Integrability and RG flow in 2d sigma models

Thesis for the degree of Doctor of Philosophy  in Physics

Nat Levine  Now at École Normale Supérieure de Paris and Institut Philippe Meyer.

Blackett Laboratory, Imperial College London,

London SW7 2AZ, UK

Motivated by the search for solvable string theories, we consider the problem of classifying the integrable bosonic 2d $\s$-models.
We  include  non-conformal $\s$-models, which have historically been a good arena for  discovering integrable models
that were later generalized to Weyl-invariant ones. General  $\s$-models  feature a quantum RG flow, given by a `generalized Ricci flow' of the target-space geometry.

This thesis is based on the conjecture that  integrable $\s$-models are renormalizable, or stable under the  RG flow.
It is widely understood that classically integrable theories are stable at the leading  1-loop order with only a few parameters  running.

Here we  address what happens at higher-loop orders. We find that integrable $\s$-models generally remain RG-stable at higher-loops provided they receive a particular choice of finite counterterms, or quantum $(\a')$ corrections to the target-space geometry. Thought to be preserving integrability at the quantum level, these corrections are analogous to those required for higher-loop conformal invariance of gauged Wess-Zumino-Witten models.

We explicitly construct the quantum corrections restoring higher-loop renormalizability for examples of integrable $\eta$- and $\lambda$-deformed $\s$-models. We then consider the integrable $G\times G$ and $G\times G/H$ models and also construct a new class of integrable
$G\times G /H$ models with .

We also reformulate the $\l$-models as $\s$-models on a “tripled” $G\times G\times G$ configuration space, where they become  automatically renormalizable  due to the decoupling of some fields.  In the limit when they become non-abelian dual (NAD) models, this suggests that the corresponding `interpolating models' for NAD are also renormalizable, with 2-loop $\b$-functions matching the  group/symmetric space $$-models.

\medskip
We then present a new and different link between integrability and the RG flow in the context of  $$-models with `local couplings' depending explicitly on 2d time. Such models are naturally obtained in the light-cone gauge
in string theory,
pointing to the possibility of a large, new class of solvable string models.

\end{abstract}

\newpage

\renewcommand{\abstractname}{Statement of originality}
\begin{abstract}
\addtocounter{page}{2}
\thispagestyle{plain}
\bigskip \noindent
I declare that, aside from properly acknowledged or referenced contributions of others, this thesis is my own work.

\bigskip \bigskip \noindent
This thesis is closely based on the following research papers. These papers were written as collaborations with co-authors who contributed equally to the work.

\bigskip \bigskip \noindent
\begin{minipage}[t]{1cm}\cite{HLTS}
\end{minipage}
\begin{minipage}[t]{\textwidth-1cm}
B.~Hoare, N.~Levine and A.~A.~Tseytlin,

\noindent
``On the massless tree-level S-matrix in 2d sigma models,''

\noindent
\doilink{J. Phys. A \textbf{52}, no.14, 144005 (2019)}{
\arxivlink{1812.02549}.
\end{minipage}
%16 citations counted in INSPIRE as of 11 Jul 2021

\bigskip \noindent
\begin{minipage}[t]{1cm}\cite{HLT1}
\end{minipage}
\begin{minipage}[t]{\textwidth-1cm}
B.~Hoare, N.~Levine and A.~A.~Tseytlin,

\noindent ``Integrable 2d sigma models: quantum corrections to geometry from RG flow,''

\noindent
\doilink{Nucl. Phys. B \textbf{949}, 114798 (2019)}{10.1016/j.nuclphysb.2019.114798}
\arxivlink{1907.04737}.
\end{minipage}

\bigskip \noindent
\begin{minipage}[t]{1cm}\cite{HLT2}
\end{minipage}
\begin{minipage}[t]{\textwidth-1cm}
B.~Hoare, N.~Levine and A.~A.~Tseytlin,

\noindent
``Integrable sigma models and 2-loop RG flow,''

\noindent
\doilink{JHEP {\bf 1912}, 146 (2019)}{doi:10.1007/JHEP12(2019)146}
\arxivlink{1910.00397}.
\end{minipage}

\bigskip \noindent
\begin{minipage}[t]{1cm}\cite{HLT2008}
\end{minipage}
\begin{minipage}[t]{\textwidth-1cm}
B.~Hoare, N.~Levine and A.~A.~Tseytlin,

\noindent
``Sigma models with local couplings: a new integrability -- RG flow connection,''

\noindent
\doilink{JHEP \textbf{11}, 020 (2020)}{10.1007/JHEP11(2020)020}
\arxivlink{2008.01112}.
\end{minipage}

\bigskip \noindent
\begin{minipage}[t]{1cm}\cite{LT2103}
\end{minipage}
\begin{minipage}[t]{\textwidth-1cm}
N.~Levine and A.~A.~Tseytlin,

\noindent
``Integrability vs. RG flow in $G \times G$ and $G \times G /H$ sigma models,''

\noindent
\doilink{JHEP \textbf{05}, 076 (2021)}{
\arxivlink{2103.10513}.
\end{minipage}

\end{abstract}

\renewcommand{\abstractname}{Acknowledgements}
\begin{abstract}
\addtocounter{page}{3}
\thispagestyle{plain}

\bigskip \noindent
I am very grateful to my advisor Arkady Tseytlin, who has taught me so much during my PhD. Thank you for your contagious interest and enthusiasm  for the subject, and for always being happy to share your knowledge. It has been a great  inspiration and  pleasure  working   with you.

\bigskip
Thank you to Ben Hoare for our enjoyable collaborations and  countless interesting discussions and explanations. Your advice and support has been extremely valuable to me.

\bigskip
I would also like to thank Tim Adamo, Lorenz Eberhardt, Falk Hassler, Sylvain Lacroix, Konstantinos Sfetsos, Alessandro Torrielli, Stijn van Tongeren, Beno\^{i}t Vicedo and Linus Wulff for interesting discussions, advice and guidance during my PhD.

\bigskip
I am grateful to EPSRC for supporting my PhD through the grant EP/N509486/1.

\bigskip
I thank my partner, family and friends for many great times during the last four years in London. A special thanks goes to my colleagues and officemates at Imperial for the amusing lunchtime discussions, `PhD meetings' and half-hearted attempts to prove the QCD mass gap. In 2020 we very sadly  lost our friend and  colleague Ed Tasker, and he will will always be dearly missed.

\end{abstract}

\newpage

\renewcommand{\abstractname}{Copyright declaration}
\begin{abstract}
\addtocounter{page}{4}
\thispagestyle{plain}
\bigskip \noindent

The copyright of this thesis rests with the author. Unless otherwise indicated, its contents are licensed under a Creative Commons Attribution 4.0 International Licence (CC BY).

Under this licence, you may copy and redistribute the material in any medium or format for both commercial and non-commercial purposes. You may also create and distribute modified versions of the work. This on the condition that you credit the author.

When reusing or sharing this work, ensure you make the licence terms clear to others by naming the licence and linking to the licence text. Where a work has been adapted, you should indicate that the work has been changed and describe those changes.

Please seek permission from the copyright holder for uses of this work that are not included in this licence or permitted under UK Copyright Law.

\end{abstract}

\newpage

\addtocounter{page}{5}
\tableofcontents

\

\bigskip \bigskip\bigskip  \bigskip\bigskip \bigskip\bigskip\bigskip\bigskip\bigskip \bigskip\bigskip  \bigskip\bigskip
{\bf \large \noindent List of tables}

\bigskip
{\bf \noindent Table 1:} Examples of integrable \sms and their time-dependent generalizations \hfill {\bf \pageref{2008tab1}}

\setcounter{section}{0}

\chapter{Introduction}
\section{Background}

The AdS/CFT correspondence is a remarkable set of dualities between
string theories in Anti-de Sitter  space AdS$_d+1$
conformal field theories CFT$_d$ in one fewer dimension \cite{Maldacena:1997re,Witten:1998qj,Gubser:1998bc}. In the Poincar\'e coordinates for AdS,
\be
ds^2 = \frac{R^2}{z^2} \big(dz^2 + dx^\m dx_\m \big) \ , \qquad \m = 1,\ldots, d \ , \la{poinc}
\ee
one may picture the CFT as existing at the ``conformal boundary'' $z=0$, where the metric formally diverges but is conformally equivalent to the flat $d$-dimensional one. Gauge-string dualities of this type are often called ``holographic'' as they conjecture a mapping  between the degrees of freedom on the boundary ($z=0$) and  those in the ``bulk'', or  interior of the AdS space. A natural first check  is  the matching of internal symmetries: indeed the isometry group $SO(2,d-1)$ of AdS$_d+1$ coincides with the conformal group of $d$-dimensional flat space.

Such a correspondence has several immediately interesting and surprising properties. First, the  $(d+1)$th dimension (the $z$ direction in \rf{poinc}) seems to  emerge  from a $d$-dimensional CFT description. Second, the string theory is gravitational while the CFT is not. Thus it is possible to reformulate a  complicated quantum theory of gravity in terms of a  simpler and better understood CFT in flat space.

In the context of string theory, the  AdS space must be embedded in a string background. A canonical example discussed in Maldacena's original proposal \cite{Maldacena:1997re} is the type IIB background obtained in the low energy limit from a stack of $N$ coincident D3-branes. The conjecture is that the low energy theory describing open strings ending on the D-branes ---  $SU(N)$ $𝒩=4$ super Yang-Mills theory in 4d  --- is dual to the theory of closed strings in the near horizon region --- type IIB strings on \adss. The theories' parameters are related by
\be
\frac{R^2}{\a'} = \sqrt{g_{\rm YM}^2 N} \ , \qquad g_s = \frac{g_{\rm YM}^2}{4\pi} \ ,
\ee
where $R$ is the AdS radius in \rf{poinc}, $'=l_s^2$ the squared string length, $g_YM$ the Yang-Mills coupling and $N$ the rank of the Yang-Mills gauge group.

\sloppy
Among the many examples of the AdS/CFT correspondence, the \adss case is notable because it is \textit{integrable}\foot{Here we give a lightning summary of integrability in the AdS/CFT correspondence, focussing on \mbox{\adss} (and missing many details and important references). For a comprehensive introduction, see the reviews \cite{adscftsol,Arutyunov:2009ga,vanTongeren:2013gva,Bombardelli:2016rwb}.}$^,$\foot{Another well-studied, integrable, holographic string background  is the type IIA background on $AdS_4 \times \mathbb{C}\mathbf{P}^3$, dual to 3d $\mathcal{N}=6$ Chern-Simons theory \cite{ABJM}; see, e.g., the review \cite{Klose:2010ki}.} in the free string ($g_s = 0$) limit, or equivalently in the planar limit ($N=∞$ with $ł= g_YM^2 N$ fixed) of the gauge theory. Here ``integrability'' is understood in the same sense of familiar integrable systems like the KdV or sine-Gordon model,  meaning aspects of the theory can be solved exactly without recourse to perturbation theory.

On the string side (strong 't Hooft coupling $ł$),  the Green-Schwarz action for strings   in the \adss background takes the Metsaev-Tseytlin form \cite{Metsaev:1998it} equivalent to a \sm on the semi-symmetric space $PSU(2,2 | 4)/[SO(1,4)×SO(5)]$ with a particular Wess-Zumino term. It is a general fact that such models are classically integrable \cite{Bena:2003wd,Das:2004hy,Young:2005jv,Magro:2008dv,Vicedo:2009sn} in the sense of admitting a flat Lax connection and associated conserved charges (see Section \ref{ci} below). In spite of classic issues preventing a direct integrable quantization due to the non-ultralocality of the classical Poisson structure \cite{maillet,Magro:2008dv}, one may just \textit{assume} the quantum theory to be integrable and bootstrap an exact integrable S-matrix  for the fully gauge-fixed theory based on  symmetries and consistency conditions (e.g. in an expansion around a BMN-type \cite{BMN} geodesic, fixing  the `uniform light-cone' gauge; see the review \cite{Arutyunov:2009ga}). This S-matrix describes scattering on a 2d plane, but the finite size (closed string) spectrum may be extracted by some Bethe ansatz equations, or more properly accounting for the ``wrapping corrections'' by double Wick rotating \cite{ZTBA} and using the Thermodynamic Bethe Ansatz (TBA) \cite{YTBA}. The TBA  (nonlinear integral) equations were reformulated as the simpler Quantum Spectral Curve \cite{Gromov:2013pga}, which is the most modern approach to computing the exact spectrum (see also the review \cite{Levkovich-Maslyuk:2019awk}).

On the gauge side (weak coupling), the structure and origin of integrability is more mysterious. It was observed that the mixing matrix for anomalous dimensions of single-trace local operators coincides with the Hamiltonian of an integrable spin chain \cite{Minahan:2002ve}. This conjures an image  of fields inside the trace corresponding to sites of a spin chain, with their R-charges corresponding to the spins. Including more and more loop orders, the corresponding spin chain was found to remain integrable, but with longer-range interactions at each order \cite{BeisertHL}.  Assuming integrability to all orders, the exact spin-chain S-matrix \cite{BeisertS} and  asymptotic Bethe ansatz equations \cite{BeisertBA} (for long operators or long spin chains) can be obtained, matching  results from the string side. The relation between integrability on the gauge and string sides can be understood qualitatively in terms effective Landau-Lifshitz actions for coherent states, with the string effection action being the direct continuum limit of the spin chain one (see, e.g., \cite{Tseytlin:2004xa} and refs.\ there).

Integrability has been extremely useful	 in understanding the holographic duality in the case of  \adss\unskip. Not only has it allowed the solution of the spectral problem for a non-trivial 4d gauge theory (with qualitative similarities to QCD), but it has also facilitated  non-trivial checks of the AdS/CFT correspondence in this example (see, e.g., \cite{Eden:2012fe}). Motivated by a wish to solve more string theories in curved space, and more 4d gauge theories, one could ask in which ways the \adss background can be \textit{deformed} while preserving integrability. More generally, we would like to know which string theories are integrable.

The question of classifiying integrable string backgrounds is already very difficult  in the simpler case of \textit{bosonic} strings, which will be our focus. Although focussing on bosonic strings takes a step back from the AdS/CFT correspondence that was our motivation, we note the historical fact that the integrable semi-symmetric space \sm governing the \adss background is a simple generalization of bosonic symmetric space \sms discussed below.
The motion of a bosonic string is described by a 2d \sm
coupled to 2d gravity,
\be \begin{aligned}
&S  = \frac{1}{4\pi\a'} \int d^2 \xi \, \L \la{csm}\\
&\L= -  \big[ \sqrt{-h} h^{\m\n} \, \GG_{mn}(x) + \e^{\m\n} \,\BB_{mn} (x) \big] \,  \del_\m x ^m \,  \del_\n x^n   - \a' \sqrt{-h}\,  R^{(2)} \p(x)  \ ,\\
&m,n=1,\ldots , D \ , \qquad \quad  \m,\n=1,2 \ .
\end{aligned} \ee
Here the scalar fields $x^m$  may be viewed as embedding coordinates for  the 2d worldsheet (traced out by a string over time) in a $D$-dimensional target space-time. The parameter $'$ plays the role of $ħ$ (counting loops) in the  $$-model. In this thesis we will  set $'=1$ for convenience, but will sometimes reinstate $'$ to emphasize the loop counting. The \sm couplings $(_mn,_mn)$ are respectively symmetric and antisymmetric and correspond to  the background values of the target space metric and NSNS 2-form. The auxiliary 2d metric $h_$ is arbitrary and $_$ is the antisymmetric 2d Levi-Civita tensor.

The model \rf{csm} is 2d diffeomorphism-invariant and classically 2d Weyl-invariant. The role of the dilaton background $ϕ(x)$,  appearing coupled to the 2d curvature $R^(2)$ at subleading order in $'$, is to compensate quantum breaking of Weyl invariance from the $(,)$ terms \cite{Fradkin:1985ys}. Preservation of the local Weyl invariance, which is necessary for the consistency of the quantum theory, imposes that the background fields satisfy certain `string equations of motion' \cite{Lovelace:1983yv} (generalizing the Einstein equation),
\be \la{se}
\begin{aligned}
&R_{mn} - \tfrac{1}{4} (\HH^2)_{mn} + 2\nabla_m \nabla_n \phi  = \O(\a') \ , \qquad \quad  (\HH_{mnp} \equiv 3 \del_{[m} \BB_{np]})  \\
&-\ha \nabla^p \HH_{mnp} + \HH_{mn}{}^p \,  \nabla_p \p   = \O(\a') \  , \\
&\tfrac{1}{6}(D-26) -\ha \nabla^2 \p + (\nabla \p)^2 - \tfrac{1}{24} \HH^2 = \O(\a') \ ,
\end{aligned}
\ee
which contain corrections at each order in $'$ (corresponding to the loop expansion of the $$-model). These equations are obtained by varying an effective gravitational action for the massless fields,
\be
S = \frac{1}{2\ka^2} \int d^D x \, e^{-2\p} \big( R - \tfrac{1}{12} \HH^2 + 4 (\del \p)^2 \big) + \O(\a')  \ .
\ee
and it has been checked
to several orders in $'$  that this action coincides with the one obtained from the tree-level string S-matrix \cite{mt,Graham:1987ep,Foakes:1987ij,Foakes:1987gg} (see also the review \cite{Tseytlin:1988rr}).

For the purposes of classical integrability, it will be sufficient to fix the `conformal' gauge $√(-h) h^ = η^$, obtaining a $$-model on flat 2d space,
\unskip\foot{The 2d light-cone derivatives are defined as $\del_\pm = \del_\t \pm \del_\s$, see Appendix \ref{conv} for the conventions.}
\be \la{13}
S = \frac{1}{4\pi\a'} \int d^2 \xi \, \L = \frac{1}{4\pi\a'} \int d^2 \xi \, \big[ \GG_{mn}(x) + \BB_{mn} (x)\big] \,  \del_+ x ^m \del_- x^n
\ ,
\ee
where we have ignored the quantum dilaton term. At the level of classical integrability, the flat and curved 2d-space $$-models are equivalent   since they are related by a redefinition $ξ^→ξ'^(ξ)$ (as the $$-model is  classically 2d Weyl invariant and all 2d metrics are locally equivalent up to a conformal factor). The Virasoro constraints ($T_++=0$, $T_–=0$) coming from the variation of the 2d metric $h_$ in \rf{csm} should not destroy classical integrability since these are just constraints on the initial data.

