chapter \ref{CT} we proved in general that the $ł$-model \rf{lam} in the formulation with unintegrated gauge field is renormalizable to all orders, by re-writing the gauge field in terms of group valued scalars and obtaining a $$-model with a ``tripled'' $G×G ×G$ configuration space. This means that the  corrections required for renormalizability of the $ł$-model in  ``standard'' configuration space may all be understood as finite counterterms arising from the ambiguous 2d gauge integral. Clearly the simple $(logM)^2$ prescription is not generally consistent with renormalizability (since it breaks down already for the $SU(2)$ case).

An interesting open problem is to find a general prescription for the  finite  local counterterms produced by integrals of this type, $∫dA exp( i ∫A_+ M A_- )$, consistent with the RG flow. This would give a general formula for the quantum corrections needed to restore higher-loop renormalizability for all $ł$-models, generalizing the $SU(2) U(1)$ and $SU(2)$ results found here. One possible approach based on the work of \cite{hassler} would be to try to construct the necessary counterterms from the scheme transformation between our standard 2-loop scheme \rf{BMT} and one that is non-Lorentz-covariant but manifestly respects Poisson-Lie symmetry,  in which the $ł$- and $η$-models are claimed to be automatically renormalizable.

For the  $G×G/H$ models in Section \ref{2103gu}, the manifest (global and local) symmetries alone restrict the theory to a finite set of couplings \rf{2103couH},\rf{2103bb1} that could run under the RG flow. The integrable theories then correspond to a ``surface'' in this space of couplings, specified by polynomial equations \rf{2103condH}. Hence we observed that the necessary counterterms to restore 2-loop stability of the integrable surface under the RG flow can be interpreted as a quantum correction to the shape of the surface. This is like a simple, finite-dimensional analog of the general case where the space of \sm couplings is essentially infinite (e.g., for the integrable deformed models above).

We also constructed  in Chapter \ref{GGS} a new ``branch'' of $H$-gauge-invariant $G×G/H$ theories in the case when $H$  is abelian, with independent WZ levels on the two copies of the group $G$. We found that a particular subfamily of this new branch is classically integrable and, in the $SU(2)×SU(2)/U(1)$ case, these correspond to a family of integrable $$-models with target space $T^1,q$ supported by a particular $$-field. These models are a generalization of the integrable $T^1,1$ models of \cite{ABL}, with the extra parameter $q$ being related to the ratio of the two WZ levels.

\

In integrable examples with enough \textit{manifest} symmetries, such as the undeformed PCM or symmetric space, renormalizability can be an automatic result of the symmetries. However, generic integrable $$-models may have no manifest global symmetries at all; rather, the integrable structure is related to an intricate structure of \textit{hidden} (non-local or higher spin) symmetries. The effect of such symmetries on the RG flow is far less obvious since, e.g., it is not clear in which way they should constrain the Riemann tensor and $$̋-field that enter the RG equations BMT. Remarkably, it appears that hidden integrable symmetries impose drastic constraints on the RG flow, restricting the 1-loop RG flow to just a flow of the couplings of the classical theory. We have observed that this property persists also at higher loop orders provided the target space geometry is suitably deformed by quantum ($\a'$) corrections. Indeed, such a deformation is  natural given the the well-known examples of  conformal $\s$-models where $\a'$ corrections are needed to restore higher-loop conformal invariance.

One interesting open  question is to understand how and why integrability  implies renormalizability, and perhaps to prove this statement, starting at the leading 1-loop order.
As we discussed in the Introduction, the rationale behind the conjectured integrability–RG flow relationship is that the stability of integrable theories under the RG flow simply corresponds to preserving their integrable structure at the quantum level. In fact, it is natural that such hidden integrable symmetries should require a particular choice of finite local counterterms to be manifest in the quantum theory, just like for standard global symmetries.
It would be interesting to verify this interpretation by checking explicitly that the finite counterterms required for renormalizability are actually restoring some quantum `Ward identities' for hidden integrable symmetries. This would give an interesting semi-classical picture of quantum integrability.
Such computations could be carried out, e.g., for the integrable-deformed models considered in Chapter <ref>, in relation to either the higher-spin charges or the non-local charges. For the $\eta$-deformed models,  studying the quantum non-local charges would also be an interesting check of the conjectured quantum group symmetry of the quantum theory <cit.>.

The integrable $G\times G$ models studied in Section <ref> are an anomaly to the discussion above. These theories are automatically stable under the 2-loop RG flow, even though (to our best knowledge) the integrability condition is not protected by any manifest symmetries. Generically we would have expected the integrable “surface” in coupling space to require some quantum deformation (like the $G\times G/H$ models) in order to remain stable beyond 1-loop. An interesting open problem is to explain this fact, which surely should not be a coincidence.

The $G\times G$ and $G\times G/H$ models demonstrate that, while integrability seems to imply renormalizability, the converse statement is not true. These models are renormalizable with a finite set of couplings but most of them are not integrable (only ones with special values of the couplings solving polynomial equations).

Classical integrability may also  become anomalous at the quantum level, and we note that renormalizability seems too coarse a measure to pick up on this.
For example the bosonic $\mathbb{C}\mathbf{P}^N$ model is well known to suffer such an anomaly <cit.> (see also <cit.>).
However, it is still 2-loop renormalizable <cit.>.For the $\mathbb{C}\mathbf{P}^N$ model it has been conjectured that quantum integrability
can be restored by including an additional free field <cit.> in the classical limit (related models have also appeared in <cit.>).
An alternative is to consider the supersymmetric $\mathbb{C}\mathbf{P}^N$ in which there is no anomaly <cit.>.

We recall that our original motivation for studying stability under the RG flow was that it might help to identify integrable $\s$-models. In a certain sense, this is what we have been doing: starting from classically integrable theories, we have been constructing particular formulations of the quantum theories (with particular finite counterterms) that are RG-stable at higher-loop orders — with the belief that these formulations are the ones consistent with integrability at the quantum level.

It would be interesting in future to apply the integrability–RG flow connection to discover new examples of integrable $\s$-models. First at the level of classical integrability, this may be carried out by imposing stability under the leading 1-loop RG flow. One possible example would be for non-symmetric $G/H$ cosets, which generally have several couplings,
Ł= -  [ J_+   ( ∑_i=1^n h_i P_𝔭_𝔦 )   J_- ]  ,
corresponding to a decomposition of the tangent space into irreducible representations $\mathfrak{p_i}$ under the adjoint action of $H$ (see Section <ref>). It may be possible in certain cases to discover integrable “surfaces” in this coupling space by demanding stability under the RG flow. Indeed, renormalizability of certain non-symmetric coset spaces was previously considered in <cit.>.

§ “TRIPLED” FORMULATION

The “tripled” formulation of the $\l$-model in Chapter <ref> may be viewed as an alternate formulation of the quantum theory where some fields decouple, likely as a manifestation of integrability. The remaining “truncated” theory is automatically stable under the RG flow to all orders due to  manifest global symmetries, without needing any quantum deformation.

The general idea of moving to an enlarged configuration space to make manifest extra symmetries (in this case integrability) is reminiscent of T-duality-invariant formulations of string theory <cit.>. Such approaches
are essentially 1st order Hamiltonian formulations of the $\s$-model, with manifest Lorentz invariance being sacrificed in favour of manifest T-duality invariance.
A related 1st order formulation of Poisson-Lie symmetric $\s$-models known as the “E-model” makes the more general Poisson-Lie duality manifest <cit.>.

In Section <ref>, we  considered  similar reformulations on enlarged configuration spaces of (i) a variant of the $\l$-model based on $G^N$ and (ii) $G \times G/H$ models. In each case the resulting model was of product type, i.e. a particular class of coupled models on some space $G^M$; and in the $G\times G/H$ case there was some “squashing”, or asymmetry between the $H$ and $G/H$ directions. Re-writing these models in this way is interesting as the corresponding enlarged models will have identical classical integrability and RG flow properties to the original ones. For the $G^N$ $\l$-model we find an identification between its RG flow and classical integrability conditions and those of an apparently simpler undeformed $G^{2N}$ model. We also learn that it is automatically renormalizable in the tripled formulation  due to its manifest symmetries (before imposing the integrability conditions of <cit.>).  For the $G\times G/H$ models we essentially discover a new class of classically integrable and RG-stable squashed $G\times G \times H$ models. It would be interesting to study the higher-loop RG flows of these theories to see if the  enlarged formulations render the integrability conditions automatically stable  at higher-loops (as for the $\l$-model).

Another interesting class of $\s$-models are those with complex homogeneous target spaces that are equivalent  <cit.>  to bosonic Gross-Neveu (GN) models <cit.> (see also the reviews <cit.>). The reformulation of these $\s$-models as GN models is essentially equivalent to the introduction of a 2d gauge field, so should be closely related to the “tripled” or “extended” formulations discussed here. It would be interesting to check stability under the 2-loop RG flow in that formulation (which was previously only checked   at 1-loop <cit.>).

An interesting future direction would be to try to construct a similar  tripled formulation of the $\eta$-deformed models. It seems plausible that such a formulation should exist since the $\eta$-models and $\l$-models are related (up to analytic continuation) by Poisson-Lie duality a <cit.>, or by limits and  T-duality <cit.>.

§ NON-ABELIAN DUALITY

In the non-abelian dual limit of the $\l$-model, the general 2-loop $\b$-functions obtained from the tripled formulation match those of the original PCM/symmetric space $$-models. This agreement is a non-trivial check of our $ł$-model $$̱-functions since the 2-loop terms are also scheme-invariant under coupling redefinitions in that single-coupling limit. It is thus natural to expect that the  interpolating models for non-abelian duality are renormalizable with the same 2-loop $\b$-functions as the original models --- justified also by their relation to the original models by just an integral over a linear Lagrange multipler imposing the flatness of the gauge field.

Hence, like the $ł$-model, the non-trivial corrections found for the NAD models based on $SU(2)/U(1)$ and $SU(2)$ in Section \ref{NADqS} should be seen as  resulting from the integral over the 2d gauge field.
In certain limits these NAD models degenerate to abelian T-duals and in those cases our corrections matched with the known corrections to the T-duality transformation.

