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I am using the method outlined in appendix C4 of a paper by Seiberg and Witten [1] to calculate the statistics of lines in $U(2)_{2, 1}$. However, this method shows that all lines are trivial.

Notation: By $U(2)_{2, 1}$, we mean a theory with $SU(2)_{1}\times U(1)_{2}$ and a Chern-Simons action of the form: $$ \frac{1}{4\pi}\int \left(A\wedge dA + \frac{2}{3} A\wedge A\wedge A\right) $$ where $A$ is a $U(2)$ gauge field. This notation matches [1], but differs from [2] (in the notation of [2] this theory is $U(2)_{1, 1}$).

Background: I am imagining the edge modes of a single integer quantum hall state where my electrons have an extra index $\psi_{i}$ that transforms under $SU(2)$ `isospin.'

$1+1$d: There is a single right-moving fermion doublet mode on the edge, $\psi_{i}$, with $(\partial_{t}+\partial_{x})\psi_{i} = 0$. We mandate that all terms in the Lagrangian respect the $U(2)$ symmetry $\psi_{i}\to U_{ij}\psi_{j}$.

If I then bosonize this system, then I get a $U(2)$ WZW theory at level 1 on the edge.

$2+1$d: In the bulk, this leads to a $U(2)_{2, 1}$ Chern Simons Theory. Following [1] Appendix C4, we can first treat this theory as $SU(2)_{1}\times U(1)_{2}$ and then take the quotient.

Beginning with $SU(2)_{1}\times U(1)_{2}$, we note that it is level-rank dual to $U(1)_{-2}\times U(1)_{2}$. Hence we have four lines: $(0, 0), (0, 1), (1, 0), $ and $ (1, 1)$. For statistics: $(1, 0)$ and $(0, 1)$ have spins $\pm ¼$ and $(1, 1)$ has spin $½$. $(1, 0)$ and $(0, 1)$ have braiding $-1$ with $(1, 1)$. All other statistics are trivial.

The problem occurs when we take the quotient.

Following [1] Appendix C4, we first identify a line which generates holonomy $-1$ in each of the $SU(2)$ and $U(1)$ factors. After using level rank, we look for the line which generates holonomy $-1$ in each of the two $U(1)$ groups. Clearly, this object is $(1, 1)$, and we force it to be trivial.

Next, we enforce a `selection rule' that only allows lines which have trivial statistics with $(1, 1)$. The only line that has trivial statistics with $(1, 1)$ is $(0, 0)$ and so we are left with no nontrivial lines!

Clearly something has gone disastrously wrong with my argument, since the bulk Chern Simons theory for this model has to be nontrivial. Can you help me find the flaw in my reasoning?

[1] N. Seiberg and E. Witten, “Gapped Boundary Phases of Topological Insulators via Weak Coupling,” PTEP, vol. 2016, no. 12, p. 12C101, 2016. https://arxiv.org/abs/1602.04251

[2] P.-S. Hsin and N. Seiberg, “Level/rank Duality and Chern-Simons-Matter Theories,” JHEP, vol. 09, p. 095, 2016. https://arxiv.org/abs/1607.07457

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  • $\begingroup$ There is no such a thing as Chern-Simons $U(2)_{2,1}$. Recall that $U(N)_{P,Q}$ is only well defined if $P-Q$ is an integer multiple of $N$. $\endgroup$ – AccidentalFourierTransform Oct 7 '18 at 2:16
  • $\begingroup$ Is there a good way to see that? Ref. [1] above gives a different consistency condition eq. C.17, that $U(2)_{P, Q}$ is consistent only when $P+2Q \in 4\mathbb{Z}$, which $U(2)_{2, 1}$ satisfies. $\endgroup$ – MDM Oct 7 '18 at 2:19
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    $\begingroup$ Ah okay there's some notation hijinks happening here. Hsin and Seiberg define $U(N)_{P,Q}$ as $SU(N)_{P}\times U(1)_{NQ}$, whereas Seiberg and Witten define $U(N)_{P,Q}$ as $SU(N)_{Q}\times U(1)_{P}$. In the Hsin-Seiberg notation this theory is $U(2)_{1,1}$. I've edited the post to be more clear. $\endgroup$ – MDM Oct 7 '18 at 3:03
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    $\begingroup$ Follow up question: If I take two copies of the above theory both constructed from right-moving edges, then in Hsin-Seiberg notation I get $U(2)_{1, 1}\times U(2)_{1, 1}$ with central charge $2$ On the other hand, if I take a copy from a right-moving edge and a copy from a left-moving edge, I get $U(2)_{1, 1}\times U(2)_{-1, -1}$ with central charge $0$. However, in the fermion dual, these are both $\{1, f\}\times \{1, f'\}$. How can one tell them apart in the fermion language? $\endgroup$ – MDM Oct 7 '18 at 5:09
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    $\begingroup$ @MDM In order to specify a topological theory, it is not sufficient to list the lines; there is extra data that defines the theory (more precisely, there is more structure to a modular tensor category than just the objects). For example, the theories $U(1)_k$ and $U(1)_{-k}$ have the same lines, but with opposite spin. $\endgroup$ – AccidentalFourierTransform Oct 7 '18 at 16:27

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