Topological holography for fermions

1. A detailed study of symmetry TFT for 1+1d fermionic invertible symmetries.
2. Construction of a new fermionic gapless SPT phase using the symmetry TFT.
3. The derivations of various results are explicit and physically well-explained.
4. The paper is clearly written.

1. Several statements are mathematically imprecise: please see the requested changes below.
2. The symmetry TFT for fermionic invertible symmetry is not conceptually new; see, e.g., Section 3 of [Gaiotto, Kulp; arXiv:2008.05960].

This paper discusses a general framework of symmetry TFT for 1+1d fermionic systems with finite invertible symmetry $G^F$, mainly focusing on the case of non-anomalous symmetries. The first half of the paper focuses on simple examples that illustrate the power of symmetry TFT. In particular, the authors constructed an example of a fermionic intrinsically gapless SPT phase using the symmetry TFT, which exemplifies the usefulness of their framework.

The latter half of the paper details the general structure of the symmetry TFT for fermionic invertible symmetry $G^F$. When $G^F$ is non-anomalous, the symmetry TFT is given by the quantum double $D(G^F)$, and the fermionic symmetry $G^F$ is realized on a fermionic topological boundary of $D(G^F)$. The authors explicitly wrote down the boundary state $|\mathcal{F}\rangle$ corresponding to this fermionic boundary and derived the $G^F$-spin structure dependence of $|\mathcal{F}\rangle$ based on the symmetry TFT analysis. Using the explicit form of $|\mathcal{F}\rangle$, the authors showed that changing the fermionic boundary of $D(G^F)$ to a bosonic one can be understood as the bosonization of the corresponding 1+1d system. The authors also studied the stacking rule of fermionic SPT phases within the framework of symmetry TFT, which generalizes the bosonic case discussed in [Turzillo, You; arXiv:2311.18782].

This paper provides a clear explanation of how the symmetry TFT works for fermionic systems. Remarkably, the authors demonstrated that the symmetry TFT is useful for constructing new fermionic gapless phases. This opens up a new method to explore quantum phases in fermionic systems. As such, the referee would recommend this paper for publication.

1. p.4, II.A, first paragraph: the last sentence is slightly misleading because the fluxes of the 2+1D symmetry TFT should be labeled by conjugacy classes rather than elements of $G$. When a flux line is pushed onto the boundary, it becomes the sum of topological lines labeled by elements of $G$.
2. p.7, below eq.(7): the algebra described there is not the group algebra, but the dual of the group algebra.
3. p.8, II.C, first paragraph: the correspondence between condensable algebras of $D(G)$ and gapped/gapless SPT phases is not one-to-one because some condensable algebras correspond to symmetry-broken phases.
4. p.10, footnote 8: the references cited there seem not to introduce local fermions in the bulk symmetry TFT. For example, Section 3 of ref.[31] discusses bosonic symmetry TFT with fermionic boundaries.
5. p.15, l.1, left column: does this statement apply only to abelian topological orders?
6. p.15, left column: similar boundary states are also discussed in, e.g., eq.(A.33) of [Delmastro, Gaiotto, Gomis], eq.(C.1) of [Kobayashi; arXiv:2203.08156], and eq.(13) of [You; arXiv:2311.01096]. It might be helpful to discuss the relation to these references.
7. p.17, III.D.2: the gapped domain wall $\mathcal{I}$ is not invertible.
8. p.19, IV, third paragraph: the first sentence is misleading because lattice models of fermionic gapless SPT phases with fermionic emergent anomaly were already constructed in the literature, e.g., Section 4.4 of [Ando; arXiv:2402.03566].
9. p.20, the fifth line of the last paragraph on the right column: if we define $a = e_1^6 m_1^2 e_2 m_2$, we have $m_1^2 = a \times (e_1^2 f_2)$ rather than $m_1^2 = a \times (e_1^6 f_2)$.
10. p.21, IV.B.d: could you clarify why the ground state subspace does not factorize into a pair of local Hilbert spaces for each edge?
11. p.24, above eq.(38): "group algebra $\mathbb{C}[G^F]$" should be the dual of the group algebra.
12. p.28, l.4, left column: what is the definition of $\delta s$?
13. p.28, left column: when a $G^F$-spin structure is denoted as $(\eta, a_b)$, do you implicitly choose a trivialization $\tau$ of $a_b^*(\rho)$? It would be helpful to clarify whether the choice of $\tau$ is a part of the data of a $G^F$-spin structure.
14. p.29, eq.(70) and below: do we need to take the trace of $R(\oint_{\gamma} a_b^*(s) + \tau)$?
15. p.33, VI.C.1: it should be mentioned that the symmetry TFT perspective on the stacking of bosonic SPT phases was discussed in [Turzillo, You; arXiv:2311.18782]
16. p.35, VII: it should be mentioned that the symmetry TFT perspective on the bosonization is discussed in [Gaiotto, Kulp; arXiv:2008.05960]
17. p.37, l.11 of the last paragraph on right column: the statement "the corresponding Hamiltonians have the same spectrum" is imprecise: they have the same spectrum only in the symmetric subspace of the total state space.
18. p.38, the last line of the first paragraph on the left column: $\sigma^2$ and $\overline{\sigma}^2$ on the boundary should be isomorphic to $1 \oplus 1$ rather than $1$. More precisely, they are isomorphic to $1 \oplus \pi$ via a bosonic isomorphism, where $\pi$ denotes a transparent fermion on the boundary; see, e.g., [Bhardwaj, Inamura, Tiwari; arXiv:2405.09754].
19. p.39, item 4 on the left column: $\mathcal{F}_b$ is not a modular extension of a super-fusion category $\mathcal{F}$.
20. p.39: the title of Appendix A seems inappropriate because condensable algebras of the quantum double are not discussed in this appendix.
21. p.40, below eq.(A6): gauge fluxes labeled by conjugacy classes of $G$ do not form a subcategory $\text{Vec}_G$ of the quantum double $D(G)$ when $G$ is non-abelian.

