Four-Dimensional Chern-Simons and Gauged Sigma Models

I would like to thank the referee for their comments. I agree with the concerns they have raised and hope that they are satisfactorily addressed in the following. I address each issue in turn, first quoting the referee and then giving my response.

The primary issue raised by the referee was:

... one has the unusual situation, which does not appear in 4D Chern-Simons (see for instance equation (2.21)), that the gauge transformations $u$, $v$ satisfy constraints (4.25) which depend on the boundary value of the fields $A$, $B$. This is not common. In such a situation, one has to be careful with the product law of gauge transformations. For instance, considering the set of solutions of constraints (4.25) for a given value of the boundary fields, generically two elements in this set cannot be multiplied together since the product would not satisfy the constraints. Thus, strictly speaking there is no gauge group.

The referee is correct to worry that the set of gauge transformations as presented does not form a closed group. Below I have shown there exists solutions to the relevant equations such that the gauge transformations of $A$ and $B$ are a group. In fact, this set is larger than that suggested in the comment:

`` Notice however that there exists a group of gauge transformations : those who go to the identity on the boundary (or maybe in some neighborhood of the boundary)and thus leave the boundary values of the fields invariant.

For the referee's ease, let me remind you of a few facts from the paper which are relevant to the following discussion. In the article I discussed field configurations of two gauge fields $A$ and $B$ which live on a manifold $\Sigma \times \mathbb{C} P^{1}$. I work in the light-cone coordinates $x^{\pm}$ on $\Sigma$ and parametrise $\mathbb{C} P^{1}$ by $z,\bar{z}$. Respectively, the fields $A$ and $B$ are valued in the Lie algebras $\textbf{g}$ and $\textbf{h}$. In the constructions $\textbf{h}$ is taken to be a subalgebra of $\textbf{g}$. In particular, I focused on the case with a decomposition $\textbf{g} = \textbf{f} \oplus \textbf{h}$, and restricted to the cases where $[\textbf{f},\textbf{h}] \subseteq \textbf{f}$. Unlike in the article, the projections into $ \textbf{f} $ and $ \textbf{h} $ are denoted by $ \pi_{\textbf{f}}$ and $\pi_ {\textbf{h}} $. This is simply to aid in writing this response.

The field configurations which I considered have boundary conditions satisfying the defect equations of motion: \begin{equation} \left( \eta_q^{(0)}+\eta_q^{(1)} \partial_z \right) \int_{\Sigma_q} \left\langle A-B \, , \, \delta A + \delta B \right\rangle = 0 \, . \end{equation} Here, $\eta_{q}^{(0,1)}$ are numbers, $\int_{\Sigma_q}$ is an integral along the surface $\Sigma$ and the subscript $q$ on $\Sigma_q$ denotes the evaluation of the expression at a point $(q,\bar{q}) \in \mathbb{C} P^1$. I presented three classes of boundary condition, these are the gauged chiral boundary conditions: \begin{equation} \pi_{\textbf{f}} (A_{-}) = 0 \, , \quad \pi_{\textbf{h}} (A_{\pm} ) = B_{\pm} \, , \end{equation} gauged anti-chiral: \begin{equation} \pi_{\textbf{f}} (A_{+}) = 0 \, , \quad \pi_{\textbf{h}} (A_{\pm} ) = B_{\pm} \, , \end{equation} and gauged Dirichlet: \begin{equation} \pi_{\textbf{f}}(A_{\pm}) = 0 \, , \quad \pi_{\textbf{h}}(A_{\pm} ) = B_{\pm} \, , \quad \partial_z ( \pi_{\textbf{h}}(A_{\pm}) - B_{\pm} ) = 0 \, , \quad \pi_{\textbf{h}}(A_{\bar{z}}) = B_{\bar{z}} \, . \end{equation} (Please note, I have dropped the condition $\partial_z ( \pi_{\textbf{h}}(A_{\bar{z}}) - B_{\bar{z}}) = 0$ from the gauged Dirichlet case, this is for reasons highlighted in report 1 which I will address in my response to that report.)

Of course, you wish for these boundary conditions to be preserved by the gauge transformations ${}^u A = u (d+A)u^{-1}$ and ${}^v B = v (d+B) v^{-1}$. To find the constraints on gauge transformations due to gauged chiral boundary conditions I defined $y = u^{-1} v$, and used $[\textbf{h},\textbf{f}] \subseteq \textbf{f}$ and $[\textbf{h},\textbf{h}] \subseteq \textbf{h}$. This leads to the following constraints upon $y$: \begin{equation} y^{-1} B_{-} y + y^{-1} \partial_{-} y = B_{-} \, , \quad \pi_{\textbf{h}}( y^{-1} A_{+} y + y^{-1} \partial_{+} y ) = B_{+} \, . \end{equation} When discussing the gauge invariance of an action you consider arbitrary field configurations satisfying the boundary condition, not just those which are on-shell, for this reason the above equation must hold for every $A$ and $B$. Thus, the first of the above equations implies the two conditions: $y \in C_{G}(\textbf{h})$, where $C_{G}(\textbf{h})$ is the centraliser of $\textbf{h}$ within $G$, and $\partial_{-} y = 0$. Using these conditions, the second equation reduces to $ \pi_{\textbf{h}}(y^{-1} \pi_{\textbf{f}}( A_{+}) y + y^{-1} \partial_{+} y) = 0$. This implies $ \pi_{\textbf{h}}(y^{-1} \partial_{+} y) = 0$ and $y \in N_{G}(\textbf{f})$, the normaliser of $\textbf{f}$ within $G$.