Our eventual goal is to understand which  string backgrounds $(,)$ (solutions of \rf{se} with some choice of $$) correspond to integrable \sms \rf{13}. At the classical level, one criterion for integrability is the 2d equations of motion taking the \textit{zero-curvature} form \cite{kdv,lax,zs,zm}
\be \la{zc}
[ \del_+ + L_+(z) , \del_- + L_-(z) ] = 0 \ ,
\ee
where the ``Lax connection'' $L_±=L_±(z; x, x)$ is a 1-complex-parameter family of connections constructed from the fields. For any given \sm couplings $(,)$, it is very difficult to know whether a zero-curvature representation exists, and moreover there is no general recipe for constructing the Lax connection. This tricky classification problem of integrable \sms is the  motivation for the thesis.

\section{Stability under the RG flow}

In this thesis we will relax the Weyl invariance condtion \rf{se} and consider general \sms on a flat 2d space, again asking the question of which choices of couplings $(,)$ correspond to integrable theories. Although non-Weyl-invariant \sms do not directly correspond to string backgrounds, their study is well-motivated by the historical observation that  non-conformal  integrable \sms such as the PCM and the symmetric space \sms have preceded the discovery of related conformal ones --- the WZW model and the semi-symmetric space \sm with WZ term. In other words, one may discover new examples of Weyl-invariant  integrable theories by first discovering non-Weyl-invariant ones.

Moreover, relaxing Weyl invariance is actually  helpful since it opens up a new `diagnostic tool' for  integrability: the \textit{renormalization group (RG) flow}. The idea is that, in order to be able to identify the integrable theories, one should thoroughly investigate their properties and distinctive features. We shall focus on  one conjectured property in particular: that \textbf{integrable \sms are  \textit{stable} under the quantum RG flow} \cite{Fateev:1992tk,fateev96,lukyanov,Fateev:2019xuq,Fateev:2018yos,Litvinov:2018bou,Valent:2009nv,Demulder:2017zhz,Itsios:2014lca,Appadu:2015nfa}. Let us now explain the meaning of this statement.

\sloppy
General \sms  feature a quantum  RG flow (breaking the scale invariance of the classical theory). The \sm can be quantized  preserving the target space covariance \cite{ecker,friedan} and,  in the simplest case with $_mn=0$, the  RG flow is given by a `generalized Ricci flow',\foot{We denote by $t=\log \m$ the 2d RG scale.}
\be \begin{aligned}
&\dt G_{mn}= \b_{mn} + \nabla_{(m} \xi_{n)} \ , \qquad \qquad \qquad (B_{mn}=0) \la{2}\\
&\b_{mn} = \a' \, R_{mn} + \a'^2 \, \ha R_{mpqr} {R_n}^{pqr} + \ldots \ .
\end{aligned} \ee
We have included the 1-loop and 2-loop terms, and there will be further contributions at each loop order. Here $ξ^n(x)$ corresponds to a choice of  renormalizations of the fields $x^n$, i.e.\ target space diffeomorphisms depending on the 2d RG scale. Now including the $$-field, the $$̱-function 2 generalizes, in a particular 2-loop subtraction schemeSee Section <ref> for discussion of scheme dependence. to <cit.> (see also <cit.>)
(_mn + _mn) = _̱mn + L_ξ(+)_mn  +  _[mł_n]   ,

_̱mn= '   R_mn +  '^2   12 [R^klp_n R_mklp - 12 R^ l p k_n R_mklp
+12 R_k mn l ^k pq ^ l_ pq ] + … ,
where $\HH_{kmn} = 3 \del_{[k} \BB_{mn]}$ and $\widehat R$ is the curvature of the generalized connection  The use of $\widehat{R}$ is simply a convenient notation and one can re-express BMT in terms of just $R$, $\HH$ and covariant derivatives: for example, $\widehat{R}_{mn} = R_{mn} - \tfrac{1}{4} \HH_{mpq}\HH_n{}^{pq} - \ha D^k \HH_{kmn}$. Here $L_\xi$ denotes the Lie derivative along a vector $\xi$, generalizing the diffeomorphism term of 2 above. The extra term $\del_{[m}\l_{n]}$ denotes a shift of the $\BB$-field by an exact 2-form depending on the 2d RG scale; this is the usual 2-form gauge freedom, and only modifies the Lagrangian 13 by a total derivative.

The string equations  se  correspond to the Weyl-invariant case when the  RG flow BMT trivializes, i.e. $\ddt (\GG_{mn}+\BB_{mn}) =0$. The diffeomorphism vector and B-gauge transformation must also take a specific form <cit.> $\xi_n = 2 \a' \del_n \phi + \O(\a'^2)$, $\l_n = \O(\a'^2)$ in terms of the dilaton, metric and B-field. The dilaton equation in se then follows from the metric and $\BB$-field equations up to a constant term, whose correct value ensures the vanishing of the Weyl anomaly coefficient.

More precisely, we may formulate the conjecture of RG stability as follows. For a family of  integrable  models $\GG_{mn}=\GG_{mn}(g_\a)$, $\BB_{mn}=\BB_{mn}(g_\a)$  parametrized by a few couplings $g_\a$, the claim is that the  RG flow BMT of the metric and $\BB$-field consistently truncates to a flow of just  the couplings $g_\a$,
g_= _̱(g)  .
In other words, integrable are   renormalizable, i.e. they are  explicit solutions of the generalized Ricci flow equation.

Let us give an intuitive explanation for this conjecture. As we shall discuss below, classically integrable feature an infinite set of commuting conserved charges, as part of an intricate Poisson structure governed by a so-called Lax operator. In general one may expect this structure to be sufficiently constraining to uniquely determine the theory in question, so that the only deformations preserving it are the coupling constants $g_\a$ of the theory. Now assuming the integrable structure to exist also in the quantum theory, then it should be respected by the quantum RG flow, and hence only the couplings $g_\a$ should run under  RG. Another explanation is that some integrable $\s$-models are understood to have exact S-matrices in the infrared, typically with a small number of parameters (see Section <ref> below). One may then expect that these parameters should somehow match up with the semiclassical $\s$-model description, and that this matching should be preserved along the RG flow.

While there are no known counterexamples
to the conjecture that  integrable $\s$-models are renormalizable,
in most examples this has only been checked to the leading 1-loop order in BMT,
and there is no general proof. In this work, we shall ask the question of what happens at higher loop orders. We will find that the target space geometry of the classical $\L = (\GG+\BB)_{mn}^{(0)} \, \del_+ x^m \del_- x^n$ must generally be deformed by quantum corrections to remain a solution of the RG equation BMT beyond 1-loop,

\begin{align}
(\GG+\BB)_{mn}^{(0)} \to (\GG+\BB)_{mn}^{(0)} + \a' (\GG+\BB)_{mn}^{(1)} + \ldots  \  . \la{qc}
\end{align}

Equivalently, one must add to the $\s$-model a particular choice of finite counterterms.
This is a scheme-independent statement because  the required counterterms will be non-trivial in arbitrary covariant subtraction schemes.

Hence, while the classical Lax integrability corresponds just to the classical (uncorrected) geometry $(\GG+\BB)_{mn}^{(0)}$,
it is the quantum-corrected geometry that seems to posses the RG properties associated with integrability at the quantum level.
These counterterms might then be understood as preserving `Ward identites' for hidden integrable symmetries at the quantum level, with the expectation that these hidden symmetries are constraining the RG flow. In this sense, one may hope to interpret the corrected as being the correct choice of bare action consistent with quantum integrability.

§ STRUCTURE OF THE THESIS

This thesis is structured as follows. The rest of  this introductory chapter is a review of integrable  2d $\s$-models with a focus on classical Lax integrability, examples of integrable  models and their quantum RG flows.

In Chapter <ref> (based on the papers <cit.>) we shall consider the higher-loop RG flow for some of the simplest non-trivial `integrable deformed' models where renormalizability does not immediately follow from manifest global symmetries.
For the $\eta$-deformation of $S^2$ (“sausage model”) or $H^2$, we explicitly construct the counterterms required for RG stability up to 3-loop order and conjecture the all-order form of the counterterms. We also study the closely related $\l$-deformation based on  the $SU(2) \ov U(1)$ coset and the $SU(2)$ group, finding the counterterms needed for 2-loop RG stability. We  point out that these counterterms continue to serve the same function of restoring renormalizability in the   non-abelian dual limit of the $\l$-model. This gives a  prescription in these examples for the leading $\a'$-corrections to the non-abelian duality transformation so that the non-abelian duality commutes with the RG flow, resolving prior confusion in the literature.

In Chapter <ref> (based on <cit.>) we consider the same question for a different class of integrable $\sigma$-models  on products of group spaces. We    first  study  the integrable $G \times G$   model   derived from the  affine Gaudin construction  (for which the 1-loop $\beta$-functions were found in <cit.>)  and show that its condition of integrability  is  automatically  preserved also  by the 2-loop  RG flow without the need for any quantum corrections.  We  then   investigate the RG flow   in the gauged $G\times G/H$  model, including  the integrable  $T^{1,1}$    model  found in <cit.>. In that case we instead find that the integrability condition relating the couplings requires a certain quantum deformation to preserve renormalizability at 2-loop order. We also  construct   a   new   class of integrable $G\times G/H$ models in the case  when the subgroup  $H$ is abelian. In the  simplest case of $G=SU(2),\ H=U(1)$  this  leads  to an integrable $\sigma$-model on the   $T^{1,q}$  space  (with a particular $\BB$-field). This  model is found to be a special case that is stable under the 2-loop  RG flow without deformation — and we  relate this   property to its  invariance  under T-duality in an isometric $U(1)$ direction. This  $T^{1,q}$ model   may be interpreted as an integrable  deformation of the GMM model  (of two coupled WZW  theories  with generic levels) away from the conformal point.

In Chapter <ref> (based on <cit.>) we derive an alternate formulation of the $\lambda$-model as a on a “tripled” configuration space $G\times G \times G $ by re-writing the 2d gauge field in terms of group-valued scalar fields.
Our central observation is that the $\l$-model quantized in this formulation is automatically renormalizable to all orders without any correction,  with only the deformation parameter $\lambda$ running.
This is in contrast to the standard $\sigma$-model found by integrating out $A_\pm$, which is 2-loop renormalizable only after adding the specific finite local counterterms discussed in Chapter <ref>.
We compute the 2-loop $\beta$-function of the $\lambda$-model for general groups
and symmetric spaces.
In the non-abelian dual limit this  yields an alternate formulation
of the interpolating models for
non-abelian duality
that is automatically renormalizable with
the same 2-loop $\beta$-function as the original PCM/symmetric space $\s$-model.

Section <ref> contains unpublished work on the construction of similar “tripled” or “enlarged” formulations of   (i) a generalization of the  $\l$-model to  product spaces  $G^N$  and (ii)  $G\times G/H$ models.

In Chapter <ref> (based on <cit.>) we study a new and different link between integrability and the RG flow. We consider several classes of integrable $\sigma$-models (on groups and symmetric spaces, $\eta$-models, $\lambda$-models) with local couplings that may depend on the 2d coordinates, e.g. on time $\tau$. We observe that (i) starting with a classically integrable 2d $\sigma$-model, (ii) formally promoting its couplings $h_\alpha$ to functions $h_\alpha(\tau)$ of 2d time, and (iii) demanding that the resulting time-dependent model also admits a Lax connection implies that $h_\alpha(\tau)$ must solve the 1-loop RG equations of the original theory with $\tau$ interpreted as RG time. This provides a novel example of an `integrability–RG flow' connection. The existence of a Lax connection suggests that these time-dependent $\sigma$-models may themselves be understood as integrable. We investigate this question by studying the possibility of constructing non-local conserved charges (see also Section 5 of <cit.> for attempts to construct local charges). Such $\sigma$-models with $D$-dimensional target space and time-dependent couplings subject to the RG flow naturally appear in string theory upon fixing the light-cone gauge in a $(D+2)$-dimensional conformal $\sigma$-model with a metric admitting a covariantly constant null Killing vector and a dilaton linear in the null coordinate. This suggests the possibility of a large, new class of solvable string models.

In Chapter <ref> we  summarize the findings of this work and comment on the open questions, possible applications and future prospects.

Appendix <ref> is an explanation of the conventions and notation used in this thesis.

In Appendix <ref>  (based on <cit.>) we discuss the effect of scheme change on the $\s$-model $\b$-functions, focussing  on the particular case with a 2d target space (relevant for the $SU(2)U(1)$ $η$- and $ł$-models considered in Chapter \ref{QCS}).

\medskip

In Appendix \ref{2103A} (based on \cite{LT2103}) we consider deriving the classical integrability condition for the couplings of $G^N$ models (discussed in Chapter \ref{GGS}) from first principles. We compare our results with the sufficient conditions for integrability obtained from affine Gaudin models. For general $N$ we find that the dimension of the integrable space of $G^N$ couplings coincides with that of the space of affine Gaudin models, and for the $G×G$ case we find that the affine Gaudin models are the only integrable ones.

\medskip

In Appendix \ref{2103B}  (based on \cite{LT2103})  we  provide the explicit formulae for the 2-loop
$$̱-functions of the $G\times G$ and $G\times G/H$ models (used in Chapter <ref>) and explain how they were derived.

The remaining appendices, based on <cit.>, pertain to the `local coupling' theories discussed in Chapter <ref>. Appendix <ref> contains details of the derivation of the RG flows from the condition of existence of Lax representations for the local coupling theories.

Appendix <ref> is about the construction of non-local charges for time-dependent $\s$-models other than the PCM (which was discussed in Section <ref>) — and in particular the time-dependent $\l$-model based on a symmetric space.

In Appendix <ref> we consider 1d harmonic oscillators with time-dependent frequency,  a special case of which was obtained in eq. 2008lt as a linearization of a 1d reduction of the time-dependent PCM.

In Appendix <ref> we explain that the sine-Gordon model also obeys the same pattern as the $\s$-models discussed above (relating its integrability with  local couplings to its RG flow).

§ REVIEW OF KEY CONCEPTS REV

In the remainder of this introductory chapter, we shall review some concepts priming the reader for the rest of the thesis.   Let us highlight Section <ref>, which  is a summary of our paper <cit.> attempting to use the tree-level $\s$-model S-matrix  as  another `diagnostic tool' for integrability.

§.§ Classical integrability ci

Most models in physics cannot be  solved exactly. One example is the Standard Model of particle physics, where there is no hope to make exact predictions, but rather computations are made by expanding perturbatively around a free theory.
A simpler example is a many-body system in Newtonian gravity, where only the 2-body case is solvable and the 3-body and higher cases are chaotic.
One could ask: what goes right for the 2-body problem? One explanation is that the 2-body problem  is integrableSee also the book <cit.> for an introduction to classical integrability. while the others are not. For mechanical 1d systems, integrability means there exist at least as many independent, Poisson-commuting conserved charges as the number of degrees of freedom. In that case it follows that the theory can be completely solved by quadratures.

For 2d models, and in particular the  that will be our focus, there are infinitely many degrees of freedom. The analogous notion of integrability in 2d is then related to the presence of infinitely many commuting conserved charges. A theory of exactly integrable PDEs in mathematical physics was developed starting from the 1960s <cit.>. The flat Lax connection is a central object, whose zero-curvature representation zc expresses the relevant non-linear equation as the consistency condition for an over-determined linear system
_±ψ= ψ  L_± ,
leading to the construction of explicit  solutions by the “inverse-scattering” method.

If a 2d field theory admits a Lax representation zc, then one can immediately build non-local conserved charges using the monodromy of the Lax connection along a constant-time contour from $\s=a$ to $\s=b$,
= 𝒫 exp∫_a^b  L_  d . Md
The path-ordered exponential in Md is defined by the properties
^-1 _b = L_|_=b   ,        |_a=b =1  . px
On the support of the equation of motion zc of zero-curvature form, it satisfies
_=   L_(b) - L_(a)    . ie

Assuming a periodicity condition $L_\t(a)=L_\t(b)$ (e.g., following from periodic boundary conditions on the fields),
then ie becomes  $\del_\t \M  = [\M , L_\t (a)]$ and hence
_[ ^n] = 0    ,     n ∈ℕ  ,
i.e. the eigenvalues of $\M$ are conserved.

If additionally the boundary value is vanishing, i.e. $L_\t(a)=L_\t(b)=0$  (for example with decaying boundary conditions on the plane  $(a,b)=(-\infty,\infty)$), then ie gives  the stronger result that all the entries of $\M$ are conserved,
_= 0  .

In either case, the number of conserved charges is actually infinite since the entries of $\M$ are  functions of the `spectral parameter' $z$ (assuming the dependence is non-trivial) so each coefficient is individually conserved in any series expansion in $z$. These charges are  `non-local' since the monodromy matrix is a non-local object (it is not an integral of a local density).

We note that a Lax connection valued in the algebra of $G$ is determined up to gauge transformations $L_\pm \to k^{-1} L_\pm k + k^{-1} \del_\pm k$,   $k(\xi) \in G$, since this transformation preserves the set of equations encoded in the zero-curvature condition zc. The monodromy matrix Md rotates covariantly under such transformations so, assuming periodicity of the gauge parameter $k$, the set of non-local  charges is preserved.

