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Three Dimensional Topological Field Theories and Nahm Sum Formulas

by Dongmin Gang, Heeyeon Kim, Byoungyoon Park, Spencer Stubbs

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Authors (as registered SciPost users): Heeyeon Kim
Submission information
Preprint Link: https://arxiv.org/abs/2411.06081v2  (pdf)
Date submitted: Sept. 4, 2025, 4:15 a.m.
Submitted by: Heeyeon Kim
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
Specialties:
  • High-Energy Physics - Theory
Approach: Theoretical

Abstract

It is known that a large class of characters of 2d conformal field theories (CFTs) can be written in the form of a Nahm sum. In \cite{Zagier:2007knq}, D. Zagier identified a list of Nahm sum expressions that are modular functions under a congruence subgroup of $SL(2,\mathbb{Z})$ and can be thought of as candidates for characters of rational CFTs. Motivated by the observation that the same formulas appear as the half-indices of certain 3d $\mathcal{N}=2$ supersymmetric gauge theories, we perform a general search over low-rank 3d $\mathcal{N}=2$ abelian Chern-Simons matter theories which either flow to unitary TFTs or $\mathcal{N}=4$ rank-zero SCFTs in the infrared. These are exceptional classes of 3d theories, which are expected to support rational and $C_2$-cofinite chiral algebras on their boundary. We compare and contrast our results with Zagier's and comment on a possible generalization of Nahm's conjecture.

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  • Provide a novel and synergetic link between different research areas.
  • Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
  • Detail a groundbreaking theoretical/experimental/computational discovery
  • Present a breakthrough on a previously-identified and long-standing research stumbling block

List of changes

  1. pg2. two more references [17,18] are added in footnote 2.

  2. pg2. text below (1.5), "A is identified with..." -> "the $r \times r$ matrix A is identified with..."

  3. pg 3. A new equation (3.4), highlighting the relation between K and the UV effective CS level.

  4. pg 8. A new paragraph called "Topological twisting" added, which explains the relation between the A/B twisting and the R-symmetry embedding more explicitly.

  5. pg9. New equations (3.17) and (3.18) and a new paragraph around them, emphasizing the relation between the specialization $\nu=\pm 1$ and the Hilbert series.

  6. pg10. A new paragraph below equation (3.22), again emphasizing the relation between the partition function of TFTs and the specialization of the indices.

  7. pg11. New paragraphs added at the beginning of section 3.4, which review our choice of boundary conditions.

  8. pg12. All of the text on this page is new, providing a detailed explanation on the specialization $\nu=\pm 1$ at the level of the half-index, including the last paragraph on the compatibility of the boundary condition B with the twisted supercharge, highlighting the open issue in our paper (A brief version of this discussion was originally in a footnote 10 of v1, which we removed in v2.)

  9. pg 14. A new footnote 12, which compares the notations with other references.

Current status:
Has been resubmitted

Reports on this Submission

Report #1 by Anonymous (Referee 2) on 2025-9-13 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2411.06081v2, delivered 2025-09-13, doi: 10.21468/SciPost.Report.11927

Report

1) I am still confused with the following description after eq.(1.5): In this context, the $r\times r$ matrix $A$ is identified with the mixed Chern-Simons level matrix $K$.'' The matrices $A$ are determined by the boundary conditions. They are intrinsically 2d data as the half-indices can enumerate the local operators at the 2d boundary. Although $A$ might be calledeffective levels'', they are not the Chern-Simons levels since the usual Chern-Simons term is defined on 3d.

2,3) There are several unclear points in the discussion on the F-maximization. In the Gang-Yamazaki paper [59], the superconformal $U(1)$ R-charge of the scalar field is fixed as $1/3$ by the F-maximization. On the other hand, in Table 2, it is stated that the superconformal R-charge of the free chiral theory $\mathcal{T}_{\Delta}$ corresponds to $R_*+\frac12 F$ with $R_*$ $=$ $0$ and $F=1$. Why do you get different R-charges? In fact, the expansions of the superconformal indices in the main text do not seem to be computed with $R_*$ $=$ $0$. For example, for the $U(1)_{3/2}+\Phi$, the expansion in section 4.1.1 begin with $1-q-(\eta+1/\eta)q^{3/2}-2q^2-...$, which agrees with the expansions found in [59] with the superconformal $U(1)$ R-charge of $\Phi$ being $1/3$.