Let $K = C_{G}(\textbf{h}) \cap N_{G}(\textbf{f})$. Hence, given gauged chiral boundary conditions the group element $y$ must satisfy the following conditions on the defect: \begin{equation} y = u^{-1} v \in K \, , \quad \partial_{-} y = 0 \, , \quad \pi_{\textbf{h}}(y^{-1} \partial_{+} y )= 0 \, . \end{equation} Using similar logic, the same conditions hold for the gauged anti-chiral case, but with $+$ and $-$ swapped.

In the gauged Dirichlet case the first two conditions as well as the last, after following the argument used above, imply $y$ satisfies the following on the defect: \begin{equation} y = u^{-1} v \in K \, , \quad \partial_{\pm} y = 0 \, , \quad \pi_{\textbf{h}}(y^{-1} \partial_{\bar{z}} y ) = 0 \, . \end{equation} The condition $\partial_{z}( \pi_{\textbf{h}}(A_{\pm}) - B_{\pm} ) = 0$ implies: \begin{equation} \partial_{z}( \pi_{\textbf{h}}(y^{-1} A_{\pm} y + y^{-1} \partial_{\pm} y) - B_{\pm}) = 0 \, . \end{equation} Which, upon expanding the $z$ derivative and using the conditions already imposed upon $y$ gives: \begin{equation} \pi_{\textbf{h}}([B_{\pm},y^{-1}\partial_{z} y] + \partial_{z} (y^{-1} \partial_{\pm} y )) = 0 \, . \end{equation} By again demanding that $y$ be independent of the field configurations you find the conditions $y^{-1} \partial_z y \in \textbf{f}$, $\partial_z (y^{-1} \partial_{\pm} y ) \in \textbf{f}$, by $[\textbf{h},\textbf{f}] \subseteq \textbf{f}$ and $[\textbf{h},\textbf{h}] \subseteq \textbf{h}$. However, $y^{-1} \partial_z y \in \textbf{f}$ combined with the constraint $y \in N_{G}(\textbf{f})$ implies $\partial_z (y^{-1} \partial_{\pm} y ) \in \textbf{f}$. This is because $y \in N_{G}(\textbf{f})$ implies $\partial_z y y^{-1} \in \textbf{f}$ on $\Sigma_q$ and thus $\partial_{\pm}( \partial_{z}y y^{-1}) \in \textbf{f}$. Which by again using $y \in N_{G}(\textbf{f})$ gives $y^{-1} \partial_{\pm}( \partial_{z}y y^{-1}) y \in \textbf{f}$. Hence, by the identity $y^{-1} \partial_{\pm} ( \partial_{z} y y^{-1} ) y = \partial_z ( y^{-1} \partial_{\pm} y)$ you have $\partial_z (y^{-1} \partial_{\pm} y ) \in \textbf{f}$. Therefore, the final condition on $y$ at the defect is: \begin{equation} y^{-1} \partial_z y |_\textbf{h} = 0 \, . \end{equation}

You might worry that the composition of two gauge transformations, each of which satisfies the above constraints, might not itself satisfy these constraints. The only potential impediment to this would be the conditions $ \pi_{\textbf{h}}(y^{-1} \partial_{z} y )= \pi_{\textbf{h}}(y^{-1} \partial_{\bar{z}} y) = 0$. Taken sequentially two gauge transformations are given by $(A,B) \rightarrow ({}^{u^{\prime}u} A, {}^{v^\prime v} B)$ where I assume $y = u^{-1} v$ and $y^\prime = {u^\prime}^{-1} v^\prime$ satisfy the above constraints. Hence, the composition of the two gauge transformations, given by $y^{\prime \prime} = (u^\prime u)^{-1} v^\prime v = u^{-1} y^\prime v$, must also satisfy the above conditions. Since $y^\prime$ commutes with $H$ by construction I have $y^{\prime \prime} = u^{-1} y^\prime v = u^{-1} v y^\prime = y y^\prime$ and thus ${y^{\prime \prime}}^{-1} \partial_z y^{\prime \prime} = {y^{\prime}}^{-1} y^{-1} \partial_{z} y y^\prime + {y^{\prime}}^{-1} \partial_{z} y^\prime$. Again, by construction I have $ \pi_{\textbf{h}}(y^{-1} \partial_{z} y) = \pi_{\textbf{h}}({y^{\prime}}^{-1} \partial_{z} y^\prime ) = 0$, which together with $y \in K$ and $[\textbf{h},\textbf{f}] \subseteq \textbf{f}$ implies $\pi_{\textbf{h}}({y^{\prime \prime}}^{-1} \partial_z y^{\prime \prime} ) = 0$. The same argument works for the $\bar{z}$ derivative. Hence the set of gauge transformations as defined is indeed closed.

