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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 submitted: | 2025-03-18 16:45 |
| Submitted by: | Wang, Siyuan |
| 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
1. 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.
2. 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.
3. 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.
4. 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.
5. 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.
6. 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.
7. 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.
8. On page 17, at the end of subsection 3.3, a paragraph is added to discuss charge excitations in the bulk.
9. On page 26, at the end of subsection 5.2, the argument for the equivalence of models with difference $R$-matrices is added.
10. Appendix C is added to list some examples.