§.§ Principal Chiral Model

Let us consider the simple example of the Principal Chiral Model (PCM),We denote the  2d indices by $\m,\n$ and define the 2d light-cone coordinates as $\xi^\pm = \ha (\t \pm \s)$ with derivatives $\del_\pm = \del_\t\pm \del_\s$. The $D$-dimensional target space indices are $m,n=1,\ldots, D$. See Appendix <ref> for conventions.
Ł= -   [ J_+ J_- ]   ,      J_= g^-1 _g  ,   g∈G  . PL
This is a special case of the general 13 with the target space taken to be a simple Lie group $G$ with the bi-invariant metric (and vanishing $\BB$-field). The model has a large global $G\times G$ symmetry corresponding to  left- and right-multiplications of the group,
g →g_L  g  g_R  ,         (g_L,g_R)∈G_L ×G_R  . gps
For this model, the global symmetry gps (or isometries of the target space) is sufficient to uniquely fix the form of the Lagrangian PL up to the overall constant $\h$. Hence the RG equations 2 are solved with just $\h$ running according to the 2-loop $\b$-function \cite{McKane:1979cm,Hikami:1980hi,friedan},\foot{In this PCM case, since the target space is isotropic, there cannot be any diffeomorphism vector $\xi$ in \rf{2} consistent with the symmetries.}
\be
\ddt \h = \cg+\ha \cg \h^{-1} + \ldots \ .  \la{Prg}
\ee
The constant $$ denotes the dual Coxeter number of the group $G$ (see Appendix \ref{conv} for conventions).

The key to noticing the model's classical integrability is that the current $J_$ is conserved on-shell, $ ^J_= 0$, and also flat, $dJ + J∧J = 0$ (since $J_=g^-1 _g$). This information can be encoded in the flatness \rf{zc} of the Lax connection \cite{zm},
\be
L_\pm \equiv L_\t \pm L_\s = \ha (1+z^{\pm 1}) \, J_\pm  \ . \la{pl}
\ee
An easy way to evaluate the monodromy matrix and extract conserved charges is by expanding in $z$ around the value $z=-1$ where $L_$ vanishes (at which point $=I$ is trivial). The expansion of the path-ordered exponential yields first the conserved Noether charge built from $J_$ itself (corresponding to the $G_R$ symmetry) and then iterated integrals of $J$ of increasing order,
\begin{align}
&\M =I + \ha (z+1) \int_a^b d\s \, J_\t \la{me} \\
&\qquad\ \ \ \
+ \tfrac{1}{4} (z+1)^2 \Big[ \int_{a<\s_1<\s_2<b} d\s_1 d\s_2 \, J_\t(\s_1)J_\t(\s_2) + \int_a^b d\s \,(J_\t-J_\s) \Big]
+ \O\big((z+1)^3\big) \no
\ .
\end{align}
Each term in the expansion (or just their eigenvalues, depending on the boundary conditions) is individually conserved by the argument above, and their  Poisson brackets satisfy a Yangian algebra \cite{Bernard:1990jw}.

\bigskip
As well as these non-local charges, integrable field theories may posses higher-spin local charges. For the example of the PCM \rf{PL},
one can build holomorphic and anti-holomorphic local currents by contracting products of $J_±$ with a symmetric invariant tensor  \cite{Evans:1999mj},
\begin{align}
\del_\pm \mathcal{J}_\mp^{(n)} = 0 \ , \qquad \qquad &\mathcal{J}_\pm^{(n)} =
d_{A_1 \cdots A_n} J_\pm ^{A_1} \cdots J_\pm^{A_n} \ \la{cur} \\
&d_{A_1 \ldots A_n} = d_{(A_1 \ldots A_n)} \ , \qquad\qquad
{f^{A}}_{B(C} d_{A_1 \cdots A_{n-1}) A_n}= 0 \ .\la{e3}
\end{align}
The $n$th currents $𝒥_∓^(n)$ are of spin $n$  and their (anti-)holomorphic conservation implies the conservation of the associated conserved charges,
\be
\del_\t Q_{\pm}^{(n)} = 0 \ , \qquad \qquad Q_\pm^{(n)} = \int d\s \, \mathcal{J}_\pm^{(n)} \ ,
\ee
which Poisson-commute with the non-local charges above \cite{Evans:1999mj}.

A canonical choice for the invariant
tensor $d_A_1 ⋯A_n$ is given by the trace,
$d_A_1 ⋯A_n = [ T_(A_1 ⋯T_A_n) ] $.
The conservation of the currents \rf{cur} follows from the equation of motion and flatness of $J_$, written in the 2d light-cone coordinates as
\be
\del_+ J_- + \del_- J_+ = 0 \ , \qquad \del_+ J_- - \del_- J_+ + [J_+, J_-] = 0 \ , \la{pe}
\ee
since these may be re-written as
\be
\del_+ J_- = -\ha [J_+, J_-] \ , \qquad \del_- J_+ = \ha [J_+, J_-] \ .
\ee

\subsection{Symmetric space sigma model \la{sssm}}
Now let us consider instead a \sm on a homogeneous coset space $G/H$, obtained  by gauging a subgroup $H$ of the $G_R$ symmetry of the PCM (here $H⊂G$ are both assumed to be simple Lie groups). The gauging procedure may be understood as (i) promoting the $H$ global symmetry to a local one, $g →g   h(ξ)^-1$, (ii) writing down the transformation of the PCM Lagrangian \rf{PL}, $Ł→-   [(J_+ - h^-1_+h)(J_-  - h^-1_-h)]$, and (iii) promoting the gauge parameter to a gauge connection, $h^-1 _±h →A_±$ (no longer fixed to be pure gauge). The resulting $G/H$ theory,
\be
\L = -\ha \h \,  \Tr[ (J_+ - A_+) (J_- - A_-) ]  \ , \qquad J_\pm = g^{-1} \del_\pm g \ , \ \  g\in G  \ , \quad  A_\pm \in \Lie(H) \la{ss1} \ ,
\ee
is invariant under the $H$ gauge transformation
\be
g \to g h \ , \quad A_\pm \to h^{-1} A_\pm h + h^{-1} \del_\pm h \ , \qquad h(\xi) \in H  \ .\la{hg}
\ee

The quadratic gauge field $A_±$  in \rf{ss1} is auxilliary (appearing without derivatives) and may be integrated out  to  obtain
\be
\L = -\ha \h \,  \Tr[ J_+\,  P_{G/H} \, J_-]  \ , \la{ss2}
\ee
where $P_G/H$ is the projector orthogonal to  $(H)$ in $(G)$ with respect to the Killing form. The $H$ gauge transformation \rf{hg} then reduces to
\be
g \to g  \, h \ , \qquad h(\xi) \in H  \ .
\ee
The theory also has a global $G$ symmetry, $g →g_L   g$,  \ $g_L ∈G_L$ corresponding to the isometry of the coset space  $G/H$.

If $G/H$ is assumed to be a \textit{symmetric space} (meaning $(H)$ is the even subalgebra under a $ℤ_2$-grading of $(G)$), then these global and local symmetries uniquely determine the theory \rf{ss1} or \rf{ss2} up to the constant $$.
Like the PCM above, this automatically makes the theory renormalizable to all orders with just $$ running. The 2-loop $$̱-function following from 2 is (see App. B of <cit.> and particular cases in <cit.>)
= 2 + 4 (-) ^-1   . sb
In contrast, for non-symmetric cosets $G/H$, the theory ss2 may not be renormalizable as
it  should be understood as part of a multi-coupling theory $\L = -\ha \Tr[ J_+ \, \big( \sum_{i=1}^n \h_i P_{\mathfrak{p}_i} \big) \, J_-]$ respecting the same symmetries, defined in terms  a decomposition of the Lie algebra
$\Lie(G) = \Lie(H) + \sum_{i=1}^n \mathfrak{p}_i$, with $\mathfrak{p}_i$ being irreducible representations under the adjoint action of $H$, mutually orthogonal under the trace. For general cosets we may have $n>1$ (i.e. multi-coupling),  while  symmetric spaces must have $n=1$ (single-coupling) if $G$,$H$  are simple.

The symmetric space condition on $G/H$ is also sufficient for classical integrability of the model ss2. The corresponding equations of motion are then of the zero-curvature form zc with the Lax connection <cit.>
L_±=B_±+ z^±1   P_± ,     B ≡P_H  J ,   P ≡P_G/H J sll
Let us note that the symmetric space is actually a generalization of the PCM, since the PCM on a group $G$ is equivalent to the on the symmetric space $G\times G/G$ (and is obtained upon fixing the gauge $g^{(2)}=1$, where $(g^{(1)},g^{(2)})\in G\times G$, and replacing $\h \to 2\h$). All of the formulas here for symmetric spaces actually apply in that case, except that $G\times G$ is not simple so the definition of $\cf$ in Appendix <ref> should be suitably modified. Indeed the symmetric space $\b$-function \rf{sb} agrees with the PCM one \rf{Prg} upon replacing $→$ and $→2$. The symmetric space Lax connection \rf{sl} in that case is related to the PCM one \rf{pl} by a gauge transformation.

Both non-local \cite{lp} and local \cite{Evans:2000qx} charges may be constructed similarly to the PCM case above. For the local charges, one now replaces $J →P$ in  \rf{cur},
\begin{align}
\del_\pm \mathcal{J}_\mp^{(n)} = 0 \ , \qquad \qquad &\mathcal{J}_\pm^{(n)} =
d_{A_1 \cdots A_n} P_\pm^{A_1} \cdots P_\pm^{A_n}   \la{cur2} \\
&d_{A_1 \ldots A_n} = d_{(A_1 \ldots A_n)} \ , \qquad\qquad
{f^{A}}_{B(C} d_{A_1 \cdots A_{n-1}) A_n}= 0 \ .\la{e32}
\end{align}
The holomorphic conservation law in \rf{cur2} follows again from the equation of motion and flatness of $J$  \cite{Evans:2000qx} (here $D^B_±≡_±+ _B_±$),
\begin{align}
\qquad &D^B_+ \, P_- + D^B_- \, P_+ = 0 \ , \qquad \del_+ J_- - \del_- J_+ + [J_+, J_-]  = 0 \ ,
\end{align}
since they now  imply that
\begin{align}
\qquad &\del_\pm \  P_\mp = [P_\mp, B_\pm ]  \ .
\end{align}

\subsection{Constructing  Lax connections for other models}
The integrable models that follow will admit the  \textit{same} flat Lax connections of the form \rf{pl} or \rf{sll} as the PCM or symmetric space (for models respectively built on groups
or  symmetric spaces).\foot{Of course, not all integrable $\s$-models will have Lax connections of this form. For example the bi-Yang-Baxter model  \cite{Klimcik:2008eq,Ki} built on a group does not possess a flat, conserved current $\mc{A}_\pm$ so its Lax connection is not of the form \rf{gl}.} The difference is that, in each case, $J_±$ will be replaced with some other model-dependent quantity in terms of the fields,
\begin{alignat}{3}
&G \ : \qquad\quad & &L_\pm = \ha (1+z^{\pm 1}) \mc{A}_\pm \la{gl} \ ,\qquad \qquad & & \mc{A}_\pm \in \Lie{(G)} \\
&G/H \ : \qquad\quad & &L_\pm = \mc{B}_\pm + z^{\pm 1} \mc{P}_\pm \la{sl} \ ,\qquad \qquad & & \mc{B}_\pm \in \Lie{(H)} \ , \ \ \mc{P}_\pm \in \Lie{(G)}/\Lie{(H)}
\end{alignat}
We note that admitting Lax connections of the same form does not imply  classical equivalence between these different theories (not being related by a canonical transformation); e.g. for group-based models, it simply means they possess a flat, conserved current $A_±$.

Non-local conserved charges are then obtained from the monodromy matrix as discussed above for the PCM. Local conserved charges are also constructed similarly to the cases discussed above, with the general form of the (anti-)holomorphic currents given by
\begin{align}
\del_\pm \mathcal{J}_\mp^{(n)} = 0 \ , \qquad \qquad &\mathcal{J}_\pm^{(n)} = \begin{cases}
d_{a_1 \cdots a_n} \mc{A}_\pm ^{a_1} \cdots \mc{A}_\pm^{a_n} \ & \text{(group} \ G) \ \la{curG}\\
d_{a_1 \cdots a_n} \mc{P}_\pm ^{a_1} \cdots \mc{P}_\pm^{a_n} \ & \text{(symmetric space}\ G/H) \
\end{cases} \\
&d_{a_1 \ldots a_n} = d_{(a_1 \ldots a_n)} \ , \qquad\qquad
{f^{a}}_{b(c} d_{a_1 \cdots a_{n-1}) a}= 0 \ .\la{e3G}
\end{align}
The conservation of the currents \rf{curG} follows from the equations of motion
\begin{align}
G \ : \qquad &\del_+ \mc{A}_- + \del_- \mc{A}_+ = 0 \ , \qquad F_{+-}(\mc{A}) = 0 \ , \\
G/H \ : \qquad &D^\mc{B}_+ \mc{P}_- + D^\mc{B}_- \mc{P}_+ = 0 \ , \qquad F_{+-}(\mc{B} + \mc{P}) = 0 \ ,
\end{align}
since these may be re-written as
\begin{align}
G \ : \qquad &\del_\pm \mc{A}_\mp = \ha [A_\mp, A_\pm] \ , \la{ni}\\
G/H \ : \qquad &\del_\pm \mc{P}_\mp = [P_\mp, B_\pm ] \ , \ \ \ \qquad \ \ F_{+-}(\mc{B}) + [\mc{P}_+, \mc{P}_-] = 0 \ . \la{ni2}
\end{align}

\subsection{T-duality and non-abelian duality \la{TS}}
For \sms with global symmetries, or isometries of the target space, one may ``dualize'' the symmetry to obtain a dual \sm\unskip, with certain observables being related between the original and  dual theories (see the review \cite{Alvarez:1994dn} and refs.\ therein). As we shall review below, the duality procedure may be carried out at the level of the path integral and applies to the quantum theory \cite{Buscher:1987qj}; in particular duality is understood to commute with the quantum RG flow, and to map Weyl-invariant  theories (string solutions) to Weyl-invariant theories. However, we stress that the procedures as discussed below are subject to quantum $'$ corrections, with the integrals producing quantum determinants.

First we shall consider \textit{abelian T-duality} for a model with a $U(1)$ isometry, which we take for simplicity to be
\be
\L = g_{ij}(x)  \, \del x^i \del x^j + f(x)  \, (\del y)^2 \la{or}
\ee
Here the isometry corresponds to constant shifts of $y$, and the $x$ coordinates are ``spectators''. Gauging the isometry $_y →_y -  p_$ and imposing the condition  $ϵ^   _p_= 0$ using a Lagrange multiplier field $ỹ$ gives a gauge-invariant theory (under $y →y + u$, $p_→p_+ _u$),
\be
\L_{\rm int} = g_{ij}(x)  \, \del x^i \del x^j + f(x) \,  (\del y -  p)^2 + \tilde{y} \, \epsilon^{\m\n}\, \del_\m p_\n  \ .
\ee
Fixing the gauge $y=0$ gives the ``interpolating model'',
\be
\L_{\rm int} = g_{ij}(x)  \, \del x^i \del x^j + f(x) \,  p^2 + \tilde{y} \, \epsilon^{\m\n}\, \del_\m p_\n  \ . \la{int}
\ee
On one hand, integrating out $ỹ$ imposes that $p_$ is pure gauge and locally given by $p_= _y$, giving back the original model \rf{or}. On the other hand, we may instead integrate out the non-dynamical quadratic field $p$ (after integrating by parts $ỹ   ϵ^    _p_→-_ỹ   ϵ^    p_)$) to get the dual \sm with the $(x,ỹ)$ directions
\be
\tilde{\L} = g_{ij}(x)  \, \del x^i \del x^j + \frac{1}{f(x)}  \, (\del \tilde y)^2  \ .
\ee
The dual theory still has a $U(1)$ isometry, corresponding now to shifts of $ỹ$. In summary, the dual pair of models are related by $f →1f$, $y →ỹ$, and they were obtained by integrating out different fields from the interpolating theory \rf{int}. This procedure is easily generalized to cases with off-diagonal metric terms and $$-fields, with the resulting transformation being known as the `Buscher rules' \cite{Buscher:1987qj}.

