$B\in\Omega^{d-2}\otimes\mathfrak{g}^{*},C\in\Omega^{d-3}\otimes\mathfrak{h}^{*},D\in\Omega^{d-4}(X)\otimes\mathfrak{i}^{*}$
implementing the aforementioned conditions.

The 3-$BF$ action (also called the BFCGDH action, but we shall not use this
name for obvious reasons) without any 3-curvature anomalies is then

$S_{\text{3BF}}=\int_{X}\langle B\wedge\mathcal{F}(A,\Sigma)\rangle+\langle
C\wedge\mathcal{G}(A,\Sigma,\Gamma)\rangle+\langle
D\wedge\mathcal{H}(A,\Sigma,\Gamma)\rangle,$ (3.13)

in which $\mathcal{F}=F-t\Sigma,\mathcal{G}=d_{A}\Sigma-t^{\prime}\Gamma$, and
$\mathcal{H}=d_{A}\Gamma+Q_{\Sigma}$ is the modified 3-curvature. Recall these
curvature quantities are covariant, (2.18), (3.6)-(3.9). For $d=3$, the 3-$BF$
theory reduces to a 2-$BF$ theory, since the dual field $D$ does not exist.

The first set of EOMs is

$\delta B\Rightarrow\mathcal{F}=0,\qquad\delta
C\Rightarrow\mathcal{G}=0,\qquad\delta D\Rightarrow\mathcal{H}=0,$

which implement precisely the 1-, 2-fake-flatness and 3-flatness conditions,
respectively. Since we also have to vary $A,\Sigma$ and the 3-connection
$\Gamma$, in addition to the maps $\Delta,t^{*}$ given in section 2.5.1, we
introduce

$\displaystyle\Delta^{\prime}:\mathfrak{i}^{\otimes 2}\rightarrow\mathfrak{g}$
$\displaystyle\,,\,$ $\displaystyle\langle D\wedge
A\wedge^{\rhd^{\prime}}\Gamma\rangle=-\langle\Delta^{\prime}(D\wedge\Gamma)\wedge
A\rangle,$ $\displaystyle\Omega:\mathfrak{i}\rightarrow\mathfrak{h},$
$\displaystyle\,,\,$ $\displaystyle\langle D\wedge
Q_{\Sigma}\rangle=-\langle\Omega(D)\wedge\Sigma\rangle,$ $\displaystyle
t^{\prime*}:\mathfrak{h}\rightarrow\mathfrak{i},$ $\displaystyle\,,\,$
$\displaystyle\langle C\wedge t^{\prime}\Gamma\rangle=\langle
t^{\prime*}C\wedge\Gamma\rangle,$

and also the dual action

$\langle z,x\rhd^{\prime}z^{\prime}\rangle=\langle
x\rhd^{\prime*}z,z^{\prime}\rangle$

for all $z,z^{\prime}\in\mathfrak{i},X\in\mathfrak{g}$. Notice that
$\Omega(D)$ is a ($d-2$)-form.

These yield the dual EOMs

$\displaystyle\delta A$ $\displaystyle\Rightarrow$ $\displaystyle dB+[A\wedge
B]^{*}-\Delta(C\wedge\Sigma)-\Delta^{\prime}(D\wedge\Gamma)=0,$
$\displaystyle\delta\Sigma$ $\displaystyle\Rightarrow$ $\displaystyle
dC+A\wedge^{\rhd^{*}}C-t^{*}B-\Omega(D)=0,$ $\displaystyle\delta\Gamma$
$\displaystyle\Rightarrow$ $\displaystyle
dD+A\wedge^{\rhd^{\prime*}}D-t^{\prime*}C=0.$

If we define, in addition to $\tilde{F}=dC+A\wedge^{\rhd^{*}}C$ and
$\tilde{K}=dB+[A\wedge B]^{*}$ as in (2.27), the quantity

$\tilde{T}=dD+A\wedge^{\rhd^{\prime*}}D,$

we see that these dual EOMs read

$(d-1)$-form:
$\displaystyle\tilde{K}=\Delta(C\wedge\Sigma)+\Delta^{\prime}(D\wedge\Gamma),$
$(d-2)$-form: $\displaystyle\tilde{F}=t^{*}B+\Omega(D),$ $(d-3)$-form:
$\displaystyle\tilde{T}=t^{\prime*}C.$ (3.14)

##### Symmetries of the action.

Similar to the 2-gauge case, we also acquire 3-gauge transformations in the
dual fields $B,C,D$. These have been developed in [33], but to write them
down, we must introduce yet more structures.

