Lax connections in ${\boldsymbol{T\bar{T}}}$-deformed integrable field theories ^*

The Lagrangian of a rank-r affine Toda field theory (for a review on Toda field theory, see [17]) is given by

$$\mathcal{L}^{(0)}\equiv \partial\vec{\phi}\cdot\bar{\partial}\vec{\phi}+V \tag{ 2 }$$

with

$$V = -\frac{m^2}{\beta^2}\sum\limits_{i = 0}^r n_i {\rm e}^{\beta \vec{\alpha}_i\cdot\vec{\phi}}, \tag{ 3 }$$

where $\vec{\phi}$ is a vector field of r components, the set of integer number $\{n_i\}$ characterizes the theory, $\{\vec{\alpha}_i,\; i =$ $1,\cdots,r\}$ are positive simple roots of the underlying Lie algebra, and $\vec{\alpha}_0 = -\sum\nolimits_i^r n_i\vec{\alpha}_i$. If in the summation the term $i = 0$ is omitted, then the theory reduces to the conformal Tada field theory. The generators of the Cartan subalgebra $\vec{H} = \{H_a,\ a = 1,2,\cdots,r\}$, and the simple roots $\{E_{\vec{\alpha}_i},E_{-\vec{\alpha}_i}, \ i = 0,1,\cdots r\}$ satisfy the standard commutation relations

$$\begin{aligned}[b] & {H_a},{H_b} = 0,\;\;\;\vec H,{E_{ \pm {{\vec \alpha }_i}}} = \pm {{\vec \alpha }_i}{E_{ \pm {{\vec \alpha }_i}}},\\ & {E_{{{\vec \alpha }_i}}},{E_{ - {{\vec \alpha }_j}}} = {\delta _{ij}}\frac{{2{{\vec \alpha }_j} \cdot \vec H}}{{|{{\vec \alpha }_j}{|^2}}},\\ & {E_{{{\vec \alpha }_i}}},{E_{{{\vec \alpha }_j}}} = \left\{ \begin{array}{l} {{\cal N}_{{{\vec \alpha }_i} + {{\vec \alpha }_j}}}{E_{{{\vec \alpha }_i} + {{\vec \alpha }_j}}},\;\;\;\;{\rm{if}}\;{{\vec \alpha }_i}{\rm{ + }}{{\vec \alpha }_j}\;{\rm{is}}\;{\rm{a}}\;{\rm{root}},\\ 0,\;\;\; \;\;\; \;\;\; \;\;\; \;\;\; {\rm{if}}\;{{\vec \alpha }_i}{\rm{ + }}{{\vec \alpha }_j}\;{\rm{is}}\;{\rm{not}}\;{\rm{a}}\;{\rm{root}}. \end{array} \right. \end{aligned} \tag{ 4 }$$

The equations of motion are simply given by

$$2\partial\bar{\partial}\vec{\phi}-\frac{\delta V}{\delta\vec{\phi}} = 0, \tag{ 5 }$$

and the Lax connections are

$$\begin{aligned}[b] & L = -\frac{\beta}{2}\partial\vec{\phi}\cdot\vec{H}-\lambda\sum\limits_{i = 0}^r m_i {\rm e}^{\beta \vec{\alpha}_i\cdot\vec{\phi}/2}E_{\vec{\alpha}_i},\\ & \bar{L} = \frac{\beta}{2}\bar{\partial}\vec{\phi}\cdot\vec{H}+\frac{1}{\lambda}\sum\limits_{i = 0}^r m_i {\rm e}^{\beta \vec{\alpha}_i\cdot\vec{\phi}/2}E_{-\vec{\alpha}_i}, \end{aligned} \tag{ 6 }$$

where $\lambda\in\mathbf{C}$ is the spectral parameter and $m_i^2 = \frac{1}{4}|\vec{\alpha}_i|^2 m^2 n_i$. For a classical integrable system, the equations of motion are equivalent to the Lax equation

$$\partial \bar{L}-\bar{\partial} L- L,\bar{L} = 0. \tag{ 7 }$$

For later convenience, we introduce two new combinations

$$\begin{aligned}[b] E_+ = & \sum\limits_{i = 0}^r m_i {\rm e}^{\beta \vec{\alpha}_i\cdot\vec{\phi}/2}E_{\vec{\alpha}_i},\\ E_- = & \sum\limits_{i = 0}^r m_i {\rm e}^{\beta \vec{\alpha}_i\cdot\vec{\phi}/2}E_{-\vec{\alpha}_i} \end{aligned} \tag{ 8 }$$

satisfying

$$\begin{aligned}[b] \left[ E_+,E_- \right] = & -\frac{\beta}{2}\vec{\nabla}{V}\cdot\vec{H},\\ \left[\vec{H},E_{\pm}\right] = & \pm\frac{2}{\beta}\vec{\nabla}{E_{\pm}}, \end{aligned} \tag{ 9 }$$

where $\vec{\nabla}{f}$ denotes $\dfrac{\delta f}{\delta \vec{\phi}}$. Then, the Lax connections (6) can be rewritten as

$$\begin{aligned}[b] L = & -\frac{\beta}{2}\partial\vec{\phi}\cdot\vec{H}-\lambda E_+,\quad \bar{L} = \frac{\beta}{2}\bar{\partial}\vec{\phi}\cdot\vec{H}+\frac{1}{\lambda}E_-. \end{aligned} \tag{ 10 }$$

The $T\bar{T}$-deformed Lagrangian of an N-scalar theory is [16, 18]

$$\mathcal{L}^{(t)} = \frac{V}{1-t V}+\frac{1}{2 t(1-t V)}(\Omega_{T}-1), \tag{ 11 }$$

where

$$\begin{aligned}[b] \Omega_{T} = & \sqrt{1+Y+Z},\quad Y = 4t(1-t V)\left(\partial\vec{\phi}\cdot\bar{\partial}\vec{\phi}\right),\\ Z = & -4 t^2 (1-t V)^2 \left(\partial\vec{\phi}\cdot\partial\vec{\phi}\right)\left(\bar{\partial}\vec{\phi}\cdot\bar{\partial}\vec{\phi}\right)-\left(\partial\vec{\phi}\cdot\bar{\partial}\vec{\phi}\right)^2 . \end{aligned} \tag{ 12 }$$