Now let us consider \textit{non-abelian duality} (NAD), where the above procedure is generalized to  non-abelian isometries \cite{Fridling:1983ha,Fradkin:1984ai,delaOssa:1992vci}. We shall focus on the examples of the PCM and symmetric space \sm (considered above), and the procedure is the same for other theories with non-abelian global symmetries. Starting from one of these  theories
\be
\L = -\ha h \, \Tr[ J_+ \, P \, J_- ]  \ , \qquad \quad P=\begin{cases}1  \qquad & \text{(PCM)} \ ,  \\ P_{G/H} \ \qquad & \text{(symmetric space)} \ , \end{cases}  \la{on}
\ee
we shall dualize the $G_L$ symmetry $g →k g$. As above, we begin by gauging the symmetry and imposing that the gauge field is pure gauge with a Lagrange multiplier field $v$ (here $F_+-(A) ≡_+ A_- - _- A_+ + [A_+, A_-]$)
\be
\L = -\ha \h \, \Tr[ (J_+ + g^{-1} A_+ g) \, P \, (J_- + g^{-1} A_- g)  + v \, F_{+-}(A)] \ .
\ee
This resulting theory is invariant under the gauge transformation $g →k g$, $A_±→k A_±k^-1 - _±k k^-1$. Fixing the gauge $g=1$, we obtain the interpolating theory
\be
\L_{\rm int} =  -\ha \h \, \Tr[  A_+  \, P \,  A_- - v \, F_{+-}(A)]  \ . \la{NADi}
\ee
On one hand, integrating out $v$ imposes that $A_±$ is flat and  locally of the form $A_±= g^-1 _±g$, giving back the original theory \rf{on}. On the other hand, after integrating by parts, $A_±$ appears quadratically without derivatives; integrating it out yields the NAD \sm depending on $v$,
\be
\tilde{\L} = -\ha \h \, \Tr[  \del_+ v \, \frac{1}{P+ \ad_{v}} \, \del_- v] \ . \la{NADt}
\ee
Unlike abelian T-duality, non-abelian duality destroys the original symmetry, meaning the original $G_L$ symmetry is not present in the NAD theory \rf{NADt}. Thus, at least within the framework of non-abelian duality, one cannot ``dualize back'' from the NAD model to the original one.\foot{In fact the $G_L$ symmetry becomes a Poisson-Lie symmetry of the NAD model \rf{NADt} (see eq.\ \rf{PLs} below) and, starting from the NAD theory, one can dualize back to original theory by Poisson-Lie duality.}

The path-integral derivations of  abelian and non-abelian duality apply in particular at the level of the  classical equations of motion (they may be formulated as canonical transformations; see, e.g., \cite{Sfetsos:1996xj} and refs.\ therein). Hence these dualities preserve the property of classical integrability. For example the NAD of the PCM (\rf{NADt} with $P=1$) inherits a flat Lax connection of the same form \rf{gl} as the PCM with $A_±=A_±(v)$ taking its value obtained in the quadratic integral in \rf{NADi},
\be
L_\pm = \ha (1+z^{\pm 1}) A_\pm(v) \ , \qquad A_\pm(v) = \frac{1}{\ad_v \mp 1} \del_\pm v \ .
\ee

\subsection{PCM with Wess-Zumino term \la{PkS}}
The PCM \rf{PL} is not the most general $G$-valued \sm with   $G_L ×G_R$ global symmetry \rf{gps}. In fact, there is also a $$-field coupling respecting this symmetry called the \textit{Wess-Zumino} (WZ) term, defined by \cite{Wess:1971yu,novikov,Witten:1983ar}
\begin{align}
\H\equiv d\BB =  \tfrac{1}{12} k \, \Tr[ J \wedge J \wedge J ] \ , \qquad  \BB \equiv \ha \BB_{mn} dx^m \wedge dx^n \ .
\end{align}
The free coefficient $k$ is called the `WZ level'.

One can check that $d=̋ 0$, implying the local existence of a $$-field satisfying $=̋d$. Although $$ is not globally defined (the expressions for $$ in different coordinate charts may be related by gauge transformations $→+ dł$), the \sm action \rf{13} is well-defined up to an additive constant. Let us take the 2d space to be the Euclidean plane $ℂ$ (after analytic continuation), which we compactify to the Riemann sphere $S^2=ℂ∪{∞}$. Then a coordinate-invariant way to write down the WZ term's contribution to the action is\foot{The factor of $i$ in \rf{iWZ} results from the Wick rotation of the antisymmetric Levi-Civita tensor $\e^{\m\n}$ in the $\s$-model \rf{csm}.}
\be
S_{\WZ} =  \frac{i}{2\pi } \int_{B_3} \H \ , \la{iWZ}
\ee
where $B_3$ is the solid ball whose boundary is $S^2$. One is instructed to continue the fields in an arbitrary way from the boundary into the interior of $B_3$ (this is always possible since Lie groups have trivial 2nd homotopy group $π_2(G)=0$). Since $$̋ is closed then its integral iWZ is independent of the choice of continuation of the fields, up to an overall topological `winding number' constant. For compact simple groupsFor non-compact groups there may be no ambiguity at all and $k$ may be any real number. this gives an ambiguous contribution $\Delta S_{\WZ} = 2 \pi i n k$ with $n$ an arbitrary integer in the homotopy group $\pi_3(G)=\mathbb{Z}$, which can be shown to classify the configurations. The Euclidean path integral  $Z = \int Dx \, e^{-S[x]}$ is then well-defined as long as the  level $k$ is quantized to be an integer.

We shall denote by `PCM$_k$' the PCM with WZ term with level $k$ (with $\L_{\WZ}$ being a local density for $S_{\WZ}$),
Ł= -  [ J_+ J_- ] + k   Ł_(g)  . pk
The PCM$_k$ is classically integrable for arbitrary values of the couplings $\h$ and $k$, with the Lax connection taking the same form gl as the PCM in terms of a modified flat, conserved current <cit.>
L_±= (1+z^±1) A_± ,         A_±= (1±kh) J_± .

It is a general fact that WZ levels do not run under the RG flow (even for non-compact groups where they are not quantized). This is because the antisymmetric $\BB$-field terms in the $\b$-functions \rf{BMT} are all globally defined,  so cannot be proportional to  WZ terms  whose $$-fields are not globally defined. The WZ term does, however, modify the RG flow of the coupling $$, and the PCM$_k$ has 2-loop $$̱-function
= (1-k^2^2) ( + ^2 (1-3k^2^2) + …)  ,     k = 0  . wzwb

§.§ Wess-Zumino-Witten model WS

The PCM$_k$ $\b$-function \rf{wzwb} admits a fixed point $= k$ (with the other fixed point $=-k$ related by 2d parity or $g→g^-1$), which is in fact an  exactly conformal theory called the  \textit{Wess Zumino Witten} (WZW) model,
\be
k \,  \L_G(g) \equiv    k \big( -\ha  \, \Tr[J_+ J_-]  + \L_{\WZ}(g) \big) \ .
\ee
For this special value of the coupling, the $G_L×G_R$ symmetry enhances to a ``chiral gauge'' symmetry \cite{Witten:1983ar},\foot{In Minkowski signature $\xi^\pm$ in \rf{cgs} are the same 2d light-cone coordinates used above. In Euclidean signature these become complex-conjugate cooordinates $z$, $\bar z$ on the complex plane.}
\be
g \to g_L(\xi^-) \, g \, g_R(\xi^+) \ , \qquad  g_L(\xi^-) \in G \ , \quad g_R(\xi^+) \in G \ . \la{cgs}
\ee
At the WZW fixed point $h=k$, the equation of motion becomes the statement of holomorphic and anti-holomorphic conservation of the Noether currents corresponding to the chiral gauge symmetries \rf{cgs},
\be
\del_- J_+ = 0  \iff  \del_+ K_- = 0 \ , \qquad J_+ \equiv g^{-1} \del_+ g \ , \quad K_- \equiv  \del_- g g^{-1} \la{WZWe} \ .
\ee
This equation is explicitly solvable in terms of arbitrary $G$-valued (anti-)holomorpic functions,
\be
g = g_1(\xi^-) \,  g_2(\xi^+) \ ,
\ee
with the currents $J_+ = J_+(ξ^+)$ and $K_-  = K_-(ξ^-)$ being holomorphic and anti-holomorphic on-shell. Computing the currents' Poisson brackets and canonically quantizing, one finds that they satisfy the affine Kac-Moody algebra of $G$ with level $k$, with the corresponding OPE,
\be\begin{aligned}
J_+^a(z) J_+^b(z')  \sim {}&{} \frac{1}{k}\Big[\frac{\ka^{ab}}{(z-z')^2} + \frac{i f^{ab}{}_c J_+^c(z')}{z-z'} \Big] \qquad  (+\text{ regular terms})
\ . \la{ca}
\end{aligned}
\ee
One can show that the affine Kac-Moody algebra contains the Virasoro algebra, i.e.\ the current algebra OPEs \rf{ca} imply that a certain  local operator $T(z) ≡k^2/2(k+)[  : J_+ J_+  : (z) ]$ satisfies the Virasoro OPE,
\be
T(z) T(z') \sim \frac{c}{2(z-z')^4} + \frac{2T(z')}{(z-z')^2} + \frac{\del T(z')}{z-z'} \ .
\ee
The antiholomorphic currents $K_-$ satisfy the same Kac-Moody algebra and generate another copy of the Virasoro algebra. Hence the WZW model is exactly conformal, being invariant under the 2d conformal algebra (two copies of the Virasoro algebra). Correspondingly the $$̱-function wzwb will vanish to all orders in a particular subtraction scheme (see the discussion around eq. 2103Pkb below). The corresponding conformal field theory, and particularly its correlation functions, were studied in <cit.>.

§.§ Gauged WZW model gWZW

There is a closely related family of 2d CFTs based on coset spaces $G/H$ <cit.> described by the gauged Wess Zumino Witten (gWZW) Lagrangian <cit.>,

k   Ł_G/H(g,A) =    k ( Ł_G(g) + [ J_+ A_- - K_- A_+ + g^-1 A_+ g A_- - A_+ A_-  ] )  ,  gws

J = g^-1 d g  ,   K = dg g^-1  ,     g ∈G  ,   A_±∈(H)  .
Like the symmetric space ss1, the gWZW model depends on a quadratic 2d gauge field (that may be integrated out to obtain a ) and is gauge-invariant under an action of the subgroup $H$,
g →h^-1 g h  ,   A_±→h^-1 A_±h + h^-1 _±h  . ggs

The classical gWZW  $\s$-model gws receives an infinite tower of $\a'$ corrections in order to be exactly conformal <cit.>. See the $SU(1,1)/U(1)$ example in eq. 6 and the review in Section <ref> below.

§.§ Integrable eta-deformation etas

Starting from an integrable like the PCM, one may consider squashing or deforming the target space geometry in a way that preserves the property of classical integrability. One such integrable deformation of the PCM, called the Yang-Baxter deformation  or $\eta$-deformation, is given by <cit.>
Ł= -  [ J_+   1/1-R̨ J_- ]   , ep
where $\k$ is the deformation parameter. The deformation is built from a linear  operator \mbox{$R:\Lie(G) \to \Lie(G)$}, antisymmetric with respect to the Killing form (i.e. \mbox{$\Tr[x R(y)] = -\Tr[R(x) y]$}) and solving the modified classical Yang-Baxter equation,
\be
[Rx,Ry]-R([Rx,y]+[x,Ry]) + c^2 [x,y] = 0 \ , \qquad x,y\in \Lie(G) \ . \la{ybe}
\ee
With $R$ satisfying these conditions, the $η$-deformed PCM \rf{ep} is classically integrable with the Lax connection again taking the the same form \rf{gl} as the PCM, now built from a different flat, conserved current \cite{Klimcik:2008eq}
\be
L_\pm = \ha (1+z^{\pm 1}) \mc{A}_\pm \ , \qquad \qquad \mc A_\pm = -(1+\k^2) \Ad_g\frac{1}{1\pm \k R} J_\pm \ .  \la{lae}
\ee

One can check that the integrability of the model (i.e.\ the equivalence between the flatness of \rf{lae} and the equations of motion of \rf{ep}) relies on $R$ being a solution of the Yang-Baxter equation \rf{ybe}. In general there may be various solutions of the Yang-Baxter equation and their classification is unknown.  We shall focus on a particular class of canonical solutions
in the ``non-split inhomogeneous'' case, i.e.\  taking $c^2>0$ in \rf{ybe}.\foot{We shall then rescale $R$ to set $c^2=1$. There are also other types of solutions with $c^2>0$.  The ``split inhomogeneous'' ($c^2<0$) solutions can only exist for non-compact groups and include  solutions of the same canonical type as \rf{DJR}; the  ``homogeneous'' ($c=0$) solutions include so-called ``abelian'' and ``Jordanian'' solutions. \la{ybs}} Given a choice of Cartan-Weyl basis ${h_i, e_, e_-}$  for $(G)$, where $h_i$ are the Cartan generators, $e_$ the positive roots and $e_-$ the negative roots, we shall assume a canonical choice of $R$-matrix (see Section 2.2 of \cite{Klimcik:2008eq} and refs.\ there) solving \rf{ybe},
\be
R(h_i) = 0 \ , \qquad R(e_\a) = - i e_\a \ , \qquad R(e_{-\a}) = i e_{-\a} \ . \la{DJR}
\ee
This $R$-matrix, and the associated $η$-deformed model \rf{ep}, depend on the choice of Cartan-Weyl basis. However, if one restricts to the case of compact simple groups, then there is only choice of Cartan-Weyl basis (up to equivalence by inner automorphisms).

The associated $η$-deformation in  \rf{ep}  breaks the manifest  global symmetry $G_L ×G_R$ of the PCM to the subgroup $G_L ×C$ where $C=U(1)^r$ is the Cartan subgroup of $G$ commuting with the $R$-matrix. The $G_R$ isometry of the target space is actually deformed to a  \textit{Poisson-Lie (PL) symmetry} \cite{Klimcik:2002zj}, with the associated Killing vectors ${v^m_a   _m }_a$ being replaced by vectors satisfying
\be
L_{v_c} (G+B)_{mn} =  - f^{ab}{}_c \, v_a^p  \, v_b^q \,  (G+B)_{mp} \, (G+B)_{qn}  \ . \la{PLs}
\ee
PL symmetry is thought to be related to a semi-classical limit of  quantum group symmetry \cite{Delduc:2016ihq}, and indeed the PL symmetries of the $η$-deformed models correspond to \textit{non-local} conserved charges  (that can be extracted from the monodromy matrix), the analogs of the leading Noether charges appearing in the expansion \rf{me} for the undeformed models. These non-local charges are found to have a classical Poisson bracket algebra obtained from a semi-classical limit of a quantum group algebra \cite{Delduc:2013fga,Delduc:2016ihq}. For this reason it is assumed that the full quantum $η$-deformed theories have quantum group symmetry, and this is the symmetry that has been assumed when bootstrapping their exact integrable S-matrices \cite{qds,Arutyunov:2015qva,ses}.

One can also simultaneously deform both the $G_L$ and $G_R$ parts of the global symmetry  to obtain the integrable `bi-Yang-Baxter' $$-model \cite{Klimcik:2008eq,Ki}.  The deformations can be generalized \cite{yosh,DMV}  to include WZ terms; additionally including `TsT transformations' \cite{betadef} gives a  4-parameter integrable deformation of the PCM \cite{Delduc:2017fib,Klimcik:2019kkf}.

Theories with  PL symmetry can be dualized in a process called \textit{Poisson-Lie duality} \cite{poissonlie},  generalizing non-abelian duality and abelian T-duality (see Section \ref{TS}). Under PL duality, the $η$-models \rf{ep},\rf{ess} are mapped  \cite{Hoare:2015gda,Klimcik:2015gba,Vicedo:2015pna,Sfetsos:2015nya,Hoare:2017ukq} to another class of integrable deformed models  called  $ł$-models, which we shall discuss in the next section.

A similar integrable $η$-deformation exists \cite{Delduc:2013fga} for the symmetric space \sm\unskip,
\be
\L = -\ha  \h  \, \Tr[ J_+ \, P_{G/H} \,  \frac{1}{1-\k \, R_g  P_{G/H}} \, J_-] \ , \qquad  \qquad R_g \equiv \Ad_g^{-1} R \, \Ad_g  \ . \la{ess}
\ee
The global $G_L$ symmetry of the symmetric space is thus deformed to a PL symmetry, with only a Cartan subgroup remaining as a manifest isometry (and the $H_R$ gauge invariance  remaining preserved).
The model is classically integrable with Lax connection of the same form \rf{sl} as the symmetric space in terms of different quantities $B, P$
\be \begin{aligned}
L_\pm = \mc{B}_\pm + z^{\pm 1} \mc{P}_\pm  \ , \qquad &\mc{B}_\pm = P_H A_\pm \ , \quad \mc{P}_\pm  = \sqrt{1+\k^2} P_{G/H} A_\pm  \\
&A_\pm = \frac{1}{1\pm \k R_g P_{G/H}} J_\pm
\end{aligned} \ee

In agreement with the conjecture that integrable \sms are stable under the RG flow, the $η$-deformed models are known to be renormalizable at the leading 1-loop order with just the couplings $h$ and $$̨ running <cit.>The renormalizability and $\b$-functions for the coset case \rf{cos} have only been directly studied in special cases, e.g.\ in \cite{Sfetsos:2015nya}. However, Poisson-Lie duality has been shown to be a symmetry of the 1-loop $$̱-functions <cit.> and, given that the $\eta$-models are understood to be related to the $\l$-models by PL duality <cit.>, the expression cos may be derived from the corresponding $\l$-model one.

\begin{align}
&G \, : \qquad   && \ddt \h = \cg(1+\k^2)^2 \ , \qquad && \ddt \k = \cg \h^{-1} \k(1+\k^2)^2 \ , \la{gro}\\
&G/H \, : \qquad  && \ddt \h  = 2 \cg (1+\k^2) \ , \qquad && \ddt \k = 2\cg \h^{-1} \k(1+\k^2) \la{cos} \ .
\end{align}

This is a non-trivial example of the integrability–RG flow relationship because the $\eta$-deformed models are not automatically RG-stable due to manifest symmetries.  Their stability is instead thought to be due to their hidden integrable symmetries.

§.§ Integrable lambda-deformation las

In <cit.> Tseytlin considered a “universal class” of models related to gWZW models $\L_{G/F}$ by deformations parametrized by a constant linear  operator $Q:\Lie(G)\to \Lie(G)$,
Ł_Tseytlin = k ( Ł_G/F(g,A) + [ A_+ (Q-1) A_- ] )  ,     g∈G  ,   A_±∈(F)  . uc
The 1-loop conformal invariance conditions were computed, taking the form of polynomial constraints on

Forgetting about conformal invariance, one may instead ask which choices of $Q$ correspond to integrable theories. A particular class of integrable examples of uc was later discovered <cit.>, called the $\l$-model based  on a either group $G$ or a symmetric space $G/H$. The model is defined by the choice $F=G$ and $Q=1-(\l^{-1}-1)P$ in uc, or
Ł_ł= k ( Ł_G/G(g,A) - (ł^-1 -1) [ A_+   P   A_- ] )  . lam
Here $k$ is a WZ level and the coupling $\l$ is a deformation parameter (we will sometimes use a redefined coupling $\ka \equiv \frac{1-\l}{1+\l}$ for convenience). The two cases of $G$ or $G/H$ are encoded in the choice of the projector $P$,
P =  1            (G case)  ,
P_G/H            (G/H case)  ,
so that  the coset case has an $H$ gauge invariance (see eq. ggs), while the group case has no gauge symmetry at all.