It would be interesting to consider more general models with non-abelian symmetries, which may have spectator coordinates (here $x$ or $u,v$), for example
\be
\L = \del_+ x \, \del_- x + f(x) \, \Tr[ J_+ J_- ]  , \qquad \qquad (J \equiv g^{-1} dg , \ \ g\in G)
\ee
or perhaps the conformal $$-model considered in \rf{2008nPCM},
\be
\L = -2\, \del_+ u \, \del_- v + u \, \Tr[ J_+ J_- ] \ .
\ee
Starting from such a theory that is integrable and renormalizable (say, for a particular choice of $f(x)$) the interpolating model for NAD is obtained by replacing $J_±→A_±$ and adding a Lagrange multiplier constraint $[v   F_+- (A)]$. One can then perform the same trick of going to a tripled formulation by re-writing the gauge field in terms of group valued scalars. It would be interesting to study the RG flows for these interpolating models in the tripled formulation and to see if they still automatically match the original models (also at 2-loops) in the presence of spectator fields.

\section{Quantum corrections and string sigma models \la{stric}}
An important application of our results would be to the related $ł$- and $η$-deformations of superstring $$-models. For example, focussing on the simplest case of \adss\unskip, the $ł$-deformed background has been extracted  from a $$-model formulation \cite{Hollowood:2014qma} and checked to be a type IIB supergravity solution \cite{sugra}. Equivalently, the string \sm is 1-loop Weyl-invariant (1-loop scale-invariance was also checked in  \cite{Appadu:2015nfa}). The $η$-deformed \adss background of \cite{hsei} was also extracted and checked to be a solution of the supergravity equations.

In general, solutions of the $d=10$ type II supergravity equations should receive $'$ corrections to become  exact string solutions, with the first corrections appearing at order $'^3$ \cite{Gross:1986iv}. In contrast with the undeformed \adss background where corrections are prohibited by manifest symmetries,\foot{Exact finiteness of  the undeformed \adss superstring $\s$-model \rf{string} follows because the WZ-type  $J^{(1)} J^{(3)}$  term is not renormalized due to general arguments and the $J^{(2)} J^{(2)}$ term may only be renormalized by an overall factor, but this cannot happen since the two terms are related by $\ka$-symmetry (see footnote 4 of \cite{Roiban:2007jf} and refs.\ there).} it is not clear whether the deformed backgrounds will require non-trivial corrections. It may be that, due to the models'  hidden integrable symmetries, the corrections can be made to vanish in some particular subtraction scheme. On the other hand, this did not happen for the bosonic analogs above, which are also integrable.

It would be interesting to study this question and  to understand the structure of the $'$ corrections to these backgrounds. For the $ł$-deformation, a systematic way to do this may come from  adapting the tripled formulation to the superstring case. This seems feasible since the $ł$-deformed \adss superstring was formulated in \cite{Hollowood:2014qma} in the language of supergroups with a 2d gauge field. In such a tripled formulation,  the $'$ corrections may be expected to  vanish due to
manifest symmetries after a decoupling of some fields
(like the bosonic case), and  the exact  finiteness  of the model may become manifest.

\section{Sigma models with local couplings}

In Chapter \ref{loc} we discussed  a new and apparently different connection between integrability and the RG flow: upon promoting the couplings of an integrable $$-model to functions of 2d time, the resulting ``local coupling'' theory remains classically integrable (with a natural ansatz for its Lax connection) only when the coupling functions solve the 1-loop RG flow, with 2d time playing the role of RG time $t=log$. This curious pattern was observed in a broad class of integrable $$-models on groups and symmetric spaces, including integrable $η$- and $ł$-deformations. It would be good to check this claim for more examples of integrable $$-models, like the $G×G$ and $G×G/H$ models. We also found the same result for the  sine-Gordon model, and it would be interesting to check more examples of  integrable theories with potentials, e.g. Toda models.

These results are surprising because it seems that the 1-loop quantum $$̱-functions are derivable in this way from a purely classical procedure (from demanding that the local-coupling theory admits a flat Lax connection). We also noted that theories with explicit time-dependence governed by the RG flow are naturally obtained in string theory upon fixing the light-cone gauge $u=\t$, starting from a class of Weyl-invariant $\s$-models <cit.> with a covariantly constant null Killing vector and a dilaton linear in the isometry direction,
Ł= -2    _+ u    _- v  + _ij(u;x) _+ x^i _- x^j  ,   _u _ij = _̱ij(x)  ,     (= v + (u,x))  . ssi
Thus we find that  — assuming the “transverse” ($u=\text{const}$) theory to be integrable — the light-cone ($u=\t$) theory is apparently classically integrable only when the string $\s$-model ssi is 1-loop Weyl invariant. This  points to a  deeper meaning or explanation of our results, and to the possibility of a large, new class of solvable string backgrounds.

What remains to be clarified is whether these time-dependent (light-cone) theories may really be viewed as solvable. Although the monodromy matrix is formally conserved, it is difficult to explicitly extract the associated conserved charges because of problems making a good expansion in the spectral parameter. We also made attempts to study higher-spin local charges in the paper <cit.>, but our results so far were  inconclusive. One possible way to demonstrate solvability would be to explicitly  construct some  classical solutions;  indeed this has already been done for the time-dependent PCM example <cit.>
by adapting the inverse scattering approach of <cit.>.

An interesting extension would be to consider the  $(2,2)$ supersymmetric $\s$-models on certain  Kähler symmetric spaces whose $\b$-functions are known to be 1-loop exact \cite{Novikov}. Supersymmetric $$-models on symmetric spaces are  classically integrable in general  \cite{Abdalla:1985nm}. Hence the pattern found above for bosonic theories may generalize to the supersymmetric ones. In that case the corresponding string backgrounds built from the 1-loop RG flow (which will be \textit{exact} string solutions) may be completely solvable.

A further question is whether the local-coupling $$-models  admitting Lax representations can be obtained from  4d Chern-Simons theory \cite{Costello:2019tri} (like all of the regular integrable $$-models considered in this thesis \cite{Vicedo:2019dej,Delduc:2019whp}). This is a natural expectation, since the 4d perspective places the 2d space-time and the complex spectral parameter on an equal footing, while the Lax connections for the local coupling theories are related to the original ones by a space-time-dependent redefinition of the spectral parameter, $z →z(w; ,)$.

\begin{appendices}

\chapter{Conventions \la{conv}}
Unless otherwise stated, the 2d space is assumed to be flat Minkowski space with signature $(-+)$, i.e. $\eta_{00}=-\eta_{11}=-1$ in terms of the 2d coordinates  $\xi^\m = (\xi^0,\xi^1) = (\t,\s)$ with 2d incides $\m,\n=1,2$. The antisymmetric Levi-Civita symbol $\e_{\m\n}$ is defined by $\e^{01}=-\e^{10}=1$ and $\e_{01}=-\e_{10}=-1$ (with indices raised/lowered by $\eta_{\m\n}$). We use 2d light-cone coordinates defined as $\xi^\pm = \ha(\t\pm\s)$ and light-cone derivatives $\del_\pm = \del_\t \pm \del_\s$.

\bigskip
\noindent The $\s$-model action is normalized as $S= \tfrac{1}{4\pi \a'} \int d^2 \xi \,  \L$ with the usual flat-space volume form $d^2 \xi = d\xi^0 \wedge d \xi^1$.
The Lagrangian is
\be
\L = - \big[ \eta^{\m\n} \, \GG_{mn}(x) + \e^{\m\n}\, \BB_{mn} (x)  \big] \,  \del_\m x ^m \, \del_\n x^n  = \big[ \GG_{mn}(x) + \BB_{mn} (x)\big] \,  \del_+ x ^m \del_- x^n \ .
\ee
The target space metric is $ds^2 = G_{mn} dx^m dx^n$. The $\BB$-field written as a 2-form is $\BB=\ha dx^m \wedge dx^n$ and its field strength is $\H=d\BB$.

\bigskip
\noindent The components of $n$-forms are normalized in the standard way, $H_{(n)} = \tfrac{1}{n!} H_{i_1 \cdots i_n} dx^{i_1} \wedge \cdots \wedge dx^{i_n}$.

\bigskip
\noindent We take $G$ to be a simple, real Lie group with generators  $T_A$ ($A=1,\ldots,\dim(G)$) satisfying $[T_A, T_B] = i {f^C}_{AB} T_C$.

\bigskip
\noindent We use a non-degenerate bilinear form defined as
\be
\Tr = \tfrac{1}{2 \, \chi_{_{G,R}}} \, \tr_{_R} \ ,
\ee
where $\tr_{_R}$ is the matrix trace in some faithful representation $R$ of $G$. The constant $\chi_{_{G,R}}$ denotes the \textit{index} of the representation $R$, defined by
\be
\tfrac{1}{2 \, \chi_{_{G,R}}} \, \tr_{_R}(T_A T_B) = -\tfrac{1}{2\, \cgg} \, f^C{}_{DA} f^D{}_{CB} \ .
\ee
Here $\cgg$ denotes the dual Coxeter number of $G$: for the classical groups these are $c_{_{SU(N)}} = N$, $c_{_{Sp(N)}} = N+1$  and $c_{_{SO(N)}}=N-2$.

\sloppy
The indices of the adjoint representations coincide with the dual Coxeter numbers, \mbox{$\chi_{_{G,adj}} =c_{_G}$}, and those of the fundamental representations of the classical groups are \mbox{$\chi_{_{SU(n),f}} = \chi_{_{Sp(n),f}} = \ha$} and  \mbox{$\chi_{_{SO(n),f}} = 1$}.

The structure constants then satisfy
\be
{f^A}_{BC} {f^B}_{AD} = -2 \, \cgg \Tr[T_C T_D] \ , \qquad {f^D}_{EA} {f^E}_{GB} {f^G}_{DC} = \cgg \, f_{ABC} \ ,
\ee
where we have defined $f_{ABC} = \Tr[T_C T_D] {f^D}_{AB}$.

