S-matrices and quantum group symmetry of k-deformed sigma models

The k-deformed theories are integrable at the classical level⁷ . To see this, we can solve for the auxiliary field A_μ in the form

$$\begin{eqnarray}&&{A}_{+}={({\rm{\Omega }}-{{\rm{Ad}}}_{{{ \mathcal F }}^{-1}})}^{-1}{{ \mathcal F }}^{-1}{\partial }_{+}{ \mathcal F },\ \ \ \ \ {A}_{-}=-{({\rm{\Omega }}-{{\rm{Ad}}}_{{ \mathcal F }})}^{-1}{\partial }_{-}{ \mathcal F }{{ \mathcal F }}^{-1}.\end{eqnarray} \tag{ 2.5 }$$

The equations-of-motion can then be written in Lax form:

$$\begin{eqnarray}&&[{\partial }_{+}+{{\mathscr{L}}}_{+}(z),{\partial }_{-}+{{\mathscr{L}}}_{-}(z)]=0,\end{eqnarray} \tag{ 2.6 }$$

where, for the k-PCM

$$\begin{eqnarray}&&{{\mathscr{L}}}_{\pm }(z)=\displaystyle \frac{z}{z\pm 1}\cdot \displaystyle \frac{2}{(1+\lambda )}{A}_{\pm },\end{eqnarray} \tag{ 2.7 }$$

while for the k-SSSM

$$\begin{eqnarray}&&{{\mathscr{L}}}_{\pm }(z)={A}_{\pm }^{(0)}+{z}^{\pm 1}{\lambda }^{-1/2}{A}_{\pm }^{(1)}.\end{eqnarray} \tag{ 2.8 }$$

For integrable theories, one way to quantize the theories directly is to intuit a form for the exact S-matrix by using a combination of the global symmetries, the standard S-matrix axioms along with factorization and the Yang–Baxter equation that result from integrability. Integrability at the quantum level is thought to follow if there are higher spin conserved, possibly non-local, charges whose conservation is non-anomalous in the quantum theory. The integrability of the PCM is known to be non-anomalous for any group F, while the absence of an anomaly for the F/G SSSM is not guaranteed in general. For instance, for compact symmetric spaces of definite signature it requires that the subgroup G is simple [31]⁸ . The quantum integrable SSSMs are then associated to either Type II symmetric spaces of the form $G\times G/G$, which provide an equivalent way to formulate the PCM for the group G, or Type I symmetric spaces of the form

$$\begin{eqnarray}&&{\rm{SU}}(N)/{\rm{SO}}(N),\,\,\,\qquad {\rm{SU}}(2N)/{\rm{Sp}}(N),\\ &&{\rm{SU}}(2)/{\rm{U}}(1)\equiv {\mathbb{C}}{P}^{1},\quad {\rm{SO}}(N)/{\rm{SO}}(N-1)\equiv {S}^{N-1},\end{eqnarray} \tag{ 2.9 }$$

plus a number of possibilities involving the exceptional groups that we shall not consider here.

We make the hypothesis that the k-deformation of these theories are also integrable at the quantum level. One way to motivate this is to find a suitable ansatz for their S-matrices which can then be subject to various consistency tests. Note that there are effectively two different kinds of k-deformations of the PCM because they can also be formulated as a SSSM.

The first issue to consider is how the couplings k and λ flow in the quantum theory. The integer level k, of course, cannot have non-trivial RG flow because it multiplies the topological Wess–Zumino term in the action. However, the coupling λ does have a non-trivial RG flow. The loop counting parameter is $1/k$ and to the one-loop level the beta function is⁹

$$\begin{eqnarray}k \mbox{-} \mathrm{PCM}:\qquad \mu \displaystyle \frac{{\rm{d}}\lambda }{{\rm{d}}\mu } & = & -\displaystyle \frac{2{c}_{2}(F)}{k}{\left(\displaystyle \frac{\lambda }{1+\lambda }\right)}^{2},\\ k \mbox{-} \mathrm{SSSM}:\qquad \mu \displaystyle \frac{{\rm{d}}\lambda }{{\rm{d}}\mu } & = & -\displaystyle \frac{{c}_{2}(F)}{k}\lambda .\end{eqnarray} \tag{ 2.10 }$$

So in both cases, the coupling λ runs to zero in the UV. In this limit, by integrating out the auxiliary field A_μ, for the PCM case, and ${A}_{\mu }^{(1)}$, for the SSSM case, the theories take the form of a WZW model deformed by a current–current operator. For the PCM this is an ordinary (i.e. non-gauged) WZW model

$$\begin{eqnarray}&&{S}_{\lambda }={S}_{{\rm{WZW}}}[{ \mathcal F }]+\displaystyle \frac{4\pi }{k\lambda }\int {{\rm{d}}}^{2}x\,{\rm{Tr}}({\hat{{\mathscr{J}}}}_{+}{\hat{{\mathscr{J}}}}_{-})\,+\,\cdots ,\end{eqnarray} \tag{ 2.11 }$$

where ${\hat{{\mathscr{J}}}}_{\pm }$ are the standard Kac–Moody currents of the WZW model:

$$\begin{eqnarray}&&{\hat{{\mathscr{J}}}}_{+}=-\displaystyle \frac{k}{2\pi }{{ \mathcal F }}^{-1}{\partial }_{+}{ \mathcal F },\qquad {\hat{{\mathscr{J}}}}_{-}=\displaystyle \frac{k}{2\pi }{\partial }_{-}{ \mathcal F }{{ \mathcal F }}^{-1}.\end{eqnarray} \tag{ 2.12 }$$

In this case the deformation is marginally relevant, as is also clear from the form of the beta function (2.10).

For the SSSM case, the UV theory is a deformation of the gauged WZW for F/G,

$$\begin{eqnarray}&&{S}_{\lambda }={S}_{{\rm{gWZW}}}[{ \mathcal F },{A}_{\mu }^{(0)}]+\displaystyle \frac{4\pi }{k\lambda }\int {{\rm{d}}}^{2}x\,{\rm{Tr}}\,({\hat{{\mathscr{J}}}}_{+}^{(1)}{\hat{{\mathscr{J}}}}_{-}^{(1)})\,+\,\cdots ,\end{eqnarray} \tag{ 2.13 }$$

where ${\hat{{\mathscr{J}}}}_{\pm }$ are the Kac–Moody currents of the F/G gauged WZW model

$$\begin{eqnarray}{\hat{{\mathscr{J}}}}_{+} & = & -\displaystyle \frac{k}{2\pi }({{ \mathcal F }}^{-1}{\partial }_{+}{ \mathcal F }+{{ \mathcal F }}^{-1}{A}_{+}^{(0)}{ \mathcal F }-{A}_{-}^{(0)}),\\ {\hat{{\mathscr{J}}}}_{-} & = & \displaystyle \frac{k}{2\pi }({\partial }_{-}{ \mathcal F }{{ \mathcal F }}^{-1}-{ \mathcal F }{A}_{-}^{(0)}{{ \mathcal F }}^{-1}+{A}_{+}^{(0)}).\end{eqnarray} \tag{ 2.14 }$$

Notice that the deformation now is relevant. In the case of a Type II symmetric space, it is worth pointing out that the WZW model involves the product group $F=G\times G$ and so there are two independent levels k₁ and k₂.

The integrable deformations of WZW models, non-gauged in (2.11) and gauged in (2.13) have been studied from a CFT perspective in [35].

There is one example of the k-deformed sigma models that has been studied in some detail and will provide the inspiration for the other cases. This corresponds to the PCM case with $F={\rm{SU}}(2)$ [36]. To start with, the PCM models associated to the classical groups have a factorized S-matrix that can be written schematically in terms of 2 particle to 2-particle matrix elements of the form [37, 38]

$$\begin{eqnarray}&&{S}^{{ab}}(\theta )={X}^{{ab}}(\theta ){\widetilde{S}}_{(\infty )}^{{ab}}(\theta )\otimes {\widetilde{S}}_{(\infty )}^{{ab}}(\theta ),\end{eqnarray} \tag{ 2.15 }$$

where $\theta ={\theta }_{1}-{\theta }_{2}$ is the rapidity difference of the incoming states. The ${X}^{{ab}}(\theta )$ are scalar factors, known also as dressing phases, that are needed to satisfy the S-matrix constraints of unitarity and crossing symmetry. The states of the theory are labelled by a and b which range over $1,2,\ldots ,r$, the rank of F. They transform in a product of representations of F, ${W}_{a}\otimes {W}_{a}$, where W_a is the ath fundamental, or antisymmetric representation, for unitary and symplectic groups, but are not irreducible for the orthogonal cases¹⁰ . The two factors reflect the ${F}_{L}\times {F}_{R}$ symmetry of the PCM which act on the group field via the left and right action: $f\to {U}_{L}{{fU}}_{R}^{-1}$. In (2.15), the ${\widetilde{S}}_{(\infty )}^{{ab}}$ are intertwiners

$$\begin{eqnarray}&&{\widetilde{S}}_{(\infty )}^{{ab}}\,:\quad {W}_{a}({\theta }_{1})\otimes {W}_{b}({\theta }_{2})\to {W}_{b}({\theta }_{2})\otimes {W}_{a}({\theta }_{1}).\end{eqnarray} \tag{ 2.16 }$$

These intertwiners are the rational solutions of the Yang–Baxter equation. The significance of the $\infty$ label will emerge. The S-matrix is covariant under the ${F}_{L}\times {F}_{R}$ symmetry of the theory, but this symmetry extends to an affine Yangian symmetry. The full spectrum of the theory can then be obtained by using the bootstrap equations. The bound states are associated to the antisymmetric representations of F although for the ${\rm{SO}}(N)$ groups these representations are sometimes reducible.

For $F={\rm{SU}}(2)$, in [36] it was conjectured a form for the S-matrix of the k-deformed theory of the form

$$\begin{eqnarray}&&S(\theta )={X}_{(k,\infty )}(\theta ){\widetilde{S}}_{(k)}(\theta )\otimes {\widetilde{S}}_{(\infty )}(\theta ),\end{eqnarray} \tag{ 2.17 }$$

where the ${\rm{SU}}{(2)}_{L}$ factor, the one that is gauged in order to construct the deformed theory, has changed from the rational ${\rm{SU}}(2)$ factor to the quantum group generalization with a quantum group deformation parameter that is a root of unity: ${q}^{2(k+2)}=1$. It is also important that the quantum group factor ${\widetilde{S}}_{(k)}(\theta )$ is taken in the so-called interaction-round-a-face' (IRF) picture, or 'restricted-solid-on-solid' (RSOS) picture. The significance of this is that the quantum numbers in this sector are naturally interpreted as kinks. So states of the theory carry a kink charge ±1 and also a vector in the two-dimensional representation of ${\rm{SU}}(2)$. The detailed form of the S-matrix is written in [36]. In the present context, the fact that the particle states are kinks can be understood as arising form the non-trivial spatial boundary conditions of the WZW models¹¹ .