In the group case there is a global $G$ symmetry, $g \to k g k^{-1}$, $A_\pm \to k A_\pm k^{-1}$, $k\in G$; whereas the coset case does not generally have any continuous global symmetries.
In both cases, the $\l$-model lam is invariant under a
$\mathbb{Z}_2$ transformation of the fields and coupling constants <cit.> (see also <cit.>),
Such a symmetry was also discussed  for the models uc in <cit.> (see footnotes 3  and 6 there).

k →- k  ,         ł→ł^-1  ,

g→g^-1  ,     A_+ →A_+ - (1-ł^-1) P A_+  ,     A_- →g A_- g^-1 - K_-  .
As we shall find below in Chapter <ref>, the preservation of this symmetry at the quantum level may require a particular choice of
regularization scheme.

One may integrate out the  2d gauge field in lam to obtain a depending on the scalar $g\in G$ (here $\L_G$ is the WZW Lagrangian),

Ł_ł= k ( Ł_G(g) + [ J_+ M(g)^-1 K_- ] )  ,        J ≡g^-1 dg  ,   K ≡dg g^-1  ,  lami

M(g) ≡Ad_g - 1  + (1-ł^-1)P  .
The $\l$-model is an integrable deformation of a  WZW model on $G$ or a gauged WZW model on $G/H$, obtained in the limit $\l=0$  when the projected parts $P A_\pm$ decouple and are set to zero by their equations.

In another interesting limit,

k →∞ ,     ł= 1 - 2k   ,     g=e^-h2kv  ,

v ∈(G)  and   fixed  ,
the $\l$-model becomes the non-abelian dual theory NADt corresponding to either the PCM or symmetric space . The theory lam with unintegrated gauge field becomes the corresponding “interpolating model” NADi.

The $\l$-models lami (after integrating out $A_\pm$) admit flat Lax connections of the same form gl,sl as the PCM/symmetric space, but with modified currents,

\begin{align}
&G \, : \qquad && L_\pm = \ha(1+z^{\pm 1}) \mc A_\pm \ , \qquad &&  \mc A_\pm = \frac{2}{1+\l} A_\pm  \ ,  \\
&G/H \, : \qquad && L_\pm = \mc B_\pm + z^{\pm 1} \mc P_\pm \ , \qquad &&   \mc B_\pm = P_H A_\pm \ , \ \ P_\pm = \tfrac{1}{\sqrt{\l}} P_{G/H} A_\pm  \ ,
\end{align}

where $A_\pm = A_\pm(g)$ are taken to be their on-shell values.

In line with the conjecture, the $\l$-model is known to be renormalizable at  1-loop  with only $\l$ runningThe 1-loop $\b$-functions \rf{la1},\rf{la2}
can also be extracted from \cite{Tseytlin:1993hm}. Note that the WZ level is not running due to general arguments.} \cite{Sfetsos:2014jfa,Appadu:2015nfa}
\begin{align}
&G \, :  && \ddt k = 0 \ , \quad \ddt \l = -  \frac{2 \cg}{k} \frac{\l^2}{(1+\l)^2}  \ ,  \la{la1} \\
&G/H \, :  && \ddt k = 0 \ , \quad \ddt \l = -  \frac{\cg \l}{k}  \ . \la{la2}
\end{align}
These $$̱-functions  respect the $\mathbb{Z}_2$ transformation z2sym, being invariant under the transformation $(k,\l) \to (-k,\l^{-1})$ of the couplings.

§.§ Integrable-deformed string backgrounds

Although we have focussed on the simplest case of bosonic theories, and relaxed the condition of Weyl invariance (i.e. that the target space is a string background), the models discussed above do admit Weyl-invariant superstring generalizations. The Green-Schwarz action for type IIB strings on with RR flux  is a on the $\mathbb{Z}_4$ “semi-symmetric” coset $PSU(2,2 | 4)/[SO(1,4) \times SO(5)]$ with a particular WZ-type $\BB$-field term (here $P^{(i)}$ is the projector onto the $i$th graded part of the algebra) <cit.>
S =  14π' ∫  d^2 ξŁ ,     Ł= -R^2 STr [√(-h) h^ J_^(2) J_^(2) + ^ J_^(1) J_^(3) ]  ,

J^(i) = P^(i) J  ,     J = g^-1 dg  ,     g∈PSU(2,2|4)  ,
This is a generalization of the symmetric space model ss2 above (replacing the $\mathbb{Z}_2$-automorphism by $\mathbb{Z}_4$) and indeed the bosonic part of the model (setting fermions to zero) is simply the symmetric space on $= {SO(2,4) \ov SO(1,4)}\times {SO(6) \ov  SO(5)}$, with $R$ being the radius of the $AdS_5$ and $S^5$ spaces.

The $\eta$-deformation  <cit.> and $\l$-deformation <cit.> can both be  generalized to integrable deformations of semi-symmetric spaces.
The $\eta$-deformed model was problematically found to be scale- but not Weyl-invariant, with the background solving only the generalized supergravity equations  <cit.>, and the perturbative light-cone S-matrix did not directly match the conjectured exact S-matrix <cit.>. It was then explained <cit.> that, to maintain Weyl invariance, one must in fact use the purely fermionic Dynkin diagram for the superalgebra $\mathfrak{psu}(2,2 | 4)$ to construct a unimodular $R$-matrix. In the recent paper <cit.>, an exact S-matrix was conjectured that is different from the previous attempt, but related by a so-called `twist';  this S-matrix indeed matches tree-level perturbative results.

Much remains to be
understood about these integrable-deformed string backgrounds.
For example one significant challenge is to understand the spectrum of the deformed theories
(a direction in which some progress was made for the $\eta$-deformation <cit.> through the TBA and  Quantum Spectral Curve).
Examining the  properties of the
bosonic $\eta$-model and $\lambda$-model
can provide
valuable insights into their superstring counterparts. For example, the
relation between the $\eta$- and $\l$-models by Poisson-Lie duality was first understood in the bosonic case <cit.>.
It is thus natural to first explore the question of quantum corrections and the higher-loop RG equations by
studying the bosonic models.
In Section <ref> below, we shall comment on the possible applications of our approach to
the integrable-deformed string backgrounds.

It is also worth emphasizing that the bosonic models are of interest in their
own right in the general context of investigating  integrable 2d theories.
The $\eta$-model has played an important role in generalizing the duality <cit.>
between the deformed $O(3)$ and $O(4)$ sigma models and massive integrable QFTs
to higher-rank groups and  cosets <cit.> and cosets of supergroups <cit.>. While the
dual theories are quantum-exact, the side of the duality is only
understood so far to leading order in the loop expansion. Thus, after
finding quantum corrections to integrable consistent with
renormalizability, it may then be interesting to study their compatibility with this duality.

There are also many other known examples of integrable string backgrounds, including those based on
lower-dimensional AdS spaces <cit.> and their integrable deformations <cit.>;
$\beta$-deformations (or TsT-deformations) <cit.>; and more general homogeneous Yang-Baxter deformations <cit.>.

§.§ Quantum integrable S-matrices QIS

The presence of higher conserved charges places strict constraints  on the possible scattering behaviour of a theory <cit.> (see also the review <cit.>). Particle production is prohibited, i.e. the number of incoming  and outgoing particles are equal; the set of initial momenta equals the set of final momenta; and the S-matrix must factorize into a product of $2\to 2$ S-matrices. The consistency of this factorization implies that the $2\to 2$ S-matrix satisfies the Yang-Baxter equation.

This special scattering behaviour is  expected for integrable theories, and has been proven assuming the presence of at least two higher-spin conserved charges satisfying certain technical conditions <cit.>. S-matrices of this type may be seen as “almost free”, and their existence is a special feature of 2-dimensions since in higher dimensions the presence of higher-spin charges implies triviality of the S-matrix (i.e. a free theory) by the Coleman-Mandula theorem.

For certain theories there are various arguments for the preservation of integrability at the quantum level. For example, in the case of certain principal chiral models, it has been  argued <cit.> using consistency with global symmetry that the higher-spin conservation laws cur cannot be modifed at the quantum level by anything other than a divergence, i.e. a quantum-corrected form of the current must still conserved to all orders in perturbation theory (see also <cit.> in the context of the $O(n)$-model, i.e. the $\s$-model on a sphere $S^{n-1}={O(n)\ov O(n-1)}$.). The non-local charges of the $O(n)$ model were also argued to persist at the quantum level <cit.> and to imply factorization of the S-matrix with no particle production.

In the important paper <cit.>, Zamolodchikov and Zamolodchikov started by assuming quantum integrability and the dynamical generation of a mass-gap for the  $O(n)$-model. Using also the model's symmetries and assuming a `minimal' solution for the phase ambiguity (with minimal number of poles on the physical sheet), they uniquely determined the exact S-matrix for non-perturbative quantum particles.

The $O(4)$-model, equivalent to the $S^3$ $\s$-model or $SU(2)$ PCM, was later  solved  explicitly <cit.> by proving equivalence to a Thirring-type fermionic model with an infinite number of flavours, finding agreement with the Zamolodchikov-Zamolodchikov S-matrix. This exact solution was later generalized   to the PCM on $SU(N)$ <cit.> and with arbitrary WZ term (PCM$_k$) <cit.>.See also <cit.>, where the exact ground state energy was computed in the large-$N$ limit.

The Zamolodchikov-Zamolodchikov `rational' S-matrices (with rational dependence on rapidities) are known to admit `trigonometric' deformations <cit.>. The 2d “sausage model” was then found <cit.> as a semi-classical $\s$-model description of the trigonometric $O(3)$ S-matrix, and this  model was later understood <cit.> to be the $SU(2) \ov U(1)$ example of the $\eta$-model. More generally, similar trigonometric S-matrices are understood to be associated to inhomogeneous Yang-Baxter deformations and the associated $q$-deformed symmetries <cit.> (see also recent papers <cit.> on the light-cone S-matrix for $\eta$-deformed string backgrounds and refs. therein).

§.§ Factorized scattering vs. tree-level sigma model S-matrix fv

Given that integrable S-matrices are so constrained, one might  hope to classify the integrable $\s$-models by (i) computing the S-matrix for a general $\s$-model in terms of the couplings $(\GG,\BB)$ (ii) imposing factorization and no particle production and (iii) systematically reading off the resulting constraints on $(\GG,\BB)$  from integrability.
Like the RG flow (which is the focus of the thesis), the S-matrix could thus serve as another `diagnostic tool' for integrability.

Here we shall summarize the paper <cit.> where we considered such a procedure for the leading tree-level S-matrix,
whose factorization should correspond to classical integrability. The factorization of the loop-corrected S-matrix should then correspond to preservation of integrability at the quantum level. For example, for the complex sine-Gordon model, we recall that the tree-level 2-body S-matrix satisfies the Yang-Baxter equation but the 1-loop-corrected S-matrix only satisfies it if the theory is supplemented by certain quantum corrections <cit.>. These corrections may be traced to the leading quantum corrections to a gauged WZW model (see eq. 6 below) — of which the complex sine-Gordon model is a massive integrable deformation. The corrections are thus expected to correspond to a particular choice of finite counterterms consistent with quantum integrability, similar to those discussed below in the context of the RG flow.

To compute an  S-matrix in perturbation theory, one must first decide on a choice of vacuum. In <cit.> we considered the `trivial' vacuum $x^m=0$ (with all fields vanishing) in an expansion of the metric and $\BB$-field  around flat space,  This is a natural choice since it is universal to any target space geometry. However, the  fields $x^m$ expanded  around this vacuum are massless 2d scalars, which are known to suffer from IR divergences at the quantum level <cit.>.  This is related to the fact that  massless particles (moving at the speed of light) on a 1d spatial line  do not separate asymptotically, so an S-matrix cannot have the usual physical interpretation. Still, massless 2d S-matrices were formally discussed for integrable theories in the context of a finite-density
TBA <cit.>. The S-matrix there retains the interpretation as the relative phase when one particle is moved past

One might still hope to define a formal tree-level S-matrix for such massless particles (e.g., from the classical action evaluated on a solution with special scattering boundary conditions), but we found  in  <cit.> that such an object still has IR singularities. The singularities are not true divergences but instead  ambiguous “0/0” contributions due to a vertex vanishing and an internal propagator  blowing up at the same time when the scattering amplitude is taken on-shell. We classified the ambiguities into two types and considered some simple approaches to regularize them (the usual $i \e$ prescription, or an alternative “massive regularization” where the particles are given a small mass). One can compute finite scattering amplitudes by regularizing the ambiguities, taking the amplitude on-shell, and finally taking the regulator to zero. However, we found that the resulting amplitudes do not have the usual connection to integrability:  in various well-known integrable examples the amplitudes exhibit particle production. These findings were in agreement with earlier attempts to compute scattering amplitudes for massless 2d scalars in integrable theories <cit.>.

Opposite conclusions have been found for massless 2d S-matrices in non-renormalizable (non scale-invariant)
Nambu-like models <cit.>. For Nambu theory with a flat target space (but not for a curved one), each scalar appears
in the action only through its derivative and thus ambiguities related to IR poles appear to be
absent, not only at the tree level but also at the loop level. Here the (naively expected) relation
between factorization of the massless S-matrix and integrability does appear to hold (with the
Nambu action being integrable beyond tree level only in special critical dimensions).
Moreover, the flat-space Nambu theory is an example of a $T\bar{T}$-deformation <cit.> of a free theory, for which integrability apparently constrains the  RG flow even in non-renormalizable cases, or rather  dictates the fixing of the counterterms  to be  consistent with the integrable S-matrix   <cit.>.

A similar S-matrix approach has been used <cit.> to rule out integrability of strings on certain (warped) AdS supergravity backgrounds  by expanding around non-trivial classical solutions (BMN geodesic <cit.>, GKP spinning string <cit.> or `null cusp' <cit.> solutions) and fixing a light-cone gauge. These vacua give a mass to at least some fields and one may then consider particular worlsheet scattering amplitudes that do not contain any ambiguous contributions from massless fields. While this approach is not easily applied to generic backgrounds (where one does not have a good understanding of classical solutions), it would be interesting in future to consider a more general class of backgrounds than AdS ones, but still assuming  enough isometries to allow a similar choice of vacuum.

§.§ Affine Gaudin models and local charges agm

There is a proposal to systematize integrable 2d theories by viewing them as realizations of affine Gaudin models <cit.> (see also <cit.>).  It so happens that all of the integrable $\s$-models described above are examples of such theories. This is indeed suggestive they may describe a large portion of the integrable theories.

The affine Gaudin-type integrable models have flat Lax connections zc. They also have a systematic construction of local, higher-spin, classically conserved charges, generalizing those constructed in curG for the PCM/symmetric space and  models with Lax connections of those same form. The classical Poisson brackets of the spatial component of the Lax connection $\L \equiv L_\s=\ha (L_+ - L_-) \in \Lie(G)$  take the  Maillet $r/s$-form <cit.> (here $z_1,z_2$ are spectral parameters and $\s_1,\s_2$ are spatial coordinates),
{ Ł_1(z_1; _1) ,Ł_2(z_2; _2) } =     ( [r_12(z_1,z_2), Ł_1(z_1; _1) + Ł_2(z_2; _1)]

+ [s_12(z_1,z_2), Ł_1(z_1; _1)- Ł_2(z_2; _1)] )  _̣_1 _2 - 2 s_12(z_1,z_2)  δ__1_2'  .
Here the underlined $\ul{1},\ul{2}$ indices in mai denote the first and second factors in a tensor product $\Lie(G) \otimes \Lie(G)$, e.g. $\L_{\ul 1} \equiv \L \otimes 1$, $\L_{\ul 2} \equiv 1\otimes \L$. The Poisson bracket mai is called `non-ultralocal' because of the presence of the $\delta'$ term. The constant $r$- and $s$-matrices $r_{\ul{12}}(z_1,z_2) = - r_{\ul{21}}(z_2,z_1)$ and $s_{\ul{12}}(z_1,z_2) =  s_{\ul{21}}(z_2,z_1)$ are respectively antisymmetric and symmetric, and they take the special “twist” form,The form of twist is slightly modified is so-called `cyclotomic' examples involving twisting by a $Z_T$ automorphism, e.g. the symmetric space $\s$-model with $T=2$; here we ignore such cases.
r_12(z_1,z_2) + s_12(z_1,z_2) = 1/(z_2-z_1) ϕ(z_2)    T_A  ⊗T^A  , twist
in terms of a model-dependent meromorphic  twist function $\p(z)$ of the spectral parameter. Here the `split Casimir' $ T_A  \otimes T^A$, built from any set of generators $T_A$  for $G$ and the dual generators $T^A$ (with respect to $\ka_{AB} \equiv \Tr[T_A T_B]$), is invariant under  change of basis.

At each zero $z_i$ of the twist function $\phi(z)$ (or more properly zeros of the 1-form $\phi(z) dz$)In particular the poles of $\L$ will be zeros of the twist function., one can automatically construct from mai an infinite tower of local, higher spin charges. Assuming for simplicity the Lax connection to be valued in a $B$, $C$ or $D$ type algebra (the $A$ type case is slightly more complicated), the following  charges $\{Q^{(2n)}(z_i)\}_{n,i}$  are all in involution <cit.> (assuming a well-defined limit of $\phi(z)\L(z)$ at $z_i$),
{ Q^(2n)(z_i), Q^(2m)(z_j) } = 0  ,     Q^(2n)(z_i) = ∫d  [ (ϕ(z) Ł(z))^2n] |_z=z_i  . trr
For realizations of affine Gaudin models, the Hamiltonian is a linear combination of some of these charges, so all of the charges are conserved (commuting with the Hamiltonian).