\bigskip
\noindent  We shall sometimes consider a Lie subgroup $H \subset G$, which is also assumed to be simple. Then we split the algebra directions as $A=(\a,a)$ where $T^\a$ are the generators of the $H$ and the remaining generators $T^a$ are orthogonal to $T^\a$ with respect to the Killing form.

The structure constants ${f^\a}_{\b\g}$ of the subgroup $H$ then satisfy
\be
{f^\a}_{\b\g} {f^\b}_{\a\d} = -2 \, \cf \, \Tr[T_\g T_\d] \, \qquad {f^\d}_{\e\a} {f^\e}_{\ka \b} {f^\ka }_{\d\g} = \cf \, f_{\a\b\g} \ ,
\ee
where $\cf$ denotes a \textit{rescaled} version of the dual Coxeter number $\tfrac{\chi_{_{H,R}}}{\chi_{_{G,R}}} \cf$ of $H$.

\chapter{3-loop \texorpdfstring{$\b$}{beta}-function in different renormalization schemes} \label{A}

In general, in quantum field theories
changes of a renormalization scheme are equivalent to local redefinitions of the coupling constants,\foot{Here we reinstate the loop-counting parameter $\a'=\hbar$.}
\be
g^i \to \hat{g}^i (g ) = g^i+ \a' \, c^i_{jk} g^j g^k + \ldots \ . \la{redef}
\ee
Under \rf{redef} the $\b$-functions $\b^i \equiv \dt g^i$ transform as a contravariant vector in the space of couplings:
\be \la{deltab}
\delta \b^i \equiv\hat{\beta}^i - \beta^i = \frac{\del \, \delta g^i}{\del g^j} \b^j - \frac{\del \b^i }{\del g^j} \delta g^j + \ldots \ , \qquad \qquad \delta g^i \equiv \hat{g}^i - g^i \ ,
\ee
where the higher order corrections are non-linear in $\delta g$. Assuming $\b^i$ and $\delta g^i$ both start at 1-loop order, i.e.
are both $\O(\a')$, then $\delta \b$ is $\O(\a'^2)$, so the 1-loop $\b$-function is scheme-invariant.
In single-coupling theories, one can further show that the 2-loop $\b$-function is also scheme-invariant.

Now let us turn to the particular case of 2d $\s$-models, for which the relevant coupling redefinitions \rf{redef} are local redefinitions of the target space geometry.
In order to preserve the manifestly covariant structure of the action \eqref{13} we restrict to covariant redefinitions.
Restricting further to include only redefinitions with a natural interpretation in terms of changing the subtraction scheme,\foot{While the term $\a'^2 \nabla^2 R_{mn}$ in \rf{metricredef} has no natural interpretation as a change of subtraction scheme starting from the minimal subtraction scheme, it is required in order that the transformation
\rf{2dredef} be closed under inversion in the case of the \sm with a 2d target space,
i.e. in order to be able to move back to the minimal subtraction scheme starting from another scheme.} we consider $\GG_{mn} \to
\hat{\GG}_{mn}$ with
\begin{align} \la{metricredef}
\hat{\GG}_{mn} = \GG_{mn} + c \, \a' R_{mn} + d \, \a'^2 (R^2)_{mn} + e \, \a'^2 \nabla^2 R_{mn} + \O(\a'^3) \ , \\
&(R^2)_{mn} \equiv R_{mabc} {R_n}^{abc} \ , \no
\end{align}
where $c, d, e$ are arbitrary coefficients (for simplicity we consider the special case with vanishing $\BB$-field; for a general discussion see \cite{mt}).

At the leading $\O(\a')$ order, both $\beta_{mn} = \a' R_{mn} + \ldots$ and $\delta \GG_{mn} = c\, \a' R_{mn}+\ldots $ only contain a single term proportional to $R_{mn}$.
Thus at the leading $\O(\a'^2)$ order in \rf{deltab}, their contributions cancel and the 2-loop $\b$-function is invariant.\foot{Note that if one also included redefinitions of the form $\a' R \GG_{mn}$ then this argument would not go through in general dimensions. However, for the special case of a 2d target space, the identity $R_{mn} = \ha R \GG_{mn}$ means these terms are proportional and the 2-loop $\b$-function remains invariant.}

Specializing further to the case of a 2d target space (relevant for the simplest $SU(2)\ov U(1)$ $\eta$- and $\l$-models considered in Chapter \ref{QCS}), the redefinition \rf{metricredef} becomes
\begin{align} \la{2dredef}
\hat{\GG}_{mn} = \GG_{mn} \Big[ 1 + \ha c \, \a' R + \ha d \, \a'^2 R^2 + \ha e \, \a'^2 \nabla^2 R \Big] \ .
\end{align}
Possible scheme dependence starts at 3-loop order,
with \rf{metricredef} or \rf{2dredef} being the relevant redefinition at this order.
The most general form of a covariant expression for a 3-loop $\b$-function
is\foot{In \rf{3loopbeta} we have dropped terms of the form $\nabla_{(m} V_{n)}$ since these may be absorbed into the diffeomorphism vector term in \rf{2}. We have excluded the term $\a'^2 \nabla^2 \nabla^2 R \GG_{mn}$ in \rf{3loopbeta} since it does not arise naturally through changes of subtraction scheme starting from the minimal subtraction scheme.}
\be \la{3loopbeta}
\b_{mn} = \Big[ \ha \a' R + \tfrac{1}{4} \a'^2 R^2 + c_1 \, \a'^3 R^3 + c_2 \, \a'^3 (\nabla R)^2 + c_3 \, \a'^3 R \nabla^2 R \Big] \GG_{mn}
+ c_4 \, \a'^3 \nabla_m R \nabla_n R \ . \ee
Not all of the coefficients $c_1,\ldots ,c_4$ can be scheme-dependent, since the space of legitimate schemes should be parametrized by the 3-dimensional space of coupling redefinitions \rf{metricredef}.
To find the values of $c_i$ that are allowed to appear in the $\b$-function in a particular scheme
let us start from the expression for the $\b$-function \rf{3loopbeta} in the
minimal subtraction scheme where \cite{Graham:1987ep,Foakes:1987ij,Foakes:1987gg}
\be
\qquad c_1 = \tfrac{5}{32} \ , \qquad c_2 = \tfrac{1}{16} \ , \qquad c_3 = 0\ , \qquad c_4=-\tfrac{1}{16}\ .
\ee
Implementing the metric redefinition \rf{metricredef} we find from \rf{deltab} that the coefficients in the
transformed $\b$-function are given by
\begin{align}
&\hat{c}_1 = \tfrac{5}{32} +\tfrac{1}{8}(c - 2d) \ , && \hat{c}_2 = \tfrac{1}{16} -\tfrac{1}{4}(c - 2d)-\tfrac{1}{8}(4e+c^2)\ , \no \\
& \hat{c}_3 = -\tfrac{1}{8}(4e+c^2)\ , && \hat{c}_4=-\tfrac{1}{16}\ . \la{genscheme}
\end{align}
We note that since \rf{genscheme} only depends on the redefinition parameters $c,d,e$ through the combinations $c-2d$ and $4e+c^2$, the choice of $c,d,e$ to give any particular values of $c_1,\ldots,c_4$ is not unique, but rather there is one free parameter. In the case of the \etm\ metric studied in Section \ref{Seta}, the ``natural'' scheme \rf{31} is given by \rf{genscheme} with, e.g., $(c,d,e)=(0, \tfrac{5}{8}, \ha)$; i.e. it is related to the minimal subtraction scheme by the redefinition
\begin{align}
\GG_{mn}^{(\rm nat)}= \big[ \GG_{mn} + \tfrac{5}{8} \, \a'^2 ({R}^2)_{mn} + \ha \, \a'^2 \nabla^2 R_{mn} \big] ^{(\rm min)}\ .
\end{align}

\renewcommand{\la}[1]{\label{2103#1}}
\renewcommand{\rf}[1]{(\ref{2103#1})}
\chapter{Deriving the  integrability conditions for the \texorpdfstring{$G^N$}{G**N} model  \la{A}}
It was shown in
\cite{DLMV}
that the coupled $G^N$ model \rf{cou} is integrable for particular choices of the couplings $(\rho_{ij}, k_i)$ corresponding to realisations of  the affine Gaudin models.
Here  we shall  try to demonstrate
the converse statement: these affine Gaudin models are the \textit{only} integrable  cases of the  coupled models \rf{cou}.

We will assume a natural ansatz \rf{Lan} for the Lax connection, valued in $\Lie(G)$
(here we  explicitly  indicate  the summation over $i=1, ..., N$)\foot{While \rf{a1} is the natural ansatz for the Lax connection arising from affine Gaudin models, it does degenerate at certain points in coupling space. For example, taking $\rho_{ij}$ to be diagonal (i.e.\ decoupled PCM$_k$ models), one instead requires a Lax connection valued in $\Lie(G)^N$. Thus it would also be interesting to consider other ansatze for the Lax connection.}
\be\la{a1}
L_+ = \sum_i \a_{i}(z) \, J_+^{(i)} \ , \qquad\qquad  L_- = \sum_i \b_{i}(z) \, J_-^{(i)} \ ,
\ee
where $z$ is the spectral parameter.
The curvature of this Lax connection takes the form
\be
F_{+-}(L) = \sum_i \Big( \b_{i}(1-\a_{i}) \  \del_+ J_-^{(i)} - \a_{i}(1-\b_{i}) \  \del_- J_+^{(i)} \Big) + \sum_{i\neq j} \a_{i} \b_{j} \ [J_+^{(i)},J_-^{(j)}] \ . \la{cur}
\ee
The equations of motion of the $G^N$  model \rf{cou}
(for arbitrary $N$)
\be \begin{aligned}
E_i \quad  \equiv {}&{}\quad \sum_j \Big(  (\rho_{ij} - \delta_{ij} k_j) \ \del_+ J_-^{(j)} + (\rho_{ij} + \delta_{ij} k_j) \ \del_- J_+^{(j)} \la{eom} \\
&\qquad \qquad \qquad \qquad \qquad +  \rho_{ij} \ [J_+^{(i)},J_-^{(j)}] + \rho_{ji} \ [J_-^{(i)},J_+^{(j)}] \Big) \quad  =\quad 0
\end{aligned} \ee
We note that \rf{cur} and \rf{eom} are the \textit{unique} ways to write these expressions without any terms of the form $[J_+^{(i)},J_-^{(i)}]$, which have been eliminated using the identity $F_{+-}(J^{(i)}) = 0$.