Please also find the following small typos.

1 p.4, l.6, right column: "$M$ is a 2+1D" $\to$ "$M$ is a 1+1D"
2. p.7, l.13, left column: "condensed, These" $\to$ "condensed. These"
3. p.7, below eq.(7): "subscript $b$" $\to$ "superscript $b$"
4. p.7, below eq.(7): "as oppose to" $\to$ "as opposed to"
5. p.7, below eq.(7): "we will introduce latter" $\to$ "we will introduce later"
6. p.15, second equation on the right column: "$-=+1$" $\to$ "$=+1$"
7. p.15, III.C, third paragraph: "$f = e \times f$" $\to$ "$f = e \times m$"
8. p.15, III.C, fourth paragraph: "$\mathcal{A}_{\text{ref}}^f = 1 + e_1 + f_1 + e_2 + f_1e_2$" $\to$ "$\mathcal{A}_{\text{ref}}^f = 1 + e_1 + f_2 + e_1f_2$"
9. p.16, above eq.(15): "with with fixed-point" $\to$ "with fixed-point"
10. p.17, below eq.(16): "$\overline{H}_b$" $\to$ "$\overline{\mathcal{H}}_b$"
11. p.21, IV.B.c, l.11: "$a = e_1^6 m_a^2 f_2$" $\to$ "$a = e_1^6 m_1^2 f_2$"
12. p.21, IV.C, l.12: "$e_1^6 m_2^2$" $\to$ "$e_1^6 m_1^2$"
13. p.21, eq.(29), first line: "$+$" is not necessary.
14. p.21, below eq.(29): "$\partial_x \varphi)_J$" $\to$ "$\partial_x \varphi_J$"
15. p.21, below eq.(29): "$\Phi$ fields" $\to$ "$\phi$ fields"
16. p.21, eq.(30): "$\phi_{L/R}$" $\to$ "$\phi_{L, R}$"
17. p.22, the caption of Figure 12: "verticle" $\to$ "vertical"
18. p.22, the caption of Figure 12: "Ths operator" $\to$ "This operator"
19. p.23, the second paragraph on the left column: "$a = m_1^2 e_2^{-2} f_2$" $\to$ "$a = m_1^2 e_1^{-2} f_2$"
20. p.23, the second paragraph on the left column: "local fermion on the physical boundary" $\to$ "local fermion on the reference boundary"
21. p.23, the second paragraph on the left column: "decorated by a local ..." is duplicated.
22. p.23, V.A.1, the first sentence: "of of $G^F$" $\to$ "of $G^F$"
23. p.24, l.6: "gauge-charge charge" $\to$ "gauge charge"
24. p.24, eq.(38) and below: "$\mathbb{C}[G]$" $\to$ "$\mathbb{C}[G^F]$"
25. p.24, eq.(43), l.3: "$\cdot |h\rangle$" $\to$ "$|h\rangle$"
26. p.25, above eq.(45): "$|\Psi_{phs}\rangle$" $\to$ "$|\Psi_{\text{phys}}\rangle$"
27. p.28, l.5, left column: "$Z^2[G^b, \mathbb{Z}_2^F]$" $\to$ "$Z^2[G^B, \mathbb{Z}_2^F]$"
28. p.29, below eq.(72): "$A_b^*(s)$" $\to$ "$A_b^*(\rho)$"
29. p.31, the second paragraph on the right column: "For latter" $\to$ "For later"
30. p.31, eq.(82): $p$ was originally denoted as $\pi$ in eq.(68)
31. p.32, the last sentence on the left column: "system can the be" $\to$ "system can then be"
32. p.34, above eq.(95): "$f^* := f \cong f$" $\to$ "$f^* := f \cong f^{\prime}$"
33. p.35, l.12 and l.13 on the left column: "$\mathbb{Z}^A$" and "$\mathbb{Z}^B$" $\to$ "$\mathbb{Z}_2^A$" and "$\mathbb{Z}_2^B$"
34. p.35, above eq.(96): subscripts $A$, $B$, and $C$ in $e_A$, $e_B$, $f_c$, etc. should be 1, 2, and 3
35. p.35, eq.(98): "$\nu_i(g) = \frac{\omega_i(g, P_f)}{\omega_i(P_g, g)}$" $\to$ "$\nu_i(g) = \frac{\omega_i(g, P_f)}{\omega_i(P_f, g)}$"
36. p.36, l.9, right column: "bosonic symmetry $G^B$" would be a typo of "bosonic symmetry $G$"
37. p.36, eq.(101): $\eta$ in the summation was originally defined as a trivialization of $w_2$ rather than a trivialization of $a_b^*(\rho)+w_2$
38. p.36, eq.(105): this expression is slightly confusing because $\tau$ was originally introduced as a trivialization of $a_b^*(\rho)$ rather than a spin structure. Could you resolve this conflict of notation?
39. p.37, l.4: "exactly tha bosonization" $\to$ "exactly the bosonization"
40. p.38, above the sentence in a box: "the conclusion" is duplicated
41. p.39, Acknowledgements: "Kantaro Omori" $\to$ "Kantaro Ohmori"
42. p.40, item c on the left column: "$\text{Rep}(\mathcal{C}_G(\sigma)$" $\to$ "$\text{Rep}(\mathcal{C}_G(\sigma))$"