Based on the understanding of the ${\rm{SU}}(2)$ k-PCM model above, it is natural to conjecture that the k-PCM for a generic group F have a S-matrix of the schematic form

$$\begin{eqnarray}&&{S}^{{ab}}(\theta )={X}_{(k,\infty )}^{{ab}}(\theta ){\widetilde{S}}_{(k)}^{{ab}}(\theta )\otimes {\widetilde{S}}_{(\infty )}^{{ab}}(\theta ),\end{eqnarray} \tag{ 2.18 }$$

with ${q}^{2(k+{c}_{2}(F))}=1$, where ${c}_{2}(F)$ is the dual Coxeter number of F. As in the ${\rm{SU}}(2)$ case, the quantum group intertwiner ${\widetilde{S}}_{(k)}^{{ab}}(\theta )$ is defined in the IRF or RSOS representation. These S-matrices in general were first written down in [35] and discussed in detail in [40]. Note that the scalar factors ${X}_{(k,\infty )}^{({ab}}$ here, and below, depend implicitly on the actual group F. Using TBA techniques, this conjecture has been checked for SU(2) in [36] and, more recently, for ${SU}(N)$ in [51].

The intertwiners ${\widetilde{S}}_{(k)}^{{ab}}(\theta )$ are invariant under the action of an affine quantum group ${U}_{q}(\hat{{\mathfrak{f}}})$ in the following sense. There is an associated co-product Δ which describes how the generators act on a tensor product: ${\rm{\Delta }}(u){W}_{a}({\theta }_{1})\otimes {W}_{b}({\theta }_{2})$, $u\in {U}_{q}(\hat{{\mathfrak{f}}})$. The theory is invariant under the affine quantum symmetry in the sense that

$$\begin{eqnarray}&&{\rm{\Delta }}(u){S}^{{ab}}({\theta }_{12})={S}^{{ab}}({\theta }_{12}){\rm{\Delta }}(u),\qquad u\in {U}_{q}(\hat{{\mathfrak{f}}}).\end{eqnarray} \tag{ 2.19 }$$

So the S-matrix elements lie in the commutant of ${U}_{q}(\hat{{\mathfrak{f}}})$ acting on a tensor product representation. In the limit, $k\to \infty$, the quantum group reduces to the Yangian symmetry of the undeformed PCM.

Now we turn to the SSSM. For the Type II cases, there is an obvious form that the S-matrix should take. One simply generalizes (2.18) to

$$\begin{eqnarray}&&{S}^{{ab}}(\theta )={X}_{({k}_{1},{k}_{2})}^{{ab}}(\theta ){\widetilde{S}}_{({k}_{1})}^{{ab}}(\theta )\otimes {\widetilde{S}}_{({k}_{2})}^{{ab}}(\theta ),\end{eqnarray} \tag{ 2.20 }$$

which means that the symmetry involves a tensor product of quantum groups ${U}_{{q}_{1}}(\hat{{\mathfrak{f}}})\otimes {U}_{{q}_{2}}(\hat{{\mathfrak{f}}})$.

For the Type I theories, a bespoke construction is needed in each case based on the S-matrices that were constructed for the undeformed theories [41]. However, the form of the undeformed S-matrix leads to a natural conjecture in each case.

${S}^{N-1}\ (\mathrm{including}\ {\mathbb{C}}{P}^{1}\equiv {S}^{2})$: for this series the S-matrix has the form originally found by Zamolodchikov and Zamolodchikov [42]

$$\begin{eqnarray}&&S(\theta )={{X}_{(\infty )}(\theta ){\widetilde{S}}_{(\infty )}(\theta )| }_{{\rm{SO}}(N)},\end{eqnarray} \tag{ 2.21 }$$

with particles transforming on the N-dimensional defining representation of $F={SO}(N)$, which we show explicitly. We conjecture that the k-deformed theory has an S-matrix of the form

$$\begin{eqnarray}&&S(\theta )={{X}_{(k)}(\theta ){\widetilde{S}}_{(k)}(\theta )| }_{{\rm{SO}}(N)},\end{eqnarray} \tag{ 2.22 }$$

with ${q}^{2(N-2+k)}=1$.

${\rm{SU}}(N)/{\rm{SO}}(N)$: for this series the S-matrix has the form

$$\begin{eqnarray}&&S(\theta )={X}_{(\infty )}(\theta ){\widetilde{S}}_{(\infty )}(\theta ){| }_{{\rm{SU}}(N)}{\otimes }_{{\rm{s}}}{\widetilde{S}}_{(\infty )}(\theta ){| }_{{\rm{SU}}(N)},\end{eqnarray} \tag{ 2.23 }$$

where ${\otimes }_{{\rm{s}}}$ indicates the symmetric product and the particles transform in the symmetric $N(N+1)/2$-dimensional representation of $F={SU}(N)$ that acts on both factors. In this case, there are no bound states. There is an analogous set of anti-particles transforming in the conjugate representation [38]. We conjecture that the k-deformed theory has an S-matrix of the form

$$\begin{eqnarray}&&S(\theta )={X}_{(k)}(\theta ){\widetilde{S}}_{(k)}(\theta ){| }_{{\rm{SU}}(N)}{\otimes }_{{\rm{s}}}{\widetilde{S}}_{(k)}(\theta ){| }_{{\rm{SU}}(N)},\end{eqnarray} \tag{ 2.24 }$$

with ${q}^{2(N+k)}=1$.

${\rm{SU}}(2N)/{\rm{Sp}}(N)$: for this series the S-matrix has the form

$$\begin{eqnarray}&&{S}^{{ab}}(\theta )={X}_{(\infty )}^{{ab}}(\theta ){\widetilde{S}}_{(\infty )}(\theta ){| }_{{\rm{SU}}(2N)}{\otimes }_{{a}}{\widetilde{S}}_{(\infty )}(\theta ){| }_{{\rm{SU}}(2N)},\end{eqnarray} \tag{ 2.25 }$$

where ${\otimes }_{{a}}$ indicates the antisymmetric product. For each a, the particles transforms in a product of fundamental representations ${V}_{{a}}\otimes {V}_{{a}}$, one for each factor. Then the ${\otimes }_{{a}}$ indicates that these are projected onto the anti-symmetric term of the ${\rm{SU}}(2N)$ which acts diagonally on both factors. We conjecture that the k-deformed theory has an S-matrix of the form

$$\begin{eqnarray}&&{S}^{{ab}}(\theta )={X}_{(k)}^{{ab}}(\theta ){\widetilde{S}}_{(k)}(\theta ){| }_{{\rm{SU}}(2N)}{\otimes }_{{a}}{\widetilde{S}}_{(k)}(\theta ){| }_{{\rm{SU}}(2N)},\end{eqnarray} \tag{ 2.26 }$$

with ${q}^{2(2N+k)}=1$.

The action (2.2) is invariant under the discrete ${{\mathbb{Z}}}_{2}$-transformation¹²

$$\begin{eqnarray}&&{\rm{\Pi }}({ \mathcal F },{A}_{+},{A}_{-},\lambda ,k)=({{ \mathcal F }}^{-1},{{\rm{Ad}}}_{{{ \mathcal F }}^{-1}}({A}_{+}+{\partial }_{+}{ \mathcal F }{{ \mathcal F }}^{-1}),{\rm{\Omega }}{A}_{-},{\lambda }^{-1},-k)\end{eqnarray} \tag{ 3.1 }$$

which, in particular, allows one to extend the range of the parameter λ from $[0,1]$ to $[0,\infty )$. Then, after integrating out the fields ${A}_{\pm }$, the resulting effective action turns out to be invariant under

$$\begin{eqnarray}&&{\rm{\Pi }}({ \mathcal F },\lambda ,k)=({{ \mathcal F }}^{-1},{\lambda }^{-1},-k)\end{eqnarray} \tag{ 3.2 }$$

as pointed out originally in [33], where it was shown that it severely restricts the form of the RG flow equation for λ. This last symmetry will also play an important role in the following.

We now show that the theories have global Poisson–Lie group symmetries. To do this, the natural starting point is to consider left and right infinitesimal F-transformations separately. Namely,

$$\begin{eqnarray}{\delta }_{L}{ \mathcal F } & = & {\epsilon }_{L}{ \mathcal F },\qquad \delta {A}_{+}=[{\epsilon }_{L},{A}_{+}]-{\partial }_{+}{\epsilon }_{L},\qquad \delta {A}_{-}=0,\\ {\delta }_{R}{ \mathcal F } & = & { \mathcal F }{\epsilon }_{R},\qquad \delta {A}_{+}=0,\qquad \delta {A}_{-}=[{\epsilon }_{R},{A}_{-}]-{\partial }_{+}{\epsilon }_{R}\end{eqnarray} \tag{ 3.3 }$$

with ${\epsilon }_{L},{\epsilon }_{R}\in {\mathfrak{f}}$. We find

$$\begin{eqnarray}{\delta }_{L}S & = & -\displaystyle \frac{k}{\pi }\displaystyle \int {{\rm{d}}}^{2}x\,{\rm{Tr}}({\epsilon }_{L}[{\partial }_{+}+{A}_{+},{\partial }_{-}+{\rm{\Omega }}{A}_{-}]),\\ {\delta }_{R}S & = & -\displaystyle \frac{k}{\pi }\displaystyle \int {{\rm{d}}}^{2}x\,{\rm{Tr}}({\epsilon }_{R}[{\partial }_{+}+{\rm{\Omega }}{A}_{+},{\partial }_{-}+{A}_{-}]).\end{eqnarray} \tag{ 3.4 }$$