If  the  model \rf{cou}  is integrable then,\foot{Here we are assuming integrability and deriving necessary conditions on the couplings. Thus we do not need to worry about whether the $v^i(z)$ in \rf{a4} are independent functions (which would be relevant for the converse question).}
for some
we have
\be
F_{+-}(L) = \sum_i v^i(z) \, E_i \ , \la{a4} \ee
which implies that
\be \la{eqs} \begin{aligned}
&\b_{j}(1-\a_{j}) = \sum_{i} v^i (\rho_{ij} - \delta_{ij} k_j) \ ,\qquad \qquad    \a_{j}(1-\b_{j}) = \sum_{i} v^i (- \rho_{ji} - \delta_{ij} k_j) \ , \\ &\qquad   \a_i \b_j  =  (v^i - v^j) \rho_{ij}  \ , \ \ \ i\neq j\  \  (\text{no summation}) \ .
\end{aligned}
\ee
This is a system of $N + N + (N^2-N) = N^2+N$ equations. Fixing the freedom to redefine the spectral parameter by setting $v^1 = z$, there are $N+N+(N-1) =3N -1$ ``artificial'' variables, $\a_i, \b_i,
v^{i\neq 1}$.
After solving for these, there are $(N^2+N)-(3N-1)=N^2-2N+1$ remaining equations to be solved for
the $N^2+N$ variables $\rho_{ij}, k_i$. After solving all the equations, this leaves $(N^2+N)-(N^2-2N+1)=3N-1$ free parameters for the integrable theory
(including the WZ levels, which may be continuous for non-compact groups).\foot{One might worry that the integrability constraints on the couplings $\rho_{ij}, k_i$  resulting from \rf{a4} might depend on the spectral parameter $v^1 = z$. However, this will not happen because there is a rescaling ambiguity  $E_i \to c_i E_i$ in the definition of the equations of motion \rf{eom}. One may thus rescale $E_1$ to effectively set $v^1=z=1$ in \rf{a4}. Since the constraints on the couplings from \rf{a4} must be invariant under such rescalings, then they must not depend on $z$.}

Thus the space of integrable models is $(3N-1)$-dimensional,
coincides with the number of free parameters
following from the affine Gaudin  construction  \cite{DLMV}.

Specializing to the $N=2$ case of $G\times G$, this counting suggests a $5$-dimensional space of integrable models.  Then  the 6   free parameters  $(s,t,u,b ,k_1,k_2)$ in \rf{par}
should   be subject to  only one  relation to ensure  integrability.
Solving the equations \rf{eqs} in this case, one indeed obtains the condition \rf{cond}
originally found  from the affine Gaudin  construction.

To summarize,  for general $N$,  the space of integrable models has the same dimension as the space of affine Gaudin models. It remains to understand if there may still be  extra branches
of integrable theories
not corresponding to the affine Gaudin models (cf.\ the $G\times G/H$ models,  where this seems to be the case for abelian $H$, see Section \ref{2103sq}).  For the $N=2$ case of $G\times G$ models, we found  exact matching between the space of  integrable  models \rf{eqs} and the space of affine Gaudin models satisfying  the condition \rf{cond}.

\chapter{Explicit 2-loop \texorpdfstring{$\b$}{beta}-functions for \texorpdfstring{$G\times G$}{GxG}  and \texorpdfstring{$G\times G/H$}{GxG/H}  models \la{B}}

Here  we shall provide  the explicit formulae for the  2-loop $\beta$-functions  of the  general
$ G\times G$ and $G\times G/H$   models
that were
used in
the Chapter \ref{GGS}.\foot{The formulae derived in this Appendix are also available in the Mathematica file attached to the arXiv submission of the paper \cite{LT2103}.} We will also briefly explain how they were derived.

\section{\texorpdfstring{$G\times G$}{GxG} model \la{GGapp}}
For the $G\times G$ model \rf{par}, let us use the notation
\be \la{not}
\rho_{ij} = \rh_{(ij)} + b_{[ij]}
= \begin{pmatrix}
s & t \\ t& u
\end{pmatrix}
+ \begin{pmatrix}
0& b\\ -b& 0
\end{pmatrix} \ .
\ee
Let the  $2\times 2$ matrix $n_{ij}$ be the
``square root'' of $\rh_{ij}=\rh_{(ij)}$, and let $m_{ij}$ be its inverse,
\be
n_{ik}\, n_{jk} = \rh_{ij} \ , \qquad m_{ik} \, n_{kj} = \delta_{ij} \ , \qquad \qquad n_{ij} = n_{(ij)}, \quad \ m_{ij} = m_{(ij)} \ . \la{not2}
\ee
The target space metric of the \sm \rf{par} is ``diagonalized'' by the vielbein
\unskip\foot{See Appendix \ref{conv} for the group theory conventions.
The generators $T_A$ satisfy $[T_A, T_B] = i {f^C}_{AB} T_C$ and we define $f_{ABC} = \Tr[T_C T_D] {f^D}_{AB}$. For simple groups $G$,  the structure constants satisfy
${f^A}_{BC} {f^B}_{AD} = -2 \cgg \Tr[T_C T_D]$
and ${f^D}_{EA} {f^E}_{GB} {f^G}_{DC} = \cgg f_{ABC}$, where $\cgg$ is the dual Coxeter number of $G$.}
\be \begin{aligned}
&E^{Ai} = n_{ik} \, J^{(k)A}
\ , \qquad A=1,\ldots,\dim{G} \ , \ \ i = 1,2 \ ,  \la{viel} \\
&J^{(k)} \equiv T_A J^{(k)A}=   \big( g^{(k)} \big)^{-1}  d g^{(k)} \ .
\end{aligned} \ee
Then  the coefficients of  the metric $ds^2 = \GG_{Ai, Bj} E^{Ai} E^{Bj}$ and the   3-form   $\H = d \BB =  \tfrac{1}{6} \H_{Ai, Bj, Ck} E^{Ai} \wedge E^{Bj} \wedge E^{Ck}$ are given by\foot{The overall factor of $i$ in  \rf{Hf} simply reflects the fact  that the vielbein \rf{viel} is imaginary. This makes no difference and could be eliminated by just multiplying $E^{Ai} \to i E^{Ai}$.}
\begin{align}
&\GG_{Ai, Bj} = -\ha \, \Tr[T_A T_B] \, \delta_{ij} \  , \la{Gf}\\
&\H_{Ai, Bj, Ck} = \tfrac{i}{2} f_{ABC} \big[ k_l  \, m_{il} \, m_{jl} \, m_{kl} + b_{lp}(  m_{ip} \, m_{jl} \, m_{kl} +  m_{il} \, m_{jp} \, m_{kl} +  m_{il} \, m_{jl} \, m_{kp}) \big]  \ . \la{Hf}
\end{align}
From  Cartan's structure equation $d E^{Ai} + \widehat{\w}^{Ai}{}_{Bj} \wedge  E^{Bj} = T^{Ai}$ with torsion $T^{Ai} = \ha  {\H^{Ai}}_{Bj, Ck} E^{Bj} \wedge E^{Ck}$, we obtain the torsionful spin connection,
\begin{align}
&\widehat{\w}^{Ai}{}_{Bj} = \tfrac{i}{2} {f^A}_{BC} M_{ijk} E^{Ck}\ , \\
&M_{ijk} \equiv  m_{il}m_{jl}n_{kl} -  m_{il}n_{jl}m_{kl} -  n_{il}m_{jl}m_{kl}  \la{me}\\
&\qquad\quad  +  k_l  \, m_{il} \, m_{jl} \, m_{kl} + b_{lp}(  m_{ip} \, m_{jl} \, m_{kl} +  m_{il} \, m_{jp} \, m_{kl} +  m_{il} \, m_{jl} \, m_{kp})  \ . \no
\end{align}
The torsionful Riemann curvature $\widehat{R}^{Ai}{}_{Bj} \equiv \ha \widehat{R}^{Ai}{}_{Bj,Ck,Dl}  E^{Ck} \wedge E^{Dl} = d \widehat{\w}^{Ai}{}_{Bj} + \widehat{\w}^{Ai}{}_{Ck} \wedge \widehat{\w}^{Ck}{}_{Bj} $ is then found in terms of $M_{ijk}$ to be
\be \begin{aligned}
\widehat{R}^{Ai}{}_{Bj,Ck,Dl} ={}& \tfrac{1}{4} \big[  2{f^A}_{BE} {f^E}_{CD} M_{ijp} n_{pq} m_{kq} m_{lq}  \\
&\qquad \qquad + {f^A}_{CE} {f^E}_{BD} M_{ipk}M_{pjl} - {f^A}_{DE} {f^E}_{BC} M_{ipl}M_{pjk} \big] \ . \la{Ri}
\end{aligned}\ee

It is then straightforward to substitute \rf{Hf},\rf{me},\rf{Ri} into the 2-loop $\beta$-functions in the \GB scheme \eqref{BMT}, obtaining  explicit formulae   for
the RG equations    $\ddt \rho_{ij} = \b_{ij}(n_{11}, n_{12}, n_{22}, b, k_1, k_2)$ depending on the components of $n_{ij}$. Using a computer symbolic algebra package (e.g.\ Mathematica) it is easy to rewrite these expressions in terms of the components $s,t,u$ of the ``squared'' coupling  $\rh_{ij} = n_{ik} n_{jk}$ in \rf{not}, with all  the square roots cancelling out
as the Riemann tensor and the $\H$-field must clearly be rational functions of $\rho_{ij}$. We thus obtain the $\beta$-functions
in the  form  given  in \rf{2c},
\be \begin{aligned}
& \qquad\qquad \ddt \rho_{ij}  = \a' \, \b_{ij}^{(1)} + \a'^2\,  \b_{ij}^{(2)} + \ldots \ , \\
\b_{ij}^{(1)} = {}& \ccg (su-t^2)^{-2} \, F^{(4)}_{ij}(s,t,u,b ,k_1,k_2) \ ,  \qquad
\b_{ij}^{(2)} = {\rm c}_{_G}^2 (su-t^2)^{-5} \, F^{(9)}_{ij}(s,t,u,b ,k_1,k_2) \ ,
\end{aligned} \ee
where   the explicit form of the homogeneous polynomials  $F^{(4)}_{ij}$   and $F^{(9)}_{ij}$ is:
\ \\

