This question originates from the following statement in Bonderson's thesis: Link to Thesis page 16 or pdf-page 23:

The quantum dimension $d_a$ of an anyon of charge $a$ satisfies $d_a \geq 1$ with equality iff a is Abelian.

Here, the quantum dimension is defined as $$ d_a = d_{\bar{a}} = |[F^{a\bar{a}a}_a]_{1,1}|^{-1}$$ and a charge $a$ is abelian if $\sum_c N^c_{ab}=1$ for all $b$.

Of course, if $a$ is abelian, this $F$-matrix is one-dimensional and by unitarity then $[F^{a\bar{a}a}_a]_{1,1} \in U(1)$, so that $d_a=1$. I have trouble proving the converse: Given that $d_a=1$, how to prove that $a$ has to be abelian?

What I know so far: $d_a$ is the Frobenius-Perron eigenvalue to $N_a$ with components $(N_a)_{bc}=N^c_{ab}$. Thus, if for $d_a=1$ all eigenvalues of $N_a$ are of absolute value smaller or equal 1.

  • $\begingroup$ maybe you could provide a link or a full reference to the thesis as right now there is minimal context to your question. $\endgroup$ – ZeroTheHero Apr 30 at 11:23
  • $\begingroup$ Of course, I forgot to put it, have done so now. $\endgroup$ – Marsl Apr 30 at 11:26

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