We define the following maps121212The Peiffer pairing defines two maps
$\mathfrak{h}\rightarrow\mathfrak{h}^{*}\times\mathfrak{i}$ by
$\mathpzc{Pf}_{y}=\\{y,\cdot\\}_{\text{Pf}}$ and its conjugate
$\overline{\mathpzc{Pf}}_{y}=\\{\cdot,y\\}_{\text{Pf}}$. Then
$\omega_{1}(\cdot,Y)=-\mathpzc{Pf}_{y}^{*}$ is the dual and
$\omega_{2}(\cdot,y)=-\overline{\mathpzc{Pf}}_{y}^{*}$ is the conjugate dual.

$\displaystyle\omega_{1,2}:\mathfrak{i}\times\mathfrak{h}\rightarrow\mathfrak{h}$
$\displaystyle\Rightarrow$ $\displaystyle\langle
z,\\{y,y^{\prime}\\}_{\text{Pf}}\rangle=-\langle\omega_{1}(z,y),y^{\prime}\rangle=-\langle\omega_{2}(z,y^{\prime}),y\rangle,$
$\displaystyle\mathcal{Y}:\mathfrak{i}\times\mathfrak{h}^{\otimes
2}\rightarrow\mathfrak{g}$ $\displaystyle\Rightarrow$ $\displaystyle\langle
z,\\{x\rhd
y,y^{\prime}\\}_{\text{Pf}}\rangle=-\langle\mathcal{Y}(z,y,y^{\prime}),x\rangle,$

for each $Z\in\mathfrak{i},Y,Y^{\prime}\in\mathfrak{h},X\in\mathfrak{g}$.
Notice that since $\\{y,y\\}_{\text{Pf}}=Q_{y}$, we have

$\langle\omega_{1,2}(z,y),y\rangle=\langle\Omega(z),y\rangle,$

and that $\omega_{1}\neq\omega_{2}$ or $\omega_{1}\neq-\omega_{2}$ in general,
as $\\{\cdot,\cdot\\}_{\text{Pf}}$ is not symmetric or skew-symmetric.

The dual 3-gauge transformations are given by [33]

$\displaystyle\lambda$ $\displaystyle:$
$\displaystyle\begin{cases}B\rightarrow B^{\lambda}=B+[\lambda,B]\\\
C\rightarrow C^{\lambda}=C+\lambda\rhd C\\\ D\rightarrow
D^{\lambda}=D+\lambda\rhd^{\prime}D\end{cases},$ $\displaystyle L$
$\displaystyle:$ $\displaystyle\begin{cases}B\rightarrow
B^{L}=B+\Delta(C\wedge L)+\mathcal{Y}(D\wedge L\wedge L)\\\ C\rightarrow
C^{L}=C+\omega_{1}(C\wedge L)+\omega_{2}(C\wedge L)\\\ D\rightarrow
D^{L}=D\end{cases},$ $\displaystyle\mathcal{L}$ $\displaystyle:$
$\displaystyle\begin{cases}B\rightarrow
B^{\mathcal{L}}=B+\Delta^{\prime}(D\wedge\mathcal{L})\\\ C\rightarrow
C^{\mathcal{L}}=C\\\ D\rightarrow D^{\mathcal{L}}=D\end{cases},$ (3.15)

which preserves the 3-$BF$ action (3.13) when performed alongside the 3-gauge
transformations (3.10).

