According to appendix C5 in this paper, $\text{SO}(N)_1$ Chern-Simons theory has no topological order. On the other hand, in this paper, Prof. Wen showed $\text{SO}(5)_m$ CS theory has the ground state degeneracy (GSD) on torus $= (m+1)(m+2)/2$ and thus $\text{GSD}=3$ for $\text{SO}(5)_1$ theory. This computation is consistent with this post. I wonder how to resolve the contradiction. How do you compute GSD for $\text{SO}(N)_1$ CS theory to show $\text{GSD}=1$?
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1$\begingroup$ Please consider spelling out abbreviations (GSD) and explaining what you mean by the subscript on the $\mathrm{SO}(N)$ to make this post more accessible. $\endgroup$– ACuriousMind ♦Commented May 18, 2017 at 10:38
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1$\begingroup$ Minor comment to the post (v3): In the future please link to abstract pages rather than pdf files. $\endgroup$– Qmechanic ♦Commented May 18, 2017 at 19:22
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Many field theories are not well defined. The $SO(N)_1$ CS theory in one paper may not be the same as the $SO(N)_1$ CS theory in another paper.
In my paper, the full theory is fermions coupled to $SO(5)$ gauge field. After integrating out the fermions, we get an effective $SO(5)_1$ CS theory. Its properties are given here https://www.math.ksu.edu/~gerald/voas/mtc/kmB2_1.html .
answered May 24, 2017 at 21:07
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$\begingroup$ Thank you, Prof. Wen. I am confused about the gauge group of (69) in your paper, where the gauge group is generated by sp4 Lie algebra. At the algebra level, sp4~so(5) but at the group level, SO(5) is not simply connected while Sp4 is. I wonder why the resulting theory after integrating the fermions is SO(5) CS theory instead of Sp4 CS theory. $\endgroup$– CharlesCommented Jun 5, 2017 at 20:22