2d dualities from 4d

1- New two-dimensional dualities between non-abelian gauge theories with low (0,2) supersymmetry: SU(N) analogues of known U(N) trialities.

2- Derivation of these dualities from 4d N=1 Seiberg duality.

3- Proposals for new dualities between (2,2) quiver theories based on twisted reductions of class S theories.

4- The strategy is sound and well-explained.

1- When discussing dualities between 2d theories with non-compact vacuum moduli space, there is a well-known order of limits issue regarding whether the IR limit is taken at a fixed place in the vacuum moduli space, or going to infinity on it (see https://arxiv.org/abs/1611.02763 by Aharony-Razamat-Seiberg-Willett). This order of issue is not considered in this paper, making it difficult to know the precise meaning of the proposed dualities in the non-compact cases.

2- In section 2 the absence of superpotential seems wrong, especially given that in the explicit theory with $g=n=1$, there is a non-trivial superpotential (2.3).

3- The (2,2) dualities in section 2 only have quite weak evidence. These dualities are simply consequences of abelian gauging in a hypothetical duality between the $g=1$ theory and $n$ copies of the $g=n=1$ theory, and the authors do not explain why they propose the dualities with abelian gauging as dualities, but not the ungauged version of the duality.

4- Throughout the paper, equalities of elliptic genera are checked up to some unspecified order in the $q$-expansion, and for unspecified gauge group ranks, rather than making attempts at obtaining them analytically, or at least specifying how much they have been tested.

5- The method used in section 3, together with more elaborate R-charge assignments, should give a triality symmetry, not just a duality.

6- In section 4 the central charges etc are given without explanations, so that it is not clear how rigorously they are obtained.

The authors approach of twisted reduction of 4d dualities down to 2d is sound, and I find particularly interesting their 2d (0,2) duality derived from Seiberg duality. The paper constitutes good work.

That said, given how selective SciPost physics is, I would suggest seeking another venue for publishing this article, such as JHEP, EPJC, etc.

1- Introduction. 'Through the computation of elliptic genera, we explicitly demonstrate that these theories are independent of the frame or complex structure of the Riemann surface.'
The elliptic genus being an index, it cannot be sensitive to the complex structure, so the calculation does not show independence of the theory itself. It might be possible to argue that deformations of the complex structure are irrelevant hence do not affect the IR limit of the 2d theories, or at least check that the deformations are suitably Q-exact in the 2d theory so that they do not affect observables of interest (beyond just the elliptic genus). Maybe this Q-exactness is established in https://arxiv.org/abs/1703.08201 by Amariti-Cassia-Penati, but I didn't read that paper.

2- Eq (2.3). Equation (2.3) includes a ${\cal N}=(2,2)$ superpotential term $Tr[\phi,\phi^\dagger]^2$, and this contradicts the comment below (2.2) stating that the authors do not include any ${\cal N}=(2,2)$ superpotential. In these kinds of 2d ${\cal N}=(2,2)$ quiver theories I would expect a cubic superpotential coupling bifundamental chiral multiplets that are charged under the same gauge group. But it is not clear whether such a superpotential indeed arises in the twisted dimensional reduction. Cf comment about eq (2.8) below.

3- Eq (2.6). In equation (2.6), the second line is obtained by picking up poles in the integral; this could usefully be mentioned (not necessarily detailed). It is not clear why the ratio of four theta function gets rewritten as a larger product of factors, because it is not clear what is special about the list of scalars $Tr(\phi^i)$, $Tr(\sigma^i)$, $Tr((\phi\sigma)^{i-1})$. What about a more general $Tr(\phi^i\sigma^j)$? Is it eliminated by an F-term condition?

4- Eq (2.8). In analogy to the last expression in (2.6) it seems important to figure out which chiral ring generators contribute which factor in the product. To understand why other (gauge-invariant) polynomials in the fields $\phi_i$ and $\sigma_i$ do not contribute, the F-term relations are probably essential, which forms a good test of what the superpotential could be.

5- Eq (2.11). I disagree with the logic of saying that the equality of elliptic genera suggests the two theories are dual. Following the same logic, (2.8) would suggest that the quiver of Figure 2 is dual to $n$ decoupled theories. Is that the case? I'm especially uncertain about these 2d dualities because the target space metric is important as explained in https://arxiv.org/abs/1611.02763 by Aharony-Razamat-Seiberg-Willett.

6- Eq (3.7). If the sign is an overall sign, write $I_A=-I_B$ I suppose; if it is an undetermined sign, maybe say ``up to an undetermined overall sign''. It would be good to specify to which order in $q$ the identity has been checked. Same comment for (2.8), and which values of $N$ and $n$ as well. Same comment for footnote 1.