The equations of motion are given by

$$\vec{A_e}\equiv \partial_{\mu}\frac{\delta \mathcal{L}^{(t)}}{\delta \partial_{\mu}\vec{\phi}}-\frac{\delta \mathcal{L}^{(t)}}{\delta \vec{\phi}} = 0. \tag{ 13 }$$

Substituting (11) and (12) into (13), one can obtain

$$\begin{aligned}[b] {{\vec A}_e} = & \left\{ {\frac{1}{{{\Omega _T}}}[\bar \partial \vec \phi - 2t(1 - tV)(\partial \vec \phi (\bar \partial \vec \phi \cdot \bar \partial \vec \phi ) - \bar \partial \vec \phi (\partial \vec \phi \cdot \bar \partial \vec \phi ))]} \right\}\\ & + \bar \partial \left\{ {\frac{1}{{{\Omega _T}}}[\partial \vec \phi - 2t(1 - tV)(\bar \partial \vec \phi (\partial \vec \phi \cdot \partial \vec \phi ) - \partial \vec \phi (\partial \vec \phi \cdot \bar \partial \vec \phi ))]} \right\}\\ & - \frac{{\vec \nabla V}}{{4{\Omega _T}{{(1 - tV)}^2}}}\left[ {{{\left( {{\Omega _T} + 1} \right)}^2} - Z} \right]. \end{aligned} \tag{ 14 }$$

Given these equations of motions, we propose a simple ansatz for the Lax connection:

$$\begin{aligned}[b] L = & -\frac{\beta}{2}\vec{a}_1\cdot\vec{H}-\lambda b_1 E_++\frac{1}{\lambda} c_1 E_-,\\ \bar{L} = & \frac{\beta}{2}\vec{a}_2\cdot\vec{H}-\lambda b_2 E_++\frac{1}{\lambda} c_2 E_-, \end{aligned} \tag{ 15 }$$

where $\vec{a}_1, b_1, c_1, \vec{a}_2, b_2, c_2$ are the functions of $\vec{\phi}$ and their derivatives and will be determined by imposing the Lax equation and the equations of motion. Notice that in our ansatz (15), the Lax connection depends uniformly on the simple roots $E_{\vec{\alpha}}$. Plugging (15) into (7) directly gives rise to a set of linear differential equations $\vec{A}_H, A_+',A_-',$ corresponding to the components $\vec{H}$, $E_+$, and $E_-$, respectively. In principle, $\vec{A}_H, A_+',A_-',$ should vanish separately. However, because terms such as $\nabla E_\pm$ are not uniformly dependent on the simple roots $E_{\vec{\alpha}}$, we require that the coefficients of terms such as $\nabla E_\pm$ vanish separately. Consequently, we obtain five sets of linear equations

$$\left\{ \begin{aligned} & \vec{A}_H \equiv \partial\vec{a}_2+\bar{\partial}\vec{a}_1-\vec{\nabla}{V}(b_1 c_2 -b_2 c_1) = 0,\\ & A_+ \equiv -\partial b_2+\bar{\partial} b_1 = 0,\\ & A_- \equiv \partial c_2-\bar{\partial} c_1 = 0,\\ & \vec{A}_{p+} \equiv -\partial\vec{\phi} b_2+\bar{\partial}\vec{\phi} b_1-(\vec{a}_1 b_2+\vec{a}_2 b_1) = 0,\\ & \vec{A}_{p-} \equiv \partial\vec{\phi} c_2-\bar{\partial}\vec{\phi} c_1-(\vec{a}_1 c_2+\vec{a}_2 c_1) = 0.\\ \end{aligned} \right. \tag{ 16 }$$

To solve these equations, we make another assumption that they can be written as linear combinations of the equations of motion, i.e.,

$$\begin{aligned}[b] & \vec{A}_H = f_H\vec{A}_e,\quad A_+ = \vec{f}_+\cdot\vec{A}_e,\quad A_- = \vec{f}_-\cdot\vec{A}_e,\\ & \vec{A}_{p+} = f_{p+}\vec{A}_e,\quad \vec{A}_{p-} = f_{p-}\vec{A}_e. \end{aligned} \tag{ 17 }$$

To ensure the equivalence between the Lax equation (16) and the equations of motion (13), there should be no common zero of $f_H, \vec{f}_{+}, \vec{f}_{-}, f_{p+}, f_{p-}$. Indeed, we are making quite strong assumptions here, but we will show that a consistent solution does exist.

For the undeformed theory, by (10), one can find that

$$f_H = 1,\quad f_{p+} = 0,\quad f_{p-} = 0, \tag{ 18 }$$

and there are no $\vec{f}_{+}, \vec{f}_{-}$ terms. We assume that (18) is still true for the deformed theory and observe that if we take

$$\vec{f}_+ = -t\bar{\partial}\vec{\phi}, \tag{ 19 }$$

then

$$\vec{f}_+\cdot\vec{A}_e = -\partial \left[ \frac{t}{\Omega_{T}}\left(\bar{\partial}\vec{\phi}\cdot\bar{\partial}\vec{\phi}\right)\right]+\bar{\partial} \left[\frac{(\Omega_{T}+1)^2-Z}{4\Omega_{T}(1-t V)}\right] \tag{ 20 }$$

suggesting that we can identify

$$b_1 = \frac{(\Omega_{T}+1)^2-Z}{4\Omega_{T}(1-t V)},\quad b_2 = \frac{t}{\Omega_{T}}\left(\bar{\partial}\vec{\phi}\cdot\bar{\partial}\vec{\phi}\right) \tag{ 21 }$$

up to some constants, which can be fixed to be zero after considering other equations. Similarly, by taking $\vec{f}_{-} = -t\partial\vec{\phi},$ we can determine $c_1$ and $c_2$

$$c_1 = \frac{t}{\Omega_{T}}\left(\partial\vec{\phi}\cdot\partial\vec{\phi}\right),\quad c_2 = \frac{(\Omega_{T}+1)^2-Z}{4\Omega_{T}(1-t V)}.\\ \tag{ 22 }$$

Finally, from $\vec{f}_{H}\vec{A}_e$, we fix all the remaining functions in our ansatz

$$\begin{aligned}[b] & \vec{a}_1 = \frac{1}{\Omega_{T}} \left[\partial\vec{\phi}-2t(1-t V)\left(\bar{\partial}\vec{\phi}\left(\partial\vec{\phi}\cdot\partial\vec{\phi}\right)-\partial\vec{\phi}\left(\partial\vec{\phi}\cdot\bar{\partial}\vec{\phi}\right)\right)\right] ,\\ & \vec{a}_2 = \frac{1}{\Omega_{T}} \left[ \bar{\partial}\vec{\phi}-2t(1-t V)\left(\partial\vec{\phi}\left(\bar{\partial}\vec{\phi}\cdot\bar{\partial}\vec{\phi}\right)-\bar{\partial}\vec{\phi}\left(\partial\vec{\phi}\cdot\bar{\partial}\vec{\phi}\right)\right)\right]. \end{aligned} \tag{ 23 }$$

Plugging (21), (22), and (23) into (16), one can check that (16) is indeed equivalent to the equations of motion (13).