The following are typos which can be fixed in a revised version. In particular, $u$ should die off to the identity and the cubic term in the Lax connection should in fact be $\left\langle \tilde{g} \mathcal{L} \tilde{g} \, , \, \tilde{g} \mathcal{L} \tilde{g}^{-1} \wedge \tilde{g} d \tilde{g}^{-1} \right\rangle$.

1. After (2.26), $u$, which is a group element, cannot die off to zero.
2. In the first line of page 10, the word ”pole”is missing
3. In the third line of the second paragraph of 3.2, a ”the” should be replaced by a ”that”.
4. In the fourth line of the first paragraph of 3.3, a ”of” is missing.
5. In (3.26) and in the subsequent sentence, it is completely unclear how a term cubic in the Lax matrix may appear.

I would like to thank the referee for their response. In the following I've addressed the conceptual issues and comments they have raised, first quoting them and following up with a comment. Please note, I have change the notation for projection into $\textbf{f}$ and $\textbf{h}$ to $\pi_{\textbf{f}}$ and $\pi_{\textbf{h}}$, respectively. This is simply to make writing this response easier.

The first conceptual issue was:

In section 4.1 boundary conditions are identified to solve equation (4.9). To me it seems that equation (4.12) is weaker that what (4.9) implies. In fact, the variations of A and B are independent. I would therefore expect to read 3 independent equations: one for $\pi_{\textbf{f}}(\delta A_k)$ (i.e. eq. (4.11), one for $\pi_{\textbf{h}}(\delta A_k)$ and one for $\delta B_k$. I would like to know if the author agrees with this. In that case, one should check if the boundary conditions presented in the paper correctly solve (4.9).

I disagree with the referee here.

This is because boundary or defect equations of motion can be satisfied either because the fields obey a dynamical equation of motion (for example Neumann conditions) or because the fields obey a constraint (such as Dirichlet conditions) or, as in the article, a mixture of the two. The requirement is \begin{equation} \left( \eta_q^{(0)}+\eta_q^{(1)} \partial_z \right) \int_{\Sigma_q} \left\langle A-B \, , \, \delta A + \delta B \right\rangle = 0 \, . \end{equation} The referee is correct to say that one could take $\delta A$ and $\delta B$ to be unconstrained and independent of each other, but that is not what I have done - I have instead chosen boundary conditions in which $\pi_{\textbf{h}}(A_\pm)$ and $B_\pm$ satisfy a constraint which implies that $\pi_{\textbf{h}}(\delta A_{\pm})$ and $\delta B_{\pm}$ are not independent. This means the variation of the action can be made to vanish if we consider field configurations satisfying (4.11) and (4.12). To be explicit, all the gauged boundary conditions in the paper satisfy the constraint $\pi_{\textbf{h}}(A_{\pm})-B_{\pm}=0$. If you then view $\delta$ as a derivative acting on the space of field configuration and assume $\pi_{\textbf{h}}(\delta A_{\pm}) - \delta B_{\pm} = \delta (\pi_{\textbf{h}}(A_{\pm}) - B_{\pm}) \neq 0$ it follows that somewhere on defect the condition $\pi_{\textbf{h}}(A_{\pm}) - B_{\pm}=0$ would not hold. Hence, $\pi_{\textbf{h}}(\delta A_{\pm}) - \delta B_{\pm} = 0$ is imposed for consistency. Note, an example of a boundary condition which is a dynamical equation of motion in the paper occurs in gauged Dirichlet boundary conditions where you have $\partial_z (\pi_{\textbf{h}}(A_{\pm}) - B_{\pm}) = 0$. This ensures the defect equations of motion are satisfied once you impose the constraints $\pi_{\textbf{h}}(A_{\pm})-B_{\pm}=0$ and $\pi_{\textbf{h}}(\delta A_{\pm}) - \delta B_{\pm} = 0$.

The second potential conceptual issue was:

After (4.52) it is claimed that in order to reach the vanishing of $I^q_{2,\text{sing}}$ the author makes the requirement $y \partial_{+} y^{-1} \in \textbf{f}$. What is the mechanism to reach this requirement? Is this a restriction on $y$? Isn't it an issue that $y$ is not generic?

The referee is correct to raise this issue. As has been discussed in the response to report 3, this condition arises naturally from the boundary conditions by requiring gauge transformations be independent of the field configurations. I will summarise this here, and point the referee to the response to report 3 for more details.