7- Section 3. Triality usually involves theories with three flavour symmetry groups $SU(N_i)$, corresponding to Fermi, fundamental chiral, and antifundamental chiral multiplets (plus determinant matter, but that is irrelevant in the $SU$ case). The gauge group is $U((N_1+N_2+N_3)/2 - N_a)$ in the $a$-th theory of the triality, see https://arxiv.org/abs/1310.0818 (by Gadde-Gukov-Putrov) figure 2. The natural guess for an $SU$ version is to throw away the determinant matter but keep the three kinds of matter fields. Here the authors only consider two kinds of matter, which amounts to taking $N_3=0$ in the triality. Then it is clear that one of the three gauge group ranks becomes negative, which is enough to explain why there is no third theory in the triality. An obvious question then is: is there an $SU$ triality with all three kinds of matter? The obvious attempt is to change (3.8) to a more general charge assignment $r=(0,...,0,1,...,1,2,...,2)$ for $Q$ and $\tilde{Q}$, but maybe anomaly cancellation forbids this?

8- Equation (4.5) is not consistent with the fact that $X$ is antisymmetric and $Y$ fundamental. Presumably the twisted compactification should give anti-fundamental (or anti-antisymmetric, by which I mean $\Lambda^2\overline{\Box}$) chiral multiplets so that $YXY$ can be gauge-invariant. This problem can also be seen when trying to show the identity (4.4) as a result of picking up certain residues: as written, the $1/\vartheta_1(a_kb_l)$ factor should have poles at $a_k=1/b_l$ so that $1/\vartheta_1(a_ia_j)$ would become $1/\vartheta_1(1/(b_ib_j))$, but if we replace fundamentals by antifundamentals, then the contribution of $Y$ becomes $1/\vartheta_1(b_l/a_k)$ so that poles are at $a_k=b_l$ which correctly leads to $1/\vartheta_1(b_ib_j)$ factors. Also, (4.5) is antisymmetric in $i,j$ so I think there are only $(N+1)(N+2)/2$ chirals, not twice that number. The same counting issue shows up for the next model, where there should be $(N-2)(N-1)/2$ instead of $(N-2)(N-1)$.

9- Below (4.8), it is proposed that the $\vartheta_1$ factor in the numerator is due to a dynamically-generated Fermi multiplet. This requires more justification. My guess would be that it is given by $(\Lambda^{n-1}X)_{\alpha\beta}(\Lambda^{n-2}Y)^{\beta\gamma}\Upsilon^\alpha{}_\gamma$ where $\Lambda$ denotes antisymmetrization, so that $\Lambda^{n-1}X$ lives in the representation with Young diagram consisting of two columns of $n-1$ boxes each, while $\Lambda^{n-2}Y=\bigwedge_{i=1}^{n-2}Y^i$ lives in the conjugate of the antisymmetric representation, and where $\Upsilon$ is the gauge field strength Fermi multiplet, in the adjoint representation of $SU(N)$. The guess could be checked by tracking down where the numerator $\vartheta_1$ comes from in a vector multiplet one-loop determinant, and by checking which JK residue does give the final numerator $\vartheta_1(x^{N-1}\prod_{i=1}^{N-2}b_i)$.

10- From the dual LG point of view, the fact that fugacities $b_ib_j$ appear, and not more general fugacities $c_{ij}$, deserves explanations. Usually, such fine-tuning of meson/hadron flavour charges comes from a superpotential coupling, so I would expect a superpotential coupling like (4.11), (4.14), (4.17) also in the models discussed around (4.5) and (4.8). I don't immediately guess the correct superpotential for these two models. Relatedly, (4.17) does not fix all fugacities, namely does not break enough symmetries, so that the LG model has too much flavour symmetry: indeed, the superpotential does not involve the $N-2$ fields $\Phi_{ii}$, which are thus free. Maybe the explanation in all these cases is that some of the chiral multiplets become free in the IR?

11- Appendix A. Concerning c-extremization, I think (without proof) that in the case of a non-compact vacuum moduli space, the chiral fields spanning non-compact directions must have zero R-charge, but some other directions could be compact, in which case we can probably perform c-extremization on them. Correspondingly, the central charge is a sum of the non-compact dimensions and a compact contribution obtained by extremization. This might be discussed in the original c-extremization papers.

12- Misc comments. It would be interesting to consider also quiver tails that arise with certain patterns of puncture data in class S theories. It might make sense to cite https://arxiv.org/abs/1609.07144 by Franco-Lee-Seong, but I am not sure: it seems to exhibit trialities of 2d (0,2) quiver theories as well, but only Abelian ones I think.

Ask for major revision

The manuscript investigates new dualities for two-dimensional supersymmetric gauge theories, with either $\mathcal{N}=(2,2)$ or $\mathcal{N}=(0,2)$ supersymmetry. This is achieved by studying the compactification of mainly 4d N=2 SCFTs, but also the 4d N=1 Seiberg duality, on a two-sphere with a suitable topological twist. The results of this manuscript enrich our current understanding of the dynamics of 2d SQFTs and their connection with four-dimensional physics. Notably, the perspective of the 4d to 2d compactification allowed the authors to discover a new family of 2d N=(2,2) theories labelled by Riemann surfaces with punctures and to show that these enjoy non-trivial infrared dualities, which can be nicely understood from the 4d perspective as different degeneration limits of the two-dimensional surface. Although the results of this manuscript are already interesting on their own, the topic demands for further study and I encourage the authors to pursue this line of research. For these reasons, I recommend this paper for publication after the next more specific comments are addressed.