In the k-PCM case, the vector action of the group corresponds to taking ${\epsilon }_{R}=-\epsilon$, ${\epsilon }_{L}=\epsilon$, and then adding both variations. This leads to

$$\begin{eqnarray}&&{\delta }_{V}S=-\displaystyle \frac{k(1-\lambda )}{\pi \lambda }\displaystyle \int {{\rm{d}}}^{2}x\,{\rm{Tr}}\,(\epsilon ({\partial }_{+}{A}_{-}+{\partial }_{-}{A}_{+}))=-\displaystyle \frac{k(1-\lambda )}{2\pi \lambda }\displaystyle \int {{\rm{d}}}^{2}x\,{\rm{Tr}}\,(\epsilon \,{\partial }_{\mu }{A}^{\mu }).\end{eqnarray} \tag{ 3.5 }$$

Then, for global transformations, ${\delta }_{V}S=0$ provided that the fields ${A}_{\pm }$ vanish at infinity. This shows that the action is invariant off-shell under global vector transformations, as expected, and provides conserved Noether charges. As a consequence, the target space geometry of the deformed PCM is invariant under the vector action of the group F. In the k-SSSM case, we get instead

$$\begin{eqnarray}&&{\delta }_{V}S=-\displaystyle \frac{k(1-\lambda )}{\pi \lambda }\displaystyle \int {{\rm{d}}}^{2}x\,{\rm{Tr}}\,\{\epsilon \,([{\partial }_{+}+{A}_{+}^{(0)},{A}_{-}^{(1)}]+[{\partial }_{-}+{A}_{-}^{(0)},{A}_{+}^{(1)}])\},\end{eqnarray} \tag{ 3.6 }$$

which vanishes off-shell for in the Lie algebra of G. This exhibits the invariance of the action under G gauge transformations, but also that the k-SSSM has no other obvious off-shell (Noether) symmetries. However, since these theories are classically integrable, this is not the end of the story.

Going back to the separate transformations, (3.4), we note that both can be put in the form

$$\begin{eqnarray}&&{\delta }_{{\rm{LP}}}S\sim \displaystyle \int {\epsilon }^{a}\left\{{{\rm{d}}{J}}^{a}+\displaystyle \frac{1}{2}{{\widetilde{f}}^{a}}_{\mathrm{bc}}{J}^{b}\wedge {J}^{c}\right\},\end{eqnarray} \tag{ 3.7 }$$

for some current J and some structure constants ${{\widetilde{f}}^{a}}_{{bc}}.$ On shell, each current is flat and so there exists conserved quantities Γ, the non-abelian moments, such that $J={{\rm{\Gamma }}}^{-1}{\rm{d}}{\rm{\Gamma }}$. It is well-known that the existence of conserved non-local charges such as these have played an important role proving the integrability of sigma models and finding their exact S-matrices; for example in [38].

To be more explicit, let us recall that the invariance of the action under generic field transformations provides the equations-of-motion which, in our case, can always be written in the Lax form (2.6). Then, the monodromy matrix¹³

$$\begin{eqnarray}&&m(z)={\rm{Pexp}}\left[-{\int }_{{S}^{1}}{\rm{d}}{x}\,{\mathscr{L}}(x,z)\right],\qquad {\mathscr{L}}={{\mathscr{L}}}_{+}-{{\mathscr{L}}}_{-},\end{eqnarray} \tag{ 3.8 }$$

is conserved for generic values of z, which provides an infinite number of, generally non-local, conserved quantities. Remarkably, the left and right variations (3.4) can be written in terms of the Lax pair evaluated at specific values of the spectral parameter z:

$$\begin{eqnarray}{\delta }_{L}S & = & -\displaystyle \frac{k}{\pi }\displaystyle \int {{\rm{d}}}^{2}x\,{\rm{Tr}}({\epsilon }_{L}[{\partial }_{+}+{{\mathscr{L}}}_{+}({z}_{-}),{\partial }_{-}+{{\mathscr{L}}}_{-}({z}_{-})]),\\ {\delta }_{R}S & = & -\displaystyle \frac{k}{\pi }\displaystyle \int {{\rm{d}}}^{2}x\,{\rm{Tr}}({\epsilon }_{R}[{\partial }_{+}+{{\mathscr{L}}}_{+}({z}_{+}),{\partial }_{-}+{{\mathscr{L}}}_{-}({z}_{+})]),\end{eqnarray} \tag{ 3.9 }$$

where in the k-PCM case

$$\begin{eqnarray}&&{z}_{\pm }=\pm \displaystyle \frac{\lambda +1}{\lambda -1},\end{eqnarray} \tag{ 3.10 }$$

and in the k-SSSM case

$$\begin{eqnarray}&&{z}_{+}={\lambda }^{-1/2},\qquad {z}_{-}={\lambda }^{1/2}.\end{eqnarray} \tag{ 3.11 }$$

In both, the significance of ${z}_{\pm }$ is that

$$\begin{eqnarray}&&{{\mathscr{L}}}_{+}({z}_{+})={\rm{\Omega }}{A}_{+},\quad {{\mathscr{L}}}_{-}({z}_{+})={A}_{-},\quad {{\mathscr{L}}}_{+}({z}_{-})={A}_{+},\quad {{\mathscr{L}}}_{-}({z}_{-})={\rm{\Omega }}{A}_{-}.\end{eqnarray} \tag{ 3.12 }$$

The equations (3.9) relate the conservation of $m({z}_{+})$ and $m({z}_{-})$ to the invariance of the action under left and right transformations, respectively, in the following sense. ${\delta }_{L/R}S=0$ implies that ${{\mathscr{L}}}_{\pm }({z}_{\mp })$ is flat, which provides the conserved quantities $m({z}_{\mp })$.

Notice that the special points z_± are uniquely determined by the deformation parameter λ and, furthermore, they are mapped into each other under the discrete transformation (3.2). Below, we will see that these points are precisely the poles of the so-called twisting functions that underlies the Poisson structure of these theories.

The Hamiltonian structure of the k-PCM and the k-SSSM can be written in terms of the canonical currents [18]

$$\begin{eqnarray}{{\mathscr{J}}}_{+} & = & -\displaystyle \frac{k}{2\pi }({{ \mathcal F }}^{-1}{\partial }_{+}{F}+{{ \mathcal F }}^{-1}{A}_{+}{ \mathcal F }-{A}_{-}),\\ {{\mathscr{J}}}_{-} & = & \displaystyle \frac{k}{2\pi }({\partial }_{-}{ \mathcal F }{{ \mathcal F }}^{-1}-{ \mathcal F }{A}_{-}{{ \mathcal F }}^{-1}+{A}_{+}),\end{eqnarray} \tag{ 3.13 }$$

which obey the classical Kac–Moody algebra¹⁴

$$\begin{eqnarray}&&\{{{\mathscr{J}}}_{\pm }^{a}(x),{{\mathscr{J}}}_{\pm }^{b}(y)\}={f}^{{abc}}{{\mathscr{J}}}_{\pm }^{c}(x){\delta }_{{xy}}\mp \displaystyle \frac{k}{2\pi }{\delta }^{{ab}}{\delta }_{{xy}}^{\prime },\quad \{{{\mathscr{J}}}_{+}^{a}(x),{{\mathscr{J}}}_{-}^{b}(y)\}=0.\end{eqnarray} \tag{ 3.14 }$$

On-shell, we can write these currents as

$$\begin{eqnarray}&&{{\mathscr{J}}}_{+}=-\displaystyle \frac{k}{2\pi }({\rm{\Omega }}{A}_{+}-{A}_{-}),\text{}{{\mathscr{J}}}_{-}=-\displaystyle \frac{k}{2\pi }({\rm{\Omega }}{A}_{-}-{A}_{+}).\end{eqnarray} \tag{ 3.15 }$$

The key observation for the following analysis if that the Kac–Moody currents are related to the Lax connection in a very simple way; namely,

$$\begin{eqnarray}&&{\mathscr{L}}({z}_{+})=-\displaystyle \frac{2\pi }{k}{{\mathscr{J}}}_{+},\ \ \ \ \ {\mathscr{L}}({z}_{-})=\displaystyle \frac{2\pi }{k}{{\mathscr{J}}}_{-},\end{eqnarray} \tag{ 3.16 }$$

where ${\mathscr{L}}={{\mathscr{L}}}_{+}-{{\mathscr{L}}}_{-}$ is the spatial component of the Lax connection. This simple observation will allow us to deduce below that the k-deformed field theories have a quantum group symmetry. Then, from (3.16) we find that the conserved quantities associated to the global ${F}_{L}\times {F}_{R}$ Poisson–Lie group action in both models take a rather universal form

$$\begin{eqnarray}&&\ m({z}_{+})={\rm{Pexp}}\left[\displaystyle \frac{2\pi }{k}{\int }_{{S}^{1}}{\rm{d}}{x}\ {{\mathscr{J}}}_{+}(x)\right],\ \ \ \ \ m({z}_{-})={\rm{Pexp}}\left[-\displaystyle \frac{2\pi }{k}{\int }_{{S}^{1}}{\rm{d}}{x}\,{{\mathscr{J}}}_{-}(x)\right].\end{eqnarray} \tag{ 3.17 }$$

Applying (3.2) to either (3.13) or (3.15) we see that the effect of the discrete transformation is to exchange the two Kac–Moody currents

$$\begin{eqnarray}&&{\rm{\Pi }}{{\mathscr{J}}}_{\pm }={{\mathscr{J}}}_{\mp }.\end{eqnarray} \tag{ 3.18 }$$

Then, $m({z}_{\pm })$ are mapped into each other under (3.2), so that one of the Poisson–Lie symmetries, say F_R, can be generated by a combination of F_L and Π. In section 4 below we turn to the quantum version of the algebra generated by $m({z}_{\pm })$.