Analogous to the 2-gauge case in section 2.5.1, we see that the 3-gauge group
$\operatorname{Gau}_{3}=\mathscr{E}\rtimes\Omega^{1}(X)\otimes\mathfrak{g}$
acts on the dual fields $B,C,D$. As such, one would expect the data
$(\Delta,\Delta^{\prime},\omega^{1,2},\mathcal{Y})$ emergent from the dual
EOMs (3.14) to define a strict coadjoint representation
$\operatorname{ad}^{*}:\mathscr{G}\rightarrow\operatorname{End}\mathscr{G}^{*}[2]$
on the dual three-term algebra complex

$\mathscr{G}^{*}[2]:\mathfrak{g}^{*}\xrightarrow{t^{*}}\mathfrak{h}^{*}\xrightarrow{t^{\prime*}}\mathfrak{i}^{*},\qquad
t^{\prime*}t^{*}=(tt^{\prime})^{*}=0.$

Unfortunately, the duality theory of Lie 3-algebras has not been studied in
the literature, and the notion of a "3-Manin triple" has yet to be developed.
We leave this task to the ambitious reader.

######  Remark 3.2.

Notice that, in order for the dual fields $B,C,D$ to have the right degree-
count to serve as a "dual 3-connection" $(D,C,B)$, we must have
$d=\operatorname{dim}X=5$ in contrast to the case in 2-BF theory (where
$d=4$). As such, it seems that the "3-Manin triple" most naturally provides
the symmetry structure of the 3-BF theory in 5D.

####  3.3.2 Anomaly resolution: 2-monopoles and the string 2-crossed-module

The key idea explained in section 2.5.3 is that a curvature anomaly (ie. the
monopole) can be absorbed by introducing a higher-gauge structure. In this
section, we demonstrate this also for a 2-curvature anomaly arising from a
Postnikov class, by introducing a 3-gauge structure to resolve it. We shall
take $G$ to be a connected, simply connected, compact simple Lie group of rank
$\geq 3$, eg. the spin group $G=\operatorname{Spin}(n)$ for $n\geq 3$.

##### Loop and path algebras.

We define the based path group $PG$ of $G$ as the space of maps
$\gamma:[0,1]\rightarrow G$ with basepoint $\gamma(0)=1\in G$ fixed. $PG$ is
equipped with the compact-open topology, and group multiplication is defined
point-wise

$(\gamma\gamma^{\prime})(\tau)=\gamma(\tau)\gamma^{\prime}(\tau),\qquad\gamma,\gamma^{\prime}\in
PG,$

where $\tau\in[0,1]$. As a manifold, $PG$ fits in the principal path fibration

$\Omega G\hookrightarrow
PG\xrightarrow{\pi}G\rightarrow\ast,\qquad\pi(\gamma)=\gamma(1)$

over $G$, where $\Omega G$ is the loop group over $G$ defined as the space of
maps $\alpha:S^{1}\rightarrow G$ with a distinguished basepoint
$\alpha(0)=\alpha(1)=1$.

By passing to the Lie algebra, we similarly define the path/loop algebras
$P\mathfrak{g},\Omega\mathfrak{g}$ consisting of maps from $[0,1]$ or $S^{1}$
into $\mathfrak{g}$, respectively, with initial values based at
$0\in\mathfrak{g}$. The Lie brackets are defined point-wise:

$[p,p^{\prime}](t)=[p(t),p^{\prime}(t)],\qquad p,p^{\prime}\in P\mathfrak{g},$

where $t\in[0,1]$, and similarly for the loop algebra $\Omega\mathfrak{g}$. We
may without loss of generality rescale the loop $S^{1}$ to have unit geodesic
length.

As $G$ is simple, all of its central extensions are trivial. This is not the
case, however, for its loop group $\Omega G$. Due to a result of Garland [76,
35], the loop algebra has the Lie algebra cohomology

$H^{2}(\Omega\mathfrak{g},\mathbb{R})\cong\mathbb{R},$

whence there is a one-dimensional space of non-trivial central extensions

$0\rightarrow\mathbb{R}\rightarrow\widehat{\Omega_{k}\mathfrak{g}}\rightarrow\Omega\mathfrak{g}$
(3.16)

spanned by $k\in\mathbb{R}$, called the affine Lie algebra of $\mathfrak{g}$.
Note we may write
$\widehat{\Omega_{k}\mathfrak{g}}\cong\Omega\mathfrak{g}\oplus\mathbb{R}$ as
vector space.