\bigskip
\hspace{-1.26cm}\includegraphics[scale=0.37]{figures/1loop11}
\medskip

\hspace{-1.26cm}\includegraphics[scale=0.37]{figures/1loop12}
\medskip

\hspace{-1.26cm}\includegraphics[scale=0.37]{figures/1loop21}
\medskip

\hspace{-1.26cm}\includegraphics[scale=0.37]{figures/1loop22}
\medskip

\hspace{-1.3cm}\includegraphics[scale=0.39]{figures/2loop11}
\medskip

\hspace{-1.3cm}\includegraphics[scale=0.39]{figures/2loop12}
\medskip

\hspace{-1.3cm}\includegraphics[scale=0.39]{figures/2loop21}
\medskip

\hspace{-1.3cm}\includegraphics[scale=0.39]{figures/2loop22}

\section{\texorpdfstring{$G\times G/H$}{GxG/H}  model \la{GGHapp}}
The computation of the $\beta$-functions for the gauge invariant $G\times G/H$ model \rf{couH},\rf{bb1} is similar to the $G\times G$  case  above, except that one has to correctly handle the gauge invariance.

We shall again use the notation \rf{not} and \rf{not2},
with  the symmetric ``square root'' of
$\rh_{ij} = \rho_{(ij)}$
being  $n_{ij}$, and its inverse being  $m_{ij}$.
In the computation below, we shall denote certain combinations of $n_{ij}$ and $m_{ij}$ by
\be \begin{aligned}
&\hij  = m_{1i}m_{1j} + m_{2i}m_{2j} \ , \quad & &\nu_{ij} = n_{1i}m_{1j} - n_{2i} m_{2j}  \ , \\
&\Hij = \tfrac{k}{2} \hij  + \tfrac{b}{2}(m_{1i} m_{2j} + m_{2i} m_{1j}) \ , \quad &  &\l_{ij} =  m_{1i}m_{1j} - m_{2i}m_{2j} \ .
\end{aligned} \ee
We shall split  up the generators  $T_A$ of $G$ into $T_\a \in \Lie(H)$ and $T_a\in \Lie(G)/\Lie(H)$ (which are orthogonal with respect to the Killing form).

\renewcommand{\ol}{\overline}
Assuming from the beginning that the matrix $r_{ij}$ satisfies the gauge invariance condition \rf{bb1},\foot{Alternatively, one could obtain the same results by starting with $r_{ij}$ unconstrained, i.e.\ without gauge invariance imposed. One could  first  compute the torsionful Riemann tensor for the target space geometry \rf{couH} with general $r_{ij}$, $\rho_{ij}$. The gauge invariance condition \rf{bb1}  would
then  be  imposed and the resulting Riemann tensor   projected onto the non-degenerate directions of the metric $\GG_{MN}$.}  the target space metric of the \sm \rf{couH} is diagonalized by the
vielbein,\foot{The index  $M$   denotes  all tangent space directions.
In the $G\times G$ case  in \rf{viel}  we had  $M= (Ai)$,   while here
M = (\a, \bar \a, a i)$. Both $G$ and $H$ are assumed to be simple.
\be \begin{aligned}
&E^M = (e^\a, e^{\ol \a} , e^{ai}) = \Big(\sqrt{r}(\B^{(1)\a} -\B^{(2)\a}) \, ,\  \B^{(1)\a} +\B^{(2)\a}\, ,\  n_{ik} P^{(k)a} \Big) \ , \\
& \B^{(k)} \equiv  T_\a \B^{(k)\a} = P_H \big[ \big(g^{(k)}\big)^{-1} d g^{(k)} \big] \ , \qquad P^{(k)} \equiv  T_a P^{(k)a} = P_{G/H} \big[ \big(g^{(k)}\big)^{-1} d g^{(k)} \big]  \ .
\end{aligned}\ee
In this frame, the metric $ds^2 = \GG_{MN} E^{M} E^{N}$ and  the 3-form   $\H = \tfrac{1}{6} \H_{MNP} E^{M} \wedge E^{N} \wedge E^{P}$ have the following non-zero components
\begin{alignat}{3}
&\GG_{\a\b} = -\ha \, \Tr[T_\a T_\b ] \ , \qquad &&\GG_{ai, bj} = -\ha \, \Tr[T_a T_b] \  ,\\
&\H_{\a\b\g} = \tfrac{i}{2} k r^{-3/2} f_{\a\b\g} \ , \qquad &&\H_{\a,bi,cj} = i r^{-1/2} \Hij  f_{\a bc} \ . \la{HH}
\end{alignat}

The $H$ gauge invariance is reflected in the vanishing of all $\ol \a$ components of $\GG_{MN}$ and $\H_{MNP}$, and,  in particular,  the fact that $\GG_{MN}$ is degenerate as a result. One could explicitly fix a gauge, eliminating some target space directions and removing this degeneracy. Instead, we find it more convenient to lift the degeneracy with a small parameter $\eps$ acting as a regulator,\foot{\sloppy
The use of the ``regulator'' $\eps$ is a short-cut for the following gauge-fixing procedure. Fixing an ``axial'' gauge ${i X^\m (\B^{(1)}_\m + \B^{(2)}_\m) = y(\xi) \in \Lie{H}}$ ($\m=1,2$  is the 2d index), the path integral  should be
independent of the choice of the constant 2d vector $X^\m$ and the algebra-valued  function  $y(\xi)$  of the 2d coordinates. Inserting the $\delta$-function  of the gauge fixing condition into the path integral   and then
integrating over $X^\m$ and $y$ with a Gaussian measure, i.e.
$ e ^{-\ha  X^\m X_\m  - \ha \eps \int d^2 \xi  \, \Tr[y y ] }$,  the result  should be  independent of $\eps$ (here we assume
Euclidean 2d signature but the  same  is true  also in Minkowski signature after an  analytic continuation).
Integrating  first over $y$ we get
$ \int d^2 X \exp \big[ {- \ha X^\m X^\n \big( \d_{\m\n} -  \eps \,   \Tr[(\B^{(1)}_\m + \B^{(2)}_\m) (\B^{(1)}_\n + \B^{(2)}_\n)] \big)}\big]$.
Integrating over $X_\m$ restores the 2d Euclidean invariance  and  the  result  to leading order in the $\e\to 0$ limit
is equivalent to simply adding the regulator term $\Delta \L = -\ha \eps \, \Tr[ (\B^{(1)}_\m + \B^{(2)}_\m)^2]$  corresponding to  \rf{reg}. \la{fgs}}
\be
\GG_{\ol \a \ol \b} = -\ha \eps \, \Tr[T_\a T_\b ] \ . \la{reg}
\ee
Computing the torsionful Riemann tensor as in  Section
\ref{2103GGapp},
one finds that it has a finite $\eps\to 0$ limit.
This means that the resulting Riemann tensor for $\eps=0$ is unambiguous (since there are no divergences that could create finite-term ambiguities). Finally,  we project out the $\ol \a$ directions to obtain the non-zero components
\be \la{RR}
\begin{aligned}
&\widehat{R}^\a{}_{\b\d\e} = -\tfrac{1}{4r} f^\a{}_{\b\g} f^\g{}_{\d\e} + \tfrac{k^2}{4r^3} ( f^\a{}_{\d\g} f^\g{}_{\b\e}  - f^\a{}_{\e\g} f^\g{}_{\b\d}  ) \ , \\
&\widehat{R}^\a{}_{\b,dk,el} = (\tfrac{k}{2r} \l_{kl} -\ha p_{kl}) f^\a{}_{\b\g} f^\g{}_{de} + \tfrac{1}{r} A_{kj}A_{lj}( f^\a{}_{dc} f^c{}_{\b e}  - f^\a{}_{ec} f^c{}_{\b d}  ) \ ,\\
&\widehat{R}^\a{}_{b i,\d,el} = -\tfrac{1}{2r} A_{ji} g_{jl} f^\a{}_{bc} f^c{}_{\d e} - \tfrac{k}{2r^2} A_{li} f^\a{}_{\d \g} f^\g{}_{b e}  + \tfrac{1}{r} A_{lj}C_{ji} f^\a{}_{ec} f^c{}_{b \d}  \ , \\
&\widehat{R}^{ai}{}_{bj,dk,el} = (C_{ij} \l_{kl} -\ha \d_{ij} p_{kl}) f^a{}_{b\g} f^\g{}_{de} + \tfrac{1}{r} A_{ki} A_{lj} f^a{}_{d\g} f^\g{}_{b e}  -  \tfrac{1}{r} A_{li} A_{kj} f^a{}_{e\g} f^\g{}_{b d}    \ , \\
&\widehat{R}^{ai}{}_{bk,\d,\e} = - \tfrac{1}{4r} \d_{ik} f^a{}_{b\g} f^\g{}_{\d\e} + \tfrac{1}{r} C_{ij} C_{jk} (f^a{}_{\d c} f^c{}_{b \e}  -  f^a{}_{\e c} f^c{}_{b \d}  )  \ , \\
&\qquad A_{ij} \equiv \tfrac{1}{4}(\nu _{ji}-\nu _{ij}) - \chi_{ij} + \tfrac{r}{2} \l_{ij} \ ,  \qquad C_{ij} \equiv  -\tfrac{1}{4}(\nu_{ji}+\nu_{ij}) + \chi_{ij} + \tfrac{r}{2} \l_{ij} \ . \\
\end{aligned}
\ee

