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Fourier-transformed gauge theory models of three-dimensional topological orders with gapped boundaries
by Siyuan Wang, Yanyan Chen, Hongyu Wang, Yuting Hu, Yidun Wan
Submission summary
| Authors (as registered SciPost users): | Siyuan Wang |
| Submission information | |
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| Preprint Link: | scipost_202406_00062v3 (pdf) |
| Date accepted: | June 12, 2025 |
| Date submitted: | March 18, 2025, 4:45 p.m. |
| Submitted by: | Siyuan Wang |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
In this paper, we apply the method of Fourier transform and basis rewriting developed in [JHEP02(2020)030] for the two-dimensional quantum double model of topological orders to the three-dimensional gauge theory model (with a gauge group G) of three-dimensional topological orders. We find that the gapped boundary condition of the gauge theory model is characterized by a Frobenius algebra in the representation category Rep(G) of G, which also describes the charge splitting and condensation on the boundary. We also show that our Fourier transform maps the three-dimensional gauge theory model with input data G to the Walker-Wang model with input data Rep(G) on a trivalent lattice with dangling edges, after truncating the Hilbert space by projecting all dangling edges to the trivial representation of G. This Fourier transform also provides a systematic construction of the gapped boundary theory of the Walker-Wang model. This establishes a correspondence between two types of topological field theories: the extended Dijkgraaf-Witten and extended Crane-Yetter theories.
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Author comments upon resubmission
List of changes
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On page 4, a footnote is added, where we have mentioned the relationship between the Fourier transform and this change of basis at the Hamiltonian level of non-abelian lattice gauge theory.
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On page 4, the fourth paragraph of subsection 1.2, a sentence is added to further explain the role of the tails in the Hilbert space. Reference 40 is added there.
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On page 6, the third paragraph of section 2, a sentence is added to clarify that only some of the gapped boundaries, that is, those boundaries with trivial twist, will be discussed in our paper.
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On page 6, the last sentence of section 2, we change the phrase "elementary excitations" to "loop-like excitations". Reference 45, studying 3-loop braiding in 3DTO's, is also added there.
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On page 12, below eq.(32), we add some text to compare our Fourier transform with the basis transformation between the electric and magnetic bases of the non-abelian lattice gauge theory.
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On page 16, below eq.(40), a sentence is added to emphasize that any two edges that cross in this projection will pick up an $R$ matrix when evaluating a particular state.
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On page 16, at the end of subsection 3.2, a sentence is added to indicate that the argument for the equivalence of models with difference $R$-matrices can be found in subsection 5.2, and another sentence is added to indicate that some examples can be found in appendix C.
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On page 17, at the end of subsection 3.3, a paragraph is added to discuss charge excitations in the bulk.
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On page 26, at the end of subsection 5.2, the argument for the equivalence of models with difference $R$-matrices is added.
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Appendix C is added to list some examples.
Published as SciPost Phys. 19, 018 (2025)