2d dualities from 4d

The authors have significantly revised the manuscript so to address the comments from my previous report and the one of the other referee, and also to include numerous new results. Most changes look good to me, however since the manuscript has changed substantially I have some more comments.

- I am also confused (like the other referee) by the superpotentials that the authors have added (upon request from the previous reports) to the 2d (0,2) dualities of section 3.

1) Compared to the first version of the manuscript, the authors have expanded the duality between $SU(2N)$ with one antisymmetric $X$ and $2N+2$ fundamental chirals $Y$ and the LG model of one $U(2N+2)$ antisymmetric chiral $A$ to a triality, by exploiting the fact that the latter theory is known to be dual also to the $Sp(2N)$ gauge theory with $2N+2$ chirals. However, for this known duality to hold one also have to add a Fermi field $\Psi$ to the LG theory which flips $\mathrm{Pf}\,A$, so such a deformation should also be turned on in the $SU(2N)$ gauge theory. In the first version, the authors identified $A$ with $YXY$, so shouldn't the superpotential (3.12) be of the form $\Psi\, \mathrm{Pf}(YXY)$?

2) In the SO duality on pages 14-15 there is now an additional Fermi singlet $\Psi$ appearing in the superpotential (3.22) compared to the first version of the manuscript. Which of the two versions is the correct one? I share the concern of the other referee on the superpotential (3.22), but if there was not such a Fermi field then there would be no need for a superpotential.

- The logic in section 3.3 is a bit confusing and hard to follow. What is the purpose of going through the complicated dualizations of figure 6 if the goal is just to explain that the SU theory Higgses to the Sp theory when $x\to 1$ (I would write that $x\to 1$ is achieved by giving a VEV to X)? These dualizations would be useful to get to the Sp theory without studying explicitly the Higgsing, if somehow the duality in figure 7 was already known by some other means, but this doesn't seem to be the case. Also the last paragraph of this section 3.3 is quite cryptic. I would suggest to the authors to improve the presentation of this section and clarify its logic and purpose.

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1- Many new two-dimensional dualities/trialities between non-abelian gauge theories with low (0,2) supersymmetry and various gauge groups including Sp, SO, SU, and Landau-Ginzburg models.

2- Derivation of these dualities from 4d N=1 and 4d N=2 dualities. Dimensional reduction is a robust way to anchor the dualities in the vast network of checks of higher-dimensional dualities.

3- Proposals for new dualities between (2,2) quiver theories.

1- Superpotentials in (3.12), (3.22) seem to be incorrectly determined as certain Pfaffians or determinants simply vanish. See specific comments in the 'Changes' section of this report. If the hope was for these superpotentials to force various fugacities to be equal, then this hope breaks down. However, for some of these theories I don't see why the superpotentials would be needed in the first place.

2- Relatedly, the mechanism responsible for eliminating some fugacities, such as in (3.11), is not understood since the superpotential does not do this. There is an attempt around (3.33) for one such fine-tuning of fugacities (x→1) but it is not clear what triggers the Higgsing. Concretely, this means that symmetries of the putative dual theories are not shown to be the same in these cases. In principle this could be seen at the appropriate level in the index.

3- Since the analysis of dualities is centered around elliptic genera, which do not depend on moduli, it is impossible to specify the dictionary between 4d parameters (masses, gauge couplings) and 2d parameters (masses, R-charges, FI parameters, theta angles).

4- The paper could be clearer somewhere about the list of 4d dualities that are being dimensionally reduced. The current presentation makes it appear quite ad hoc.

The authors approach of twisted reduction of 4d dualities down to 2d is sound, and I find particularly interesting their 2d (0,2) dualities and trialities derived from Seiberg duality. The paper advances significantly the state of knowledge of 2d dualities with low supersymmetry and constitutes good work.

1- Intro: 'of the duality frame' would be clearer than 'of the frame'

2- Section 2: regarding 'if these charges are non-negative integers', the situation (see the end of section 2 in ref [9]) seems to be that if *all* charges are non-negative then the 4d theory reduces cleanly to a single 2d theory. Under this condition it is confusing to talk about getting (1-r) chiral multiplets for r less than 1, since the only non-negative integer less than 1 is simply zero.