To summarize, the Lax connections of the $T\bar{T}$-deformed (affine) Toda field theories are of the forms expressed in (15), with the functions being given by (21), (22), and (23). We want to stress that after we assume (18) and (19), the solutions can be determined directly, without solving any other equations.

To compare with the existing results in the literature, let us consider some specific examples. The first one is the Liouville field theory, which corresponds to the Toda field theory of $sl_2$ Lie algebra with parameters

$$\beta = \frac{1}{2},\ \ \ \ m_0 = 0,\ \ \ \ m_1 = -\frac{\sqrt{\mu}}{2}. \tag{ 24 }$$

The undeformed Lagrangian is

$$\mathcal{L}^{(0)} = \partial \phi \bar{\partial}\phi-\mu {\rm e}^\phi. \tag{ 25 }$$

The $T\bar{T}$-deformed Liouville field theory was studied in [19], where infinite conserved currents were constructed from some ansatz without using the Lax connection. From the discussion in the last subsection, we can present the deformed Lax connections explicitly

$$\begin{aligned}[b] L = & -\frac{1}{4}\frac{\partial\phi}{\Omega_{T}}H+ \sqrt{\mu}\lambda B {\rm e}^{ \phi/2}E_{\alpha_1}- \frac{\sqrt{\mu}}{\lambda}(\partial\phi)^2 C {\rm e}^{ \phi/2}E_{-\alpha_1},\\ \bar{L} = & \frac{1}{4}\frac{\bar{\partial}\phi}{\Omega_{T}}H- \frac{\sqrt{\mu}}{\lambda} B {\rm e}^{ \phi/2}E_{-\alpha_1}+\sqrt{\mu}\lambda (\bar{\partial}\phi)^2 C {\rm e}^{ \phi/2}E_{\alpha_1}, \end{aligned} \tag{ 26 }$$

where

$$\begin{aligned}[b] B = & \frac{(\Omega_{T}+1)^{2}}{8 \Omega_{T}(1-t V)}, \quad C = \frac{t}{2 \Omega_{T}},\\ \Omega_{T} = & \sqrt{1+4t(1-t V)\left(\partial\phi\bar{\partial}\phi\right)}. \end{aligned} \tag{ 27 }$$

Let the generators of $sl_2$ Lie algebra be

$$H = \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \\ \end{array} \right),\ \ E_{\alpha_1} = \left( \begin{array}{cc} 0 & 1 \\ 0 & 0 \\ \end{array} \right),\ \ E_{-\alpha_1} = \left( \begin{array}{cc} 0 & 0 \\ 1 & 0 \\ \end{array} \right), \ \ \tag{ 28 }$$

then, if we take the undeforming limit, $t \rightarrow 0$, the Lax connections become

$$L = \left( \begin{array}{cc} -\dfrac{1}{4}\partial\phi & \dfrac{\sqrt{\mu}\lambda }{2} {\rm e}^{ \phi/2} \\ 0 & \dfrac{1}{4}\partial\phi \\ \end{array} \right),\ \ \bar{L} = \left( \begin{array}{cc} \dfrac{1}{4}\bar{\partial}\phi & 0 \\ -\dfrac{\sqrt{\mu} }{2\lambda} {\rm e}^{ \phi/2} & -\dfrac{1}{4}\bar{\partial}\phi \\ \end{array} \right),\ \ \tag{ 29 }$$

which are the Lax connections of the Liouville field theory.

Our next example is the sine-Gordon model, which corresponds to the affine Toda field of affine $sl_2$ algebra with parameters

$$\beta = \frac{\rm i}{2},\ \, m_0 = m_1 = -\frac{\rm i}{2},\ \, n_0 = n_1 = 1, \alpha_0 = -2,\ \ \alpha_1 = 2. \tag{ 30 }$$

The undeformed Lagrangian is given by

$$\mathcal{L}^{(0)} = \partial \phi \bar{\partial} \phi -2\cos{\phi }. \tag{ 31 }$$

By setting

$$E_{\alpha_{0}} = E_{-\alpha_{1}},\ \ \ \ E_{-\alpha_{0}} = E_{\alpha_{1}}, \tag{ 32 }$$

we find that the deformed Lax connections are

$$\begin{aligned}[b] L = & -\frac{\rm i}{4}\frac{\partial\phi}{\Omega_{T}}H+\left( {\rm i}\lambda B {\rm e}^{{\rm i}\phi/2}+\frac{1}{{\rm i}\lambda}(\partial\phi)^2 C {\rm e}^{-{\rm i} \phi/2}\right) E_{\alpha_1}\\ & + \left({\rm i}\lambda B {\rm e}^{-{\rm i}\phi/2}+\frac{1}{{\rm i}\lambda}(\partial\phi)^2 C {\rm e}^{{\rm i} \phi/2}\right)E_{-\alpha_1},\\ \bar{L} = & \frac{\rm i}{4}\frac{\bar{\partial}\phi}{\Omega_{T}}H+\left( \frac{1}{{\rm i}\lambda}B {\rm e}^{-{\rm i}\phi/2}+{\rm i}\lambda(\bar{\partial}\phi)^2 C {\rm e}^{{\rm i} \phi/2}\right) E_{\alpha_1}\\ & + \left( \frac{1}{{\rm i}\lambda}B {\rm e}^{{\rm i}\phi/2}+{\rm i}\lambda(\bar{\partial}\phi)^2 C {\rm e}^{-{\rm i} \phi/2}\right) E_{-\alpha_1} , \end{aligned} \tag{ 33 }$$