All that remains is to substitute \rf{HH},\rf{RR} into the 2-loop $\beta$-functions \eqref{BMT} in the \GB scheme. The resulting expression is hard to evaluate as it contains hundreds of terms, each proportional to a contraction of the form $(f f f f)_{\a\b}$ or
$(f f f f)_{ab}$
where each $f$ denotes a component $f_{\g\d\e}$ or $f_{\g d e}$ of the structure constants, and indices are contracted using the Killing form $\Tr[T_A T_B]$. One can show, however, that there are only 11 independent  such contractions after accounting for the antisymmetry of the structure constants.
This  allows for
the efficient evaluation of the resulting $\beta$-functions for $r$ and $\rho_{ij}$. These are first obtained depending on $n_{11}, n_{12}, n_{22}$ but, as discussed in subsection \ref{2103GGapp}, they may be rewritten in terms of $s,t,u$ with all square roots cancelling.  As a result, we obtain the 2-loop generalization of the 1-loop $\beta$-functions in \rf{1H}--\rf{fH},
\be
\ddt \lh_p  = \a' \b_{\lh_p}^{(1)}+ \a'^2 \b_{\lh_p}^{(2)} \ , \qquad \qquad \lh_p\equiv (r,s,t,b,u) \ ,
\ee
where $\b_{\lh_p}^{(2)}$ are given by the following expressions:
\ \\

\bigskip
\hspace{-1.3cm}\includegraphics[scale=0.39]{figures/2lr}
\medskip

\hspace{-1.3cm}\includegraphics[scale=0.39]{figures/2ls}
\medskip

\hspace{-1.3cm}\includegraphics[scale=0.39]{figures/2lt}
\medskip

\hspace{-1.3cm}\includegraphics[scale=0.39]{figures/2lb}
\medskip

\hspace{-1.3cm}\includegraphics[scale=0.39]{figures/2lu}
\medskip

\renewcommand{\la}{\label}
\renewcommand{\rf}[1]{(\ref{#1})}

\renewcommand{\la}[1]{\label{2008#1}}
\renewcommand{\rf}[1]{(\ref{2008#1})}

\chapter{Derivation of RG flow from existence of Lax connection for time-dependent sigma models}\la{A}

Here we shall provide some further details of the derivation of the RG flow
from the existence of a Lax representation for the $(\t,\s)$-dependent models in Section \ref{loc}.

\section{Derivation of the equations \rf{alg},\rf{als}}\la{det}

To derive \rf{alg},\rf{als}, we shall ignore terms proportional to the derivatives of the couplings $\del_\pm h_\a$ and match the other terms between the flatness condition of the Lax connection and the equation of motion. The algebraic equations \rf{alg},\rf{als} follow essentially because these `non-derivative'
terms do not change upon the introduction of space-time dependence of the couplings.

Let us consider
the cases of models associated to a group and to a symmetric space
For the group space case, the original equations of motion following from the flatness of Lax in \rf{gl} are the flatness
and the conservation of the current $\mc{A}_\pm$.
In the $(\t,\s)$-dependent model \rf{sth}, these equations may only be modified by $\O(\del h)$ terms
\be
F_{+-}(\mc{A})= \O(\del h) \ , \qquad\qquad \del_+ \mc{A}_- + \del_- \mc{A}_+ = \O(\del h) \ . \la{gc1}
\ee
They should follow from the flatness of the Lax connection ansatz \rf{glt},
\be
F_{+-} (\widehat{L}) = p_+ p_- F_{+-} (\mc{A}) + \big[p_-(1 - p_+) \del_+ \mc{A}_- - p_+(1-p_-) \del_- \mc{A}_+\big]+ \O(\del p) \ . \la{gc2}
\ee
Comparing first
the terms with $\del_\pm \mc{A}$ and $\mc{A}^2$, i.e.\ neglecting the $\O(\del h)$ and $\O(\del p)$ terms in \rf{gc1},\rf{gc2},
the matching of \rf{gc1} and \rf{gc2} then implies
\be
p_-(1 - p_+) = - p_+(1-p_-) \ .
\ee
This leads to the algebraic equation \rf{alg}.

For the symmetric space case, the original equations of motion following from the vanishing of the curvature of \rf{sl} are the flatness condition for $(\mc{B}+\mc{P})_\pm$ and the equations $D^{\mc{B}}_\pm \mc{P}_\mp = 0$, where $D^{\mc{B}}_\m$ is the covariant derivative with respect to the connection $\mc{B}_\m \in \Lie (H)$.
In the $(\t,\s)$-dependent model \rf{sth} these equations are modified, as in \rf{gc1},
\be
F_{+-}(\mc{B+P})= \O(\del h) \ , \qquad D^\mc{B}_\pm \mc{P}_\mp = \O(\del h) \ . \la{sc1}
\ee
These should follow from the flatness of the ansatz for the Lax pair in \rf{slt},
\begin{align}
F_{+-} (\widehat{L}) ={} &q_+ q_- F_{+-}(\mc{B+P}) + (r_- - q_+ q_-) D^\mc{B}_+ \mc{P}_- - (r_+ - q_+ q_-) D_-^\mc{B} \mc{P}_+
\no \\
& \ + q_-(1-q_+)\del_+ \mc{B}_- - q_+(1-q_-)\del_- \mc{B}_+ + (r_+ r_- - q_+ q_-)[\mc{P}_+, \mc{P}_-] \no \\
& \ + r_- (q_+ -1)[ \mc{B}_+, \mc{P}_-] - r_+ (q_- -1) [ \mc{B}_-, \mc{P}_+]
+ \O(\del q,\del r) \ . \la{sc2}
\end{align}
Comparing the terms in \rf{sc1} and \rf{sc2} that contain
$\del \mc{B}$, $\del \mc{P}$, $\mc{B}^2$, $\mc{P}^2$ and $\mc{BP}$, i.e.\ neglecting the $\O(\del h)$ terms in \rf{sc1} and $\O(\del q, \del r)$ terms in \rf{sc2},
we conclude in particular that the coefficients of the extra $\del\mc{B}$, $[\mc{B,P}]$ and $[\mc{P,P}]$ terms in $\rf{sc2}$ must vanish. Assuming $q_\pm,r_\pm$ are not all zero (so that the Lax connection \rf{slt} is not identically zero) the only solution is $q_+ = q_- = 1$ and $r_+ r_- = 1$, i.e.\ the conditions in \rf{als}.

\section{RG flow in PCM case}\la{fdr}

The matching of the $\O(\del h)$ terms then forces the coupling functions to solve the RG flow equations. The general structure of this argument is explained in eqs.\ \rf{pd1}--\rf{RGs2}, but here we shall run through it explicitly for the simplest PCM example.

Together, \rf{glt} and \rf{alg} lead to the following ansatz for the Lax connection (cf.\ \rf{lpl})
\be
\widehat{L}_\pm = \ha \Big(1+[z(\t,\s)]^{\pm 1}\Big) J_\pm \ , \la{laxh}
\ee
whose curvature is ($F_{+-}(J)=0$ since $J=g^{-1} dg$)
\begin{align}
F_{+-}(\widehat{L}) &= \tfrac{1-{z}^2}{	4{z}} (\del_+ J_- + \del_- J_+) - \tfrac{\del_+ z}{2z^2} J_- - \tfrac{\del_- z}{2} J_+
\ . \la{lah}
\end{align}
The equation of motion for the PCM with coupling $h(\t,\s)$ is (generalizing \rf{pe1})
\be
h (\del_+ J_- + \del_- J_+) + \del_+ h J_- + \del_- h J_+ = 0 \ . \la{eoh}
\ee
For this to be equivalent to the vanishing of the curvature \rf{lah}
the ratios
of the different
coefficients should match.
This leads to the equations of the form \rf{pd1}, i.e.
\be\la{a9}
\del_+ z = - \tfrac{z}{2} (1-z)^2 \,\tfrac{\del_+ h}{h} \ , \qquad \del_- z = - \tfrac{1}{2z} (1-z)^2 \,\tfrac{\del_- h}{h}\ .
\ee
The consistency condition $\del_+(\del_- z) - \del_- (\del_+ z) = 0$ gives
\be
\tfrac{(1-z^2)^2}{2h z} \ \del_+ \del_- h = 0 \ .
\ee
It is remarkable that the $z$ dependence has totally factored out
(a term proportional to $\del_+ h \del_- h$ is absent here due to a special cancellation).
Excluding the trivial cases $z(\t,\s) =\pm 1 $, which would not encode the correct equation of motion, we find that the Lax connection \rf{laxh} only matches the correct equation of motion if $\del_+ \del_- h =0$, i.e.\ if
$h = f^+ (\xi^+) + f^-(\xi^-)$.
Any such solution is related to the PCM's 1-loop RG flow, $h=c\, \t$, by a 2d conformal transformation.

\section{RG flow for theories with multiple couplings}\la{mr}

In Section \ref{2008s3} we explained the derivation of the RG flow
focussing on the case with only one coupling (see eqs.\ \rf{pd1}--\rf{RGs2}).
The same conclusion also holds for the group space $\eta$-model with two couplings $h$ and $\k$ (see Table \ref{2008tab1}), and more generally is expected to be true for multi-coupling theories.

There are multiple independent structures in the equation of motion (for the $\eta$-model these involve different
powers of the $R$-matrix) and correspondingly in the flatness of the Lax connection. Matching
the coefficients of these structures in the $\eta$-model case yields \textit{two} pairs of equations (cf.\ \rf{pd1}),
\begin{align}
&\del_\t z = U_\t(z;h,\k) \ , \qquad \del_\s z = U_\s(z;h,\k) \ , \la{sy1}\\
&\del_\t z = V_\t(z;h,\k) \ , \qquad \del_\s z = V_\s(z;h,\k) \ . \la{sy2}
\end{align}
In general, for an $N$-coupling theory we would expect to find $N$ pairs of equations.