For the group-based models above (PCM, PCM$_k$, group $\eta$-model, group $\l$-model) with Lax connections of the form gl in terms of a flat, conserved current $\mc A_\pm$, let us re-define the spectral parameter $z =\tfrac{1+w}{1-w}$ for simplicity so that the Lax matrix $\L=L_\s$ has simple poles at finite values  $w=\pm 1$. These models' twist functions all have simple zeros at these points $\pm 1$, and take the form $\phi(w) = \tfrac{(1-w)(1+w)}{p(w)}$ where $p$ is a polynomial not vanishing at $\pm 1$. The resulting charges at $w_1 =1$ and $w_2 =-1$ are proportional to the  local charges obtained from the (anti-)holomorphic currents in curG above,We seem to have produced only half as many charges in chag as in curG, but in fact all of the odd currents $J^{(2n+1)}$ vanish for $B$, $C$ and $D$ type groups <cit.>. Formally this procedure applies to the complexifed Lie group and one specifies a real form by some reality conditions. This does not affect the charges in this case since the zeros $w=\pm 1$ of the twist function are real and so the charges chag are real.

\begin{align} \la{chag}
\ddt Q^{(2n)}(\pm 1) = 0 \ , \qquad Q^{(2n)}(\pm 1) \sim \int d\s \, \mc J^{(2n)}_\pm \ , \qquad \mc J^{(2n)}_\pm = \Tr[ \mc A_\pm^{2n} ] \ .
\end{align}

Here we have used the trace but, as in curGe3G above, we could have instead  used any symmetric, invariant tensor in trr.

Thus one learns from trr that the higher-spin charges chag are all in involution.
It also follows from the Poisson bracket mai that these charges are in involution with the monodromy matrix and its non-local conserved charges <cit.> as constructed in Section <ref>. A similar construction of local, conserved charges applies  to models based on symmetric spaces (or other $Z_T$-cosets).

As well as re-interpreting known models, the affine Gaudin approach has the power to generate new integrable theories, essentially by picking new twist functions $\phi(z)$. For example, the $\eta$-deformations of  symmetric space <cit.> and semi-symmetric space <cit.> $\s$-models  were discovered in this way (based on the previously known group space case); more recent examples include the integrable $G^N$ models  <cit.>, deformed $G^N$ models <cit.> and $G\times G/H$ models <cit.> (see Chapter <ref>).

The affine Gaudin construction systematizes certain reformulations of these integrable $\s$-models.  For a certain class of twist functions, the affine-Gaudin-type models are equivalent <cit.>  to “E-models” <cit.> (a first order formulation with manifest Poisson-Lie symmetry), and can also be obtained  <cit.> from 4d Chern-Simons theory <cit.>.

Finally we remark that the affine Gaudin formalism could lead to a more systematic understanding of the RG flow of integrable $\sigma$-models, or perhaps even a first-principles proof of their renormalizability. Along these lines, a recent paper <cit.> noted a pattern in the 1-loop RG flow for several integrable affine-Gaudin-type $\s$-models: in each case it may be written in the same universal way as a flow of the twist function. It would be interesting to try to derive this statement, or to check it beyond the leading 1-loop order. One complication is that, beyond the  1-loop order, we should expect the integrable theories to require a quantum deformation to remain RG-stable (see below); it is not yet clear how these quantum deformations would be encoded in the affine Gaudin language.

§.§ On scheme dependence SD

Under covariant scheme transformations, or redefinitions of the couplings $(\GG,\BB)$ (in terms of constants $c,d,e$),
_mn →_mn + ' (c   R_mn + d    _mpq_n^pq) + … ,   _mn →_mn + '   e   ∇^p _pmn + … , csc
the $\b$-function $ _̱mn = (+)_mn$ transforms as a contravariant vector in the space of couplings. Simple power counting implies that the leading 1-loop $$̱-function is invariant while, in general, scheme dependence will begin at the 2-loop order.For the special case of vanishing $\BB=0$, the may be viewed as a “single-coupling” theory and its 2-loop $\b$-function \rf{2} is also invariant (see Appendix \ref{A}).}

In this thesis, we will generally use the
2-loop $β$-functions given in \rf{BMT},
which were computed  in a  special 2-loop   scheme
that effectively  treats
$_mn$ and $_mn$  as symmetrically as possible  \cite{Bos:1987mw,mt} (see also \cite{Hull:1987pc,Ketov}).
Indeed,  the
$β$-functions   for $$ and $$ in \rf{BMT} are given by the symmetric and antisymmetric parts of a single tensor expression.
We shall refer to this $$-$$ symmetric scheme   as  the
``\GB scheme'' (see the introduction to Chapter \ref{GGS} for discussion of the properties of this scheme).

It is important to emphasize, however, that our conclusions will be scheme-independent. In particular, the statement that `a model is not renormalizable' is understood to mean that it is not renormalizable in any covariant subtraction scheme. We shall find certain finite counterterms that restore higher-loop renormalizability and, while the form of these counterterms will change in different subtraction schemes, they cannot be eliminated by covariant scheme changes such as \rf{csc}.

\chapter{Quantum corrections to geometry  from  RG flow \la{QCS}}

As discussed in the Introduction, it was observed for some years that  integrable \sms appear to be stable under the leading 1-loop RG flow, and our purpose here is to investigate what happens at higher-loop orders. The general finding is that the integrable theories remain stable under the higher-loop RG flow provided they are supplemented with particular finite counterterms, i.e.\ the classical target space geometry $(+)_mn^(0)$ receives certain quantum corrections,
\begin{align}
(\GG+\BB)_{mn} = (\GG+\BB)_{mn}^{(0)} + \a' (\GG+\BB)_{mn}^{(1)} + \ldots  \  . \la{qc2}
\end{align}

One might first try to study the integrability--RG flow relationship  for a canonical integrable example such as the PCM,
\be
\L_{\PCM} = -\ha \h  \Tr[ J_+ J_- ]    \ . \la{PCMk}
\ee
However, the question trivializes for this simple example, because the renormalizability of the PCM follows immediately from its   large $G ×G$ global symmetry (and parity invariance)
\be
g \to g_L \,  g  \, g_R \ , \qquad (g_L, g_R )  \in G_L  \times  G_R \ . \la{PCMs}
\ee
Only the curvature radius $√(h)$ can run because the theory \rf{PCMk} does not admit any deformations preserving these symmetries.
This conclusion is consistent with the conjecture that integrable models are renormalizable, but it is not good evidence for it: we  only used manifest global symmetry, without relying on the  hidden integrable symmetries.

In order to really probe the role of integrability in constraining the RG flow, we shall consider integrable theories with less global symmetry. As we shall observe,  integrability appears to be sufficient for renormalizability,  even in cases without any manifest global symmetries.
In this chapter we shall focus on  the integrable $η$- and $ł$-deformations introduced in Sections \ref{etas} and \ref{las}, which
are classically integrable but have reduced global symmetry groups.  In particular, these models are not fully constrained by their global symmetries, and renormalizability does not follow from global symmetries alone.

First in Section \ref{Seta}, we shall consider the simplest $η$-deformed model
based on the coset $SU(2) U(1)$.  Then in secton \ref{Slam}, we shall consider $ł$-deformed models
based on the coset $SU(2) U(1)$ and the group $SU(2)$. As we reviewed above, these models are already known to be stable under the leading 1-loop RG flow; we shall determine the quantum corrections
required to solve the 2- and higher-loop RG equation \rf{BMT}.

\

\noindent This chapter is based on the papers \cite{HLT1,HLT2} and closely follows the presentation there.

\section{On quantum corrections in sigma models \la{qcs}}
Before proceeding, let us provide  justification for the  presence of quantum corrections \rf{qc2}  in $$-models
by  recalling some familiar examples where such corrections are known to occur.

One class of  examples are
the special integrable \sms  \rf{gws} corresponding to gauged WZW conformal field theories.
The classical $$-models \rf{gws} are indeed scale-invariant, i.e.\ fixed-point solutions of the RG equation \rf{BMT}, at the leading 1-loop order.
However, to remain conformally invariant beyond 1-loop, the target space metric and $$-field
require non-trivial $'$ corrections at each order,
which may be derived exactly
the underlying quantum gWZW theory
(for discussion and examples see \cite{Tseytlin:1993df,Horowitz:1994ei,Tseytlin:1995fh}).
The simplest example is provided by the exact counterpart \cite{Dijkgraaf:1991ba} of the
familiar classical $SU(1,1)/U(1)$ gWZW metric (or 2d Euclidean black hole) \cite{Bardacki:1990wj,Witten:1991yr}
\begin{align}
ds^2 = k(dr^2 + \tanh^2 r \, dy^2) \ \ \to \ \
ds^2 = (k-2) \Big[ dr^2 + \big( \coth^2 r - \tfrac{2 }{k}\big)^{-1} dy^2\Big] \la{6} ~.
\end{align}
Here the inverse level $k^-1$ (and later the inverse coupling $h^-1$) plays the role of the loop-counting parameter $'$  in \rf{13}

As was checked directly in \cite{Tseytlin:1991ht}, the leading $k^-1$ correction in \rf{6}
is  required to solve the 2-loop scale invariance equation following from \rf{2}
(see also \cite{Jack:1992mk} for a 4-loop test of \rf{6}).\foot{In this Weyl-invariant case
the diffeomorphism  vector $\xi_n$ can be written as $ 2 \del_n \Phi$ where the exact dilaton field $\Phi$ is given by
$e^{-2 \Phi}= \sinh 2 r \ \big( \coth^2 r - \tfrac{2 }{k}\big)^{1/2}$. \la{f4}}
The integrable cases that follow will be a direct generalization of this principle: instead of ensuring complete
scale-invariance, the quantum corrections will ensure invariance under the RG flow with just a few couplings running.

It is also well-known that the classical Buscher rules for abelian T-duality are subject to quantum corrections \cite{Tseytlin:1991wr,Panvel:1992he,Kaloper:1997ux,Parsons:1999ze}, so geometries resulting from T-duality will contain non-trivial $'$ corrections. For example,
given the metric
$ds^2 = [dx^i dx^i + (x) dy^2] $  with an isometry direction $y$, the standard T-duality rule $→= ^-1$ is known to be
modified at the 2-loop level \cite{Tseytlin:1991wr} (see also \cite{Haagensen:1997er,Haagensen:1997bh})
\be
\wt \cM = \cM^{-1} \big( 1 + \tfrac{1}{2} \hh^{-1} \, \del_i \log \cM\, \del^i \log \cM \big) \ . \la{5}\ee
In an example below in Section \ref{Slam}, this quantum correction to the T-duality transformation will be reproduced by special limits of the corrections for the $ł$-model. We shall also produce similar corrections to the transformation rules of non-abelian duality in some special examples.

\section{\texorpdfstring{$\eta$}{eta}-model \la{Seta}}

The metric of the 2d ``sausage'' model\foot{We note that $\hh$ in this chapter is equivalent to $4h$ in \cite{HLT1}.}
\be
ds^2 = \tfrac{1}{4} \hh \Big[ \frac{dr^2}{(1-r^2)(1+\k^2 r^2)} + \frac{1-r^2}{1+\k^2 r^2} d\p^2 \Big] \ , \la{21}
\ee
was originally written down in \cite{Fateev:1992tk} as the leading semi-classical approximation to the \sm corresponding to the massive integrable trigonometric S-matrix of \cite{Zamolodchikov:1980ku}. It was discovered as a solution to the 1-loop RG equation (the leading Ricci flow term in \rf{2}) and conjectured to be classically integrable. Its classical integrability was later shown in \cite{lukyanov}. In \cite{Hoare:2014pna} this model was identified as the $SU(2) U(1)$ example of the $η$-deformation \cite{Klimcik:2002zj,Delduc:2013fga}, introduced in Section \ref{etas}.\foot{It is interesting to note that the metric \rf{21} is formally self T-dual, i.e. invariant under
$\phi\to \td \phi $ and $\td \phi \to i \k^{-1} \phi$,\ \ $r \to i \k^{-1} r^{-1}$. \la{fstd}}

The metric \rf{21} is an integrable deformation of the round $S^2$ ($=̨0$) of radius $√(/4)$, with
$$̨ being the deformation parameter.
The usual $\eta$-deformation  correspnds to the case when $\varkappa$ is real, however the authors of  <cit.> considered another regime of interest with  $\varkappa$  imaginary, in which case the is UV stable with a free UV fixed point at $\k^2 = -1$.
Alternatively, we can send $\k^2\to -1$ while simultaneously expanding around $r^2 = 1$.
Taking the limit in this way, the metric 21 reduces to the classical metric of the $\frac{SU(1,1)}{U(1)}$ gWZW model (cf 6),
r^2 = 1 - (1+^̨2) sinh^2  ,       ^̨2 →-1 22 ,     ds^2→14( d^2 + tanh^2   d^2)  , with $\h/4$ becoming the level.
Another useful limit is the maximal deformation limit $\k \to \infty$, $\hh\to \infty$ with $\hh' \equiv \tfrac{\hh}{\k^2}$ fixed, which yields
<cit.> the undeformed hyperbolic space $H^2$  with radius $\sqrt{\hh'/4}$ (here we set $r=\tfrac{1}{\cosh{ \theta}}$)
ds^2 = '/4 r^2 [ dr^2/1-r^2 + (1-r^2) d^2 ] = 14 ' [ d θ^2 + sinh^2 θ  d^2 ]  . 222

The metric 21 solves the 1-loop RG equation 2 with $\hh$ and $\k$ running as (eq.\ \rf{cos} with $c__SU(2)=2$)
\be \la{23}
\dt \hh = 4 (1+\k^2) + \OO(\hh^{-1}) \ , \qquad \qquad
\dt \k = 4 \hh^{-1} \k (1+\k^2) + \OO(\hh^{-2}) \ . \ee
There is an RG-invariant combination of couplings
\be
\dt \nu =0\ , \qquad \qquad \la{244}
\nu \equiv {\hh \ov 4 \k} + \O(\hh^0) \ . \ee
In agreement with the above comments,
$^̨2=-1$ is a UV fixed point (corresponding to either a free theory or a gWZW model depending how the limit is taken).

However,  including also the 2-loop term in the RG equations \rf{2}, the $$-model on the classical sausage geometry \rf{21} appears not to be renormalizable: it does not solve the 2-loop RG equations with only $$ and $$̨ running. This is true in any covariant subtraction schemeAs discussed in Appendix <ref>, for 2d target spaces, the scheme dependence does not begin until 3-loops. and with any diffeomorphism vector (here one only needs to consider diffeomorphism vectors  of the form $\xi^m=\xi^m(r)$ since $\phi$ is an isometry direction).

The relation to the gWZW model suggests that,
for 21 to remain a solution to the RG equation at higher loops with only $\hh$ and $\k$ running,
it should similarly be modified by quantum (i.e. $1/$) corrections.
Inspired by the analogy with the form of the gWZW metric in \rf{6} we propose
the following conjecture for the exact generalization of \rf{21}
\begin{align}
ds^2 = (\tfrac{1}{4} \hh- 1+\k^2 ) \Big[ \frac{dr^2}{(1-r^2)(1+\k^2r^2)} + \big(
\frac{1+ \k^2r^2}{1- r^2} + \frac{8\k^2}{\hh} \big)^{-1}
d\p^2 \Big] \ . \la{24}
\end{align}
The metric \rf{24} is expected to solve the RG equations to all loop orders
(in a particular scheme and modulo coordinate redefinitions) with $$ and $$̨ running
according to a generalization of 23.
The conjecture passes some obvious tests:

* The metric 24 reduces to $S^2$ when setting $\k=0$, now with
shifted radius-squared coupling
${\tfrac{1}{4}\hh-1}$, which is simply related to the classical one $\tfrac{1}{4}{\hh}$  by a finite coupling redefinition.
Alternatively, one may shift $\h\to \h+4(1-\k^2)$ in 24 to obtain
$ds^2 = \tfrac{1}{4} \h \Big[ \frac{dr^2}{(1-r^2)(1+\k^2r^2)} + \big(
\frac{1+ \k^2r^2}{1- r^2} + \frac{2\k^2}{\tfrac{1}{4} \h+1-\k^2} \big)^{-1}
d\p^2 \Big] ,$
so that the radius-squared of $S^2$ remains $\tfrac{1}{4}\hh$ in the $\k=0$ limit.

* The metric remains flat for $\k^2=-1$.

* In the non-trivial $\k^2 \to -1$ limit 22 it reduces to the exact gWZW metric in 6 with level $k=\h/4$.

* In the maximal deformation limit 222 it reduces to $H^2$, now with shifted radius-squared coupling  $\h'/4+1$.