- At the beginning of section 2 the authors review the field content of the 2d theories obtained by compactifying a 4d N=1 theory on $S^2$ with a twist by a specific choice of R-symmetry. However, there is a subtlety discussed in Ref. [26] that the authors did not mention. In general the result of a 4d gauge theory will lead to a direct sum of distinct 2d theories labelled by the value of the gauge flux through the $S^2$. However, as discussed in [26], one can obtain a single 2d theory if the R-symmetry chosen for the twist is such that all chiral fields have non-negative R-charge, and not just integral. Since this is a crucial assumption for the construction used in the present manuscript, I think the authors should state it.

- I think the $U(1)_c$ symmetry of the minimal puncture of the 4d trinion theory considered on page 3 corresponds to the $U(1)_f$ involved in the topological twist from the general discussion on the previous page. If that is the case, I would suggest the authors to state this more clearly.

- I would also suggest to add a reminder of how the 2d N=(2,2) left and right moving R-symmetries (or equivalently the vector and axial R-symmetries) are embedded inside the 4d R-symmetry and how the fields transform under them. This would make more clear the expressions for the 2d central charges and the elliptic genus. In particular, it is not clear in (2.1) and (2.2) which of these symmetries do the fugacities q and y correspond to.

- I would suggest the authors to add a reference to the Appendix A when calculating the central charges, especially in the first few examples considered in (2.5) and (2.7). Moreover, some of the theories considered in the manuscript possess abelian flavor symmetries (the $U(1)_{c_j}$) that in principle might mix with the R-symmetry in the IR, so I would add a brief explanation as to why this does not happen. In the case of the torus with one puncture for example this is due to the extended supersymmetry. What about the case of a torus with multiple punctures?

- Above (2.3) I would specify that the puncture of this model is minimal. Moreover, in the theory (B) of the duality stated at the beginning of Section 3 I would specify that M is a chiral field.

- I would suggest to expand more on why in the computation of the central charges around (3.2)-(3.3)-(3.4) one should set the R-charges of the Fermi fields to 1 and those of the chirals to 0. The authors correctly say that the issue is the non-compactness of the target space, but I would expand a bit more on this since it might not be so immediate to most readers. The point is that the R-symmetry cannot mix with the flavor symmetries associated to the non-compact directions. This is done by identifying which chiral operators parametrize such directions and setting their R-charges to zero. Since such operators are constructed from all the chirals in the two theories, we then need to set their R-charges to zero. The R-charge of the Fermi field $\Psi$ is consequently set to one by requiring that the superpotential has R-charge 1.

- An additional check of the duality of Section 3 is the matching the anomalies for possible finite symmetries in the theory. In fact, there is a symmetry acting with charge 1 on all the chirals of theory (A) which is broken by anomalies to a finite group. It should be possible to identify such symmetry also in theory (B) and to match the anomalies for it. See Ref. [21] for a similar discussion.

- Below (3.7) the authors state that the agreement of the elliptic genera has been checked through a power expansion in $q$. It would be good to specify for which values of the parameters $N_+$, $N_-$, $N_1$, $N_2$ this has been done and also to which order in $q$. Similarly for similar checks that have been done in the rest of the manuscript. Moreover, what does it mean that they have been matched “up to sign”? Do the authors have an understanding of this sign?

- Is the statement below (3.11) that the $SU(N_c)$ theories do not enjoy the triality completely correct? Indeed, in Ref. [26] it has been shown that the duality corresponding to $N_2\leftrightarrow N_3$ is still valid and that it can be obtained from the reduction of the Seiberg duality. The point is that in the $SU(N_c)$ case we only get a duality and not a triality.

- In Section 4, the authors identify many $N=(0,2)$ GLSM’s that are IR dual to LG models. The latter consist of chiral fields with some superpotential interaction, however the authors do not identify the superpotentials for all of the proposed dualities, but only for some. I think the authors should add the missing ones.

- I suspect some of the dualities in Section 4 can be obtained from $S^2$ reduction of the 4d $N=1$ S-confining dualities classified in arXiv:hep-th/9612207. For example, the duality for $Sp(N)+(2N+2)F$ was obtained in Ref. [26] as the dimensional reduction of a duality due to Intriligator and Pouliot arXiv:hep-th/9505006. Similarly, I suspect that the duality for $SO(N)+(N-2)F$ can be obtained from dimensional reduction of the Intriligator—Seiberg duality arXiv:hep-th/9503179, while those for the theories with antisymmetric matter as reduction of the S-confining dualities that can be found in the paper mentioned above. It would be really interesting if the authors could recover the dualities of this section in this way, however I do not consider it as necessary for publication.

- At the end of page 17 the authors state that when the target space is non-compact, then central charge can be computed as three times the complex dimension of the moduli space. It would be good if the authors could expand more on this point, specifying under which assumptions this is true. A reference for the stated fact would also be good.

I have also found a couple of typos:

- Above (2.2): “form” -> “from”.

- At the end of second to last paragraph on page 11: “…obtained from of the Lagrangian…”, remove “of”.

Ask for minor revision