As a system of equations for $z$, \rf{sy1},\rf{sy2} is clearly overdetermined. In two combinations of these equations the $z$ dependence cancels to give relations between $h(\t,\s)$ and $\k(\t,\s)$
\begin{align} \la{a13}
\del_\t( \k^{-1} h ) = \del_\s ( \k^{-1} h ) = 0 \ .
\end{align}
Eq.\ \rf{a13}
implies that $\nu \equiv  \frac{h}{4\k} $ is a constant; this coincides precisely with the 1-loop RG invariant
of the $\eta$-model (see Table \ref{2008tab1}).
For an $N$-coupling theory, we may expect to obtain $(N-1)$ RG invariants $\nu_r$ in this way.

the system \rf{sy1},\rf{sy2} reduces to just two equations --- effectively returning to the single-coupling case of equations \rf{pd1}. Again, the consistency condition for the two remaining equations takes the remarkable form \rf{2w}, where the $\b$-function $\b(h) \equiv \beta^h (h, \nu )$ is understood as a function of the coupling $h$ and the RG invariant. As in eqs.\ \rf{cfs},\rf{RGs2} it then follows (modulo a conformal transformation) that $h(\t,\s)$ depends only on $\t$ and follows the 1-loop RG flow.
The same should generalize to the $N$-coupling case with
$(N-1)$ independent RG invariants $\nu_r$ that can be chosen as constants.

\chapter{On non-local charges in time-dependent sigma models

In Section \ref{2008s4} we discussed the construction of non-local charges in the time-dependent PCM.
Here we shall comment on the other time-dependent models in Table \ref{2008tab1},
and, in particular, on the symmetric space $\l$-model.

The construction of the conserved monodromy matrix satisfying \rf{lp} works similarly, although for the models built on symmetric spaces (symmetric space $\s$-model and symmetric space $\l$-model), it is only the eigenvalues of the monodromy matrix that are conserved (because the boundary value of $L_\t$ does not vanish). A sufficient boundary condition in all cases is that $g(\t,\s) \to g_0$ at spatial infinity and that $(g-g_0)$ decays sufficiently fast so
that the monodromy converges at spatial infinity. As in the PCM example it is hard to evaluate the conserved charges explicitly and thus to verify that they are infinite in number (i.e.\ depend non-trivially on the spectral parameter $w$).

Except for the symmetric space $\l$-model, all the other models have global symmetries.
As for the PCM in \rf{noet}, the associated
charges can be obtained by expanding the monodromy around $w=\infty$.\foot{This applies also for the symmetric space $\s$-model after applying the gauge transformation $L_\pm \to g L_\pm g^{-1} - \del_\pm g g^{-1}$ to the Lax connection $L = J^H_\pm + z^{\pm 1} J_\pm^{G/H}$, obtaining the alternate Lax connection $L_\pm = \ha (1-z^{\pm 1}) (-2 g J_\pm^{G/H} g^{-1} )$ of the `group space' form \rf{gl}, instead of \rf{sl} (modulo sign reversal of $z$).}
In the remainder of this appendix, we shall consider
the trivial reduction of the time-dependent symmetric space $\l$-model, which has no manifest global symmetries.
In general, the eigenvalues of the monodromy matrix would be conserved on an infinite spatial line;
however, in the ``trivial'' reduction where the fields depend only on time (see Section \ref{2008s4}), the monodromy matrix does not converge at spatial infinity since $L_\s$ does not vanish there.
Below we will try
to shed some light on this issue.

Let us first recall what happens for geodesics in the usual time-independent case, where the Lax connection is $L_\pm = A_\pm^H + z^{\pm 1} \tfrac{1}{\sqrt{\l}} A_\pm^{G/H}$. In the trivial reduction we have $A_\pm = A_\pm (g(\t))$ so
the periodicity condition \rf{pe} is satisfied and the eigenvalues of the monodromy conserved even on a finite interval of length $a $. The path-ordered exponential trivializes to give $\M = \exp{ (a \, L_\s )}$ and hence, equivalently,
the eigenvalues of $L_\s = \ha (A_+^H - A_-^H + \tfrac{z}{\sqrt{\l}} A_+^{G/H} - \tfrac{z^{-1}}{\sqrt{\l}} A_-^{G/H})$ are conserved. For example,
let us consider the simplest $SU(2) \ov U(1)$ $\l$-model (with the  $U(1)$ subgroup generated by $\s_2$), parametrized after gauge fixing as (cf.\ eqs.\ \eqref{42},\eqref{43})
\begin{align}
&g = e^{i \a \s_3} e^{i \b \s_2} \ , \qquad\qquad \cos{\a} = \sqrt{p^2+q^2} \ , \qquad \ \tan{\b}=\frac{p}{q} \ , \\
&\L = \frac{k}{1-\l^2} \frac{(1-\l)^2 (\del p)^2 + (1+\l)^2 (\del q)^2}{1-p^2-q^2} \ . \la{2dl}
\end{align}
The trivial 1d reduction, or geodesic motion, is described by the Lagrangian ($\dot{} \equiv \del_\t$)
\be
L = \frac{k}{1-\l^2} \frac{(1-\l)^2 \dot{p}^2 + (1+\l)^2 \dot{q}^2}{1-p^2-q^2} \ . \la{t1d}
\ee
There is only one independent eigenvalue $l$ of $L_\s$ since it is a traceless $2\times 2$ matrix. Its expansion around, e.g.,
$z=1$ gives (at least) two independent conserved charges for the geodesic motion
\begin{align}
&\dot{Q}_1 = \dot{Q}_2 = 0 \ , \qquad \qquad l = \tfrac{1}{2(\l^2-1)}\sqrt{Q_1} + (z-1)^2 \tfrac{\l}{\l^2-1} \frac{Q_2}{\sqrt{Q_1}} + \ldots \ ,\\
&\quad Q_1 = \frac{- (\l-1)^4q^2\dot{p}^2 + 2(\l^2-1)^2 p q \dot{p}\dot{q}  + (1+\l)^2[4\l(q^2-1)-(\l-1)^2p^2]\dot{q}^2 }{(1-p^2-q^2)^2} \ , \la{q1} \\
&\quad Q_2 = \frac{ (1-\l)^2 \dot{p}^2+ (1+\l)^2\dot{q}^2 }{1-p^2-q^2} \ , \la{q2}
\end{align}
with $Q_2$ proportional to the Hamiltonian.

Returning to the time-dependent case, the monodromy is not defined on an infinite line, and on any finite interval the periodicity condition is not satisfied due to the explicit $\s$ dependence of the Lax connection for generic values of $w$. However, at certain special values of $w$ satisfying $\exp{(\tfrac{c}{k}w)}=\pm \infty$, the $\s$ dependence disappears to give flat connections
\begin{align}
&[\del_+ + A_+^H + \tfrac{1}{\l(\t)} A_+^{G/H} , \ \del_- + A_-^H + A_-^{G/H} ]=0 \ , \la{f1} \\
&[\del_+ + A_+^H + A_+^{G/H} , \ \del_- + A_-^H + \tfrac{1}{\l(\t)} A_-^{G/H} ] = 0 \ , \qquad \qquad \l(\t) = \exp{(-\tfrac{c}{k} \t)} \ , \la{f2}
\end{align}
generalizing the same expressions from the time-independent case ($\l(\t) \to \l$). At these values the periodicity condition \textit{is} satisfied on any finite interval. The monodromy trivializes as in the time-independent case to give $\M = \exp{(a\, L_\s)}$, so the eigenvalues of $L_\s$ are again conserved.

In fact, the two flat connections \rf{f1},\rf{f2} are related by a gauge transformation, so their conserved charges are the same. Hence, the maximum number of independent charges obtained from the flat connections \rf{f1},\rf{f2} is $r = {\rm rank}(G)$
for a symmetric space $G/H$. This number is generally less than the number of fields, $\dim{G/H} =\dim{G}-\dim{H}$ (e.g.\ for $SO(n+1)/SO(n)$ we get $r= n-1 < \dim{SO(n+1)/SO(n)} = n$), so this is not sufficient for integrability.

For example, in the ${SU(2) \ov U(1)} \sim {SO(3) \ov SO(2)}$ case \rf{2dl}, where there are 2 fields, we only obtain $r=1$ conserved charge,
\be
Q = \frac{ [\l(\t)-1]^4(p^2-1)\dot{p}^2 + 2[\l(\t)^2-1]^2 p q \dot{p}\dot{q}+ [\l(\t)+1]^4 (q^2-1) \dot{q}^2 }{[\l(\t)^2-1]^2(1-p^2-q^2)^2}  \ , \la{cnt}
\ee
for the trivial 1d reduction, which is  a time-dependent generalization of \rf{t1d},
\be
L = \frac{k}{1-\l^2} \frac{(1-\l(\t))^2 \dot{p}^2 + (1+\l(\t))^2 \dot{q}^2}{1-p^2-q^2} \ ,\quad  \qquad \l(\t) = \exp{(-\tfrac{c}{k} \t)}  \ .
\ee

In the time-independent limit $\l(\t) \to \l$ (obtained, e.g.,
by shifting $\t \to - \tfrac{k}{c}\log{\l} + b \t$ and taking $b\to 0$), this charge becomes a particular combination of the charges \rf{q1},\rf{q2} in the time-independent theory,
\be
Q \to Q_1 - (\l-1)^2 Q_2 \ .
\ee
The presence of the charge \rf{cnt} in the time-dependent 1d theory is a non-trivial property
However, at least upon restricting consideration to special values of $w$, we have not found
enough charges for the integrability of the geodesics.
Since the construction of charges is subtle, depending on the value of $w$,  boundary conditions and the choice of the spatial domain
(the periodicity condition \rf{pe} must be satisfied and, if the interval is infinite, the monodromy must converge at infinity),
it is possible that the monodromy constructed on some special domain would yield further conserved charges,
but this remains to be clarified.