Another concern is about the axial symmetries. In general for each $\mathcal{N}=2$ charged chiral multiplet there will be the axial symmetry factor. If the authors consider the $U(1)^r$ gauge theory with $r$ charged chirals, there will be $U(1)^r$ axial charges. There may be mixing between them with the R-symmetry. How are they dealt with in the F-maximization? The axial symmetry is suddenly discussed in the paragraph \textbf{Toplogical twisting} on page 8 and denoted by $A$. What is the relation between the $\mathcal{N}=2$ axial symmetry factors and $A$?

Besides, I am not sure whether the superconformal R-charge in the presence of the boundary is simply determined via the F-maximization of the 3d bulk theory as discussed. The system is defined on the 2d boundary that may involve different $U(1)$ global symmetries so the determination of the superconformal R-charge will depend on the choice of boundary conditions in general. In the boundary conditions the authors are considering, the deformed Dirichlet boundary condition $D_c$ can be obtained by choosing the Dirichlet boundary condition $D$ first and then introducing the boundary term involving the 2d Fermi multiplet so that the chiral multiplet scalars can take the non-zero constant values. Then one also needs to consider a mixing of the boundary global $U(1)$ symmetries as the broken gauge group with the R-symmetry at the first step. But such mixing depending on the boundary condition is not discussed in the text.

4) I cannot follow the logic. But let us assume that the bulk 3d theories have the enhanced $\mathcal{N}=4$ supersymmetry in the IR. Then if I understand correctly, the R-charge of the scalar field will be fixed as $1/3$ as discussed in the Gang-Yamazaki paper [59]. But the authors seem to choose different R-charges as mentioned above. In fact, the Dirichlet-half-index in eq.(3.35) is computed with R-charge $0$.

5) The CS level in the $U(1)_{-1/2}$ quantization can be used for the bulk theory itself if you want as in the paper [62]. But it looks an abuse of notation in the presence of the boundary. What do you mean by the bare CS levels in the $U(1)_{-1/2}$ quantization in the presence of the boundary? Related to the point 1), the Chern-Simons term is defined on 3d. If the authors would like to call $K$ the CS levels, they should be defined in the 3d bulk which should be intrinsically independent of the UV boundary conditions. If $K$ are defined as the 3d data, then they should not be identified with the matrices $A$.

6) If the identification of the Dirichlet half index of the $U(1)_{3/2}+\Phi$ was already made in [19], it should be better to cite it to make it clear that the identification is not a new result. But neither citations nor comments are added in the modified draft.

There are further comments.

7) On page 10: While the Wilson line is discussed in section 3.3, what symmetry is associated with the charges $\vec{Q}$? When the deformed Dirichlet boundary condition $D_c$ is chosen, the boundary global $U(1)^r$ symmetries corresponding to the gauge group will be broken.

8) On page 11: ...boundary torus. [53-56]'' will be...boundary torus [53-56].''

9) There are many expansions of the superconformal indices in section 4. I could not get the same answers with $R=0$ as discussed in section 3. Also it is briefly stated that some of the indices are the same as some other indices. But are they only checked by looking at the first few terms in the expansions or are they proved analytically?

10) On footnote 18; ``which is arises'' is a typo.

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Author:  Heeyeon Kim  on 2025-09-13  [id 5810]

(in reply to Report 1 on 2025-09-13)

I would like to thank the referee for the comments.

Regarding comments 1) and 5). As stated in eq (3.4), the UV CS level for the bulk theory is $K-\frac12 I$. The proposal of [56], which we review on page 11-12, is that the matrix $A$ encoding the 2d boundary anomaly is in fact identical to $K$. In the convention of section 3.4.1 of [56], their UV CS level is denoted by (lowercase) $k$ and one finds that

$$ A = \text{(anomaly from the CS term)} + \text{(anomaly from the chiral multiplet with Dirichlet)} = k+1/2 = (K-1/2)+1/2 = K. $$
It is unclear to us whether the referee disagrees with this statement itself, or the comparison of our notation to that of [56], and we would appreciate clarification. 

We fully acknowledge that $K$ and $A$ are a priori distinct physical quantities (the former defining the bulk 3d CS level and the latter the boundary anomaly) but the proposal of [56] is precisely that they coincide under the Dirichlet boundary condition. This proposal has been verified in numerous examples, and we adopt it here as our starting point.