which are the same as those found in [18]. In the undeforming limit, $t \rightarrow 0$, the Lax connections reduce to the Lax connections of the sine-Gordon model

$$L = \left( \begin{array}{cc} -\dfrac{\rm i}{4}\partial\phi & \dfrac{{\rm i} \lambda }{2} {\rm e}^{ {\rm i}\phi/2} \\ \dfrac{{\rm i} \lambda }{2} {\rm e}^{-{\rm i}\phi/2} & \dfrac{\rm i}{4}\partial\phi \\ \end{array} \right),\ \ \bar{L} = \left( \begin{array}{cc} \dfrac{\rm i}{4}\bar{\partial}\phi & \dfrac{1}{2 {\rm i}\lambda} {\rm e}^{-{\rm i} \phi/2} \\ \dfrac{1}{2 {\rm i}\lambda} {\rm e}^{{\rm i} \phi/2} & -\dfrac{\rm i}{4}\bar{\partial}\phi \\ \end{array} \right). \tag{ 34 }$$

The PCM is a field theory whose field takes values in some Lie group manifold. Its action is ^②

$$S_0 = \int {\rm d} x^2 g^{\mu\nu} \text{Tr}\left(g^{-1}\partial_{\mu}g g^{-1}\partial_{\nu}g\right),\quad g\in G. \tag{ 35 }$$

Usually, the Lie group is chosen to be semisimple, but we will leave it to be arbitrary because our focus is on integrability. The model has symmetry group $G_L\times G_R$. The equation of motion of the PCM is simply

$$\partial^{\mu} \left(g^{-1}\partial_{\mu}g\right) = 0, \tag{ 36 }$$

which is equivalent to the current conservation equation

$$\partial_\mu j^\mu = 0,\quad j^\mu \equiv g^{-1}\partial^{\mu}g. \tag{ 37 }$$

Here, $j^\mu$ is the conserved current corresponding to the $G_R$ symmetry. In addition to (37), the conserved current also satisfies the flatness condition:

$$\partial_0 j_1-\partial_1 j_0 = - [j_0, j_1] . \tag{ 38 }$$

Equations (37) and (38) are equivalent to the Lax equation with the Lax connections

$$\begin{aligned}[b] L_0 =; -\frac{1}{\lambda^2+1}(\lambda j_1+ j_0),\quad L_1 = -\frac{1}{\lambda^2+1}(-\lambda j_0+j_1), \end{aligned} \tag{ 39 }$$

where $\lambda \in \mathbf{C}$ is the spectral parameter.

The $T\bar{T}$-deformed Lagrangian of the PCM is given by [16]

$$\mathcal{L}^{(t)}_{\rm PCM} = \frac{1}{2 t} \left(-1+\Omega_{P}\right), \tag{ 40 }$$

where

$$\begin{aligned}[b] \Omega_{P} = & \sqrt{1+4 t \text{Tr}\left(g^{-1}\partial_{\mu}g g^{-1}\partial_{\mu}g\right)+8 t^2 \epsilon^{\mu\nu}\epsilon^{\rho\sigma}\text{Tr}\left(g^{-1}\partial_{\mu}g g^{-1}\partial_{\rho}g\right)\text{Tr}\left(g^{-1}\partial_{\nu}g g^{-1}\partial_{\sigma}g\right)}\\ = & \sqrt{1+4 t \text{Tr}\left(j_{\mu}j^{\mu}\right)+8 t^2 \epsilon^{\mu\nu}\epsilon^{\rho\sigma}\text{Tr}\left(j_{\mu}j_{\rho}\right)\text{Tr}\left(j_{\nu}j_{\sigma}\right)}. \end{aligned} \tag{ 41 }$$

The equation of motion, $A_{e\rm PCM} = 0$, can also be cast into the form of a conservation law:

$$A_{e\rm PCM}\equiv \partial_{\mu}\frac{\delta \mathcal{L}^{(t)}_{\rm PCM}}{\delta \partial_{\mu}\vec{\phi}}-\frac{\delta \mathcal{L}^{(t)}_{\rm PCM}}{\delta \vec{\phi}} = 2(\partial_{\mu} J^{\mu}) g^{-1}. \tag{ 42 }$$

Here, the conserved current $J^{\mu}$ is defined as

$$J^{\mu} = \frac{1}{\Omega_{P}}\left(j^{\mu}+4 t \epsilon^{\mu\nu}\epsilon^{\rho\sigma}j_{\rho}\text{Tr}\left(j_{\nu}j_{\sigma}\right)\right), \tag{ 43 }$$

which satisfies the following useful identities

$$\begin{aligned}[b] [J_0, J_1] = & [j_0, j_1],\\ [J_0, j_0] = & \frac{1}{\Omega_{P}}4 t \text{Tr}\left(j_{0}j_{1}\right)[j_0, j_1],\\ [J_0, j_1] = & \frac{1}{\Omega_{P}}\left(1+4 t \text{Tr}\left(j_{1}j_{1}\right)\right)[j_0, j_1],\\ [J_1, j_0] = & -\frac{1}{\Omega_{P}}\left(1+4 t \text{Tr}\left(j_{0}j_{0}\right)\right)[j_0, j_1],\\ [J_1, j_1] = & -\frac{1}{\Omega_{P}}4 t \text{Tr}\left(j_{0}j_{1}\right)[j_0, j_1]. \end{aligned} \tag{ 44 }$$

Note that the current $j_\mu$ still satisfies the flatness condition (38), so a reasonable ansatz for the Lax connections could be that they are the linear combination of the new conserved current (43) and $j_\mu$:

$$\begin{aligned}[b] L_0 = & a_0 J_1+b_0 j_0+c_0 j_1,\\ L_1 = & a_1 J_0+b_1 j_0+c_1 j_1, \end{aligned} \tag{ 45 }$$

where $a_0, a_1, b_0, b_1, c_0, c_1$ are constants to be determined. Again, we assume that the Lax equation is linearly dependent on the equation of motion:

$$\begin{array}{l} A_{\rm LPCM}\equiv \partial_0 L_1-\partial_1 L_0 - [ L_0, L_1],\\ A_{\rm LPCM} = A_{e\rm PCM} \cdot f_{\rm PCM}. \end{array} \tag{ 46 }$$