In this section, we exploit the Kac–Moody algebraic structure (3.14) of the k-deformed theories to compute the Poisson brackets of the spatial component of the Lax connection. In particular, we are interested in extracting the twisting function ${\varphi }_{\epsilon }(z)$ induced by the ${\mathfrak{r}}$/${\mathfrak{s}}$ form and study its pole structure. We begin with the k-PCM theories, and then consider the k-SSSM theories, which require a slightly different treatment.

k-PCM case:

We start by writing the spatial component of the k-PCM Lax connection for generic values of the spectral parameter in the form

$$\begin{eqnarray}&&{\mathscr{L}}(z)={f}_{-}(z){{\mathscr{J}}}_{+}+{f}_{+}(z){{\mathscr{J}}}_{-},\end{eqnarray} \tag{ 3.19 }$$

with

$$\begin{eqnarray}&&{f}_{\pm }(z)=-\displaystyle \frac{z}{1-{z}^{2}}(p\pm {qz}),\ \ p=\displaystyle \frac{4\pi \lambda }{k(1-{\lambda }^{2})},\ \ q=\displaystyle \frac{4\pi \lambda }{k{(1+\lambda )}^{2}}.\end{eqnarray} \tag{ 3.20 }$$

Then, the Kac–Moody algebra (3.14) provides the Maillet ${\mathfrak{r}}$/${\mathfrak{s}}$ representation of the Poisson bracket

$$\begin{eqnarray}\{\mathop{{\mathscr{L}}}\limits^{1}(x,z),\mathop{{\mathscr{L}}}\limits^{2}(y,w)\} & = & [{\mathfrak{r}},\mathop{{\mathscr{L}}}\limits^{1}(x,z)+\mathop{{\mathscr{L}}}\limits^{2}(y,w)]{\delta }_{{xy}}\\ & & -[{\mathfrak{s}},\mathop{{\mathscr{L}}}\limits^{1}(x,z)-\mathop{{\mathscr{L}}}\limits^{2}(y,w)]{\delta }_{{xy}}-2{\mathfrak{s}}{\delta }_{{xy}}^{\prime }.\end{eqnarray} \tag{ 3.21 }$$

In this kind of representation, the equation is to be understood as acting on some tensor product $V\otimes V$, where V is some faithful representation of the Lie algebra ${\mathfrak{f}}$, and the index on top show which of the factors the quantity acts on, e.g. $\mathop{{\mathscr{L}}}\limits^{1}(V\otimes V)=({\mathscr{L}}V)\otimes V$ and $\mathop{{\mathscr{L}}}\limits^{2}(V\otimes V)=V\otimes ({\mathscr{L}}V)$, etc. Quantities like ${\mathfrak{s}}$ and ${\mathfrak{r}}$ defined below act on both factors.

In (3.21), the ${\mathfrak{r}}$/${\mathfrak{s}}$ matrices are given by

$$\begin{eqnarray}&&{\mathfrak{r}}(z,w)=\left[\displaystyle \frac{w{\varphi }_{\epsilon }{(z)}^{-1}+z{\varphi }_{\epsilon }{(w)}^{-1}}{z-w}\right]{\mathfrak{C}},\quad {\mathfrak{s}}(z,w)=\left[\displaystyle \frac{w{\varphi }_{\epsilon }{(z)}^{-1}-z{\varphi }_{\epsilon }{(w)}^{-1}}{z-w}\right]{\mathfrak{C}},\end{eqnarray} \tag{ 3.22 }$$

where

$$\begin{eqnarray}&&{\varphi }_{\epsilon }(z)=2p\cdot \displaystyle \frac{1}{z}\cdot \displaystyle \frac{1-{z}^{2}}{{q}^{2}{z}^{2}-{p}^{2}}\end{eqnarray} \tag{ 3.23 }$$

is the so-called twisting function and ${\mathfrak{C}}={\sum }_{a}{T}^{a}\otimes {T}^{a}$ is the tensor Casimir. Notice that ${\varphi }_{\epsilon }(z)$ has two poles located precisely at at ${z}_{\pm }$

$$\begin{eqnarray}&&{z}_{\pm }=\mp \displaystyle \frac{p}{q}=\pm \displaystyle \frac{\lambda +1}{\lambda -1}.\end{eqnarray} \tag{ 3.24 }$$

These poles are exactly the evaluation points displayed in (3.10) used to extract the Lie–Poisson group charges out of the monodromy matrix. The extra pole at z = 0 has no physical significance and can be removed by working with $z\to 1/z$ instead. In this case, the poles are inverted ${z}_{\pm }\to {z}_{\pm }^{-1}$ and the (3.22) can be rewritten in a more suggestive way

$$\begin{eqnarray}&&{\mathfrak{r}}(z,w)=-\left[\displaystyle \frac{{\phi }_{\epsilon }{(z)}^{-1}+{\phi }_{\epsilon }{(w)}^{-1}}{z-w}\right]{\mathfrak{C}},\quad {\mathfrak{s}}(z,w)=-\left[\displaystyle \frac{{\phi }_{\epsilon }{(z)}^{-1}-{\phi }_{\epsilon }{(w)}^{-1}}{z-w}\right]{\mathfrak{C}},\end{eqnarray} \tag{ 3.25 }$$

where¹⁵

$$\begin{eqnarray}&&{\phi }_{\epsilon }{(z)}^{-1}=-\displaystyle \frac{k}{2\pi }{pq}[\phi {(z)}_{{\rm{PCM}}}^{-1}+{\epsilon }^{2}\phi {(z)}_{{\rm{FR}}}^{-1}]\end{eqnarray} \tag{ 3.26 }$$

and

$$\begin{eqnarray}&&\phi {(z)}_{{\rm{PCM}}}=\displaystyle \frac{1}{{z}^{2}}-1,\ \ \phi {(z)}_{{\rm{FR}}}=1,\ \ {\epsilon }^{2}=-\displaystyle \frac{1}{4\lambda }{(1-\lambda )}^{2},\end{eqnarray} \tag{ 3.27 }$$

are the twisting functions of the undeformed PCM and the FR model, and is the deformation parameter in the ${\epsilon }^{2}\lt 0$ branch.

The ${\mathfrak{r}}$/${\mathfrak{s}}$ matrices satisfy a modified classical Yang–Baxter equation, mCYBE, a fact which was proven explicitly in [44]. Notice that in the FR limit $\lambda \to 0$, we have

$$\begin{eqnarray}&&{\mathfrak{s}}(z,w)=0\end{eqnarray} \tag{ 3.28 }$$

and the non-ultra-locality is completely removed. This is the celebrated ultra-localization of the PCM introduced by hand in [1]. Consequently, the k-PCM can be thought as the Lagrangian implementation of the FR ultra-localization mechanism.

The values of the s matrix at the poles are given by

$$\begin{eqnarray}&&{\mathfrak{s}}({z}_{\pm },{z}_{\pm })=\pm \displaystyle \frac{\pi }{k},\ \ \ \ \ {\mathfrak{s}}({z}_{\pm },{z}_{\mp })=0.\end{eqnarray} \tag{ 3.29 }$$

Using these values in the Maillet bracket (3.21), we recover the Kac–Moody current algebra (3.14) in tensorial form

$$\begin{eqnarray}&&\{{\mathop{{\mathscr{J}}}\limits^{1}}_{\pm }(x),{\mathop{{\mathscr{J}}}\limits^{2}}_{\pm }(y)\}=\displaystyle \frac{1}{2}[{\mathfrak{C}},{\mathop{{\mathscr{J}}}\limits^{1}}_{\pm }(x)-{\mathop{{\mathscr{J}}}\limits^{2}}_{\pm }(x)]{\delta }_{{xy}}\mp \displaystyle \frac{k}{2\pi }{\mathfrak{C}}{\delta }_{{xy}}^{\prime }.\end{eqnarray} \tag{ 3.30 }$$

The values of ${\mathfrak{r}}(z,w)$ at the same poles depends on how we take the limit but are always proportional to the tensor Casimir and hence do not contribute to the Poisson bracket. However, the values at different poles vanish identically.

We end this section, by rewriting (3.21) in a slightly different way which will be useful when we consider the k–F/G theories. By defining

$$\begin{eqnarray}&&{\mathfrak{r}}(z,w)=\displaystyle \frac{1}{2}({\mathfrak{R}}(z,w)-{{\mathfrak{R}}}^{t}(z,w)),\quad {\mathfrak{s}}(z,w)=-\displaystyle \frac{1}{2}({\mathfrak{R}}(z,w)+{{\mathfrak{R}}}^{t}(z,w)),\end{eqnarray} \tag{ 3.31 }$$

we can write the Maillet bracket as

$$\begin{eqnarray}&&\{\mathop{{\mathscr{L}}}\limits^{1}(x,z),\mathop{{\mathscr{L}}}\limits^{2}(y,w)\}=[{\mathfrak{R}},\mathop{{\mathscr{L}}}\limits^{1}(x,z)]{\delta }_{{xy}}-[{{\mathfrak{R}}}^{t},\mathop{{\mathscr{L}}}\limits^{2}(y,w)]{\delta }_{{xy}}+({\mathfrak{R}}+{{\mathfrak{R}}}^{t}){\delta }_{{xy}}^{\prime },\end{eqnarray} \tag{ 3.32 }$$

where

$$\begin{eqnarray}&&{\mathfrak{R}}(z,w)=-\displaystyle \frac{2{\phi }_{\epsilon }{(w)}^{-1}}{z-w}{\mathfrak{C}}.\end{eqnarray} \tag{ 3.33 }$$

k-SSSM case:

Inspired by (3.26), this time we take the following inverse twisting function

$$\begin{eqnarray}&&{\varphi }_{\epsilon }{(z)}^{-1}=c[\varphi {(z)}_{\sigma }^{-1}+{\epsilon }^{2}\varphi {(z)}_{{\rm{FR}}}^{-1}],\end{eqnarray} \tag{ 3.34 }$$

where

$$\begin{eqnarray}&&\varphi {(z)}_{\sigma }=\displaystyle \frac{4{z}^{2}}{{(1-{z}^{2})}^{2}},\ \ \varphi {(z)}_{{\rm{FR}}}=1,\ \ {\epsilon }^{2}=-\displaystyle \frac{1}{4\lambda }{(1-\lambda )}^{2}\end{eqnarray} \tag{ 3.35 }$$

and c is a constant yet to be fixed by the evaluation of ${\mathfrak{s}}(z,w)$ at the poles of ${\varphi }_{\epsilon }(z)$.

The deformed twisting function is

$$\begin{eqnarray}&&c\,{\varphi }_{\epsilon }(z)=\displaystyle \frac{4}{{z}^{2}+2(2\,{\epsilon }^{2}-1)+{z}^{-2}}\end{eqnarray} \tag{ 3.36 }$$

which has four poles given by

$$\begin{eqnarray}&&{z}_{+}^{2}={\lambda }^{-1},\ \ {z}_{-}^{2}=\lambda .\end{eqnarray} \tag{ 3.37 }$$

The two poles corresponding to the positive branch of the square root are the evaluation points (3.11) used to extract the Lie–Poisson group charges out of the monodromy matrix¹⁶ . These poles are important for establishing the quantum group symmetry of the theory, as we shall see below.

