Of course, this predates the present understanding of generalized global symmetries; Polyakov and ’t Hooft did not articulate the Yang–Mills centre symmetry in terms of a one-form symmetry.

Here $V[k]$ denotes the degree-shift of a graded vector space $V={⨁}_i{V}^i$ by $k\in \mathbb{Z}$, where $V[k{]}^i≔V^{i+k}$. Note that $\mathfrak{g}$ and $\mathfrak{h}$ are regarded as graded vector spaces concentrated in degree zero, i.e. $\mathfrak{g}={\mathfrak{g}}^0$, so $\mathfrak{h}[1]\oplus \mathfrak{g}[0]$ has non-trivial degrees $-1$ and 0.

Note that, by abuse of notation, we overload the symbols $s$, $\rho$ and $\kappa$ for both the finite (crossed module of Lie groups) and infinitesimal (crossed module of Lie algebras) cases; the intended meaning should be clear from the context.

If, however, $\kappa$ is antisymmetric, then $\delta H=0$ and $\mathrm{d}H=0$, even if $\mathfrak{h}$ is non-Abelian;

in that case, $H$ is the Noether current for a ($d-4$)-form symmetry.

In fact, since $\gamma$ is valued in the group $Z(G)\subset G$ whereas $\lambda$ is valued in $Z(\mathfrak{h})\subset \mathfrak{h}$, this is better regarded as the centre of a differential graded Lie group with body $G$ and with nonzero-degree coordinates corresponding to $\mathfrak{h}$ obtained as a partial linearization of the 2-group $\mathcal{G}$.

In practice, however, it quickly becomes impractical to spell out the $p$-group coherence relations as $p$ grows.

We do not need periodicity; an aperiodic tiling will also do.

We neglect such complications as when some but not all of centre vortices condense, corresponding to a breaking of ${\mathbb{Z}}_N$ into a subgroup;

see [1] [App. E] for a detailed discussion.