For the undeformed theory, using (39) and the definitions of $A_{\rm LPCM}$ and $A_{e\rm PCM}$, we obtain

$$f_{\rm PCM} = \frac{1}{2}\frac{\lambda}{\lambda^2+1}g. \tag{ 47 }$$

Assuming that (47) is still true in the deformed case, we end up with

$$A_{\rm LPCM} = \frac{\lambda}{\lambda^2+1}\partial_{\mu} J^{\mu}. \tag{ 48 }$$

Plugging (45) into (46) and matching it with (48), we have

$$a_0 = -a_1 = -\frac{\lambda}{\lambda^2+1},\quad b_0 = c_1 = -\frac{1}{\lambda^2+1},\quad b_1 = c_0 = 0, \tag{ 49 }$$

where we have used the identity, $\partial_0 j_1-\partial_1 j_0 = - [j_0, j_1] .$

In summary, the Lax connection of the $T\bar{T}$-deformed PCM is given by

$$\begin{aligned}[b] L_0 = -\frac{1}{\lambda^2+1}(\lambda J_1+ j_0),\quad L_1 = -\frac{1}{\lambda^2+1}(-\lambda J_0+j_1), \end{aligned} \tag{ 50 }$$

where $J_{\mu}$ has been defined by (43). This result is expected considering the identities (44). Given the Lax connection, we can define the monodromy matrix as the holonomy along a constant time slice

$$M(x^0;\lambda) = \mathcal{P}\exp\left(\int_{-\infty}^{\infty} {\rm d} x^1\, L_1(x^0,x^1,\lambda)\right). \tag{ 51 }$$

The set of (non-local) infinite conserved charges can be generated by expanding the monodromy matrix with respect to the spectral parameter as

$$\begin{aligned}[b] M(\lambda) = & \exp \left(\sum\limits_{n = 1}^\infty \frac{Q_n}{\lambda^n}\right) = 1+\frac{1}{\lambda} \int_{-\infty}^{+\infty} {\rm d}x^1 J_0\\ & -\frac{1}{\lambda^2}\left(\int_{-\infty}^{+\infty} {\rm d}x^1 \, j_1- \int_{-\infty}^{+\infty}{\rm d}x^1 \int_{-\infty}^{x^1}{\rm d}y^1 J_0(x)J_0(y)\right)\\ & +\mathcal{O}\left(\frac{1}{\lambda^3}\right).\end{aligned} \tag{ 52 }$$

For the undeformed PCM, these non-local charges span the classical Yangian algebra [20]. Under the $T\bar{T}$-deformation, the algebra becomes deformed in a very complicated way.

Consider the free scalar with the Lagrangian

$$L(\mathbf{w}) = \partial_w \phi \partial_{\bar{w}} \phi. \tag{ 66 }$$

The model is integrable with the trivial Lax pair

$$\mathcal{L}_w = \partial_w \phi,\quad \mathcal{L}_{\bar{w}} = -\partial_{\bar{w}} \phi \tag{ 67 }$$

such that the Lax equation

$$\partial_{\bar{w}}\mathcal{L}_w-\partial_w \mathcal{L}_{\bar{w}} = 2\partial_w \partial_{\bar{w}}\phi = 0 \tag{ 68 }$$

coincides with the equation of motion. The stress-energy tensor is simply

$$\begin{aligned}[b] & T_2(\mathbf{w}) = -\frac{1}{2} (\partial_w \phi)^2,\quad \Theta_0(\mathbf{w}) = 0,\\ & \Delta = 1-4t^2 T_2(\mathbf{w})\bar{T}_2(\mathbf{w}), \end{aligned} \tag{ 69 }$$

which leads to the following transformation

$$\partial_w \phi = \partial \phi-\frac{1}{4\tau}\left(\frac{-1+\Omega_{T}}{\bar{\partial}\phi}\right)^2 \bar{\partial}\phi, \tag{ 70 }$$

$$\partial_{\bar{w}}\phi = \bar{\partial} \phi-\frac{1}{4 t}\left(\frac{-1+\Omega_{T}}{{\partial}\phi}\right)^2\partial {\phi}, \tag{ 71 }$$

with $\Omega_{T} = \sqrt{1+4 t\partial\phi\bar{\partial}\phi}.$ Therefore, the deformed Lax connection is given by

$$\mathcal{L}(\mathbf{z},\tau) = \frac{\partial_w \phi(\mathbf{w}(\mathbf{z}))+2 t \partial_{\bar{w}}\phi(\mathbf{w}(\mathbf{z}))T_2(\mathbf{w}(\mathbf{z}))}{1-4t^2 T_2(\mathbf{w}(\mathbf{z}))\bar{T}_2(\mathbf{w}(\mathbf{z}))} = \frac{\partial\phi}{\Omega_{T}}, \tag{ 72 }$$

$$\bar{\mathcal{L}}(\mathbf{z},\tau) = \frac{-\partial_{\bar{w}} \phi(\mathbf{w}(\mathbf{z}))-2t \partial_{w}\phi(\mathbf{w}(\mathbf{z}))\bar{T}_2(\mathbf{w}(\mathbf{z}))}{1-4t^2 T_2(\mathbf{w}(\mathbf{z}))\bar{T}_2(\mathbf{w}(\mathbf{z}))} = -\frac{\bar{\partial}\phi}{\Omega_{T}}. \tag{ 73 }$$

Indeed, the Lax equation matches the equation of motion of the $T\bar{T}$-deformed free scalar:

$$\partial\left(\frac{\bar{\partial}\phi}{\Omega_{T}}\right)+\bar{\partial}\left(\frac{\partial\phi}{\Omega_{T}}\right) = 0. \tag{ 74 }$$

Next, we turn to the $T\bar{T}$-deformed sine-Gordon model, whose Lax pair has been given in [18]. As a first step, we need to find the Jacobian (60), which is determined by the stress-energy tensor in $\mathbf{w}$ space-time. The Lagrangian of the sine-Gordon model is given by adding the potential ^④

$$V = 4 \sin^2\left(\frac{\phi}{2}\right) \tag{ 75 }$$

to the free scalar Lagrangian (66). From the standard procedure, one can find the expression of the stress-energy tensor

$$\begin{aligned}[b] & T_2(\mathbf{w}) = -\frac{1}{2}(\partial_w \phi)^2,\quad \bar{T}_2(\mathbf{w}) = -\frac{1}{2}(\partial_{\bar{w}} \phi)^2,\\ & \Theta_0(\mathbf{w}) = -2\sin^2\left(\frac{\phi}{2}\right), \end{aligned} \tag{ 76 }$$

which leads to the following transformations

$$\begin{aligned}[b] & \partial_w \phi = \frac{-1+\Omega_{T}}{2t \bar{\partial}\phi},\quad \partial_{\bar{w}} \phi = \frac{-1+\Omega_{T}}{2t \partial\phi},\\ & \Omega_{T} = \sqrt{1+4t(1-tV)\partial \phi \bar{\partial}\phi}. \end{aligned} \tag{ 77 }$$