The calculation we need to find ${\mathfrak{r}}$ and ${\mathfrak{s}}$ was already done in [43] for the branch $0\leqslant {\epsilon }^{2}\lt 1$ and also holds in the present case as the compatibility of the Poisson brackets is independent of the sign of ${\epsilon }^{2}$. The R-matrix corresponding to the Poisson bracket in the form (3.32) is¹⁷

$$\begin{eqnarray}&&{\mathfrak{R}}(z,w)=-\displaystyle \frac{2}{{z}^{2}-{w}^{2}}\left[\displaystyle \sum _{j=0,1}{z}^{j}{w}^{2-j}{{\mathfrak{C}}}^{(j,2-j)}\right]{\varphi }_{\epsilon }{(w)}^{-1},\end{eqnarray} \tag{ 3.38 }$$

from which follows the ${\mathfrak{r}}$/${\mathfrak{s}}$ matrices in our present case

$$\begin{eqnarray}{\mathfrak{r}}(z,w) & = & -\displaystyle \frac{1}{{z}^{2}-{w}^{2}}\displaystyle \sum _{j=0,1}[{w}^{j}{z}^{2-j}{\varphi }_{\epsilon }{(z)}^{-1}+{z}^{j}{w}^{2-j}{\varphi }_{\epsilon }{(w)}^{-1}]{{\mathfrak{C}}}^{(j,2-j)},\\ {\mathfrak{s}}(z,w) & = & -\displaystyle \frac{1}{{z}^{2}-{w}^{2}}\displaystyle \sum _{j=0,1}[{w}^{j}{z}^{2-j}{\varphi }_{\epsilon }{(z)}^{-1}-{z}^{j}{w}^{2-j}{\varphi }_{\epsilon }{(w)}^{-1}]{{\mathfrak{C}}}^{(j,2-j)}.\end{eqnarray} \tag{ 3.39 }$$

However, it is important to recall that the expressions for ${\mathfrak{r}}/{\mathfrak{s}}$ written right above follows by using the following modified Lax operator

$$\begin{eqnarray}&&{\mathscr{L}}(z)={f}_{-}(z)\left({{\mathscr{J}}}_{+}^{(0)}+\displaystyle \frac{{\lambda }^{-1/2}}{z}{{\mathscr{J}}}_{+}^{(1)}\right)+{f}_{+}(z)\left({{\mathscr{J}}}_{-}^{(0)}+\displaystyle \frac{{\lambda }^{1/2}}{z}{{\mathscr{J}}}_{-}^{(1)}\right),\end{eqnarray} \tag{ 3.40 }$$

where

$$\begin{eqnarray}&&{f}_{\pm }(z)\equiv -\displaystyle \frac{2\pi }{k}\displaystyle \frac{\lambda }{(1-{\lambda }^{2})}({z}^{2}-{\lambda }^{\mp 1}).\end{eqnarray} \tag{ 3.41 }$$

It corresponds to the extension of the Lax operator out of the constraint surface by the only first class constraint present for this case

$$\begin{eqnarray}&&{{\mathscr{J}}}_{+}^{(0)}+{{\mathscr{J}}}_{-}^{(0)}.\end{eqnarray} \tag{ 3.42 }$$

See [18] for details on the Hamiltonian approach. The Maillet bracket then follows as a simple consequence of the underlying Kac–Moody algebra structure.

Concentrating on the ${\mathfrak{s}}(z,w)$ matrix, after some simplification we find it can be put in the form

$$\begin{eqnarray}&&{\mathfrak{s}}(z,w)=-\displaystyle \frac{c}{4}[{z}^{2}+{w}^{2}+2(2\,{\epsilon }^{2}-1)]{{\mathfrak{C}}}^{(00)}+\displaystyle \frac{c}{4{zw}}[1-{z}^{2}{w}^{2}]{{\mathfrak{C}}}^{(11)}.\end{eqnarray} \tag{ 3.43 }$$

Written in this way we can easily evaluate this function at the poles

$$\begin{eqnarray}&&{\mathfrak{s}}({z}_{\pm },{z}_{\pm })=\mp \displaystyle \frac{c}{4\lambda }(1-{\lambda }^{2}){\mathfrak{C}},\ \ {\mathfrak{s}}({z}_{\pm },{z}_{\mp })=0.\end{eqnarray} \tag{ 3.44 }$$

The normalization of the central term in the Kac–Moody algebra imposes the condition

$$\begin{eqnarray}&&-2{\mathfrak{s}}({z}_{\pm },{z}_{\pm })=\mp \displaystyle \frac{2\pi }{k}{\mathfrak{C}},\end{eqnarray} \tag{ 3.45 }$$

which fixes the so far unspecified constant c to be

$$\begin{eqnarray}&&c=-\displaystyle \frac{4\pi \lambda }{k(1-{\lambda }^{2})}.\end{eqnarray} \tag{ 3.46 }$$

This value for c can be verified independently by direct use of the extended Lax operator (3.40). In the limit $\lambda \to 0,$ we have

$$\begin{eqnarray}&&{\mathfrak{s}}(z,w)=-\displaystyle \frac{\pi }{k}{{\mathfrak{C}}}^{(00)},\end{eqnarray} \tag{ 3.47 }$$

from which follows that in the coset case the non-ultra-locality can be only partially tamed [2]. As in the k-PCM case, from the Maillet bracket we recover the Kac–Moody algebra in tensor form.

We work in the de-compatified limit, in which the monodromy matrix takes the general form¹⁸

$$\begin{eqnarray}&&m={\rm{Pexp}}\left[-{\int }_{a}^{b}{\rm{d}}{x}\,{\mathscr{L}}(x)\right]=\exp \left[\sum _{n=1}^{\infty }{(-1)}^{n}{F}_{n}\right],\end{eqnarray} \tag{ 3.48 }$$

where for the first three terms we get

$$\begin{eqnarray*}{F}_{1} & = & {\displaystyle \int }_{a}^{b}{\rm{d}}{x}\,{\mathscr{L}}(x),\\ {F}_{2} & = & \displaystyle \frac{1}{4}{\displaystyle \int }_{a}^{b}{\rm{d}}{x}\,{\rm{d}}{y}\,[{\mathscr{L}}(x),{\mathscr{L}}(y)]{\epsilon }_{{xy}},\\ {F}_{3} & = & \displaystyle \frac{1}{12}{\displaystyle \int }_{a}^{b}{\rm{d}}{x}\,{\rm{d}}{y}\,{\rm{d}}{z}\,[{\mathscr{L}}(x),[{\mathscr{L}}(y),{\mathscr{L}}(z)]]{\epsilon }_{{xy}}{\epsilon }_{{yz}}.\end{eqnarray*}$$

The limit a and b will subsequently be taken to $\pm \infty$, respectively. Expanding ${\mathscr{L}}(x)$ around a chosen point in the complex plane and reorganizing the series in powers of the spectral parameter z, not to be confused with the integration variable z, we find the corresponding non-local charges associated to that particular point.

Let us focus first on the k-PCM theory. When expanded around the poles ${z}_{\pm }=\mp p/q,$ we find the following expression for the spatial Lax operator

$$\begin{eqnarray}&&{\mathscr{L}}(x)={j}_{(0)\pm }(x)+{j}_{(1)\pm }(x){\widehat{z}}_{\pm }+{j}_{(2)\pm }(x){\widehat{z}}_{\pm }^{2}+{j}_{(3)\pm }(x){\widehat{z}}_{\pm }^{3}+{ \mathcal O }({\widehat{z}}_{\pm }^{4}),\end{eqnarray} \tag{ 3.49 }$$

where ${\widehat{z}}_{\pm }=z-{z}_{\pm }$ and

$$\begin{eqnarray}{j}_{(0)\pm } & = & \pm \displaystyle \frac{1}{4\lambda }(1+\lambda )[(1-\lambda ){I}_{0}\pm (1+\lambda ){I}_{1}]=\mp \displaystyle \frac{2\pi }{k}{{\mathscr{J}}}_{\pm }={\mathscr{L}}({z}_{\pm }),\\ {j}_{(1)\pm } & = & \displaystyle \frac{1}{8{\lambda }^{2}}{(1-\lambda )}^{2}[(1+{\lambda }^{2}){I}_{0}\pm (1-{\lambda }^{2}){I}_{1}],\\ {j}_{(2)\pm } & = & \pm \displaystyle \frac{1}{16{\lambda }^{3}}{(1-\lambda )}^{3}[(1+{\lambda }^{3}){I}_{0}\pm (1-{\lambda }^{3}){I}_{1}],\\ {j}_{(3)\pm } & = & \displaystyle \frac{1}{32{\lambda }^{4}}{(1-\lambda )}^{4}[(1+{\lambda }^{4}){I}_{0}\pm (1-{\lambda }^{4}){I}_{1}].\end{eqnarray} \tag{ 3.50 }$$

The expansion for the Lax operator around the poles (3.49) starts with terms of order ${\widehat{z}}_{\pm }^{0}$. These terms can be removed by judicious gauge transformations,

$$\begin{eqnarray}&&m^{\prime} (z)=u(b)m(z)u{(a)}^{-1},\ \ {{\mathscr{L}}}^{\prime }=u({\mathscr{L}}+{\partial }_{x}){u}^{-1},\end{eqnarray} \tag{ 3.51 }$$

where the gauge transformation takes the form

$$\begin{eqnarray}&&u(x,{x}_{0},{z}_{\pm })={\rm{Pexp}}\left[-{\int }_{{x}_{0}}^{x}{\rm{d}}{y}\,{\mathscr{L}}({z}_{\pm })\right]\equiv u{(x)}_{\pm }\end{eqnarray} \tag{ 3.52 }$$

around each pole, respectively. In this way, (3.48) can be written as follows

$$\begin{eqnarray}&&m(z)=u{(b)}_{\pm }{\rm{Pexp}}\left[-{\int }_{a}^{b}{\rm{d}}{x}\,{\mathscr{L}}^{\prime} {(x)}_{\pm }\right]u{(a)}_{\pm }^{-1},\end{eqnarray} \tag{ 3.53 }$$

with a transformed Lax operator

$$\begin{eqnarray}&&{\mathscr{L}}^{\prime} {(x)}_{\pm }={j}_{(1)\pm }^{\prime }(x){\widehat{z}}_{\pm }+{j}_{(2)\pm }^{\prime }(x){\widehat{z}}_{\pm }^{2}+{j}_{(3)\pm }^{\prime }(x){\widehat{z}}_{\pm }^{3}+{ \mathcal O }({\widehat{z}}_{\pm }^{4})\end{eqnarray} \tag{ 3.54 }$$

defined in terms of the non-local dressed currents

$$\begin{eqnarray}&&{j}_{(n)\pm }^{\prime }(x)=u{(x)}_{\pm }^{-1}{j}_{(n)\pm }(x)u{(x)}_{\pm },\text{}n\geqslant 1.\end{eqnarray} \tag{ 3.55 }$$

