Exactly Solvable Lattice Hamiltonians and Gravitational Anomalies

1. They construct explicit lattice Hamiltonian models of (4+1)D beyond cohomology invertible phases, in the case with Z2 0-form symmetry or without any global symmetry. They also obtain the boundary theory given by Z2 gauge theory in (3+1)D.
2. In the case without global symmetry, the boundary (3+1)D Z2 gauge theory of the beyond-cohomology invertible phase (with effective action w2w3) supports a magnetic loop excitation phrased as a 'fermonic loop' in literature. They discovered a non-trivial statistical property of the fermonic loop excitation based on the concrete lattice model and effective field theory.

This paper gives concrete lattice Hamiltonian models for the beyond-cohomology invertible phases in (4+1)D. In particular, their construction of the w2w3 phase without symmetry is quite clever, where they express the w2w3 phase in terms of a higher-form Z2 gauge theory whose action does not have explicit dependence of Stiefel-Whitney classes w2, w3. This makes possible to realize a simple lattice model without employing explicit cochain representatives of w2, w3 on lattice. They also study the statistical property of the fermonic loop excitation of the (3+1)D Z2 gauge theory on its boundary, and found that its orientation-reversing defect supports a fermion.
It clarifies the existing arguments about fermionic loops in literature, and regarded as an important contribution about understanding of fermionic loops. Overall, this is an excellent paper and I can safely recommend this article for publication.

They provide a disentangler of the lattice model for w2w3 phase, and claim that the disentangler realizes a non-trivial quantum cellular automaton (QCA) in (4+1)D. This claim should be a conjecture, based on the other primary conjecture that the (3+1)D Z2 gauge theory w/ fermonic particle and a fermonic loop cannot be realized by a commuting projector Hamiltonian.
However, I do not know if anyone studied or even talked about this conjecture, while it is widely believed that the chiral topological phase in (2+1)D cannot be realized by a commuting projector model, which leads to a construction of non-trivial QCA in (3+1)D.
I think it would benefit community, if the author can add some comment about the conjecture about commuting projector model described above. But this is an additional comment which is not necessarily required for publication. I can already recommend this article for publication.

1- Explicit construction of Hamiltonian model gapped boundary of beyond group cohomology SPTs.
2- The idea of integrating the Lagrange multiplier and regarding an invertible system as a finite gauge theory, then realize it as a Hamiltonian system.

The manuscript describes the construction of projector hamiltonian models of gapped boundaries of SPTs that are beyond group cohomology. The idea is to realize the SPT as a finite gauge theory with a suitable DW twist, and to realize it as a Hamiltonian system. Then the gapped boundary is obtained by truncating the lattice system.
The construction as a Hamiltonian system is new and will be helpful in the study of SPT and topological phases, and thus I recommend publishing it in SciPost after the authors consider the suggestions below.

1- Although the construction of Hamiltonian system is new, as for a spacetime system it is discussed in https://arxiv.org/1905.05391 . The bosonic cases it is briefly discussed in Section 3. (Despite the name of the title, time-reversal symmetry is not necessarily in the construction.) It would be ideal to discuss the relationship to this construction.
2- Typo in the title of section 2.1: Beyong -> Beyond.
3- The sentence “If we denote the background..” on page 10 does not finish.