In order to integrate this algebra extension sequence (3.16) to the group
level, it was shown [76] that it must satisfy a certain integrality condition.
In particular, the level $k\in\mathbb{Z}$ must be quantized131313In addition
to this, the pairing $\langle\cdot,\cdot\rangle$ on $\mathfrak{g}$ must
satisfy $\langle\alpha,\alpha\rangle\in\frac{1}{2\pi}\mathbb{Z}$ for the
coroot $\alpha$ of the highest root $\mu$ of $G$., whence we obtain the group
extension sequence

$1\rightarrow U(1)\rightarrow\widehat{\Omega_{k}G}\rightarrow\Omega
G\rightarrow 1$

called the Kac-Moody extension $\widehat{\Omega_{k}G}$ of level $k$; see [35,
76] for an explicit construction.

##### The loop model for the string 2-algebra.

As shown in [35] and mentioned in Remark 2.6, the string 2-algebra
$\mathfrak{string}_{k}(\mathfrak{g})$ admits a description in terms of a non-
trivial crossed-module $\mathfrak{l}_{k}$. We have
$\mathfrak{l}_{k}=(t:\widehat{\Omega_{k}\mathfrak{g}}\cong\Omega\mathfrak{g}\oplus\mathbb{R}\rightarrow
P\mathfrak{g},\rhd)$, and

  1. 1.

the map $t$ is given by the affine projection
$\widehat{\Omega_{k}\mathfrak{g}}\rightarrow\Omega\mathfrak{g}$ composed with
the inclusion $\Omega\mathfrak{g}\hookrightarrow P\mathfrak{g}$,

  2. 2.

$\operatorname{ker}t\cong\mathbb{R}=\\{(0,c)\in\Omega\mathfrak{g}\oplus\mathbb{R}\\}$,
and $\operatorname{coker}t\cong
P\mathfrak{g}/\Omega\mathfrak{g}\cong\mathfrak{g}$ is isomorphic to the
constant paths,

  3. 3.

the action is defined as

$p\rhd(\ell,c)=([p,\ell],2k\int_{S^{1}}\langle p,\dot{\ell}\rangle),\qquad
p\in P\mathfrak{g},\quad(\ell,c)\in\widehat{\Omega_{k}\mathfrak{g}},$ (3.17)

whence the induced action $\rhd$ of $\operatorname{coker}t=\mathfrak{g}$ on
$\operatorname{ker}t=\mathbb{R}$ is trivial,

  4. 4.

the Postnikov class $[\kappa]\in H^{3}(\mathfrak{g},\mathbb{R})$ of
$\mathfrak{l}_{k}$ is given by the Diximier-Douady class $k[\omega]\in
H^{3}(G,\mathbb{Z})$ of $\operatorname{String}_{k}(G)$, where
$\omega=\langle\cdot,[\cdot,\cdot]\rangle$ is the fundamental
3-cocycle141414The Lie algebra cohomology $H^{\bullet}(\mathfrak{g})$
coincides with the de Rham cohomology $H^{\bullet}(G)$ [77]. This is called
the van Est map [78]. on $\mathfrak{g}$ [76, 35].

Notice that, in the Chevalley baiss $\\{T_{a}\\}_{a}$ of $\mathfrak{g}$, the
fundamental 3-cocycle $\omega$ gives just the structure constant $f_{ab}^{c}$.
The Lie bracket on $\widehat{\Omega_{k}\mathfrak{g}}$ is given by the
level-$k$ Kac-Moody extension

$[(\ell,c),(\ell^{\prime},c^{\prime})]=t(\ell,c)\rhd(\ell^{\prime},c^{\prime})=([\ell,\ell^{\prime}],2k\int_{S^{1}}\langle\ell,\dot{\ell}^{\prime}\rangle),$

which is also given by the Peiffer identity. We shall focus on the case of
level $k=1$ in the following for brevity, but the case for arbitrary values of
$k\in\mathbb{Z}$ can be treated identically.

