The 3d $A$-model and generalised symmetries, Part I: bosonic Chern-Simons theories

The paper extends the application of the 3d $A$-model to a broader class of gauge groups, specifically those obtained by gauging discrete one-form symmetries. The authors provide a meticulous comparison between the supersymmetric and TQFT approaches, identifying and accounting for subtle counterterms.

There are some statements that should be clarified.

This paper makes a significant contribution to the understanding of the 3d $A$-model and its relation to bosonic Chern-Simons theories. The extension to more general gauge groups that are not simply connected and the detailed analysis of the correspondence between different approaches are commendable. With minor revisions, the paper would be a valuable addition to the literature on topological defects in supersymmetric gauge theories.

1) Questions about the presentation (these are almost trivial. I point them out because the paper, especially section 2, has an intent of acting as a reference source) - In (2.14) the notation for the Bethe equations is $\Pi_a(\hat{u}, \nu) =1$, but it's different from that used in (2.12), and the subscript $a$ has not been defined earlier. Could the authors please clarify? - p. 11 +1 "Bethe vacua of the $\tilde{G}_k$ (?) theory are (?) the solutions" - is the color-coding used in figure (2.31) consistent with that of, e.g., (2.58)? Should the circle in (2.31) intended to be blue? - p. 17 - 6 $\tilde{G}_k$ (?) - p. 18 - 10. Hasn't the modular group already been "alluded to" also in section 2.2? - p. 21 -7 "The class of (?) geometries" - p. 36 footnote 34: could the authors add a couple of words around the equation $\mathcal{N}{\mu\nu\delta} = C$ to explain why the $\delta$ index disappeared?

2) Clarifications - What is the status of (3.91)? It is said to be equivalent to (3.90), which is one of the comparison equations between supersymmetric and TQFT approach. Has it been proved? Could the authors please clarify what they mean by "verified numerically in a number of non-trivial examples"? For instance, something along the lines of footnote 50 (p. 55). How relevant is it to the comparison of (3.92) with the TQFT answer? - p. 52 below (3.99): could the authors please clarify what they mean? Is this a check or a rewriting? What are they matching to?

Publish (meets expectations and criteria for this Journal)

1. The paper presents a complete and well-motivated formalism for implementing 1-form symmetry gauging in 3D \mathcal{N}=2 (bosonic) Chern–Simons theories via the A-model.

2. It carefully matches SUSY partition functions with TQFT computations, including explicit modular data and Seifert fibering operators.

3. The exposition is technically thorough and precise, providing a solid foundation for future work on generalised symmetries in the context of SUSY QFTs.

1. The conceptual ideas behind 1-form symmetry gauging are not new, and the main novelty lies in the detailed realization within a supersymmetric setting.

2. The restriction to bosonic CS theories with simple simply-connected group without matter limits the richness of the symmetry structure, although this is acknowledged as a first step.

3. Some derivations, especially in the later sections, could benefit from more pedagogical examples to guide readers through the formalism.

This is a carefully constructed and technically detailed paper that provides a systematic study of 1-form symmetry gauging in 3d \mathcal{N}=2 supersymmetric (pure) Chern–Simons theories, using the formalism of the 3d A-model.

While the general idea of gauging one-form symmetries in 3d is well-established, the present work stands out by formulating and executing this procedure in a supersymmetric context. It fills a gap in the literature and sets the stage for further developments involving matter fields and spin TQFTs.

The paper is clearly written, self-contained, and of high technical quality. I believe it meets the editorial criteria of SciPost and recommend publication after a few minor revisions to improve accessibility and clarity.

(Optional) While clearly beyond the scope of the current paper, it might be interesting in future work to clarify whether the gauging procedure considered here leads to codimension-1 defects associated with generalized (possibly non-invertible) 0-form symmetries, and how such structures manifest in the 3d A-model framework. Also, It may be interesting in future work to comment on potential mathematical connections to quantum K-theory.

1. Clarify the off-shell versus on-shell distinction for the Seifert fibering operators in Sections 2.3–2.4. A brief comment on their significance beyond the matching with TQFT would be helpful.

2. Add more concrete examples in Section 3.2.5 to illustrate the gauging procedure in familiar low-rank cases (e.g., SU(2), SU(3)). This could be helpful for the reader not familiar with the context.

3. Further comments on the counterterm (2.101) would be appreciated. Specifically, can the cancellation of the framing anomaly be understood in terms of an anomaly polynomial or a supersymmetric completion of a gravitational Chern–Simons term involving the background supergravity fields?

4. (Optional) A brief remark in the conclusion about future directions—such as matter fields or spin TQFTs—would help contextualize this work as part of a broader program.

Ask for minor revision