In the paper I consider gauge transformations $(A,B) \rightarrow ({}^u A, {}^v B) = (u(d+A)u^{-1},v(d+B)v^{-1})$ which preserve the boundary conditions. This constraint implies a series of conditions on the group element $y = u^{-1} v$. For example, the gauged chiral boundary conditions: \begin{equation} \pi_{\textbf{f}}(A_{\pm}) = 0 \, , \quad \pi_{\textbf{h}}(A_{\pm}) = B_{\pm} \, , \end{equation} imply: \begin{equation} y^{-1} B_{-} y + y^{-1} \partial_{-} y = B_{-} \, , \quad \pi_{\textbf{h}}( y^{-1} A_{+} y + y^{-1} \partial_{+} y ) = B_{+} \, , \end{equation} by demanding $y$ be independent of $A$ and $B$ we find the following conditions on $y$ at the defect: \begin{equation} y = u^{-1} v \in K \, , \quad \partial_{-} y = 0 \, , \quad \pi_{\textbf{h}}(y^{-1} \partial_{+} y ) = 0 \, . \end{equation} Here $K = C_{G}(\textbf{h}) \cap N_{G}(\textbf{f})$, where $C_{G}(\textbf{h})$ is the centraliser of $\textbf{h}$ in $G$ and $N_{G}(\textbf{f})$ is the normaliser of $\textbf{f}$ in $G$. It is not an issue that $y$ is not a generic element of $G$. This is because $y = u^{-1} v$ is the freedom leftover in choosing the group element $u$ given a fixed $v$. The constraint on the space time dependence encoded in $\pi_{\textbf{h}}(y^{-1} \partial_{+} y )= 0$ is perfectly natural in four-dimensional Chern-Simons. For example, if $\textbf{h} = 0$ and $\textbf{f} = \textbf{g}$ then $y=u^{-1}$ and the above conditions are those associated to chiral boundary conditions in standard four-dimensional Chern-Simons.

I agree with the following suggested changes and will be made in a revised version:

-In $\eta^l_q$ the index $l$ may be mistaken for a power exponent. I would suggest to modify the notation to avoid confusion (for example using $\eta^{(l)}_q$). - In section 2.1 there is a lot of material that is repeated from appendix A. I suggest to reduce the material presented in section 2.1, leaving only the relevant points. - After equation (2.26) "left hand side" should be replaced by "right hand side". - In equation (2.27) I would suggest to remove the $=0$ on the right hand side. In fact, the vanishing of $\delta S_q$ is something that is not found at that stage yet, it is demonstrated later. - When writing the boundary conditions, the notation never makes it explicit that the equations are understood as evaluated at fixed values of z. I do not want to ask to introduce a new notation where this information is made more explicitly, because the notation would become quite heavy. But there are cases in which this "light notation" may create confusion in the reader, for example in (4.31) (and later) where the equations without the $\partial_z$ would naively imply those with $\partial_z$. I would suggest to add a comment in the text around those equations reminding that they are always meant at fixed values of $z$, so that to avoid confusion. - In the text after (5.64) there is a misplaced sentence "If we define $\tilde{g}$ as in section 3.2." - In the text after (3.8) there is a reference to (3.20) which I believe is misplaced. The reference should probably be to (2.4). - In the text after (3.16) there is a reference to (5.4) which I believe is misplaced. The reference should probably be to (3.7). - After (2.26), totaly $\rightarrow$ total. - In the text after (5.3), $y$ should be $u$. - In the text after (5.12) there is an $h$ missing a hat. - In the Conclusion "sectio" $\rightarrow$ "section".''

The following is actually the result of a typo, $\left\langle \tilde{g} \mathcal{L} \tilde{g} \, , \, \tilde{g} \mathcal{L} \tilde{g}^{-1} \wedge \tilde{g} \mathcal{L} \tilde{g}^{-1} \right\rangle$ should be $\left\langle \tilde{g} \mathcal{L} \tilde{g} \, , \, \tilde{g} \mathcal{L} \tilde{g}^{-1} \wedge \tilde{g} d \tilde{g}^{-1} \right\rangle$. The text then relates to both terms.

In the text just after (3.26), the expression $A^\prime \wedge A^\prime \wedge \hat{A}$ is related to the last term in (3.26), but it should be related to the first term there instead.

I would like to thank the referee for their very thorough report. They were correct that the highlighted issues affected the generality of the construction. In the following I've quoted each issue and followed it with the relevant discussion. As in the other responses, I will use $\pi_{\textbf{f}}$ and $\pi_{\textbf{h}}$ to denote the projections into $\textbf{f}$ and $\textbf{h}$, respectively. I have also changed the notation for the Lax connections of $A$ and $B$ to $\mathcal{A}$ and $\mathcal{B}$. This is only to aid in writing this response.

I begin by addressing the following:

(1) The first comment concerns the passage to the archipelago gauge in section 5.3 and more precisely the proof that the gauge transformation by $\tilde{u}$ respects the boundary condition (5.17) at a double pole $q$ of $\omega$. Below this equation, it is argued that this is the case using $\pi_{\textbf{h}}(A_{\bar{z}}) = \pi_{\textbf{h}}(A_{\bar{z}}) = 0$ at the point $(z,\bar{z}) = (q,\bar{q})$, which together with $A_{\bar{z}} = \hat{g}^h \partial_{\bar{z}} (\hat{g}^h)^{-1}$ imply $\pi_{\textbf{h}}(\hat{g}^h \partial_{\bar{z}} (\hat{g}^h)^{-1}) = \pi_{\textbf{h}}(\hat{g}^h \partial_{z} (\hat{g}^h)^{-1}) = 0$ at $(z,\bar{z}) = (q,\bar{q})$. It seems to me that this last equality is wrong (for a simple counter-example, take for instance $q=0$ and $\hat{g}^h = \exp(z X)$, with $X \in \textbf{h}$). I agree with the referee, for this reason I have dropped the boundary condition $\partial_{z}(\pi_\textbf{h}(A_{\bar{z}}) - B_{\bar{z}}) = 0$.