##### 2-gauge structure of the loop model.

With the loop model crossed-module structure $\mathfrak{l}_{1}$, let
$(A,\Sigma)$ denote the associated 2-connection with
$\Sigma=(\sigma,c)\in\Omega^{2}(X)\otimes(\widehat{\Omega\mathfrak{g}_{1}})\cong(\Omega^{2}(X)\otimes\Omega\mathfrak{g})\oplus\Omega^{2}(X)$,
such that $t\Sigma=\sigma$. To derive the 2-curvature anomaly $\kappa(A)$, we
compute the 2-curvature as

$K=d_{A}\Sigma=(d_{A}\sigma,dc+2\int_{S^{1}}\langle
A\wedge\dot{\sigma}\rangle),$

where the prefactor of $2$ comes from the crossed-module action (3.17). On-
shell of the fake-flatness condition $\mathcal{F}=F-t\Sigma=0$, we then have

$K\sim(d_{A}F,dc+2\int_{S^{1}}\langle
A\wedge\dot{F}\rangle)=(0,dc+2\int_{S^{1}}\langle A\wedge\dot{F}\rangle),$

where we have used the 1-Bianchi identity $d_{A}F=0$. Thus we see $K$ is
valued in $\operatorname{ker}t$, consistent with what we have discussed in
section 2.3.

The 2-form connection is given in components by $\Sigma=(\sigma,c)$ with
$\sigma$ valued in $\Omega\mathfrak{g}$, the loop algebra. Since the loops are
by definition based at $0\in\mathfrak{g}$, we have in particular
$\sigma(1)=\sigma(0)=0$. The fake-flatness condition $F=t\Sigma=\sigma$ then
forces $F$ to be valued in the loop algebra $\Omega\mathfrak{g}$ as well:

$t\Sigma(1)=\sigma(1)=F(1)=dA(1)+\frac{1}{2}[A(1)\wedge A(1)]=0.$ (3.18)

Notice that this does not necessarily force $A$ to also be valued in
$\Omega\mathfrak{g}$, meaning that $A(1)$ does not have to be zero. With this
in mind, we compute that

$\langle\dot{A}\wedge[A\wedge
A]\rangle=\frac{1}{3}\frac{\partial}{\partial\tau}\langle A\wedge[A\wedge
A]\rangle$

is a total derivative in the path parameter $\tau\in[0,1]$, whence by a direct
computation we have (recalling that $A(0)=0$)

$\displaystyle\int_{S^{1}}\langle A\wedge\dot{F}\rangle$ $\displaystyle=$
$\displaystyle-\int_{0}^{1}\langle\dot{A}\wedge
dA\rangle-\frac{1}{2}\int_{0}^{1}\langle\dot{A}\wedge[A\wedge A]\rangle$
$\displaystyle=$ $\displaystyle-\langle A(1)\wedge
dA(1)\rangle-\frac{1}{2\cdot 3}\langle A(1)\wedge[A(1)\wedge
A(1)]\rangle+\int_{0}^{1}\langle A\wedge d\dot{A}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{3}\langle A(1)\wedge[A(1)\wedge
A(1)]\rangle+\int_{0}^{1}\langle A\wedge d\dot{A}\rangle,$

where we have used (3.18) in the last equality. Using (3.18) once again, we
can compute

$\displaystyle d\int_{0}^{1}\langle A\wedge\dot{A}\rangle$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\langle dA\wedge\dot{A}\rangle-\langle A\wedge
d\dot{A}\rangle$ $\displaystyle=$ $\displaystyle\langle dA(1)\wedge
A(1)\rangle-2\int_{0}^{1}\langle A\wedge d\dot{A}\rangle$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\langle A(1)\wedge[A(1)\wedge
A(1)]\rangle-2\int_{0}^{1}\langle A\wedge d\dot{A}\rangle,$

which when recombined with the above expression yields the on-shell
2-curvature

$K\sim(0,2(\frac{1}{3}-\frac{1}{4})\langle A(1)\wedge[A(1)\wedge
A(1)]\rangle+d(c-c_{1}))=(0,\frac{1}{6}\langle A(1)\wedge[A(1)\wedge
A(1)]\rangle+dc^{\prime}),$

where $c_{1}=\int_{0}^{1}\langle A\wedge\dot{A}\rangle$ and
$c^{\prime}=c-c_{1}$.