Recall that the undeformed Lax connection is

$$\mathcal{L}_w = -\frac{\rm i}{4}\partial_w\phi H+\frac{\lambda}{2}{\rm e}^{{\rm i}\frac{\phi}{2}}E_++\frac{\lambda}{2}{\rm e}^{-{\rm i}\frac{\phi}{2}}E_-, \tag{ 78 }$$

$$\mathcal{L}_{\bar{w}} = \frac{\rm i}{4}\partial_{\bar{w}}\phi H+\frac{1}{2\lambda}{\rm e}^{-{\rm i}\frac{\phi}{2}}E_++\frac{1}{2\lambda}{\rm e}^{{\rm i}\frac{\phi}{2}}E_-. \tag{ 79 }$$

The deformed Lax connection can be expanded with respect to these three generators as

$$\begin{aligned}[b] \mathcal{L}(\mathbf{z},t) = & \mathcal{L}^0 H+\mathcal{L}^+ E_++\mathcal{L}^- E_-,\\ \bar{\mathcal{L}}(\mathbf{z},\tau) = & \bar{\mathcal{L}}^0 H+\bar{\mathcal{L}}^+ E_++\bar{\mathcal{L}}^- E_-. \end{aligned} \tag{ 80 }$$

From transformation (65), we have the deformed Lax connections

$$\begin{aligned}[b] \mathcal{L}^0 = & -\frac{{\rm i}\partial \phi}{4 \Omega_{T}},\quad \bar{\mathcal{L}}^0 = \frac{{\rm i}\bar{\partial} \phi}{4 \Omega_{T}},\\ \mathcal{L}^+ = & \frac{{\rm e}^{-{\rm i}\frac{\phi}{2}}}{\lambda}\frac{(\partial\phi)^2t}{2\Omega_{T}}+\lambda {\rm e}^{{\rm i}\frac{\phi}{2}}\frac{(\Omega_{T}+1)^2}{8\Omega_{T}(1-t V)},\\ \bar{\mathcal{L}}^+ = & \lambda{{\rm e}^{{\rm i}\frac{\phi}{2}}}{}\frac{(\bar{\partial}\phi)^2t}{2\Omega_{T}}+\frac{ {\rm e}^{-{\rm i}\frac{\phi}{2}}}{\lambda}\frac{(\Omega_{T}+1)^2}{8\Omega_{T}(1-t V)},\\ \mathcal{L}^- = & \frac{{\rm e}^{{\rm i}\frac{\phi}{2}}}{\lambda}\frac{(\partial\phi)^2t}{2\Omega_{T}}+\lambda {\rm e}^{-{\rm i}\frac{\phi}{2}}\frac{(\Omega_{T}+1)^2}{8\Omega_{T}(1-t V)},\\ \bar{\mathcal{L}}^- = & \lambda{{\rm e}^{-{\rm i}\frac{\phi}{2}}}{}\frac{(\bar{\partial}\phi)^2t}{2\Omega_{T}}+\frac{ {\rm e}^{{\rm i}\frac{\phi}{2}}}{\lambda}\frac{(\Omega_{T}+1)^2}{8\Omega_{T}(1-tV)}, \end{aligned} \tag{ 81 }$$

which coincide with those found in [18].

The Lagrangian of the classical Liouville field theory is

$$\mathcal{L}(\mathbf{w}) = \partial_w \phi \partial_{\bar{w}}\phi-\mu {\rm e}^\phi,\quad V = -\mu {\rm e}^{\phi} \tag{ 82 }$$

with the Lax connection

$$\begin{aligned}[b] \mathcal{L}_w = & -\partial_w\phi H+2\lambda\sqrt{\mu}{\rm e}^{\frac{\phi}{2}}E_+,\\ \mathcal{L}_{\bar{w}} = & {\partial_{\bar{w}}}\phi H-\frac{1}{2\lambda}\sqrt{\mu} {\rm e}^{\frac{\phi}{2}}E_-. \end{aligned} \tag{ 83 }$$

The field transformation is also given by (77). Decomposing the Lax connection as (80), we again find

$$\begin{aligned}[b] \mathcal{L}^0 = & -\frac{ \partial \phi}{ \Omega_{T}},\quad \bar{\mathcal{L}}^0 = \frac{\bar{\partial} \phi}{ \Omega_{T}},\\ \mathcal{L}^+ = & \frac{\lambda\sqrt{\mu}{\rm e}^{\frac{\phi}{2}}(1+\Omega_{T})^2}{2\Omega_{T}(1-tV)},\quad \bar{\mathcal{L}}^+ = \frac{2t\lambda\sqrt{\mu} (\bar{\partial} \phi)^2 }{\Omega_{T}}, \\ \mathcal{L}^- = & -\frac{t\sqrt{\mu} {\rm e}^{\frac{\phi}{2}}(\partial\phi)^2 }{2\lambda \Omega_{T}},\quad \bar{\mathcal{L}}^- = -\frac{\sqrt{\mu}{\rm e}^{\frac{\phi}{2}}(1+\Omega_{T})^2}{8\lambda \Omega_{T}(1-tV)}. \end{aligned} \tag{ 84 }$$

These differ from the ones in (25) up to numerical factors because of the different conventions.