Ask for minor revision

1) The paper includes many illustrative examples and clear explanations. There is a nice review of symtft for bosonic systems which the authors then build on to extend to fermionic systems.
2) The calculations are very explicit and detailed and the authors use different approaches to rederive results.
3) The paper fills in an obvious hole in the literature.

1) The results are all for unitary symmetries. For these cases, we already have a good understanding of symmetries and anomalies for fermionic systems in 1+1D. The symtft framework is somewhat more useful for non-invertible symmetries, where one one cannot use an SPT (at least in a tensor product Hilbert space) in one higher dimension. Noninvertible symmetries are mentioned briefly in the discussion (see last question in the report).
2) Bosonization is discussed rather far down in the paper. Since it is a fundamental topic, perhaps it should be discussed earlier, at least before the systematics and technical details. One can also make a connection to bosonization/fermionization on the lattice, but I understand that that may be beyond the scope of this paper because this paper is not meant to deal with details of lattice models.

This paper is clearly written and provides a nice approach for incorporating fermionic symmetries in the symtft framework (by introducing local fermions on the reference boundary). The paper has several novel results, including an interpretation of bosonization within the symtft framework and new intrinsically fermionic phases.

The paper should also mention the relation to https://arxiv.org/abs/2405.0961. I would also appreciate more details comments comparing/contrasting with the references in footnote 8 (esp 55).

I have some other questions/comments:
1) At the end of the first paragraph, how do you define symmetry? If symmetries are given by operators that act on the Hilbert space, how is the dual symmetry a symmetry? I thought what you meant by "all such generalized symmetries" would be a fusion category in 1+1D (this I think would instead be a multifusion category).
2) On the right side of p2, aren't all such topological manipulations combinations of gauging & stacking with invertible phases?
3) Left side of p3: small comment but "doubled Ising" is usually used, not "double Ising."
4) Bottom left of p4: symtft also works for nonanomalous symmetry. Why do you specify anomalous here?
5) Top right of p4: should it be that M is a 1+1D (closed,...) and for point 3, B_{phys} can be varied to describe different possible G symmetric and SSB phases? "and SSB" is also neglected in various other places.
6) Left side of p7: "In general a composite anyon of the original phase becomes a simple anyon of the new phase" anyons of the original theory can also split into multiple anyons in the new phase, i.e. if you condense 1+\psi\bar{\psi} in doubled Ising.
7) Bottom of p9: probably a similar story holds for general anomaly free fusion category except there is no canonical choice for the vacuum. Maybe this can be mentioned?
8) Top left of p15: why true that $W^t_a|A\rangle=|A\rangle$? For example take $|A\rangle$ for doubled fib to be $|1\rangle+|\tau\bar{\tau}\rangle$. Then we get $\tau\bar{\tau}(|1\rangle+|\tau\bar{\tau}\rangle)=|\tau\bar{\tau}\rangle+|1\rangle+|\tau\rangle+|\bar{\tau}\rangle+|\tau\bar{\tau}\rangle$ according to the definition on the bottom right of p14?
9) Bottom right of p17: why is this domain wall invertible? You condense $\psi\bar{\psi}$ on this domain wall. It is not an automorphism, so I don't think it's invertible (more concretely, one can calculate the fusion rules for this domain wall).
10) Right side of p19: this "gauging the SPT to get the symtft" really only works for unitary symmetries. Comment on this?
11) Left of p20: you can reference PhysRevB.104.245130 for the boundary hilbert space structure. Isn't the gauged SPT just doubled Ising?
12) Left of p39 (pt 2): should I think of the fact that the bulk is bosonic as coming from the fact that you can always gauge the fermion parity in the bulk i.e. by minimal modular extension?
13) Left p39 (pt 4): what do you mean by $Z[F_b]$? Isn't modular extension to make a theory have modular braiding? But the fusion category that is input into the center construction does not need to be braided?

Please respond to the questions/comments above and make appropriate revisions.

Ask for minor revision