Taking ${x}_{0}=a=-\infty$ and $b=\infty$, the expansion of the monodromy matrix around the poles ${z}_{\pm }$ now takes the form

$$\begin{eqnarray}&&m(z)=m({z}_{\pm })\exp \left[\sum _{n=1}^{\infty }{q}_{(n)\pm }{\widehat{z}}_{\pm }^{n}\right].\end{eqnarray} \tag{ 3.56 }$$

As a result of the gauge transformation, all the charges ${q}_{(n)\pm }$ are non-local and the first three take the explicit form

$$\begin{eqnarray}{q}_{(1)\pm } & = & -{\displaystyle \int }_{-\infty }^{\infty }{\rm{d}}{x}\,{j}_{(1)\pm }^{\prime }(x),\\ {q}_{(2)\pm } & = & \displaystyle \frac{1}{4}{\displaystyle \int }_{-\infty }^{\infty }{\rm{d}}{x}{\rm{d}}{y}[{j}_{(1)\pm }^{\prime }(x),{j}_{(1)\pm }^{\prime }(y)]{\epsilon }_{{xy}}-{\displaystyle \int }_{-\infty }^{\infty }{\rm{d}}{x}\,{j}_{(2)\pm }^{\prime }(x),\\ {q}_{(3)\pm } & = & -\displaystyle \frac{1}{12}{\displaystyle \int }_{-\infty }^{\infty }{\rm{d}}{x}{\rm{d}}{y}{\rm{d}}{z}[{j}_{(1)\pm }^{\prime }(x),[{j}_{(1)\pm }^{\prime }(y),{j}_{(1)\pm }^{\prime }(z)]]{\epsilon }_{{xy}}{\epsilon }_{{yz}}\\ & & +\displaystyle \frac{1}{2}{\displaystyle \int }_{-\infty }^{\infty }{\rm{d}}{x}{\rm{d}}{y}\,[{j}_{(1)\pm }^{\prime }(x),{j}_{(2)\pm }^{\prime }(y)]{\epsilon }_{{xy}}-{\displaystyle \int }_{-\infty }^{\infty }{\rm{d}}{x}\,{j}_{(3)\pm }^{\prime }(x).\end{eqnarray} \tag{ 3.57 }$$

Notice that under $\lambda \to {\lambda }^{-1}$, which corresponds to the transformation (3.2), we do indeed have ${q}_{(n)\pm }\to {q}_{(n)\mp }$ as claimed. This follows from our earlier observation that we can combine the F_L group action with (3.2) to generate F_R and vice versa. If we evaluate these charges at the, for us, unphysical value of the coupling constant ${\lambda }^{\prime }=-1$, then the two poles coalesce at ${z}_{\pm }=0$ and the expansion of the monodromy produces charges that coincide with the first three Yangian charges found in [44]

$$\begin{eqnarray}&&{q}_{(1)\pm }({\lambda }^{\prime })\equiv {Q}_{(0)},\ \ {q}_{(2)\pm }({\lambda }^{\prime })\equiv {Q}_{(1)},\ \ {q}_{(3)\pm }({\lambda }^{\prime })\equiv {Q}_{(2)}.\end{eqnarray} \tag{ 3.58 }$$

On the one hand, the dominant term for the expansion around z = 0 is the local Noether charge ${Q}_{(0)}$ corresponding to the global vector action of F and given by (3.5). The Poisson algebra generated by ${Q}_{(0)}$ is isomorphic to ${\mathfrak{f}}$ and the higher charges ${Q}_{(n)}$, $n\gt 0$ enhance this Poisson algebra to a Yangian, see [44].

On the other hand, the dominant terms for the expansions around ${z}_{\pm }=\mp p/q$ are the non-abelian moments (3.17) associated to the global Poisson–Lie group action ${F}_{L}\times {F}_{R}$. They will be shown to generate the finite-dimensional quantum group ${U}_{q}({\mathfrak{f}})$ after quantization. It is then natural to expect that higher charges enhance ${U}_{q}({\mathfrak{f}})$ to an affine quantum symmetry to match the symmetries of the S-matrices that we have conjectured to describe these theories in section 2. However, explicit computations of the algebra of higher charges is complicated by the non-trivial dressing factors $u{(x)}_{\pm }$, hence we expect to report about this problem in the near future. Fortunately, some evidence that this enhancement may occur in our case comes from a different kind of deformation of the ${\rm{SU}}(2)$ PCM [45–47], in which a classical q-deformed Poisson algebra of symmetries induced by the squashing is studied for this particular case. What suggests this is that, technically, the situation in those references is essentially the same as ours, namely, the generators of the q-deformed symmetry group are extracted from the expansion of the monodromy matrix at the poles of the twisting function.

For the k-SSSM theories, a similar story holds. We have exactly the same expansion as given by (3.49) but with different expressions for the current components

$$\begin{eqnarray}{j}_{(0)\pm } & = & \mp \displaystyle \frac{2\pi }{k}{{\mathscr{J}}}_{\pm }={\mathscr{L}}({z}_{\pm }),\\ {j}_{(1)\pm } & = & -\displaystyle \frac{2\pi }{k}\displaystyle \frac{{\lambda }^{\mp 1/2}}{{\lambda }^{-1}-\lambda }[2({{\mathscr{J}}}_{+}^{(0)}+{{\mathscr{J}}}_{-}^{(0)})+(1+{\lambda }^{\pm 2}){{\mathscr{J}}}_{\pm }^{(1)}+2{\lambda }^{\pm 1}{{\mathscr{J}}}_{\mp }^{(1)}],\\ {j}_{(2)\pm } & = & -\displaystyle \frac{2\pi }{k}\displaystyle \frac{1}{{\lambda }^{-1}-\lambda }[{{\mathscr{J}}}_{+}^{(0)}+{{\mathscr{J}}}_{-}^{(0)}-{\lambda }^{\pm 1}({\lambda }^{\pm 1}{{\mathscr{J}}}_{\pm }^{(1)}+{{\mathscr{J}}}_{\mp }^{(1)})],\\ {j}_{(3)\pm } & = & -\displaystyle \frac{2\pi }{k}\displaystyle \frac{{\lambda }^{\pm 3/2}}{{\lambda }^{-1}-\lambda }[{\lambda }^{\pm 1}{{\mathscr{J}}}_{\pm }^{(1)}+{{\mathscr{J}}}_{\mp }^{(1)}],\end{eqnarray} \tag{ 3.59 }$$

where ${\widehat{z}}_{\pm }=z-{\lambda }^{\mp 1/2}.$ An intuitive way to seek for Yangian symmetries is to find a value of λ for which ${j}_{(0)\pm }=\,0$, see the first equation of (3.50) with $\lambda =-1$ in the k-PCM case. Notice, however, that such a solution does not exist for the k-SSSM theories, hence we do not expect any Yangian symmetries in this case. This is in agreement with the absence of global Noether charges as noticed above around (3.6).

Then, the expansion of the monodromy matrix around the poles takes the rather universal form (3.56). The model dependence shows up in the higher charges ${q}_{(n)\pm },$ $n\geqslant 1.$ In the k-PCM they are all Lie-algebra valued while in the k–F/G they are all coset valued except for the extension associated to the first class constraint.

We now proceed to quantize the monodromies $m({z}_{\pm })$ and find their quantum algebra.

Let us recall that the equations of motion of the k-deformed theories can be re-cast in Lax form as the zero curvature condition (2.6). This form manifests an infinite-dimensional group of gauge transformations generated by the group elements ${\rm{\Gamma }}(x,z)$

$$\begin{eqnarray}&&{{\mathscr{L}}}_{\pm }^{\prime }(z)={\rm{\Gamma }}(z){{\mathscr{L}}}_{\pm }(z){\rm{\Gamma }}{(z)}^{-1}-{\partial }_{\pm }{\rm{\Gamma }}(z){\rm{\Gamma }}{(z)}^{-1}.\end{eqnarray} \tag{ 4.1 }$$

From (3.16) we conclude that, when evaluated at the poles ${z}_{\pm }$, this group becomes the ordinary loop group ${g}_{\pm }(x)={\rm{\Gamma }}(x,{z}_{\pm })$ of gauge transformations of the Kac–Moody currents

$$\begin{eqnarray}&&{{\mathscr{J}}}_{\pm }^{\prime }(x)={g}_{\pm }(x){{\mathscr{J}}}_{\pm }(x){g}_{\pm }{(x)}^{-1}\pm \displaystyle \frac{k}{2\pi }{\partial }_{x}{g}_{\pm }(x){g}_{\pm }{(x)}^{-1},\end{eqnarray} \tag{ 4.2 }$$

which are symmetries of the Kac–Moody algebra (3.30).