Such exact forms $dc^{\prime}$ can be absorbed into a 2-gauge
transformation151515This is because, as $P\mathfrak{g}$ acts trivially on
$\operatorname{ker}t$, a 2-gauge transformation parameterized by
$L=(L_{\ell},L_{c})\in\Omega^{1}(X)\otimes\widehat{\Omega_{1}\mathfrak{g}}$
acts by the shift $c\rightarrow c+dL_{c}$ by an exact 1-form., hence we
achieve the 2-curvature anomaly

$\kappa(A)\equiv(0,\frac{1}{3!}\langle A(1)\wedge[A(1)\wedge
A(1)]\rangle)\in\operatorname{ker}t=\mathbb{R}.$ (3.19)

This $\kappa(A)=\frac{1}{3!}\omega(A(1),A(1),A(1))$ is given precisely in
terms of the fundamental 3-cocycle $\omega=\langle\cdot,[\cdot,\cdot]\rangle$,
which is consistent with the fact that the Postnikov class of
$\mathfrak{l}_{1}$ coincides with $\omega$ in this case [35].

It is easy to see that $\kappa(A)$ is invariant under 2-gauge transformation
$A\rightarrow A+tL$. The terms we acquire are cyclic permutations of

$\langle A(1)\wedge[A(1)\wedge tL(1)]\rangle+\langle A(1)\wedge[tL(1)\wedge
tL(1)]\rangle+\langle tL(1)\wedge[tL(1)\wedge tL(1)]\rangle,$

which all vanish because $tL$ is valued in $\Omega\mathfrak{g}$, whence
$tL(1)=0=tL(0)$.

Since we have a non-trivial $\kappa(A)$, we know from section 2.4.2 that the
1-gauge transformations of $\Sigma$ need to be modified by the first
descendant $\zeta(A,\lambda)$. Let $\lambda\in P\mathfrak{g}$ denote a 1-gauge
parameter. As $P\mathfrak{g}$ acts trivially on the kernel $\mathbb{R}$, we
see that the descent equation (2.25) for $\kappa(A)$ reads

$d\zeta(A,\lambda)=\kappa(A^{\lambda})-\kappa(A)=\langle
d_{A}\lambda(1),[A(1)\wedge A(1)]\rangle$

to first order in $\lambda$. To find $\zeta(A,\lambda)$ that solves this
equation, we first notice that the total skew-symmetry of the pairing form
$\langle\cdot,[\cdot,\cdot]\rangle$ implies that

$\langle[A(1),\lambda(1)]\wedge[A(1)\wedge
A(1)]\rangle=-\langle\lambda(1),[A(1)\wedge[A(1)\wedge A(1)]]\rangle=0,$

which vanishes by the Jacobi identity in $\mathfrak{g}$, hence
$\omega(d_{A}\lambda(1),A(1),A(1))=\omega(d\lambda(1),A(1),A(1))$. Now by
Leibniz rule and (3.18),

$\displaystyle d(\omega(\lambda(1),A(1),A(1)))$ $\displaystyle=$
$\displaystyle\omega(d\lambda(1),A(1),A(1))+\langle\lambda(1),[dA(1)\wedge
A(1)]\rangle-\langle\lambda(1),[A(1)\wedge dA(1)]\rangle$ $\displaystyle=$
$\displaystyle\omega(d\lambda(1),A(1),A(1))+2\langle\lambda(1),[dA(1)\wedge
A(1)]\rangle$ $\displaystyle=$
$\displaystyle\omega(d\lambda(1),A(1),A(1))-\langle\lambda(1),[[A(1)\wedge
A(1)]\wedge A(1)]\rangle,$

the last term of which vanishes once again by the Jacobi identity in
$\mathfrak{g}$, so we can identify $\zeta$,

$d\zeta(A,\lambda)=d(\omega(\lambda(1),A(1),A(1))),\qquad\zeta(A,\lambda)=\langle\lambda(1),[A(1)\wedge
A(1)]\rangle.$ (3.20)

With such $\zeta$, we do have a proper 2-gauge theory based on the loop model
$\mathfrak{l}_{1}$ as discussed in section 2.4.2. We intend now to absorb this
2-curvature anomaly $\kappa(A)$ by describing a 3-gauge theory encoding the
same information.

