Topological Defect Lines in Two Dimensional Fermionic CFTs

Dear Editor and referee,

We thank you for the detailed and instructive reviews, which helped clarify some unclear points and improve the quality of the manuscript. We have reviewed all points raised by the referees, and make updates accordingly.

Best,
Chi-Ming, Jin and Fengjun

In the following, we will list the referee's comments, followed by our replies:

1.
In well-defined non-chiral fermionic theories one can gauge the fermion parity symmetry to obtain bosonic theories, and the topological lines will become the topological lines previously studied.
This method is extensively used to study fermionic SPT phases.
Can the author comment on such approach to understanding topological lines in fermionic theories?

We add a few paragraphs to comment on this point in the end of Summary and discussions section 6:

``Finally, let us comment on the relation between the fermionization of CFTs and the fermion condensation of the TDLs. A bosonic CFT ${\mathcal T}_{\rm b}$ can be fermionized to a fermionic CFT ${\mathcal T}_{\rm f}$ if ${\mathcal T}_{\rm b}$ has a non-anomalous $\mathbb Z_2$ symmetry $\eta$. More explicitly, the fermionization is a procedure of tensoring ${\mathcal T}_{\rm b}$ with an Arf TQFT and gauging the diagonal $\mathbb Z_2$ symmetry, i.e. ${\mathcal T}_{\rm f}=({\mathcal T}_{\rm b}\times \text{Arf TQFT})/\mathbb Z_2$. From this procedure, it is not obvious what the TDLs in ${\mathcal T}_{\rm f}$ are and what the super fusion category describing the TDLs is. We do not have a full answer to these two questions, but from our analysis of the fermionic minimal models in Section 5, we can deduce some rules:

1. When $\eta$ has integer topological spin (as in the case of $m=1,\,2\mod 4$), the fermionic CFT ${\mathcal T}_{\rm f}$ has the same number of TDLs as ${\mathcal T}_{\rm b}$ and with the same fusion rules (as in (5.9)).

2. When $\eta$ has half-integer topological spin (as in the case of $m=0,\,3\mod 4$), the fermionic CFT ${\mathcal T}_{\rm f}$ would have a genuine super fusion category, which in particular could contain fermionic junctions or q-type TDLs. In fact, a subcategory of the full super fusion category can be obtained by the fermion condensation [70, 72, 78], which is a procedure applied on a modular tensor category with an abelian fermionic anyon (a $\mathbb Z_2$ TDL of half-integer topological spin) to get a super fusion category. Importantly, the $(-1)^F$ TDL is not inside this subcategory, and is ``emergent" in this sense.
For example, in the Ising CFT after condensing $\eta$, the non-invertible duality TDL becomes the q-type TDL $(-1)^{F_L}$ [46], but the $(-1)^F$ TDL has no predecessor in the Ising CFT.

On the other hand, one can get back the bosonic CFT ${\mathcal T}_{\rm b}$ by bosonizing the fermionic CFT ${\mathcal T}_{\rm f}$, more precisely, by gauging the $(-1)^F$ symmetry. However, we do not know a general procedure to apply on the super fusion category of ${\mathcal T}_{\rm f}$ to get (at least a subcategory of) the fusion category of the bosonic CFT ${\mathcal T}_{\rm b}$."

2.
The manuscript discussed changing the normalization of the topological line operators. However, the fusion coefficients of topological operators are quantized (described by well-defined lower dimensional TQFTs), and such change of normalization does not seem compatible with the topological structure. Can the author clarify this?

We thank the referee for pointing out the relation between fusion coefficients and lower dimensional TQFTs. We add a sentence below eq.(2.47):

``which is a specialization of the more general fusion eq.(2.45) without distinguishing $1_{\rm b}$ and $1_{\rm f}$. The factor of 2 can also be understood as the dimension of the Hilbert space of two 1d Majorana fermions, each coming from $\mathcal L_{i_{\rm q}}$ and $\overline\mathcal L_{i_{\rm q}}$ respectively."

to clarify that the factor of 2 corresponds to the 1d TFT from gapping two 1d Majorana fermions.