Before introducing a lattice regularization of the theory we can infer, via a semi-classical analysis, that the k-deformation parameter of the quantum group generated by $m({z}_{\pm })$ is indeed a root of unity. This information can be extracted from the Maillet representation (3.21) evaluated at the poles. Introducing the parallel transport

$$\begin{eqnarray}&&u(x,{x}_{0})={\rm{Pexp}}\left[-{\int }_{{x}_{0}}^{x}{\rm{d}}{y}\,{\mathscr{L}}(y)\right],\end{eqnarray} \tag{ 4.3 }$$

which obeys

$$\begin{eqnarray}&&{\partial }_{x}u(x,{x}_{0})=-{\mathscr{L}}(x)u(x,{x}_{0}),\ \ {\partial }_{{x}_{0}}u(x,{x}_{0})={}u(x,{x}_{0}){\mathscr{L}}({x}_{0}),\end{eqnarray} \tag{ 4.4 }$$

and then following Maillet [19], we find in our case

$$\begin{eqnarray}&&\{\mathop{u}\limits^{1}(x,{x}_{0}),\mathop{u}\limits^{2}(y,{y}_{0})\}={\int }_{{x}_{0}}^{x}{\rm{d}}{z}{\int }_{{y}_{0}}^{y}{{\rm{d}}{z}}^{\prime }\,\mathop{u}\limits^{1}(x,z)\mathop{u}\limits^{2}(y,{z}^{\prime })\{\mathop{{\mathscr{L}}}\limits^{1}(z),\mathop{{\mathscr{L}}}\limits^{2}({z}^{\prime })\}\mathop{u}\limits^{1}(z,{x}_{0})\mathop{u}\limits^{2}({z}^{\prime },{y}_{0}).\end{eqnarray} \tag{ 4.5 }$$

In the above, for brevity, we have not indicated the spectral parameter dependence but we should keep in mind that u depend on it. In fact, we are only interested in the result above when it is evaluated at the poles ${z}_{\pm }$ in which ${\mathscr{L}}(z)$, as in (3.16), picks out the Kac–Moody currents. In this case we find that

$$\begin{eqnarray}&&\{\mathop{u}\limits^{1}(x,{x}_{0}),\mathop{u}\limits^{2}(y,{y}_{0})\}=s\,\mathop{u}\limits^{1}(x,{x}_{0})\mathop{u}\limits^{2}(y,{y}_{0})({\epsilon }_{{xy}}+{\epsilon }_{{x}_{0}{y}_{0}}){\mathfrak{C}},\end{eqnarray} \tag{ 4.6 }$$

where s is given by

$$\begin{eqnarray}&&{\mathfrak{s}}({z}_{\pm },{z}_{\pm })=s({z}_{\pm }){\mathfrak{C}},\ \ \ \ \ s({z}_{\pm })=\pm \displaystyle \frac{\pi }{k},\end{eqnarray} \tag{ 4.7 }$$

and ${\epsilon }_{{xy}}$ is the sign function and in what follows we will drop the ± indices for simplicity²⁰ .

The expression (4.6) is the classical exchange algebra we want to quantize semi-classically. We want to emphasize that at the moment we are proceeding in a rather ad hoc fashion in order to provide some inspiration for the lattice algebra defined below. By exploiting the classical commutativity of the u's and using the quantization rule²¹ $-{\rm{i}}\{\ast ,\ast \}\to [\ast ,\ast ]$, we get

$$\begin{eqnarray*}\mathop{u}\limits^{1}(x)\mathop{u}\limits^{2}(y) & = & \mathop{u}\limits^{2}(y)\mathop{u}\limits^{1}(x)(1-{\rm{i}}{s}\,{\mathfrak{C}}\,{\epsilon }_{{xy}}+\cdots )\\ & = & \mathop{u}\limits^{2}(y)\mathop{u}\limits^{1}(x){{\mathfrak{R}}}_{{xy}},\end{eqnarray*}$$

where the ellipsis denote sub-leading quantum contributions. From this we infer that, roughly speaking, we should expect something like

$$\begin{eqnarray}&&{{\mathfrak{R}}}_{{xy}}\sim {q}^{{\mathfrak{C}}{\epsilon }_{{xy}}},\quad q\sim {{\rm{e}}}^{-{\rm{i}}{s}},\quad {q}^{k}\sim 1,\end{eqnarray} \tag{ 4.8 }$$

which is an interesting expression as it relates Maillet's s matrix, the poles and the k-deformation parameter. Notice that the effect of the tranformation (3.2) is to map $q\to {q}^{-1}.$

The calculation above should not be considered as a formal one, we use it only as a source of inspiration for the type of quantum algebra we should expect. In particular, the Poisson brackets (4.6) do not obey the Jacobi identity. Indeed, we find

$$\begin{eqnarray}&&{\rm{Jacobi}}=-{s}^{2}\,\mathop{u}\limits^{1}(x)\mathop{u}\limits^{2}(y)\mathop{u}\limits^{3}(z)[{\epsilon }_{{xy}}{\epsilon }_{{yz}}+{\epsilon }_{{xz}}{\epsilon }_{{zy}}+{\epsilon }_{{zx}}{\epsilon }_{{xy}}]{f}^{{abc}}{T}^{a}\otimes {T}^{b}\otimes {T}^{c},\end{eqnarray} \tag{ 4.9 }$$

which never vanishes for distinct $x,y,z$.

After the naïve semi-classical approach, we are now ready to introduce the lattice algebra. The first thing is to discretize the spatial circle ${S}^{1}$ in terms of N points with coordinates ${x}_{n}=n{\rm{\Delta }}$, $n=1,\ldots ,N$ mod N in the order ${x}_{1}\lt {x}_{2}\lt \cdots \,\lt \,{x}_{N-1}\lt {x}_{N}$. Then, we define the transport matrix

$$\begin{eqnarray}&&t({x}_{n+1},{x}_{n})={\rm{Pexp}}\left[-{\int }_{{x}_{n}}^{{x}_{n+1}}{\rm{d}}{x}\,{ \mathcal L }(x)\right]\end{eqnarray} \tag{ 4.10 }$$

between sites. Evaluating it at the poles ${z}_{\pm }$ defines for us the lattice variables

$$\begin{eqnarray}&&{J}_{n}^{L}={\rm{Pexp}}\left[\displaystyle \frac{2\pi }{k}{\int }_{{x}_{n}}^{{x}_{n+1}}{\rm{d}}{x}{{\mathscr{J}}}_{+}(x)\right],\ \ {J}_{n}^{R}={\rm{Pexp}}\left[-\displaystyle \frac{2\pi }{k}{\int }_{{x}_{n}}^{{x}_{n+1}}{\rm{d}}{x}{{\mathscr{J}}}_{-}(x)\right].\end{eqnarray} \tag{ 4.11 }$$

Choosing a particular ordering, we can define two monodromy matrices on ${S}^{1}$,

$$\begin{eqnarray}&&{M}^{L}={J}_{N}^{L}\cdots {J}_{1}^{L}\text{,}{M}^{R}={J}_{1}^{R}\cdots {J}_{N}^{R}.\end{eqnarray} \tag{ 4.12 }$$

However, in order to keep things simpler we will concentrate on the pole ${z}_{+}$ only, drop the index L and set ${g}_{+}(x)=g(x).$

The quantum lattice Kac–Moody algebra at the pole ${z}_{+}$ is then defined by the following algebra

$$\begin{eqnarray}&&{\mathop{J}\limits^{1}}_{n+1}{{\mathfrak{R}}}^{-}{\mathop{J}\limits^{2}}_{n}={\mathop{J}\limits^{2}}_{n}{\mathop{J}\limits^{1}}_{n+1},\ \ \ \ \ {\mathop{J}\limits^{1}}_{n}{\mathop{J}\limits^{2}}_{n}={{\mathfrak{R}}}^{+}{\mathop{J}\limits^{2}}_{n}{\mathop{J}\limits^{1}}_{n}{{\mathfrak{R}}}^{-},\end{eqnarray} \tag{ 4.13 }$$

where ${{\mathfrak{R}}}^{+}={{\mathfrak{R}}}^{t}$ (transpose) and ${{\mathfrak{R}}}^{-}={{\mathfrak{R}}}^{-1}$ (inverse).

Now, the classical continuum limit is achieved by writing

$$\begin{eqnarray}&&{J}_{n}\sim 1+\displaystyle \frac{2\pi }{k}{\rm{\Delta }}{{\mathscr{J}}}_{+}(x),\ \ {\mathfrak{R}}\sim 1+{\rm{i}}\gamma \overline{{\mathfrak{r}}},\end{eqnarray} \tag{ 4.14 }$$

where ${\mathfrak{R}}$ and $\overline{{\mathfrak{r}}}$ are solutions of the quantum/classical YBE such that

$$\begin{eqnarray}&&\overline{{\mathfrak{r}}}+{\overline{{\mathfrak{r}}}}^{t}=2{\mathfrak{C}},\end{eqnarray} \tag{ 4.15 }$$

where ${\rm{\Delta }}=1/N\to 0$ is the lattice spacing and $x=n/N$. The first step is to find the classical limit of (4.13). We find

$$\begin{eqnarray}&&\displaystyle \frac{1}{\gamma }\{{\mathop{J}\limits^{1}}_{m},{\mathop{J}\limits^{2}}_{n}\}={\mathop{J}\limits^{1}}_{m}{\mathop{J}\limits^{2}}_{n}\bar{{\mathfrak{r}}}{\delta }_{{mn}}-{\bar{{\mathfrak{r}}}}^{t}{\mathop{J}\limits^{1}}_{m}{\mathop{J}\limits^{2}}_{n}{\delta }_{{mn}}-{\mathop{J}\limits^{1}}_{m}\bar{{\mathfrak{r}}}{\mathop{J}\limits^{2}}_{n}{\delta }_{m,n+1}+{\mathop{J}\limits^{2}}_{n}{\bar{{\mathfrak{r}}}}^{t}{\mathop{J}\limits^{1}}_{m}{\delta }_{m+1,n}.\end{eqnarray} \tag{ 4.16 }$$

The next step is to take the continuum limit. We find that $\gamma =-s$ and then the Kac–Moody algebra follows. The algebra (4.13) underlies the quantization of the monodromy matrix $m({z}_{+})$.