The referee is correct in their assessment here:

Although this might seem like a small technical point, this can potentially have important consequences on the paper. Indeed, this seems to suggest that the archipelago gauge cannot be attained in the presence of a gauged Dirichlet boundary condition, thus putting in jeopardy the main results of section 5, including the gauged Wess-Zumino-Witten examples. In fact, such an observation would mean that the quantity $\hat{g}^h \partial_{z} (\hat{g}^h)^{-1}$ evaluated on the 2-dimensional defect $(z,\bar{z}) = (q,\bar{q})$ contains physical degrees of freedom, which cannot be eliminated by a gauge transformation and which should thus appear in the resulting 2-dimensional theory.

I will now go into some detail and describe the resolution to this issue. In ref. [39] it shown in propositions 2.8 and 3.6 that for a group element $g \in C^{\infty}(\Sigma \times \mathbb{C} P^1, G)$ the Wess-Zumino term:

\begin{equation} I_{\text{WZ}} = \int_{\Sigma \times \mathbb{C}P^1} \omega \wedge \left\langle g^{-1} d g , (g^{-1} d g)^2 \right\rangle \end{equation}

depends upon $g$ and its $z$-derivatives evaluated at the poles of $\omega$.This is because $\left\langle g^{-1} d g , (g^{-1} d g)^2 \right\rangle$ is closed and therefore exact on the manifolds the authors (and I) consider. In Lemmas 2.4, 2.5, 2.7 and 3.5 it was shown that there is one homotopy class of group elements with the values of $g$ and its $z$-derivatives at the poles of $\omega$. Let $\tilde{g}$ be a group element in this class which is rotationally invariant in a disc around each pole and the identity outside of the discs. Since the integrals in the four-dimensional Chern-Simons action are determined by the boundary values of $g$ we are free to replace $g$ by the group element $\tilde{g}$. Therefore, as in ref. [39], we have:

\begin{equation} I_{\text{WZ}} = \int_{\Sigma \times \mathbb{C}P^1} \omega \wedge \left\langle \tilde{g}^{-1} d \tilde{g} , (\tilde{g}^{-1} d \tilde{g})^2 \right\rangle = \sum_{q \in P} \int_{\Sigma \times U_q} \omega \wedge \left\langle \tilde{g}^{-1} d \tilde{g} , (\tilde{g}^{-1} d \tilde{g})^2 \right\rangle = \sum_{q \in P} \int_{\Sigma \times U_q} \omega \wedge d\left( - \int_{\gamma_z} \left\langle \tilde{g}^{-1} d \tilde{g} , (\tilde{g}^{-1} d \tilde{g})^2 \right\rangle \right) = - 2\pi i \sum_{q \in P} \sum_{l=0}^{n_q-1} \eta_q^{(l)} \int_{\Sigma \times I_q} \partial_{z}^l \left\langle \tilde{g}^{-1} d \tilde{g} , (\tilde{g}^{-1} d \tilde{g})^2 \right\rangle =: \tilde{I}_{\text{WZ}} (\tilde{g}) \end{equation}

where $I_q$ denotes the interval of the radial coordinate of the disc $U_q$. Note, I've had to integrate by parts here and sent an integral of a total derivative to zero, this requires the regularisation constraint $g^{-1} \partial_{\bar{z}} g = 0$. If I apply the replacement of $g$ argument to the localisation of the unified gauged sigma model action for $\omega$ with at most double poles, I find new additional terms. These terms include a new scalar field $\Delta_q = (\hat{g}^h_q)^{-1} \partial_{z} \hat{g}^h_q \in \textbf{g}$ which is found at double poles of $\omega$. Note, here $ \partial_{z} \hat{g}^h_q$ means the evaluation of $\partial_z \hat{g}^h$ at $z=(q,\bar{q})$. The new action is:

\begin{equation} S_{USGM} (g^h, \mathcal{A} ,\mathcal{B} ,\Delta )= - \frac{i}{\hbar} \sum_q \int_{ \Sigma_q} \left( \left\langle res_q \left( \omega \wedge \mathcal{A} \right) , (g^h_q)^{-1} d_{\Sigma} g^h_q \right\rangle + res_q \left( \omega \wedge \left\langle - d_{\Sigma} g^h_q (g^h_q)^{-1} + g^h_q \mathcal{A} (g^h_q)^{-1} , \mathcal{B} \right\rangle \right) \right) \end{equation} \begin{equation} - \frac{i}{\hbar} \sum_{q \in P} \eta_q^{(1)} \int_{\Sigma_q} \left( \left\langle \mathcal{A} , d_{\Sigma} \Delta \right\rangle + \left\langle (g^h_q)^{-1} d_{\Sigma} g^h_q , [\mathcal{A}, \Delta] \right\rangle - \left\langle d_{\Sigma} \Delta + [\mathcal{A} ,\Delta], (g^h_q)^{-1} \mathcal{B} g^h_q \right\rangle \right) + \tilde{I}_{\text{WZ}}(\tilde{g}^{h}) \quad \quad (1) \end{equation}