More importantly,
one can directly check that 24 with the leading $h^{-1}$ correction
ds^2    = [ dr^2/(1-r^2)(1+^̨2r^2) (1/4-1-^̨2)

+ 1- r^2/1+ ^̨2 r^2 (1/4 - 1-^̨2 - 2 ^̨2   1- r^2/1+ ^̨2 r^2 + …) d^2 ]  , 277
indeed solves the 2-loop RG equation
2, which becomes in this case with 2d target space,In 2 dimensions the curvature tensor can be written as
$R_{mnkl} = \ha R ( \GG_{mk} \GG_{nl} - \GG_{nk} \GG_{ml})$. Note that any $\BB$-field on a 2d space is trivial (with zero field strength $\H=d\BB$) so we use the $\b$-function \rf{2} for the case of vanishing $$-field (also taking $ł_m=0$).}
\be \la{27}
\dt \GG_{mn} = (\ha R + \tfrac{1}{4} R^2 ) \GG_{mn} + \ldots + \nabla_{(m} \xi_{n)} \ ,
\ee
for a particular diffeomorphism vector $ξ_n$ with components,\foot{We note that the diffeomorphism vector \rf{2777} begins at 2-loop order; the 1-loop diffeomorphism vanishes in these coordinates.}
\be \la{2777}
\xi_\p = 0 \ , \qquad \quad \xi_r = \frac{8\k^2 (1+\k^2) r}{\h (1+\k^2 r^2)^2} \ .
\ee
The $β$-functions \rf{23} receive the following 2-loop corrections
\begin{align}
\la{28}
\dt \h = (1+\k^2)&\big[4 + 16 \, \h^{-1} (1-\k^2) + \OO(\h^{-2})\big] \ , \\
\la{288} \dt \k = \h^{-1} \k (1+\k^2)&\big[4 + 16\,  \h^{-1} (1-\k^2) + \OO(\h^{-2}) \big] \ .
\end{align}
The RG-invariant quantity in \rf{244} remains RG-invariant without 2-loop corrections,
\be
\dt \nu =0\ , \qquad \qquad
\nu \equiv {\h \ov 4 \k} + 0 + \mathcal{O}(\h^{-1}) \ . \la{2888}\ee
Furthermore, the ansatz \rf{24} also solves the 3-loop RG equation \rf{2} with
\be \la{29}
\b_{mn} = \Big[ \ha R + \tfrac{1}{4} R^2 + c_1 R^3 + c_2 (\nabla R)^2 + c_3 R \nabla^2 R \Big] \GG_{mn}
+ c_4 \nabla_m R \nabla_n R \ , \ee
in a particular ``natural'' renormalization scheme corresponding to
\be \qquad c_1 = 0 \ , \qquad c_2 = \tfrac{1}{8} \ , \qquad c_3 = -\tfrac{1}{4} \ , \qquad c_4=- \tfrac{1}{16}\ ,
\la{31}
\ee
in which the 3-loop $$̱-functions for $\h$ and $\k$ take the form
\begin{align}
\la{32}
\dt { \h} =(1+\k^2)&\Big[4 + 16 \, \h^{-1} (1-\k^2) + 64\,  \h^{-2} (1-\k^2)^2 + \O(\h^{-3}) \Big] \ , \\
\la{33} \dt { \k} = \h^{-1} \k (1+\k^2)&\Big[ 4 + 16\,  \h^{-1} (1-\k^2) + 64\,  \h^{-2} (1-\k^2)^2 +\O(\h^{-3})\Big] \ .
\end{align}
The diffeomorphism vector in \rf{2777} receives the following 3-loop correction
\be \la{300}
\xi_\p = 0 \ , \qquad \quad \xi_r = \frac{8\k^2 (1+\k^2) r}{\h (1+\k^2 r^2)^2} \Big[ 1 + \frac{8\k^2 (r^2-1)}{\h(1+\k^2 r^2)} \Big] \ .
\ee
This scheme \rf{31} is related to the minimal subtraction scheme in \cite{Graham:1987ep,Foakes:1987ij,Foakes:1987gg},
\be \la{30}
\qquad c_1 = \tfrac{5}{32} \ , \qquad c_2 = \tfrac{1}{16} \ , \qquad c_3 = 0\ , \qquad c_4=-\tfrac{1}{16}\ ,
\ee
by the covariant coupling 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}
where $_mn^(nat) ≡_mn$ is the metric \rf{24} in the ``natural'' scheme and $_mn^(min)$ is the corresponding metric in the minimal scheme (see Appendix \ref{A}).

We call the scheme \rf{31} ``natural'' because the RG-invariant quantity in \rf{2888} remains invariant with no 3-loop corrections:
\be
\dt \nu =0\ , \qquad \qquad
\nu \equiv {\h \ov 4\k} + 0 + 0 + \mathcal{O}(\h^{-2}) \ . \ee
This prompts us to conjecture that, in a natural choice of
subtraction scheme at each loop order, the  quantity $ν≡$ is an exact RG-invariant.
This suggests that $ν$ should be the parameter that appears in the exact quantum trigonometric S-matrix
that generalizes the non-perturbative massive S-matrix of the $O(3)$-invariant $S^2$ \sm
\cite{Zamolodchikov:1978xm}.

Relatedly, we extrapolate from the obvious pattern in \rf{32},\rf{33} to conjecture that, in the same natural scheme, the all-loop $$̱-functions of $h$ and $\k$ are
\begin{align}
\dt {\h} &=\frac{1+\k^2}{\tfrac{1}{4} \h-(1-\k^2) } \, \h \ , \la{exactb1} \\
\dt { \k} &=\frac{1+\k^2}{\tfrac{1}{4} \h- (1-\k^2)} \, \k \ . \la{exactb2}
\end{align}
Here $^̨2=-1$
remains a fixed point, as it should to all orders since it corresponds both to flat space and the gWZW limit.
For $=̨0$ we get $ R^2_S^2 =1+R^-2_S^2$ for the $S^2$ radius $R_S^2=√(14-1)$, which agrees with the 1-loop and 2-loop coefficients in the $$̱-function of the
$S^2$ model. In the maximal deformation limit 222 we get $\dt { R^2_{H^2}} =-1+R^{-2}_{H^2}$ for the $H^2$ radius $R_{H^2} = \sqrt{\tfrac{1}{4}\h'+1} = \sqrt{\tfrac{1}{4}\h\k^{-2}+1}$. The fact that this is correctly related to the $S^2$ $\b$-function by the analytic continuation $R^2_H^2 = - R^2_S^2$ is a
check of the consistency of the conjectured exact metric \rf{24} and exact $$̱-functions exactb1,exactb2. Moreover,
if there is a natural scheme where exactb1,exactb2 are exact,
then the $S^2$ and $H^2$ $\b$-functions are 2-loop exact in this scheme
(this may be possible in a special non-minimal scheme  since the 3- and higher-loop  $$̱-function coefficients are scheme-dependent).

Let us mention that
in the case of the
(1,1) supersymmetric generalization of the 13, the first potential correction to the 1-loop $\b$-function
at 4 loops ($∼(3) R^4$ in the minimal scheme \cite{Grisaru:1986px}, with this particular invariant actually vanishing in the case of 2d target space).
There is also
no deformation of the super-gWZW metric (to all orders) and it is thus natural to expect that the same may apply to the
supersymmetric $$-model on the sausage geometry \rf{21} (i.e.\ only $$ and $$̨  running without adding corrections to the metric).

One may also repeat the above discussion for the
$\eta$-deformation of hyperbolic space $H^2= \frac{SU(1,1)}{U(1)}$
(or the Lorentzian signature symmetric spaces $dS_2$ and $AdS_2$).
The corresponding metric is related to 21 by the
formal analytic continuation,
r→i r ,       →i  ,       →̨i ,       →-  .i.e. we findftdLet us note that the metric 34 is self T-dual, i.e. invariant under
$\phi\to \td \phi $ and $\td \phi \to \k^{-1} \phi$,  $r \to \k^{-1} r^{-1}$.
ds^2 = 14 [ dr^2/(1+r^2)(1+^̨2r^2) + 1+r^2/1+^̨2 r^2 d^2 ]  . This inhomogeneous  Yang-Baxter deformation of $H^2$ is based on the split $R$-matrix of $\Lie(SU(1,1))$ (see footnote <ref>).
On the other hand, if we had not continued $\k\to i \k$ in \rf{344}, we would have found Yang-Baxter deformation based on the non-split $R$-matrix.
For $S^2$ only the latter case exists as a Yang-Baxter deformation in a strict sense since there is no split $R$-matrix of $(SU(2))$ (see, e.g., \cite{BDCGR}).

$=̨0$ \rf{34} becomes the $H^2$ metric, while for $^̨2=1$ it is flat.
The limit analogous to \rf{22}, i.e.
$→̨1$ with $ r^2→-1 + (1-^̨2) sinh^2 t$, is now a formal limit
and gives the metric
$ ds^2 = 14( - dt^2 + tanh^2 t  d^2)$, which is the classical metric of the
$SU(1,1)/ℝ$ gWZW model.

conformally-flat coordinates the metric \rf{34} may be written as
\be \la{35}
ds^2 = {\h \ov 4(p^2 + \k^2 q^2)} ( dp^2 + dq^2) \ , \qquad \qquad
r = \frac{q}{p} ~, \quad \phi = \ha \log({p^2+q^2})\ ,
\ee
where the scaling symmetry $(p, q) →ł(p, q)$ is the counterpart of
$$-shift isometry of \rf{34}.

The analog of the exact metric \rf{24} is found by the analytic continuation \rf{344},
\be \la{36}
ds^2 = (\tfrac{1}{4}\h+1+\k^2) \Big[ \frac{dr^2}{(1+r^2)(1+\k^2r^2)} + \big(
\frac{1+ \k^2r^2}{1+ r^2} + \frac{8\k^2}{\h} \big)^{-1}
d\p^2 \Big]\ . \ee
Again, formally taking $ r^2 →-1 + (1-^̨2) sinh^2 t$, the limit
$^̨2 →1$ is the exact metric of the $SU(1,1)/ℝ$ gWZW model.

\sloppy
Interestingly the exact metric \rf{36} may have different properties compared to the classical metric \rf{34}.
For example, analytically continuing further \mbox{($\k \to  i \k, \     r \to  \rho, \    \phi \to  i t $)} to the non-split $η$-deformation of AdS$_2$,
there is a singularity at
$ρ^2 = - 8 ^̨2 ^̨2 (+8)$ that is always  present in the classical limit $≫1$
but   absent  for $<  8 ^̨2$.

Written in the conformally-flat coordinates as in \rf{35}, the quantum-corrected metric \rf{36} is
\be \la{37}
ds^2 = (\tfrac{1}{4} \h+ 1+\k^2 ) \Big[ { dp^2 + dq^2 \ov p^2 + \k^2 q^2} - \frac{\k^2}{2 \h} \, \frac{ \big[ d (p^2+q^2)\big]^2 }{(p^2+\k^2 q^2)^2( \tfrac{1}{4} + {2\k^2 \ov \h} { p^2+ q^2\ov p^2 + \k^2 q^2})} \Big]\ .
\ee
Expanding the metric \rf{37} to first subleading order in small $^-1 $ one obtains
\be \la{38}
ds^2 = \tfrac{1}{4} \h\, { dp^2 + dq^2 \ov p^2 + \k^2 q^2} + (1+\k^2){ dp^2 + dq^2 \ov p^2 + \k^2 q^2} - \frac{\k^2}{2} \, \frac{ \big[ d (p^2+q^2)\big]^2 }{
( p^2 + \k^2 q^2)^2 } + \OO(\h^{-1})
\ . \ee

\section{\texorpdfstring{$\l$}{lambda}-model \la{Slam}}

Let us now turn to the  $ł$-deformed \sms\unskip.
As we reviewed in Section \ref{las}, the $ł$-model may be
constructed by starting with the $G/G$ gauged WZW action and adding
a deformation  quadratic in the gauge field, breaking some or all of the gauge symmetry,
\be \begin{aligned} \la{40}
&\L = \kk\, \Big( \L_G(g)  + \Tr\big[ g^{-1} \partial_+ g A_- -  A_+ \partial_- g g^{-1} \\
&\qquad \qquad \qquad  \qquad \qquad + g^{-1} A_+ g A_-  - A_+ A_- + (\l^{-1} -1) A_+ P A_- \big]\Big) \ .
\end{aligned} \ee
One obtains a \sm by integrating out the  gauge field $A_±$ (here $Ł_G$ is the WZW Lagrangian, see Section \ref{WS})
\begin{equation}\la{lamc}
\L = k \, \Big( \L_G (g) + \Tr[ J_+\, M^{-1}\, K_- ] \Big) \ ,\quad \qquad M = \operatorname{Ad}_g - I +(1-\lambda^{-1}) P \ .
\end{equation}
Here the integration over the gauge field  was done classically so the resulting \sm \rf{lamc} should only be understood as a classical approximation that will be subject to  quantum corrections (cf.\ the classical gWZW geometry in eq.\ \rf{6}).
In what follows we shall construct such corrections in a manner consistent with 2-loop renormalizability  in the simplest examples based on $SU(2) U(1)$ and $SU(2)$.

\subsection{\texorpdfstring{$SU(2)/U(1)$}{SU(2)/U(1)} example}
First let us consider the $ł$-model with 2d target space  based on the symmetric space $GH = SU(2) U(1)$.
We fix the $U(1)$ gauge symmetry (generated by $_2$) by choosing the following parametrization of the coset element
\be \la{42}
\te g = \exp(i \alpha \s_3) \exp({i} \beta \s_2) \ , \ \ \ \ \ \ \cos  \alpha = \sqrt{p^2+q^2}\ ,\ \ \ \ \ \tan \beta = \frac{p}{q} \ . \ee
The  \lam\   \rf{lamc} then becomes
\be \la{43}
{\L} = \frac{\kk}{1-p^2-q^2}\big(\ka\, \partial_+p \partial_-p + \ka^{-1}\partial_+ q\partial_-q\big)\ , \qquad \qquad
\ka \equiv \frac{1-\l}{1+\l}
\ . \ee

\sloppy
The model \rf{43} again admits several useful limits.
In general, in a ``boost'' limit that infinitely rescales the Cartan directions, the $ł$-model gives the T-dual of the \etm\ on an analytically continued coset \cite{Hoare:2015gda} (equivalent to the Poisson-Lie duality relating the $η$- and $ł$-models discussed in  Section \ref{etas}).
In this special 2d case, the $η$-deformed $S^2$ model is self T-dual (see footnote \ref{fstd}) so the T-duality may be skipped:
infinitely rescaling the coordinates,\foot{This limit takes the coordinates $(p,q)$ out of their proper coordinate range $p^2+q^2<1$ in \rf{43}. This may be avoided by instead considering the $\l$-model on the analytically continued coset $SU(1,1) \ov U(1)$.}
\be (p,q) \to \g(p, \kappa q) \ , \qquad \g \to \infty \la{resc} \ee
the metric becomes equivalent \cite{Hoare:2015gda} to the \etm \ metric \rf{35} on the analytically continued coset $SU(1,1) U(1)$.
Changing coordinates $(p,q) →(r,) = (q/p, log(p^2+q^2))$
and analytically continuing $(r,) →(ir, i)$, we obtain again the $η$-model \rf{21} on $SU(2) U(1)$, with the couplings related by
\be
\h = 4 k \ka \ , \qquad \k = i \ka \ . \la{cre}
\ee

The limit $=1$ (or $ł=0$) corresponds to the classical metric of the $SU(2)U(1)$ gWZW model:
\be \la{45}
ds^2 = \frac{k}{1-p^2- q^2 }( { dp^2 + dq^2 }) = \kk(d\a^2 + \cot^2 \alpha\, d \b^2) \ , \ \ \quad (p,q) = \cos\a (\sin \b, \cos \b)
\ . \ee
As discussed in Section \ref{qcs}, the gWZW metric  \rf{45} is known to be subject to quantum corrections at each order. Thus it is already clear that the $ł$-deformation \rf{42} of this model should be subject to  non-trivial quantum corrections.

The \lam\ is known to be 1-loop renormalizable with only $$ running \cite{Itsios:2014lca,Appadu:2015nfa} (eq.\ \rf{la2} with $c__SU(2)=2$ and $≡1-ł1+ł$):
\begin{equation}
{\dt }\kk = 0+ \mathcal{O}(\kk^{-1}) \ , \qquad \qquad
\dt{\ka} = - \kk^{-1}(1-\ka^2) + \mathcal{O}(\kk^{-2}) \ .\la{310}
\end{equation}
The main observation here is that one may ensure the 2-loop renormalizability of this model by modifying the classical action
\rf{43} with a particular quantum correction.
A natural guess is that the necessary counterterms  originate from a determinant of integration over the 2d gauge field, which appears quadratically in \rf{40}.
Such integrals are not well-defined \cite{Schwarz:1992te}, with the resulting  local terms depending on a choice of regularization. One particular prescription is to write the  gauge field in terms of two scalars, $A= d u + d*v$ and integrate over $u,v$ with measures $M$ \cite{Tseytlin:1991wr,Schwarz:1992te} (see also \cite{Gerasimov:1990fi}), obtaining the result
\be \la{7}
\int [dA] \, e^{ i \int d^2 \xi\, \sqrt{-h}
\, (-M \, A^\m A_\m)} = \exp \Big[\frac{i}{4 \pi} \int d^2 \xi  \, \sqrt{-h} \, \big(
\ha (\del \log M)^2 + \ha R^{(2)} \log M \big) \Big]\ .
\ee
For generality we assumed a curved 2d background with curvature $R^(2)$, and we have ignored a trivial $M$-independent factor in \rf{7}.
The $R^(2) logM$ term in \rf{7} is related to the familiar dilaton shift found in
\cite{Buscher:1987qj} (see also \cite{DeJaegher:1998pp}), and vanishes on a flat 2d background.

However, the $(logM)^2$ term remains non-trivial on a flat 2d background.
In this case  $M=M(g)$ is the matrix that appears in the part of the $ł$-model action \rf{40} that is quadratic in the gauge field, i.e. $Ł= …+ A_+ M(g)A_-$. It generally depends on the group element $g$, taking the the form $M ∝1 + (ł^-1 -1) P - _g$ (written as an operator $(G) →(G)$). Hence the prescription \rf{7}
suggests the
correction to the classical Lagrangian \rf{43} of the $SU(2)U(1)$ \lam,\foot{Recall that the Lagrangian $\L = \GG_{mn} \del_+ x^m \del_- x^n = - \GG_{mn} \del_\m x^m \del^\m x^n$ has a minus sign relative to the target space metric when written in covariant 2d notation. This explains the relative minus sign between the two lines in \rf{47}.}
\be\begin{aligned} \la{47}
&\Delta \L =  \ha ( \del \log \det M )^2  =   \ha \big[ \del \log (1-p^2 - q^2)\big]^2 \ ,\\
&\Delta(ds^2) = -\ha \big[ d \log (1-p^2 - q^2)\big]^2 \ .
\end{aligned} \ee
Somewhat surprisingly, in this case the 1-loop correction \rf{47} happens to be independent of the deformation parameter $ł$.

