Three Dimensional Topological Field Theories and Nahm Sum Formulas

We would like to thank the referee for carefully reading our manuscript. Here’s the reply to the comments in the order in which they appear in the report. All the page/citation/equation numbers below refer to those in the first version of the manuscript.

1) The matrix A depends on r. More precisely, r is chosen to be the rank of A, as stated explicitly below eq.(1.3). With the choice described in eq. (1.5), and together with the supersymmetric Dirichlet boundary condition (see point 2 and 3 below for its precise definition), the matrix A encodes the boundary ’t Hooft anomaly, which is identified with the matrix K appearing later in the manuscript. In the second version of the manuscript, we emphasized this once again in the text below equation (1.5).

2,3) The boundary condition adopted throughout the paper is the supersymmetric Dirichlet boundary condition that preserves 2d N=(0,2) supersymmetry, as discussed in the paper [54]. This uniquely determines the boundary conditions for all elementary fields in the theory (up to the boundary value of the scalar field which does not affect the index calculation). Additional data, in particular the combination ($\mu_0-a$), is further fixed by the F-maximization and by the choice of topological twist, as discussed in page 7. We acknowledge that this point was not stated clearly in the main text. In the revised version of the manuscript, we have included a more self-contained review of the boundary conditions.

4) The logic behind this specialization is as follows.

The bulk abelian gauge theory is expected to flow to a super-conformal fixed point with enhanced supersymmetry (a “rank-zero theory”), a conjecture motivated by F-maximization. At the super-conformal point, the superconformal R-charge is identified with $R_{\mu_0}$ in eq. (3.12).

Since the IR theory enjoys N=4 supersymmetry, it admits two topological twists, that we denote by $\nu=1$ and $\nu=-1$. Eq (3.12) justifies this definition: For $\nu=1$ (resp. -1), the R-symmetry used to define the twisted spin is given by $2J^C$ (respt. $2J_H$), which corresponds to the usual B-twists (respt. A-twist). We emphasize that the specialization $\nu=\pm 1$ is part of the definition to construct the semi-simple TFTs that can support the rational VOAs of interest. While one may also study boundary VAs for the untwisted superconformal theory at $\nu=0$, in general, these do not yield rational VOAs.

Now, consider the supersymmetric Dirichlet boundary conditions in the UV theory (as defined above). This boundary condition flows in the IR to some boundary condition of the superconformal theory, which in turn maps to a boundary condition of the topologically twisted theory. An important and non-trivial consistency check is whether the BRST supercharge of the twisted theory ($Q_A$ or $Q_B$) is compatible with this boundary condition. This is very subtle to check explicitly, especially due to the presence of monopole super potentials, as we discuss briefly on page 11 and in footnote 10 in the first version of the manuscript. A more careful discussion on this subtlety is given in our earlier works [19,21]. In the present work, we do not attempt a detailed analysis of the $Q_{A/B}$-invariance of the boundary condition in the IR topological theory. Instead, we focus on analyzing the expressions (3.27) for the theories $T [K, {O_I }]$ that are expected to flow to rank-zero SCFTs, and propose candidate boundary VOAs supported by the corresponding TFTs when the conditions are satisfied. We indeed find in section 4 that there are examples (see e.g., 3-6, 3-7 and 3-8) where the index identically vanishes, suggesting that the the boundary condition B may not preserve the twisted supercharges. We acknowledge that this subtlety was not explained very clearly in the main text. In the revised version, we have added an extended paragraph that highlights the open issue.

This choice fixes the data of the topological field theory and hence completely determines the boundary vacuum character. Additional insertions of line operators can also be considered, as discussed in section 3.3. In particular, there exists a class of line operators whose insertions shift the q^(m-linear) factors in the Nahm sum formula, which superficially mimics a change of topological twist. However, following the logic I described above, (essentially via the F-maximization) one can clearly distinguish the origin of such contributions.

5) Here $K$ denotes the bare CS level in the so called “$U(1)_{-1/2}$ quantization”. The UV effective Chern Simons level is $K - \frac12 I$, which is 3/2 in the simplest case that the referee mentions. In this particular example, the theory is mirror dual to its orientation reversal, so it coincides with the description involving $U(1)_{-3/2}$. In v2, we added a comment on our convention and also a new footnote comparing the notations with other references.

6) The identification of the Dirichlet half index of the $U(1)_{3/2} + \Phi$ theory was first established in [19] by three of the authors of this manuscript, prior to the work mentioned by the referee. The reasoning behind our specific specialization is the same as explained above in point 4. While the vacuum character of the $\nu=-1$ twisted theory coincides with the index of the $\nu=1$ twist with a line operator inserted (a fact that is closely related to the self-mirror property of the theory, as explained in [21] by the second author of this manuscript), the correct identification of the VOA vacuum character should be made with the index without any line operator insertion. The explicit boundary OPE computations done in [21] support this identification.

We would like to thank the referee for carefully reading our manuscript. We added two references that the referee mentioned in the second version of the manuscript.