I will now address the construction of the gauged Wess-Zumino-Witten model using this new action. As in the paper I take $\omega$ to be: \begin{equation} \omega = \frac{(z-k)}{z}dz \, , \end{equation} with poles at $z=0$ and $z=\infty$. I impose gauged chiral boundary conditions at $z=0$: \begin{equation} A_{-} = B_{-} \, , \quad \pi_{\textbf{h}} (A_{+}) = B_{+} \, , \end{equation} gauged Dirichlet conditions at $z=\infty$: \begin{equation} A_{\pm} = B_{\pm} \, , \quad z^2\partial_{z} (\pi_{\textbf{h}} (A_{\pm}) - B_{\pm}) = 0 \, , \quad \quad (2) \end{equation} and insert a $(0,1)$ defect in $\mathcal{A}$ at $z=k$. I take $\mathcal{B}$ to be regular. Hence, the Lax connections are of the form:

\begin{equation} \mathcal{A} = \mathcal{A}^c_{+} dx^+ + \left( \mathcal{A}^c_{-} + \frac{\mathcal{A}^k_{-} }{z-k}\right) dx^- \, , \quad \mathcal{B} = B_{+} (x^\pm) dx^{+} + B_{+} (x^\pm) dx^{-} \, . \quad \quad (3) \end{equation}

As in the paper, the right redundancy is used to set $\tilde{g}^{h} = 1$ at $z=\infty$ and we denote $\tilde{g}^h_{0} = g^h_{0} = g$. Using the equations

\begin{equation} A = \tilde{g}^h d_{\Sigma} (\tilde{g}^h)^{-1} + \tilde{g}^h \mathcal{A} (\tilde{g}^h)^{-1} \, , \quad \mathcal{B} = B \end{equation}

and the boundary conditions we can solve for the Lax connections. The first equation of the gauged Dirichlet boundary conditions (2) implies $\mathcal{A}^c = B$ while the second equation reduces to:

\begin{equation} \lim_{z\to\infty} \; z^2 \pi_{\textbf{h}} \left( \partial_z \mathcal{A} - d_{\Sigma} \Delta_{\infty} - [\mathcal{A} ,\Delta_\infty] \right) = 0 \, . \end{equation}

Upon substituting $\mathcal{A}$ from equations (3) this simplifies further to the equations:

\begin{equation} \pi_{\textbf{h}} ( d_{\Sigma} \Delta_{\infty} + [ B ,\Delta_{\infty} ]) = 0 \, , \quad \pi_{\textbf{h}}(\mathcal{A}^{k}_{-} ) = 0 \, , \end{equation}

\begin{equation} \pi_{\textbf{h}} ([\Delta_{\infty}, \mathcal{A}^{k}_{-}]) = 0 \end{equation}

These equations will be a subset of the equations of motion of the sigma model which appears on the defects. The gauged WZW model appears from this model as a "gauge fixing" or "truncation", $\Delta_{\infty} = 0$ in very much the same way that affine Toda field theory appears from conformal affine Toda field theory by setting a subset of the fields to zero. From here you can follow the argument in the paper and recover the gauged WZW model using equation (1).

Given the above we can remove the discussion highlighted by the comment below from the paper. This is because the condition $\pi_{\textbf{h}}(\mathcal{A}_{-}^k)=0$ now follows naturally as a solution to the equations of motion and no other justification is required.

(2) The second comment concerns the discussion around and below equation (5.39), whose goal is to argue that $\pi_{\textbf{h}}(\mathcal{A})$ and $\mathcal{B}$ have no pole at $z=k$. The approach followed in the article is to consider the limit $k \rightarrow \infty$ and ask that the analytical structure of $\mathcal{A}$ and $\mathcal{B}$ for finite $k$ is compatible with the presence of a gauged anti-chiral boundary condition at $z=\infty$ in this limit. I have several reservations on this argument. First of all, it is not clear to me why an analysis of the pole structure of $\pi_{\textbf{h}}(\mathcal{A})$ and $\mathcal{B}$ in the limit $k \rightarrow \infty$ would have exact consequences for $k$ finite: rather, I would think that it can at most establish some results up to corrections which vanish when $k\rightarrow \infty$.

The rest of the referees comments are dealt with below.

(4) In the introduction, it is said that the only gauged sigma-models that have been obtained from standard 4-dimensional Chern-Simons theory are the ones on symmetric spaces in [12]. Let me note that there are various existing works in the literature extending this construction to $\eta$-deformed and $\lambda$-deformed coset models, including .... Of particular relevance for the present paper are the works on coset $\lambda$-models, whose conformal points are gauged WZW models, showing that the latter also arise in the context of standard 4-dimensional Chern-Simons theory (with the restriction that the subgroup $H \subset G$ is the set of fixed points of an involutive automorphism).

Could the referee please confirm which paper(s) they intended to refer to in `...' in the above comment? I assume this at least includes 1909.13824. I will change the introduction to clarify this issue.

(5) Before equation (2.10), it is said that $\Lambda$ is supposed here to be a 3-form depending meromorphically on $z$ and that $\text{CS}(A)$ is an example of such 3-form. It seems to me that this is not true, as $\text{CS}(A)$ is not meromorphic in $z$ (even after the gauge transformation sending $A$ to $\mathcal{L}$). In fact, it seems to me that in most cases where (2.10) is applied, the 3-form $\Lambda$ is not meromorphic in $z$.