With these deformed Lax connections, one can derive infinite conserved charges. Conversely, the (anti)-holomorphic currents are simply given by taking powers of the modified traceless stress-energy tensor

$$T_{2n} = \left((\partial_w\phi)^2-2\partial_w^2 \phi \right)^n,\quad \bar{T}_{2n} = \left((\bar{\partial}_w\phi)^2-2\bar{\partial}_w^2 \phi \right)^n. \tag{ 85 }$$

From (77) and (62), one can read the deformed currents

$$\begin{aligned}[b] & T_{2n}(\mathbf{z}) = -\frac{\Omega_{T}+(2t(1-tV)\partial\phi \bar{\partial}\phi+1)}{2\Omega_{T}(1-\tau V)}T_{2n}(\mathbf{w}(\mathbf{z})),\\ & \Theta_{2n}(\mathbf{z}) = \frac{t(\bar{\partial}\phi)^2}{\Omega_{T}}T_{2n}(\mathbf{w}(\mathbf{z})). \end{aligned} \tag{ 86 }$$

The explicit expressions of these currents have been derived in [19] using a different method.

To construct the deformed Lax connections for the (affine) Toda field theories, let us first consider N free scalars with arbitrary potential [16, 18]

$$\mathcal{L}_N = \sum\limits_i ^N \partial_w \phi_i \partial_{\bar{w}}\phi_i+V(\phi_i). \tag{ 87 }$$

From the relationships

$$\begin{aligned}[b] & \frac{\partial x^1}{\partial y^1} = 1+t T^2_{\; 2}(\mathbf{y}),\quad \frac{\partial x^2}{\partial y^2} = 1+t T^1_{\; 1}(\mathbf{y}),\\ & \frac{\partial x^1}{\partial y^2} = \frac{\partial x^2}{\partial y^1}- = -t T^1_{\; 2}(\mathbf{y}), \end{aligned} \tag{ 88 }$$

we can compute the inverse of the Jacobian

$$\begin{aligned} \mathcal{J}_N^{-1} = \begin{pmatrix} \partial _w z & \partial_w \bar{z}\\ \partial_{\bar{w}}z & \partial_{\bar{w}}\bar{z} \end{pmatrix} = \begin{pmatrix} 1-t V & -t \sum\limits_i (\partial_w \phi_i)^2\\ -t \sum\limits_i (\partial_{\bar{w}}\phi_i)^2 & 1-t V \end{pmatrix}. \end{aligned} \tag{ 89 }$$

The main technical difficulty of this method is to solve $\partial_w \phi_i$ and $\partial_{\bar{w}}\phi_i$ from

$$\begin{pmatrix} \partial_w \phi_i \\ \partial_{\bar{w}} \phi_i \end{pmatrix} = \mathcal{J}_N^{-1}\begin{pmatrix} \partial \phi_i \\ \bar{\partial} \phi_i \end{pmatrix} \tag{ 90 }$$

in terms of $\partial\phi_i$ and $\bar{\partial}\phi_i$. For this particular example, we find the following solution

$$\partial_w \phi_i = \frac{1}{2t}\frac{\bar{\partial}\phi_i(-1+\Omega_{T})+\tilde{t} \dfrac{\partial B}{\partial \partial\phi_i}}{\bar{{K}}}, \tag{ 91 }$$

$$\partial_{\bar{w}}\phi = \frac{1}{2t}\frac{{\partial}\phi_i(-1+\Omega_{T})+\tilde{t} \dfrac{\partial B}{\partial \bar{\partial}\phi_i}}{{{K}}}, \tag{ 92 }$$

with

$$\tilde{t} = t(1-t V),\quad \Omega_{T} = \sqrt{1+4\tilde{t}(\mathcal{L}^{(0)}-\tilde{t}B)}, \tag{ 93 }$$

$$\mathcal{L}^{(0)} = \sum\limits_{i = 1}^N \partial\phi_i \bar{\partial}\phi_i,\quad K = \sum\limits_i^N (\partial \phi_i)^2,\quad \bar{K} = \sum\limits_i^N (\bar{\partial}\phi_i)^2, \tag{ 94 }$$

$$B = \sum\limits_{i = 1}^N (\partial\phi_i)^2\sum\limits_{j = 1}^N(\bar{\partial}\phi_j)^2-\left(\sum\limits_{i = 1}^N \partial\phi_i\bar{\partial}\phi_i\right)^2. \tag{ 95 }$$

The stress-energy tensor is given by

$$K_w = \sum\limits_i^N (\partial_w \phi_i)^2,\quad \bar{K}_{\bar{w}} = \sum\limits_i^N (\partial_{\bar{w}}\phi_i)^2, \tag{ 96 }$$

$$T_2 = -\frac{1}{2}K_w,\quad \bar{T}_2 = -\frac{1}{2}\bar{K}_{\bar{w}},\quad \Theta_0 = -\frac{1}{2}V. \tag{ 97 }$$

Therefore, the deformed Lax connection is directly given by (65)

$$\mathcal{L} = \frac{(1-\tau V)\mathcal{L}_{w}+\tau K_w \mathcal{L}_{\bar{w}}}{(1-\tau V)^2-\tau^2K_w\bar{K}_{\bar{w}}}, \tag{ 98 }$$

$$\bar{\mathcal{L}} = \frac{(1-\tau V)\mathcal{L}_{\bar{w}}+\tau \bar{K}_{\bar{w}} \mathcal{L}_{w}}{(1-\tau V)^2-\tau^2K_w\bar{K}_{\bar{w}}}. \tag{ 99 }$$

Using the identities (90), we can find the relations among $K_w$, $\bar{K}_{\bar{w}}$, K, and $\bar{K}$

$$K_w = (1-\tau V)^2 K+\tau^2K_{w}^2\bar{K}-2\tau K_w(1-\tau V)\mathcal{L}^{(0)}, \tag{ 100 }$$

$$\bar{K}_{\bar{w}} = (1-\tau V)^2\bar{K}+\tau^2K_{\bar{w}}^2 K-2\tau \bar{K}_{\bar{w}}(1-\tau V)\mathcal{L}^{(0)}. \tag{ 101 }$$

These are quadratic equations, whose solutions are ^⑤

$$K_w = \frac{2\tilde{t}\mathcal{L}^{(0)}+1-\Omega_{T}}{2t^2 \bar{K}},\quad \bar{K}_{\bar{w}} = \frac{2\tilde{t}\mathcal{L}^{(0)}+1- \Omega_{T}}{2\tau^2 {K}}, \tag{ 102 }$$

where we used the identity

$$B = K\bar{K}-\mathcal{L}^{(0)}\mathcal{L}^{(0)}. \tag{ 103 }$$

Substituting (102) into (98) gives

$$\mathcal{L} = -\frac{\Omega_{T}+ (2\tilde{t}\mathcal{L}^{(0)}+1)}{2\Omega_{T}(1-t V)}\mathcal{L}_w-\frac{t K}{\Omega_{T}}\mathcal{L}_{\bar{w}}, \tag{ 104 }$$

$$\bar{\mathcal{L}} = -\frac{\Omega_{T}+ (2\tilde{t}\mathcal{L}^{(0)}+1)}{2\Omega_{T}(1-t V)}\mathcal{L}_{\bar{w}}-\frac{t \bar{K}}{\Omega_{T}}\mathcal{L}_{w}. \tag{ 105 }$$

