More generally, these states fall into different equivalence classes under stochastic local operations and classical communication (SLOCC).

Although these theories agree at the level of the action, the measures of Chern-Simons theory and three-dimensional gravity do not agree. See [30] for recent progress in this direction.

Their definition of W-like is essentially not-GHZ like: we will instead adopt the latter terminology.

The Chern-Simons action $I_{CS}$ can be shifted by multiples of $2\pi$ by gauge transformations that are not continuously connected to the identity. This implies that the level $k$ must be an integer, to ensure that $Z[M]$ is well-defined.

This is called a Heegaard splitting of $M$.

An isotopy is a diffeomorphism which is continuously connected to the identity, i.e., a “small diffeomorphism.”

The modular $S$ transformation arises because the vertical Wilson lines of Fig. 2 are related to the boundary cycles of ${\mathrm{\Sigma }}_S$ by an $S$ transformation.

Furthermore, to avoid an edge case, we ignore the case $(g,n)=(0,3)$. $\mathrm{Mod}(0,3)$ is isomorphic to the symmetric group $S_3$, and so has no pseudo-Anosov elements [43]. By the Nielsen-Thurston classification [41], this means there are no three component hyperbolic links with genus 0. This is consistent with the examples $K_3$ and $H_3$ of [22].

A bounded function is a function with a constant $K$ such that $|F(x)|\le K$ for all $x\in \mathrm{Mod}(g,n)$.

One definition of an amenable group is essentially that $⟨F{⟩}_{\text{naive}}$ exists for bounded functions.

One might worry that $K_f[J]$ is only a projective representation, so the additional phase factor may affect the averages we take. But this extra phase is $J$ independent [4] and therefore cancels out of $\mathrm{\Psi }$ itself, and won’t affect the averaging we do.

To see the connection to Schur’s lemma, if $K_g[J]=K_g[L]$, this would say that ${\mathcal{O}}_{\chi \xi }$ commuted with the group action. This restricts ${\mathcal{O}}_{\chi \xi }$ to be proportional to the identity. One way to phrase the reason for this is because the space of intertwiners from an irreducible representation to itself is one dimensional, with a basis vector given by the identity map.

One might worry that this Schur’s lemma argument doesn’t apply because $\mathrm{Mod}(g,n)$ is not compact, but because ${\mathcal{O}}_{\chi \xi }$ is a map between finite dimensional representations, we don’t need to worry about that here.

$\overline{L}$ is the conjugate representation to $L$.

This can be proven by using the fact that $[N_0{]}_a^b={\delta }_a^b$ and that $N_{abc}$ is symmetric.

For more details about the justification of this procedure, see, e.g., [79, 80, 81].

The exception is if $\mathrm{\Omega }(c)$ is independent of $c$, which is false, e.g., for all non-abelian $G_k$ fusion categories.

A bounded function is a function $F$ with the property that there exists a constant $K$ such that for all $[f]\in \mathrm{Mod}(g,n)$, $F([f])\le K$.

In fact, von Neumann conjectured that any non-amenable group contains $F_2$ as a subgroup. This turns out to be false [92], but is true in many examples, including the mapping class group for most $(g,n)$.

This is even true for amenable groups, if we took the limit via a subsequence ${\mathrm{\Delta }}_M$, with $M\sim 2^N$. For amenable groups, however, this class of convergent functions is broader than for non-amenable groups.

This is how we constructed the counterexample $F_{\pm }$: it was designed to oscillate strongly near the boundary $\partial B_N(e)$ as we vary $N$.