(7) What are the assumptions on $\xi$ in equation (2.12)? It is said in the following sentence that $\omega \wedge \xi$ is meromophic in $z$ but it seems to me that, similarly to the case of $\Lambda$ in the point 5 above, this identity is mostly used in the case where $\xi$ is in fact not meromorphic (for instance when $\xi = \langle A, d A + \rangle$). Moreover, the notation $\Sigma_q$ should be explained in this equation. Finally, after (2.12), it seems to me that it is the integral of $\bar{\partial} (\omega \wedge \xi)$ which is send to zero rather than $\bar{\partial} (\omega \wedge \xi)$ itself.

These comments were a reference to an old calculation and should not have appeared in the paper. The referee is correct that meromorphicity is not true in most cases where (2.10) is used. I will add a comment making it clear that $\Sigma_q = \Sigma \times (q,\bar{q})$ before (2.12). You are correct that it is the integral of $\bar{\partial} (\omega \wedge \xi)$ which is sent to zero, this is the only assumption made of $\xi$.

(6) After equation (2.12), it is said that one can remove $\partial \psi$ from the action of the model. If I understand well, this is relevant for the case $\Lambda = \text{CS}(A)$, to search under which condition the action is finite: it would seem useful to explain this context explicitly, as in my opinion, the goal of the discussion is not stated in the current version, making the understanding of this part difficult.

I will add an explanation into the paper addressing the point above.

(9) Technically, the proof that (2.33) vanishes due to $\xi_{+-} = 0$ on the defect seems to only treat the case of simple poles, since a double pole would require also $\partial_z \xi_{+-} = 0$. It would seem relevant to also explain how the case of double poles with Dirichlet conditions is treated or to mention that one restricts to this simple poles for simplicity.

(19) In (4.49), it is required that $\partial_{\pm} y = 0$ and $\partial_z (\partial_{\pm} y) = 0$ at a double pole to ensure the vanishing of $E(y)$ and $\partial_{z}E(y)$. It seems to me that the first condition is enough, since the various terms in the derivative $\partial_z E(y)$ are always proportional to either $\partial_+ \zeta$ or $\partial_{-} \zeta$.

I agree with point 19 here, whose content addresses point 9. In particular, $\partial_z \xi_{+-}$ will always be proportional to either $\partial_+ \zeta$ or $\partial_{-} \zeta$ meaning the Dirichlet boundary conditions $\partial_{\pm} u = 0$ ensure gauge invariance.

(10) At the end of page 10 (beginning of section 3), it is said that working in the archipelago gauge requires that $\omega F_{\bar{z} \pm} = 0$. It is not clear to me why this is the case and I would thus like to ask the author to clarify this point (is that condition used rather for the next step, \textit{i.e.} extracting a Lax connection from $A$?).

The condition in the above does indeed refer to extracting the Lax connection, and should come after the following sentence.

(11) The discussion in section 3.2 relies extensively on the fact that a gauge transformation $A \mapsto A^u$ acts on $\hat{g}$ by $\hat{g} \mapsto u \hat{g}$. As far I can see, this is not mentioned at this point in the paper, making the discussion potentially difficult to follow.

The referee is correct in the above. I will insert a few sentences in section 3.1 explaining the origin of the gauge transformation law of $\hat{g}$ and make the required changes to section 3.3 as a result.

(12) It seems to me that the proposed form (3.6) of $\tilde{g}$ is in general not smooth at the boundary of $U_q$ and $V_q$ and thus that the interpolating path between $g_q$ and $1$ should be reparametrised using a map that has constant plateau around this boundaries.

The above is true and I will add in the required clarification. It has no impact on the construction of actions as all integrals are determined by the data at the poles of $\omega$.

(16) In (3.12), the right redundancy acts on $\tilde{g}$ by $h$, whereas the notation used before in (3.2) was $k_g$: it would be useful to use uniform notations. Moreover, the discussion here is a bit confusing since it is explained early in this subsection that we will always work with the right-redundancy fixed by $\tilde{g}_{\infty} = 1$: the discussion around (3.12) now seems to relax this assumption, which is then imposed again around (3.15).

I agree that the change in notation is confusing and will fix this. However, it is necessary to include a discussion of the action of the right redundancy on $\mathcal{L}$ since a generic (boundary condition preserving) gauge transformation will not leave $\mathcal{L}$ invariant, as shown in equation (3.17). I will restructure this argument to avoid removing and reimposing the fixing of the right redundancy as highlighted.

(21) Should the last equations in (5.32) and (5.33) be supplemented with their analogues including a derivative with respect to $z$ when $q$ is a double pole? Otherwise, it would seem that these equations are not enough to deduce the flatness of $\mathcal{A}$ and $\mathcal{B}$.

This is true, and now occurs for the revised gauged sigma model action given above.

(23) After (5.53), I would say that $B$ and $C$ should not be seen as valued in the adjoint representations of $\textbf{n}^+$ and $\textbf{n}^-$, since the latter are highly non faithful (for instance, the highest-root vector of $\textbf{n}^+$ vanishes in the adjoint representation of $\textbf{n}^+$). Rather, it seems to me that $A$, $B$ and $C$ should all be seen in the adjoint representation of $\textbf{g}$ (or alternatively as elements of the abstract Lie algebra $\textbf{g}$).