This leads to a 1-loop corrected metric for the \lam\  \rf{43}  given by (cf. \rf{38})
\be \la{48}
ds^2 = \kk\, {\ka\, dp^2 + \ka^{-1}d q^2 \ov 1-p^2-q^2} - \frac{1}{2} \Big[\frac{d ( 1- p^2 - q^2 )}{1-p^2 - q^2 }\Big]^2
+ \OO ( k^{-1}) \ .
\ee
One can then check directly that the metric \rf{48} indeed solves the 2-loop RG equations %\rf{2},
\rf{27}
with  (recall that $≡1-ł1+ł$
\be \la{49}
{\dt }\kk = 0 + \kk^{-1}(\ka^{-1}-\ka)^2 + \mathcal{O}(\kk^{-2}) \ , \qquad \qquad
\dt{\ka} =  \kk^{-1}(1-\ka^2) + 0 + \mathcal{O}(\kk^{-3}) \ ,
\ee
and the components of the
diffeomorphism vector $ξ_m$ given by
\begin{align}
\xi_p &= \frac{2p}{1-p^2-q^2}\big[1 + {1 \ov \kk} \frac{\kappa (p^2-1) -\kappa^{-1} p^2 }{1- p^2-q^2} \big] \ ,
\no \\
\xi_q &= \frac{2q}{1-p^2-q^2}\big[1 + {1 \ov \kk} \frac{\kappa^{-1} (q^2-1) - \kappa q^2)}{1- p^2-q^2} \big] \ . \la{50}
\end{align}
The $$̱-function for the deformation parameter $\ka$
does not receive a 2-loop correction.
Surprisingly, the parameter $k$ (which has the interpretation of the level in the gWZW model) starts running
at 2-loop order.
Still, there is an RG invariant (cf. 2888)
=0  ,        ≡+ (κ^-1 + κ) + 𝒪(^-1)  .
This suggests that it is $\rk$ rather than $k$ that should be identified with level of the gWZW model in 40.
However, in this 2d case  the WZ term vanishes upon gauge fixing (any $\BB$-field is a total derivative in 2d) so the question of  which is the WZ level becomes somewhat meaningless.

These results are consistent with the limits of the  discussed above.
Setting (cf. 42,45)
(p,q) = (1 + ^-1) cos(sin,̱ cos)̱
in 48 and taking the gWZW limit, $\kappa =1$, we find
ds^2 = (dα^2 + ^2 α  dβ^2)
+ 2 (dα^2 + ^2αsin^2α  d β^2) + 𝒪(^-1)  , which precisely matches the large $\kk$ expansion of the exact gWZW metric (cf. 6), i.e.
ds^2 = (+2)(dα^2 + ^2 α/1-2/^2α   dβ^2)  . Note that in this limit the RG invariant in 51 becomes $\rk =\kk+2$, i.e. the
shifted level of the $SU(2)/U(1)$ gWZW model.In the $\ka = 1$ limit
the diffeomorphism vector 50 becomes a gradient of the exact dilaton of the gWZW model:
$\xi_n = 2 \del_n \Phi$, $ \Phi = - \log (1-p^2 - q^2 ) - { 1 \ov 2 k} { 1 \ov 1-p^2 - q^2 } + \OO ( k^{-2}) $ (cf.
footnote <ref>).

In the  limit, the  corrected  metric 48 reduces to the leading correction in the  metric  24  if one modifies the scaling limit resc and coupling relation cre by certain quantum corrections,

\begin{align}
& p\to \g p \ ,\qquad \qquad q \to -i\g \k \big[1- 4 \h^{-1} ({1+\k^2}) \big]\, q \ , \qquad \qquad \g\to\infty \ , \la{54}\\
& \ka = -i \k \big[1-8\h^{-1} (1+\k^2)\big] \ , \qquad \qquad \tfrac{1}{4}\h-1+\k^2= -i\kk \k \ ,\la{53}
\end{align}

followed by again making the same change of variables $(r,\p) = (q/p, \ha \log (p^2+q^2))$ and  continuing $(r,\p) \to (ir, i\p)$.
In this limit the RG equations 49 reproduce the corresponding ones for the  couplings $\k$ and $$ in \rf{28},\rf{288}
with the RG invariant \rf{51} being mapped to an analytic continuation of the $η$-model RG invariant \rf{2888},
\be
\rk = \kk + ({\ka}^{-1} + {\ka}) + \OO(\kk^{-1}) = i \frac{\h}{4\k} + \OO(\h^{-1}) =i \nu +\OO(\nu^{-1}) \ . \la{56}
\ee
This may be viewed as a hint that a quantum deformation parameter
that appears
in the corresponding exact S-matrix should be $q= e^-  i    = e^-1 $ (cf. \cite{Hoare:2015gda}),
which becomes  $q= e^ -  i  +2   $ at the gWZW point $=1$.

\renewcommand{\la}[1]{\label{1910#1}}
\renewcommand{\rf}[1]{(\ref{1910#1})}
\renewcommand{\kk}{{\rm k}}
\renewcommand{\sm}{$\s$-model}
\renewcommand{\sm}{$\s$-model}
\renewcommand{\lam}{$\l$-model}
\renewcommand{\etm}{$\eta$-model}

\iffa
\section{Renormalization of \texorpdfstring{$\lambda$}{lambda}-model: standard
configuration space}\la{standard}

In section \ref{1910extended} we demonstrated the 2-loop renormalizability of the $λ$-model \rf{1} for general groups $G$ and symmetric spaces $G/$.
It is then natural to ask what
this implies
for the model on the standard or
physical configuration space, i.e. the \sm{} found by integrating
out $A_±$ in \rf{1}.
Doing so classically gives the following Lagrangian
\begin{equation}\la{lamphys}
\L = k \, \Big( {\PCM}(g) + {\WZ}(g) + \Tr[ J_+\, M^{-1}\, K_- ] \Big) \ ,\quad \qquad M = \operatorname{Ad}_g - I +(1-\lambda^{-1}) P \ .
\end{equation}
Similarly, for the NAD model \rf{14} we find
\begin{equation}\la{32}
\L = -\tfrac12 \hh\,\Tr\big[ \partial_+ v\, \M^{-1}
\, \partial_- v \big] \ , \quad \qquad \M
= \ad_v + P \ .
\end{equation}
The integration over $A_±$ may also give rise to quantum counterterms required to preserve the renormalizability of \rf{lamc} at 2 loops \cite{HLT1}.
It is natural to expect that, since the term quadratic in $A_±$ in the
Lagrangian \rf{1} has the form $[ A_+ M A_-]$, these corrections may depend on the matrix $M$ in \rf{lamc},
but determining their form in general
appears to be
Here we will focus on the
examples of the \lam{} for the $SU(2) U(1)$ symmetric space and $SU(2)$ group space.

\subsection{\texorpdfstring{$SU(2)/U(1)$}{SU(2) \ov U(1)}}

The $λ$-model for $SU(2) U(1)$ is related by analytic continuation to that of $SU(1,1)/U(1)$, which was studied in detail in \cite{HLT1}.
Here we briefly summarize certain key points of the discussion there.
Fixing the $U(1)$ gauge symmetry by choosing the following parametrization of the coset element
\begin{equation}
g = \exp(i\alpha\sigma_3)\exp(i \beta \sigma_2) \ , \qquad \cos \a = \sqrt{p^2 + q^2} \ , \qquad \tan \b = \frac{p}{q} \ ,
\end{equation}
the \sm{} \rf{lamc} yields the following classical metric
(the $$-field is trivial in 2d target space and $$ is defined in \rf{133})
\begin{equation}\la{g0coset}
\GG_0 = \frac{k }{1-p^2-q^2}(\ka\, dp^2 + \ka^{-1} dq^2) \ .
\end{equation}
The observation in \cite{HLT1} was that this metric should be modified by a particular quantum correction
from the determinant \cite{Schwarz:1992te} resulting from integrating over $A_±$
\begin{equation}
\delta \GG = -\frac{1}{2}\big(d \log \det M\big)^2 = - \frac{1}{2} \big[d \log (1-p^2-q^2)\big]^2 \ . \la{su2u1counter}
\end{equation}
The 1-loop corrected background $=_0 + δ$
then solves the 2-loop RG equation \rf{rg} with
\begin{equation}\begin{split}\la{0oldcoset}
\frac{d}{dt} k & = 0 + \frac{1}{k} \frac{(1-\ka^2)^2}{ \ka^2} \ , \qquad\qquad
\frac{d}{dt} \ka = \frac{1}{k} (1-\ka^2) \ ,
\end{split}\end{equation}
\begin{equation}
\xi^p = -\frac{p}{k\ka}\Big[1-\frac{\ka}{k}+\frac{\ka^{-1}p^2 + \ka q^2}{k(1-p^2-q^2)}\Big]
\ , \quad
\xi^q = -\frac{q\ka}{k}\Big[1-\frac{1}{k\ka}+\frac{\ka^{-1}p^2 + \ka q^2}{k(1-p^2-q^2)}\Big]
\ , \quad \l_{p,q} = 0.
\end{equation}
In this analysis the symmetry \rf{z2sym}
of the 1-loop RG equation in \rf{0oldcoset}
survives at the 2-loop level.
Indeed, while the leading 1-loop terms in \rf{0oldcoset} agree with the RG equations
\rf{6d2loopbetacoset} found from the analysis on the extended configuration space,
they deviate at the 2-loop order.
Since the 2-loop terms in a two-coupling theory are generally scheme-dependent,
we can, in fact, match the two $$̱-function expressions by
redefining the parameters in g0coset as follows
The most general redefinition achieving this is ($C_1$ and $C_2$ are free constants)
$ k \to k - \tfrac{(1 -\ka)^2}{\ka} + 2 C_1$ and

$\quad \ka \to \ka + \frac{4}{k} \big[(1-\ka)\big(1-(1+\ka)C_2\big) - 2 C_1(1-\ka^2)\operatorname{arctanh}\ka\big]$.

\begin{equation}\la{38}
k \to k - \frac{(1 -\ka)^2}{\ka} \ , \qquad \qquad \ka \to \ka + \frac{4(1-\ka)}{k} \ .
\end{equation}

Note that in the $\ka \to 1$ limit the level $k$ remains unmodified, in agreement with this limit corresponding to the $SU(2) \ov U(1)$ gWZW model.

§.§ SU(2) example

Now let us turn to the  for the group $SU(2)$
Parametrizing the group element as
g = exp[ -i arcsin  (cos   _2 + sin(cos   _3 - sin  _1))]  ,
we obtain from (<ref>) the following 3-dimensional classical  metric and $\BB$-field <cit.>
Note that, in these coordinates, WZ term  $k \L_{\WZ}(g)$ in (<ref>) contributes $k(\arcsin\a - \a \sqrt{1-\a^2}) \sin{\b} \, d\b \wedge d\g$ to the $\BB$-field.

\begin{align}
(ds^2)^{(0)} & = k \big[\frac{d\a^2}{\ka(1-\a^2)} + \frac{\kappa \a^2}{\Delta} ( d\beta^2 + \sin^2\beta d\gamma^2)\big] \ , \quad\qquad
\Delta(\a) \equiv \ka^2 + (1-\ka^2)\a^2 \ , \no
\\
\BB^{(0)} & = k\big(\arcsin\a - \frac{\ka^2\a\sqrt{1-\a^2}}{\Delta} \big)\sin\beta\, d\beta\wedge d\gamma \ , \la{b1}
\\
\HH^{(0)} & =d\BB^{(0)}= \frac{k\a^2}{\sqrt{1-\a^2}\Delta^2}\big[2\ka^2+(1-\ka^2)\Delta\big] \sin\b\, d\a \wedge d\b \wedge d\g \ . \no
\end{align}

As discussed in Section <ref>,   $\l$-models are known to be 1-loop renormalizable but, again, one can check that the classical background b1 does not solve the 2-loop RG equation (<ref>) (with only $k,\ka$ running) in any covariant subtraction scheme and with any choice of diffeomorphism vector. (Here due to the $SU(2)$ global symmetry acting on $\b,\g$, the diffeomorphism vector must be of the form $\xi^\a=\xi^\a(\a)$, $\xi^{\b,\g}=0$ and the $\BB$-gauge transformation must be trivial, $\del_{[m}\l_{n]}=0$.)

However, we observe that it is possible to restore the 2-loop renormalizability of the model
by adding  special
quantum $(1/k)$ counterterms to this classical background,

\begin{equation}\begin{aligned}\la{b2}
ds^2  & = (ds^2)^{(0)} + \frac{2(1-\ka^2)^2 \a^4}{\ka^2 (1-\a^2)\Delta^2} d\a^2 \ ,
\\
\BB & = \frac{\kk}{k} \BB^{(0)} -2 \big( \arctan\frac{\a}{\ka\sqrt{1-\a^2}}
- \frac{\ka \a \sqrt{1-\a^2}}{\Delta}\big)\sin\beta\, d\beta\wedge d\gamma \ ,
\\
\H & = \frac{\kk}{k} \H^{(0)} - \frac{4\ka\a^2}{\sqrt{1-\a^2}{\Delta^2}} \sin\b\, d\a \wedge d\b \wedge d\g \ ,
\qquad \qquad \kk \equiv k + \frac{4+(1+\ka^2)^2}{4\ka} \ .
\end{aligned}\end{equation}

This corrected background b2 indeed solves the 2-loop RG equation (<ref>) withThe 1-loop terms in 0 match the general result (<ref>) with $c_{_{SU(2)}}=2$ and $\ka \equiv \frac{1-\l}{1+\l}$.

\begin{equation}\la{0}
\frac{d}{dt}k = 0 + \frac{(1-\ka^2)^3(5+3\ka^2)}{8k\ka^2} \ , \qquad
\qquad \frac{d}{dt} \ka = \frac{(1-\ka^2)^2}{2k} \Big[1 - \frac{(1-\ka^2)^2}{k\ka} \Big] \ .
\end{equation}

The only non-zero component of the diffeomorphism vector $\xi^m$ is

\begin{equation}\begin{split}\la{diffv}
\xi^\a & = \frac{\a(1-\a^2)(1-\ka^2)}{k\Delta}\Big[\ka + \frac{2\a^2\ka^2(1-\ka^2) - (3-2\ka^2+\ka^4)\Delta^2}{k\Delta^2}\Big] \ ,
\end{split}\end{equation}

and the B-gauge transformation in (<ref>) vanishes, $\l_m=0$.

Note that the coupling $\kk$ defined in b2 does not run at 2-loop order,
d/dt = 0 + 1/×0 + …knr  . This RG invariant $\kk$ is the coefficient of $\arcsin\a$ (present in $\BB_0$ in b1) in the 1-loop corrected background b2. Choosing it
to be integer-valued removes the global ambiguities arising from the $\arcsin\a$ term.
Furthermore, given that under a large transformation,
Note that $\arctan\frac{\a}{\ka\sqrt{1-\a^2}} = \operatorname{sign} \ka \, \arcsin\frac{\a}{\sqrt{\Delta}}$ and $\alpha = \frac{\alpha}{\sqrt{\Delta}}$ for $\alpha = 0,\pm1$.

\begin{equation}
\delta (\arcsin \a) = \operatorname{sign} \ka \ \delta (\arctan\frac{\a}{\ka\sqrt{1-\a^2}})\ ,
\end{equation}

and that the coefficient of $\arctan\frac{\a}{\ka\sqrt{1-\a^2}}$ in b2 is integer-valued, the $\arctan$ term does not lead to any additional ambiguities.
The quantization of the flux,
1/4π^2 ∫^̋(0) = k  ,         1/4π^2 ∫=̋ -2  ,
also supports the identification of $\kk$ as integer-valued.

In the case of the $SU(2) \ov U(1)$
it was possible to write the 1-loop corrections
in terms of a simple
$\Delta \L = -\ha (\del \log \det M)^2 = -\ha (\Tr [M^{-1} \del M ])^2$ in terms of the matrix $M$ as defined in (<ref>).
For this $SU(2)$ case, we find however that this particular counterterm is not sufficient to restore 2-loop renormalizability (in any covariant subtraction scheme) — i.e. the required corrections b2 cannot be written in this form even after a covariant scheme change.
let us also include
similar counterterms (with coefficients $c_i$)
built out of the quantity $M^{-1} \del M$.
In addition, we may also include counterterms (with coefficients $d_i$)
proportional to the terms present in the classical Lagrangian (<ref>) (and its image under parity). As a result, we are led to the following ansatzIn qccc and below, $\L_\PCM (a) \equiv -\ha \Tr[a^{-1} \del_+ a \, a^{-1} \del_- a]$ and $\L_{\WZ}$ denote the principal chiral model and Wess-Zumino Lagrangian terms (see Section <ref>).

\begin{equation}\begin{aligned}\la{qccc}
\Delta \L &= c_1 (\del_+ \log \det M) ( \del_- \log \det M)
+ c_2 \L_{\PCM}(M) + c_3 \L_{\WZ}(M)
\\ &\quad + d_1 \L_{\PCM}(g) + d_2 \L_{\WZ}(g) + d_3 \Tr[J_+ M^{-1} K_-] + d_4 \Tr[J_- M^{-1} K_+] \ .
\end{aligned}\end{equation}

We find that this matches the required 1-loop corrected background
b2 provided the constants $c_i$ and $d_i$ take the following values

\begin{align}\la{origin}
&c_1 =-\tfrac{3+\ka}{2(1-\ka)} \ , \qquad c_2 = - \tfrac{2(1+\ka)}{1-\ka} \ , \qquad c_3 = 1 \ ,
\\
&d_1 = 0 \ , \quad d_2 = {\rm k} - k - 2 \ ,\quad
\ d_3 = \tfrac{1}{2}\big[{\rm k} - k + (1+\kappa^{-1})\big] \ , \quad d_4 = -\tfrac{1}{2}\big[{\rm k}-k - 3 (1+\kappa^{-1})\big] \ .\no
\end{align}

Combining the quantum counterterms qccc with the classical Lagrangian (<ref>) allows us to represent the  corresponding to
the 1-loop corrected geometry b2 in the form

\begin{equation}\begin{aligned}\la{originlag}
&\L  = k\, \L_{\PCM}(g) + ({\rm k} -2) \L_{\WZ}(g)
\\ &\quad  \qquad