For the affine Toda theories, whose Lax connections are known, it is straightforward to read the deformed Lax connection from (105). They turn out to match the ones we derived previously in Section II.B. Furthermore, we can use the relations to derive the deformed Lax connection of the PCM if we make the following identification

$$j_\mu = j^i_\mu T_i,\quad j^i_\mu \equiv \partial_\mu \phi_i, \tag{ 106 }$$

where $T_i$ are the generators of the Lie algebra with the Killing metric $\text{Tr}(T_iT_j) = \delta_{ij}$.

As our last example, let us consider the $T\bar{T}$-deformed nonlinear Schrödinger model, which is a non-relativistic complex field theory. The $T\bar{T}$-deformed Lagrangian was recently derived in [22-24]. Here, we derive the deformed Lax connection from the dynamic coordinate transformation.

For the undeformed model, the Lagrangian is

$$\mathcal{L}_{\rm NS}(y_1,y_2) = \frac{\rm i}{2}\left(\bar{q} \partial_{y_1} q-q \partial_{y_1} \bar{q} \right)-\frac{\partial_{y_2} q \partial_{y_2} \bar{q}}{2m}-g |q\bar{q}|^2, \tag{ 107 }$$

which has the following equations of motion

$$-{\rm i}\partial_{y_1} q = \frac{1}{2m}\partial^2_{y_2}q-2g q^2\bar{q},\quad {\rm i}\partial_{y_1} \bar{q} = \frac{1}{2m}\partial^2_{y_2}\bar{q}-2g q\bar{q}^2, \tag{ 108 }$$

and the stress-energy tensor

$$T_{y_2y_2} = -\frac{1}{m}\partial_{y_2}q \partial_{y_2}\bar{q}-\mathcal{L}_{NS}(y_1,y_2), \tag{ 109 }$$

$$T_{y_2y_1} = -\frac{1}{2m}\left(\partial_{y_2}\bar{q}\partial_{y_1}q+\partial_{y_2}q\partial_{y_1}\bar{q}\right), \tag{ 110 }$$

$$T_{y_1y_2} = \frac{\rm i}{2}\left(\bar{q}\partial_{y_2}q-q\partial_{y_2}\bar{q}\right), \tag{ 111 }$$

$$T_{y_1y_1} = \frac{\rm i}{2}\left(\bar{q}\partial_{y_1}q-q\partial_{y_1}\bar{q}\right)-\mathcal{L}_{\rm NS}(y_1,y_2). \tag{ 112 }$$

The corresponding Lax connection is

$$U_{y_2} = -{\rm i}\lambda \sigma_3+ {\rm i} \sqrt{2gm}Q, \tag{ 113 }$$

$$V_{y_1} = -\frac{{\rm i}\lambda^2}{m}\sigma_3+{\rm i}\sqrt{\frac{2g}{m}}\lambda Q+\sqrt{\frac{g}{2m}}\partial_{y_2}Q\sigma_3+{\rm i}gQ^2\sigma_3, \tag{ 114 }$$

where

$$\begin{array}{c} \sigma_3 = \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix},\quad Q = \begin{pmatrix} 0 & q\\ -\bar{q} & 0 \end{pmatrix}. \end{array} \tag{ 115 }$$

Solving (88), one can find the following rules of transformation [24]:

$$\begin{aligned}[b] & \partial_{y_1}q = \frac{2m(B-S)\partial_{x_1}\bar{q}+2\tilde{t}\bar{A} C}{2t \bar{A}^2},\quad \partial_{y_2}q = \frac{2m(B-S)}{2t \bar{A}},\\ & \partial_{y_1}\bar{q} = \frac{2m(B-S)\partial_{x_1}q-2\tilde{t}A C}{2t A^2},\quad \partial_{y_2}\bar{q} = \frac{2m(B-S)}{2t A}, \end{aligned} \tag{ 116 }$$

where we have defined

$$\tilde{t} = t(1+tV),\quad C = \partial_{x_2}\bar{q} \partial_{x_1}q-\partial_{x_1}\bar{q} \partial_{x_2}q, \tag{ 117 }$$

$$B = 1+\frac{{\rm i}t}{2}(\bar{q} \partial_{x_1}q-q\partial_{x_1}\bar{q}),\quad S = \sqrt{B^2-\frac{2\tilde{t}}{m}A\bar{A}}, \tag{ 118 }$$

$$A = \partial_{x_2}q+\frac{{\rm i}t}{2}q C,\quad \bar{A} = \partial_{x_2}\bar{q}+\frac{{\rm i}t}{2}\bar{q} C. \tag{ 119 }$$

Substituting (108) and (116) into (65), and after some manipulation, we end up with the final results for the deformed Lax connection

$$\begin{pmatrix} V_{x_1}\\ U_{x_2} \end{pmatrix} = \begin{pmatrix} J_{11} & J_{12}\\ J_{21} & J_{22} \end{pmatrix}\begin{pmatrix} V_{y_1}\\ U_{y_2} \end{pmatrix}, \tag{ 120 }$$

where

$$\begin{aligned}[b] & J_{11} = \frac{tB(B+S)}{2S\tilde{t}},\quad J_{12} = -\frac{t(A\partial_{x_1}\bar{q}+\bar{A}\partial_{x_1}q)}{2mS}, \\ & J_{21} = \frac{{\rm i}t^2(B+S)(\bar{q} \partial_{x_2}q-q \partial_{x_2}\bar{q})}{4S \tilde{t}}, \end{aligned} \tag{ 121 }$$

$$J_{22} = \frac{2t A\bar{A}}{2mS(B-S)}-\frac{t(A\partial_{x_2}\bar{q}+\bar{A}\partial_{x_2}q)}{2mS}. \tag{ 122 }$$