I agree. The correct statement is that $A$, $B$ and $C$ are valued in the adjoint representation of $\textbf{g}$.

(24) In the discussion on top of page 36, it is said that the boundary conditions (5.61,5.62) are preserved by the gauge transformation by $u$ since $u$ is the identity at infinity. It seems to me that this is not true for (5.62) since the latter also involves derivatives with respect to $z$. This is similar to the aspects discussed in the point 1 of the main report and should also be addressed in the context of nilpotent gauged WZW models.

The above point has the same resolution as given to point 1.

(13) It seems to me that one should not include the term $\chi_z dz$ in (3.7) if one has redefined $d$ as in (2.9).

My intent in the above was to write the most general expression, where $d$ includes $\partial_z$. This is clearly a source of confusion, given (2.9), I will therefore remove $\chi_z dz$.

(15) In the paragraph before (3.9), it is said that the paper will now explain why $\mathcal{L}$ satisfies the property 3 of a Lax connection in the case $\Sigma = S^1 \times \mathbb{R}$. I think that this is not what is done here, as the proposed explanation only shows how to extract conserved quantities from the monodromy, following directly from the property 1. It seems to me that the main non-trivial part of the property 3 is the Poisson-commutation of these quantities, which requires further analysis, as done in [24]. Let me also mention that in the property 3, one should more technically consider the expansion in $z$ of the trace of the monodromy matrix, rather than of the matrix itself. In the discussion around equation (3.9), it would also be useful to introduce the notations $\theta$ and $t$ used here. Finally, it seems to me that in equation (3.11), $\mathcal{L}_\theta$ should in fact be $\mathcal{L}_t$.

My intent with this paragraph was not to explain the origin's of integrability, indeed, I do not in fact say I will do this. I only wished to illustrate the origins of the charges from the four-dimensional Chern-Simons prospective and to point to the relevant literature for the Poisson commutativity requirement. I agree that I should say the traces of the monodromies. You are correct that $\mathcal{L}_\theta$ should be $\mathcal{L}_t$.

(17) It is said that (3.20) holds when $\omega$ has a pole at infinity: it seems to me that it more precisely happens when $\omega$ is non-vanishing at infinity.

Yes, this is correct - all I intended to say was that it occurred in the cases I consider, in all of which $\omega$ has a pole at infinity, which clearly implies $\omega$ is non-vanishing at infinity. But I would agree that this is confusing and I will clarify this in the paper.

(25) Several paragraphs in the conclusion are devoted to the discussion of Manin doubles and triples and their use to define other types of boundary conditions. It seems to me that the notations used there can lead to some confusions. In particular, the Lie algebra $\textbf{d}$ used in these paragraphs is never defined. In the context of this conclusion, is it simply equal to the Lie algebra $\textbf{g}$ in which the gauge field $A$ is valued? This case, to the best of my understanding, is the setup considered in [10,12] (the one of [25] being slightly different, building $\textbf{d} = \textbf{g} \oplus \textbf{g}$ from a pair of simple poles). In my opinion, adding some precision in this paragraph on the definition of $\textbf{d}$ and the setup would be a useful improvement. Similarly, the following paragraph on Poisson-Lie T-duality can also lead to some confusion as my understanding is that this notion is mostly related to the exchange of the two isotropic subalgebras in a Manin triple rather than to a change in the reality conditions (although an analytic continuation is sometimes required to treat some of the examples).

I agree with the above and will add more precision.

The following are true, I will make the required changes and clarifications in a revised version. Where necessary I have included responses in brackets after the query.

(3) In various places in the paper, when dealing with a double pole of $\omega$ at $q$, one gets expressions involving the derivative $\partial_z A$ evaluated at $q$. Importantly, this is true only when $q$ is finite: if $q = \infty$, one has to replace $z$ by a local coordinate $z^{-1}$ around infinity, thus yielding derivatives $-z^2 \partial_z A$ rather than $\partial_z A$. This has important consequences when considering gauged Dirichlet boundary conditions imposed at $z = \infty$. (8) To derive equation (2.15), one has to discard the integral of an exterior derivative $d_\Sigma (...)$ along $\Sigma$: it would seem useful to mention it. (14) The paragraph after equation (3.8) refers to (3.20), which is later in the text. (This sentence will be altered to say solution to the first of equations 3.8 will be given shortly). (20) Before (5.20), should "they take the form (5.1)" be "they take the form (5.18)"? (Yes.) (18) Before (3.24), should "third term" be "fourth term"? In (3.26) and in the sentence below, one $\tilde{g} \mathcal{L} \tilde{g}^{-1}$ should be a $\tilde{g} d \tilde{g}^{-1}$. (26) In equation (C.2), shouldn't there be an additional $e^{i \theta_q}$ coming from the measure? This seems necessary to obtain the residue term in equation (C.3).

I agree with the typos found in.

(27) Below are what seem to be typos in equations and mathematical notations:

I will address the following by stating when the equation holds at a defect.

in various places, it is sometimes not clear from the notation or the surrounding text whether some equation holds only at a defect $(z,\bar{z}) = (q,\bar{q})$ or on the whole Riemann sphere, which can make some arguments harder to follow.