Regarding comments 2), 3) and 4). The statements in the references [32, 59] that the superconformal R-charge for the chiral multiplet is 1/3 can be misleading. The superconformal R-charge of a non-gauge-invariant operator cannot be uniquely determined by F-maximization, since it may always be redefined by further mixing with the gauge symmetry. Thus the different R-charge assignment ($R(\Phi) = 0$ in our case versus $R(\Phi) = 1/3$ in [32, 59]) arises simply from a different choice of mixing the R-symmetry with the gauge U(1) symmetry. Concretely,

$$ R_{GY} = R_{our} + (1/3) G,    $$
where $G$ is the U(1) gauge charge (and the subscript GY denotes the choice in [32,59]). Since the superconformal index (SCI) receives contributions only from gauge-invariant operators, this difference is irrelevant, and one is free to choose any such mixing in the SCI computation.

To be more explicit, let us consider the theory $U(1)_{k = -3/2} + \Phi$, which was studied in [32,59]. In equations (3.7) and (3.8) of our paper, we have

$$ R_{our}(V_m) = (1/2)(m + |m|) + \mu m,  $$
where $V_m$ is the bare monopole operator with monopole flux $m$, and $\mu$ is the mixing parameter of the R-symmetry with the $U(1)_{top}$ symmetry. The value of $\mu$ can be fixed by F-maximization, as explained in equation (3.20). One finds that $\mu_0=0$. (In our manuscript, we instead considered the theory $U(1)_{k = 3/2} + Φ$, for which we obtained $\mu_0 = -1$. Performing the same computation for $U(1)_{k = -3/2}$ yields $\mu_0 = 0$)

Thus, the superconformal R-charge is 

$$ R_{our}(V_m) = (1/2)(m + |m|).  $$
In [GY], they instead find 
$$ R_{GY}(V_m) = (1/3)|m|. $$
Note that these results are related by $R_{GY}(V_m) = R_{our}(V_m) + (1/3) G(V_m)$ as anticipated above, where $G(V_m) = -(3/2) m - (1/2)|m|$, as given in the first line of equation (3.6).

We emphasize again that the R-symmetry of gauge-variant operators (such as the chiral fields in our theories) cannot be uniquely determined by F-maximization, since the R-symmetry can be redefined by mixing with the gauge symmetry. F-maximization is insensitive to such an auxiliary mixing parameter, because we are integrating over the gauge fields in the path integral. At the level of the localization computation on $S^3$, this mixing amounts merely to shifting the dummy integration variable $Z \rightarrow Z + \text{const}$.

When we consider the half-index with the $D_c$ boundary condition on chiral multiplets, the gauge group is broken at the boundary, and we are further required to impose $R(\Phi) = 0$ in order to preserve the $U(1)_R$ symmetry. As explained in [56] (see the paragraph above section 2.1.1 in loc.cit.), giving a charged chiral $D_c$ boundary condition with $R(\Phi) = 0$ is part of the UV definition of the superconformal boundary conditions.

Finally, regarding the referee’s comment “If the authors consider the $U(1)^r$ gauge theory with r charged chirals, there will be $U(1)^r$ axial charges”: We do not understand what the referee means by the “axial charges” in this sentence. The theory of $r$ free chiral multiplets has $U(1)^r$ flavor symmetries, but they are all gauged. Our theory has $U(1)^r$ topological symmetries instead, but they are all explicitly broken except for one residual $U(1)$ by the super-potential deformations involving $r-1$ monopole operators. We call this remaining $U(1)$ symmetry the axial symmetry $A$. We perform the F-maximization with respect to this $A$. We emphasize this point clearly around equation (3.10).

7) There is no extra symmetry associated with $\vec Q$. Even though the deformed Dirichlet boundary condition $D_c$ breaks the boundary $U(1)^r$ symmetry, a Wilson line $W = \exp (i\int A)$ in the bulk still exists and can terminate at the boundary, defining a module of boundary VOA. The fact that the boundary condition does not preserve any global symmetry simply implies that the endpoint of this Wilson line does not transform under any boundary global symmetry.

9) We say clearly in the first paragraph of section 4 that the identification of the indices with known characters are conjectures based on the q-series expansion, except for a few cases with known Nahm sum representations.

We are happy to correct the typos mentioned in 8) and 10) and add another reference to our own work in footnote 12, as referee mentioned in 6).

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