Recall that classically we have

$$\begin{eqnarray}&&{{\mathscr{J}}}_{+}(x)=\displaystyle \frac{k}{2\pi }{\partial }_{x}u(x)u{(x)}^{-1},\end{eqnarray} \tag{ 4.17 }$$

where $u(x)$ is the 'wave-function'

$$\begin{eqnarray}&&u(x)={\rm{Pexp}}\left[\displaystyle \frac{2\pi }{k}{\int }_{{x}_{0}}^{x}{\rm{d}}{x}{{\mathscr{J}}}_{+}(x)\right].\end{eqnarray} \tag{ 4.18 }$$

The gauge transformations (4.2) acting on ${{\mathscr{J}}}_{+}(x)$ are now equivalent to the action of the loop group $u(x)\to g(x)u(x)$ on the ${u}^{\prime }s$. On the lattice, this classical expressions become²²

$$\begin{eqnarray}&&{u}_{n}={J}_{n}\cdots {J}_{1},\ \ {u}_{n}={J}_{n}{u}_{n-1},\ \ {u}_{n}\to {g}_{n}{u}_{n},\ \ \ \ \ \ {J}_{n}\to {g}_{n}{J}_{n}{g}_{n-1}^{-1}.\end{eqnarray} \tag{ 4.19 }$$

The wave functions also obey the following quantum exchange algebra relations

$$\begin{eqnarray}{\mathop{u}\limits^{1}}_{m}{\mathop{u}\limits^{2}}_{n} & = & {\mathop{u}\limits^{2}}_{n}{\mathop{u}\limits^{1}}_{m}{{\mathfrak{R}}}^{+},\text{}m\gt n,\\ {\mathop{u}\limits^{1}}_{m}{\mathop{u}\limits^{2}}_{n} & = & {\mathop{u}\limits^{2}}_{n}{\mathop{u}\limits^{1}}_{m}{{\mathfrak{R}}}^{-},\text{}m\lt n,\\ {\mathop{u}\limits^{1}}_{n}{\mathop{u}\limits^{2}}_{n} & = & {{\mathfrak{R}}}^{+}{\mathop{u}\limits^{2}}_{n}{\mathop{u}\limits^{1}}_{n}{{\mathfrak{R}}}^{-}.\end{eqnarray} \tag{ 4.20 }$$

The last relation means that the classical gauge loop group is promoted to a quantum gauge group with algebra [20]

$$\begin{eqnarray}&&{\mathop{g}\limits^{1}}_{n}{\mathop{g}\limits^{2}}_{n}{{\mathfrak{R}}}^{+}={{\mathfrak{R}}}^{+}{\mathop{g}\limits^{2}}_{n}{\mathop{g}\limits^{1}}_{n}\,\ \ \ \ \ {\mathop{g}\limits^{1}}_{m}{\mathop{g}\limits^{2}}_{n}={\mathop{g}\limits^{2}}_{n}{\mathop{g}\limits^{1}}_{m},\ \ m\ne n.\end{eqnarray} \tag{ 4.21 }$$

This quantum group leaves the Kac–Moody lattice algebra (4.13) invariant in the same way its classical counterpart leaves invariant the usual Kac–Moody algebra.

Writing $\overline{{\mathfrak{r}}}=-{\mathfrak{r}}+{\mathfrak{C}},$ with ${{\mathfrak{r}}}^{t}=-{\mathfrak{r}}$ antisymmetric which explicitly obeys the constraint (4.15), we can find the corrected version of the Poisson bracket (4.6) for the wave-function $u(x)$ (in the normalization ${x}_{0}={y}_{0})$ directly from the classical/continuum limit of the first two equations of the exchange algebra (4.20), namely,

$$\begin{eqnarray*}&&\{\mathop{u}\limits^{1}(x),\mathop{u}\limits^{2}(y)\}=s\mathop{u}\limits^{1}(x)\mathop{u}\limits^{2}(y)[{\mathfrak{r}}+{\mathfrak{C}}{\epsilon }_{{xy}}].\end{eqnarray*}$$

This Poisson bracket do obey the Jacobi identity provided the antisymmetric ${\mathfrak{r}}$-matrix is a solution of the mCYBE of the split-type [29]. See the discussion around equation (5.9) of that paper. This decomposition of $\overline{{\mathfrak{r}}}$ is consistent with the requirement that it is a solution of the CYBE.

The quantum monodromy matrix is given by $M={u}^{N}$ and satisfy the quadratic algebra

$$\begin{eqnarray}&&\mathop{M}\limits^{1}{{\mathfrak{R}}}^{-}\mathop{M}\limits^{2}{({{\mathfrak{R}}}^{-})}^{-1}={{\mathfrak{R}}}^{+}\mathop{M}\limits^{2}{({{\mathfrak{R}}}^{+})}^{-1}\mathop{M}\limits^{1}.\end{eqnarray} \tag{ 4.22 }$$

When we consider the following quantum factorization of the monodromy matrix

$$\begin{eqnarray}&&M={M}_{-}^{-1}{M}_{+},\end{eqnarray} \tag{ 4.23 }$$

the RTT quadratic relations given by its factorized components, namely,

$$\begin{eqnarray}&&{{\mathfrak{R}}}^{+}{\mathop{M}\limits^{1}}_{\pm }{\mathop{M}\limits^{2}}_{\pm }={\mathop{M}\limits^{2}}_{\pm }{\mathop{M}\limits^{1}}_{\pm }{{\mathfrak{R}}}^{+},\ \ \ \ \ {{\mathfrak{R}}}^{+}{\mathop{M}\limits^{1}}_{-}{\mathop{M}\limits^{2}}_{+}={\mathop{M}\limits^{2}}_{+}{\mathop{M}\limits^{1}}_{-}{{\mathfrak{R}}}^{+},\end{eqnarray} \tag{ 4.24 }$$

are compatible with (4.22). Notice that (4.24) is the analogue of equation (13) of [21] in the case of sl(2), which is another definition of the associated quantum group ${{Fun}}_{q}{({SL}(2)}^{* })$. The quantum monodromy matrix $M({z}_{+})$ then becomes an element of F_q. The Poisson–Lie symmetry found in the classical k-deformed theory is then promoted to a quantum group symmetry.

The continuum limit of (4.20) is

$$\begin{eqnarray}&&\mathop{u}\limits^{1}(x)\mathop{u}\limits^{2}(y)=\mathop{u}\limits^{2}(y)\mathop{u}\limits^{1}(x){{\mathfrak{R}}}_{{xy}},\ \ \ \ \ {{\mathfrak{R}}}_{{xy}}={\mathfrak{F}}\,{q}^{{\mathfrak{P}}{\epsilon }_{{xy}}}\,{{\mathfrak{F}}}^{t-1},\end{eqnarray} \tag{ 4.25 }$$

where ${\mathfrak{P}}$ is the permutation operator and ${\mathfrak{F}}$ an invertible matrix. This way of factorizing the usual R matrix comes from the theory of quasi-Hopf algebras introduced by Drinfeld²³ . It was shown in [21], that this exchange algebra underlies the quantization of the Kac–Moody algebra. Indeed, the quantum version of the classical currents (4.17), namely, the quantum operators

$$\begin{eqnarray}&&j(x)={\partial }_{x}u(x)\cdot u{(x)}^{-1}\end{eqnarray} \tag{ 4.26 }$$

obey the following algebra

$$\begin{eqnarray}&&[\mathop{j}\limits^{1}(x),\mathop{j}\limits^{2}(y)]=-\mathrm{ln}q[{\mathfrak{P}},\mathop{j}\limits^{1}(x)-\mathop{j}\limits^{2}(y)]{\delta }_{{xy}}-2\mathrm{ln}q(1+a){\mathfrak{P}}{\delta }_{{xy}}^{\prime },\end{eqnarray} \tag{ 4.27 }$$

where

$$\begin{eqnarray}&&a=-\displaystyle \frac{n\mathrm{ln}q}{{\rm{i}}\pi }\end{eqnarray} \tag{ 4.28 }$$

and n is the size of the matrices used to represent the currents, e.g., n for the fundamental representation of ${\mathfrak{sl}}(n)$, which corresponds to the present case. Taking the ansatz for the deforming parameter

$$\begin{eqnarray}&&q={{\rm{e}}}^{{\rm{i}}\pi /(k+n)},\end{eqnarray} \tag{ 4.29 }$$

and re-scaling the quantum currents (take $k\to k+n$ in (4.17)) it was shown in [21] that the rescaled currents obey the quantum Kac–Moody algebra. Notice that in the semi-classical limit we recover our previous rough result with $q={{\rm{e}}}^{-{\rm{i}}{s}({z}_{+})}$. Using the duality between the Faddeev–Reshetikhin–Takhtajan $({F}_{q})$ and the Drinfeld–Jimbo $({U}_{q}({\mathfrak{f}}))$ realizations of quantum groups, we infer that the deformed sigma models considered so far are q-deformed with deformation parameter being a root of unity.

A comment is in order. From the second equation in (3.31) and (4.15) one would naively conclude that

$$\begin{eqnarray}&&{\mathfrak{R}}({z}_{+},{z}_{+})=\displaystyle \frac{\pi }{k}\bar{{\mathfrak{r}}},\end{eqnarray} \tag{ 4.30 }$$

but the following observation shows this is not the case. The $z\to {z}_{+},w\to {z}_{+}$ limit is ambiguous as can be seen from

$$\begin{eqnarray}&&{\mathfrak{R}}(z,{z}_{+})=0,\ \ \ \ \mathop{\mathrm{lim}}\limits_{w\to {z}_{+}}\,{\mathfrak{R}}({z}_{+},w)\sim {\mathfrak{C}}\end{eqnarray} \tag{ 4.31 }$$

and even if we ignore this and work with the non-zero answer, this identification would imply that the classical limits of (4.22) and (4.24) are trivial, which is inconsistent. Then, the poles of the twisting function are to be treated as punctures in the complex plane z with a complete separate analysis. In this respect, the quantum ${\mathfrak{R}}$-matrix and its classical counterpart $\bar{{\mathfrak{r}}}$ used in (4.13) are artifacts of our quantization scheme not related to the ${\mathfrak{r}}/{\mathfrak{s}}$ operators in the Maillet bracket at all. In the classical/continuum limit, quantities should be independent of the quantization procedure but if, for instance, we consider the symmetry algebras (4.22) and (4.24) they would depend on $\bar{{\mathfrak{r}}}$ in this limit, hence the quantized monodromy matrix should remain quantum. Notice that this does not apply to its associated KM algebra because it is independent of $\bar{{\mathfrak{r}}}$ in the classical/continuum limit, as claimed in [21].

It is noteworthy that it is the central charge k of the Kac–Moody algebra instead of parameter λ that sets the deformation of the symmetry group. This is, perhaps, surprising as one might have expected q to depend on the deformation parameter λ as in the η-deformation [8]. Presumably the reason lies in the fact that the Kac–Moody currents (3.13) obey the Kac–Moody algebra no matter what the deformation parameter λ is and hence their monodromies obey a λ-independent algebra. In this respect, the k-deformations ($-\infty \lt {\epsilon }^{2}\leqslant 0$) are quite different in nature from the η-deformation (${\epsilon }^{2}\in [0,1)$). However, there is still the possibility that somehow higher spin charges appearing in the z expansion of the monodromy matrix around the poles, e.g. (3.56), introduce a λ dependence in a way yet to be understood.