\chapter{Time-dependent 1d harmonic oscillator and conserved charge}\la{C}

In Section \ref{2008s4} we came across a particular time-dependent, linear, 1d model \rf{lt}.
Starting with a general time-dependent linear 1d action\foot{We do not include the term $f(\t) \theta \dot{\theta}$
as it can be put in the form $k(\t) \theta^2$ by adding a total derivative.}
\be
S = \int d\t \, \big[ h(\t)\, \dot{\theta}^2 - k(\t)\, \theta^2 \big] \ , \qquad \dot{\theta} \equiv \del_\t \theta \ ,
\la{orig}
\ee
one may redefine $\t$ as $\t \to t(\t)$, $\dot{t}(\t) = h^{-1}(\t)$ to put all time dependence in the harmonic potential term
\be
S = \int dt \, \big[ {\theta'}^2 - m^2(t)\, \theta^2 ] \ , \qquad \qquad m^2(t) = k(\t(t)) h(\t(t)) \ , \ \ \ \theta ' \equiv \del_t \theta \ .
\ee
The corresponding equation of motion is
\be
\theta'' +m^2(t)\, \theta = 0 \ . \la{eoml}
\ee
It is easy to see that for a given function $\theta_0(t)$, the quantity
\be
Q = \theta_0 \, \theta' - {{\theta}'_0} \, \theta \ ,
\ee
is conserved on-shell if and only if $ \theta_0$ is a particular solution of the equation of motion \rf{eoml}.
such a conserved charge provides a first integral for \rf{eoml},
\be
Q = {\theta_0} \, \theta' - \theta'_0 \, \theta = C_1 = \text{const} \ \
\quad \to \quad (\tfrac{\theta}{\theta_0})' = \tfrac{C_1}{\theta_0^2} \ . \la{exact}
\ee
Integrating this first-order equation
yields the general solution of \rf{eoml} ($C_2=\const$),
\be\la{d6}
\theta(t) = C_1\, \theta_0(t) \int \tfrac{dt }{\theta_0^2(t)} +\ C_2\, \theta_0(t) \ ,
\ee
with $\theta=\theta_0$ being, of course, a special case.
Thus \rf{eoml} is solvable if a particular solution $\theta_0$ can be constructed explicitly.

Changing back to the original parametrization \rf{orig} ($t \to \t$), the conserved charge \rf{exact}
takes the form
\be
Q = h(\t) \big[ {\theta_0} \, \dot{\theta} - \dot{\theta}_0 \, \theta \big] \ , \la{lic}
\ee
where $\theta_0=\theta_0(\t)$ is a particular solution of \rf{orig}, while \rf{d6} becomes
\be
\theta(\t) = C_1\, \theta_0(\t) \int \tfrac{d\t}{h(\t) \, \theta_0^2(\t)} + C_2\, \theta_0(\t) \ . \la{gena}
\ee
The linearized theory \rf{lt} corresponds to \rf{orig} with $h(\t) = k(\t) = \t$.
The conserved charge in \rf{ql},\rf{ql2} is indeed of the form \rf{lic}.
From the monodromy matrix one finds that
$Q = \t [ \g(\t) \, \theta + \bw(\t) \, \dot{\theta} ]$ where
\be
\hspace{-0.3cm}\g(\t) = \int_{-\infty}^{+\infty} d\s \,e^{-im \s} \, \tfrac{m (s_+ s_- -w-\s)}{2 \t s_+ s_- } ,\ \ \  \bw(\t) = \int_{-\infty}^{+\infty} d\s\, \, e^{-im \s}\, \tfrac{ i}{2 s_+ s_- } , \ \ \ s_\pm \equiv \sqrt{w+\s\pm \t} \ . \la{ints}
\ee
Let us show that $\g = - \dot \bw$, as required to match \rf{lic}. Indeed, the term proportional to $\dot{\theta}$ in the leading $\O(\e)$
expansion of the flatness equation $\del_\t L_\s - \del_\s L_\t + [L_\t,L_\s] = 0$ tells us that the integrands in \rf{ints}
satisfy $
\del_\t [ \tfrac{e^{-im \s} i}{2 s_+ s_- } ] + \tfrac{e^{-im \s} m (s_+ s_- -w-\s)}{2 \t s_+ s_- } = \del_\s\, [\tfrac{e^{-im \s}\, i (s_+ s_- -w-\s)}{2 \t s_+ s_- } ] $.
Integrating over $\s$ and noting that $ \tfrac{e^{-im \s}\, i (s_+ s_- -w-\s)}{2 \t s_+ s_- } $ vanishes at $\s=\pm \infty$,\foot{This follows from the periodicity condition \rf{pe} since it is the coefficient of $\dot{\theta}$ in a component of the leading $\O(\e)$ term in $L_\t$.}
we conclude that $\gamma + \dot\bw=0$.

Since the charge in \rf{ql} is conserved on-shell (from the monodromy matrix construction),
it follows that $\bw(\t) $ is a particular solution, and thus the general solution \rf{gena} is in this case given by
\be\la{genn}
\theta = C_1\, \bw(\t) \int \tfrac{d\t}{\t\, \bw^2(\t)} + C_2\, \bw (\t)\ .
\ee

\chapter{Lax connection in time-dependent sine-Gordon model}\la{sing}

As discussed in Section \ref{2008s6}, the sine-Gordon model,
\be
\L = \tfrac{1}{g^2} \big[ \ha \del_+ x \del_- x + m^2 \cos{x}\big] \ , \la{sgma}
\ee
displays the same pattern as the $\s$-models considered above: upon promoting the couplings $(g,m)$ to functions of 2d time $\t$, the Lax connection naturally generalizes to the resulting time-dependent model only if the time dependence
is given (up to rescalings of time) by the 1-loop RG flow of the original model,
\be m^2(\t) = e^{ \b(g)\, \t} m_0^2 \ , \ \ \ \qquad g(\t) = g \ , \ \ \ \ \qquad \b(g) = - 2 + g^2 \ . \la{52a} \ee
Below we shall justify this claim in more detail.

As was noted in section \ref{2008s6}, the time-dependent theory obtained from \rf{52a}
\be
\widehat{\L} = \tfrac{1}{g^2} \big[ \ha \del_+ x \del_- x + e^{\b(g)\, \t}\, m^2_0\, \cos{x}\big]\ , \la{53a}
\ee
is clearly integrable
since the explicit $\t$-dependence in \rf{53a} can
be removed by a 2d conformal transformation getting back to \rf{sgma}.
Indeed, starting with the Lax pair \cite{Lax:1968fm} for the original
sine-Gordon model \rf{sgma} ($\s_i$ are Pauli matrices)
\begin{align}
\hspace{-2cm} L_\pm = \pm \tfrac{i}{4} \del_\pm x \ \s_3 + \tfrac{i}{2} z^{\pm 1} \, m \cos{\tfrac{x}{2}} \ \s_1 \pm \tfrac{i}{2} z^{\pm 1} \, m \sin{\tfrac{x}{2}} \ \s_2 \ , \la{sgla}
\end{align}
and applying a conformal transformation $\xi^\pm \to f^\pm (\xi^\pm)$, one obtains a Lax connection
for the time-dependent theory \rf{53a},\foot{Note that, as for the $\s$-models discussed in Chapter \ref{loc} above, the dependence on the spectral parameter $z$ in \rf{larga} is again correlated with
a constant shift of $\s$.}
\be
\widehat{L}_\pm =\pm \tfrac{i}{4} \del_\pm x \ \s_3 + \tfrac{i}{2} z^{\pm 1} \, e^{\b(g)\, \xi^\pm} \, m_0 \cos{\tfrac{x}{2}} \ \s_1 \pm \tfrac{i}{2} z^{\pm 1} \,e^{\b(g) \, \xi^\pm} \, m_0 \sin{\tfrac{x}{2}} \ \s_2 \ . \la{larga}
\ee
Note that this Lax connection follows the same ansatz \rf{lac2} as in the \sm case:
it is obtained from the original Lax connection \rf{sgla} by the replacements
$z \to \widehat{z}= e^{\frac12 \b(g)\s}z$ and $m_0 \to m(\t) = e^{\frac12 \b(g)\t} m_0$.

Conversely, suppose we replace the couplings $(g,m)$ in \rf{sgma}
with general functions of $(\t,\s)$ and then demand that the resulting theory (cf.\ \rf{sth})
\be
\widehat{\L} = \tfrac{1}{g^2(\t,\s)} \big[ \ha \del_+ x \del_- x + m^2(\t,\s) \, \cos{x}\big] \la{54a}
\ee
admits a Lax representation. Motivated by \rf{sgla}, we shall assume the following ansatz for the Lax connection (cf.\ \rf{glt},\rf{slt})
\begin{align}
&\widehat{L}_\pm = f_{\pm}(\t,\s)\ \tfrac{i}{4} \del_\pm x \, \s_3 + \, v_\pm (\t,\s) \ \tfrac{i}{2}\cos{\tfrac{x}{2}} \, \s_1 + w_\pm (\t,\s) \ \tfrac{i}{2}\sin{\tfrac{x}{2}} \, \s_2\ . \la{sga}
\end{align}
matching coefficients of various terms in the zero-curvature condition for \rf{sga} and in
the equation of motion corresponding to \rf{54a} leads to the following constraints on the coefficient functions in \rf{sga}
and the coupling functions,
\begin{align}
&f_\pm = \pm 1 \ , \qquad\qquad w_\pm = \pm v_\pm \ , \qquad\qquad \del_\mp v_\pm = 0 \ , \la{fe1} \\
&g(\t,\s) = g= \const \ , \qquad\qquad v_+ v_- = m^2(\t,\s) \ . \la{fe2}
\end{align}
It follows from \rf{fe1},\rf{fe2} that $v_\pm = \pm w_\pm = v_\pm(\xi^\pm)$, \ $m^2(\t,\s)=v_+(\xi^+)\, v_-(\xi^-)$.
Finally, applying a conformal transformation to \rf{sga},
we may set, e.g., $v_\pm(\xi^\pm) \to z^{\pm 1} \, e^{\b(g) \, \xi^\pm} \, m_0$, $m^2(\t,\s) \to e^{\b(g) \, \t} \, m_0^2$,
thus bringing the Lagrangian \rf{54a} and the Lax connection \rf{sga} to the form \rf{53a} and \rf{larga}
where the time dependence is given by the 1-loop RG flow \rf{52a} of the standard sine-Gordon model.

\renewcommand{\la}{\label}
\renewcommand{\rf}[1]{(\ref{#1})}

\end{appendices}

\newpage
\input{refs.tex}

\end{document}


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