3- 'it becomes 2d' → 'it becomes a 2d'

4- Equation (2.2) has a stray word 'equivalent'

5- It's hard to understand when reading the text that between (2.8) and Figure 3 the authors turn to a new 2d theory. Maybe a more liberal use of the paragraph LaTeX commands would help clarify, or simply a clearer sentence at the start of the list of theories stating how the section 2 is organized?

6- In (2.11) and nearby, $U_2$ should be $U_N$? In (2.11) there are two $,V$ superscripts missing.

7- The identity (2.12) is an immediate consequence of gauging $U(1)$ in the identity (2.8) for $n=2$. There is probably a TQFT interpretation of all this.

8- The authors do not explain the condition (3.3) ($N_1\geq N_2+N_3$) imposed for the 2d (0,2) SU duality. A wrong explanation is given below (3.7): the truth is that the Grassmannians are defined whenever $N_1\geq|N_2-N_3|$, weaker than (3.3). I didn't locate in the literature an analysis of the IR behaviour of 2d (0,2) SU(n) SQCD theories, for instance ref [20] seems to focus on U(n), the authors may have better luck. Maybe, contrarily to U(n) theories which can flow to two different NLSMs depending on FI, the SU(n) theories can only flow to a single NLSM, at least in the regime $N_1\geq N_2+N_3$. Beyond this bound it may be that we get several NLSMs. For instance, in the left side of Figure SU-dual, going from the large~$N_1$ regime to the $N_2\geq N_1+N_3$ regime, the NLSM should go from giving vevs to the $N_1$ fundamental chirals to giving vevs to the $N_2$ antifundamental chirals. This whole discussion is probably beyond the scope of this work, unless some reference already does it.

9- When citing [21,22] write the words 'c-extremization' to orient the reader.

10- As far as I can tell, the Pfaffian of $A$ in (3.12) vanishes, because $A$ has rank at most 2. Indeed, in $\mathbb{C}^{2N+2}$, if the $2N$ vectors $Y_\alpha=(Y_\alpha^i)_{i=1,\dots,2N+2}$ for $\alpha=1,\dots 2N$ are not linearly independent, $A$ vanishes altogether, and otherwise we check that $A_{ij}Y_\alpha^i=0$ for all $\alpha$, hence $A$ is orthogonal to a $2N$-dimensional subspace of $\mathbb{C}^{2N+2}$. Maybe the correct superpotential mixes $X$ and $A$? Same comment for (C.7)--(C.8).

11- In (3.23) $A$ vanishes I think. The combination $\epsilon^{\alpha_1\dots\alpha_N} Y_{\alpha_1}^{j_1}\dots Y_{\alpha_N}^{j_N}$ vanishes because it is antisymmetrizing $N$ vectors $Y_\alpha\in\mathbb{R}^{N-2}$. In contrast, (3.25) is fine because $A$ is a projection of an $N\times N$ symmetric matrix~$Z$ to an $(N-2)\times(N-2)$ symmetric matrix, which can very well have full rank. But (C.18)--(C.19) seems to have a problem since ${\rm rank}(A)\leq{\rm rank}(X)\leq n-2$ so ${\rm Pf}(A)=0$. Same problem for (C.23)--(C.25).

12- I don't understand what the two paragraphs after (3.57) are doing there in particular. It seems we are in the middle of studying reductions of 4d ${\cal N}=4$, and suddenly we switch back to some comments about reductions of 4d ${\cal N}=2$ theories.

13- Comments around pages 27--30 about fields that contribute to the central charge but not to the elliptic genus seem very surprising. I think that the elliptic genus is a Jacobi form of weight zero and index related to the central charges, but maybe it is a combination of central charges that is not affected by these extra fields?

14- Below (B.2) it is stated that for a non-compact vacuum moduli space, the right-moving central charge is determined by the dimension, but this is wrong: in the product of a non-compact and compact theories, the central charge should simply add up, so c-extremization has to be done in the compact part of the theory. My understanding is that the non-compact chiral multiplets are forced to have R-charge zero, and then within the class of R-symmetries that respect this, one should do c-extremization, but I am unsure.

15- Just before section C.2 the reference to Fig Sym-dual should be AS-dual maybe?

16- In various places (in appendix C) 'one fundamental chirals' is incorrectly plural.

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