######  Remark 3.3.

It is interesting to note the striking resemblance between (3.20) and (2.15),
the latter of which has appeared in [34]. It is important to note, however,
that here we are using a different 2-gauge structure as [34]. In their
language, we are in the "unadjusted" case where $K$ transforms covariantly
under the adjoint representation of $P\mathfrak{g}$, and $\mathcal{F},\Sigma$
transform independently. In the adjusted case, $K$ is invariant, and
$\mathcal{F},\Sigma$ form a $\mathfrak{g}$-multiplet.

##### 2-monopoles and anomalous 2-gauge transformations.

We now cast $\kappa(A)$ as a 2-monopole defect. Let $X$ be a spin 5-manifold.
We define the 2-monopole charge conventionally in terms of its current as

$\tilde{q}_{m}=\frac{1}{8\pi}\int_{W}\ast\tilde{j}_{m},$ (3.21)

where $\tilde{j}_{m}$ is the 1-form 2-monopole current. In analogy with the
monopole case discussed in section 2.5.3, the 2-monopoe current violates the
2-Bianchi identity (2.9) of the loop model 2-gauge theory we have described
above.

To see this, we assume $W=V\times l$ decomposes into a closed 3-cycle and the
2-monopole worldline $l$. We introduce an anomalous EOM $dK=\ast\tilde{j}_{m}$
valued in $\operatorname{ker}t=\mathbb{R}$, such that integrating across the
3-cycle $V\subset X$ yields

$\int_{V}K=\int_{W}dK\neq 0$

by Stokes’s theorem. This means that the 2-form connection $\Sigma$ acquires a
non-trivial 2-holonomy [79, 38] about $V$, which coincides with the 2-monopole
charge. On the other hand, the 2-curvature anomaly EOM $K=\kappa(A)$ implies
that this 2-holonomy is precisely the period of $\kappa(A)$ in (3.19),

$\tilde{q}_{m}=\frac{1}{8\pi}\int_{W}\ast\tilde{j}_{m}=\frac{1}{2(4\pi)}\int_{S^{3}}K=\frac{1}{8\pi}\int_{S^{3}}\kappa(A).$
(3.22)

Such periods are classified by the Postnikov class $[\kappa]=[\omega]\in
H^{3}(\mathfrak{g},\mathbb{R})$ (via pullback), therefore if $[\omega]\in
H^{3}(\mathfrak{g},\mathbb{Z})\subset H^{3}(\mathfrak{g},\mathbb{R})$ is an
integral class161616As is the case for simple Lie algebras, where the
structural constants $f_{ab}^{c}\in\mathbb{Z}$ are integral in the Chevalley
basis $\\{T_{a}\\}_{a}$ of $\mathfrak{g}$., then the 2-monopole charge
$\tilde{q}\in\mathbb{Z}$ is quantized.

We recall that the monopole defect studied in section 2.5.3 can be constructed
from clutching an anomalous 1-gauge transformation. We do the same for the
2-monopole defect here. By a similar excision trick as in section 2.5.3, we
take a tubular neighborhood about the 2-monopole worldline $l$ and remove it
from the 5-manifold $X$. This yields a new 5-manifold $X^{\prime}$ with
boundary $\partial X^{\prime}\cong S^{3}\times l$, away from which $K$ is
regular. We acquire two components $L_{0},L_{1}$ in a 2-gauge transformation

$\Sigma\rightarrow\Sigma+d_{A}L_{0}+\frac{1}{2}L_{0}^{2}+d_{A}L_{1}+\frac{1}{2}L_{1}^{2},$

for which $L_{0},L_{1}$ are regular on $X^{\prime}$, but $L_{0}$ can be
smoothly extended into $X$ while $L_{1}$ cannot. We call $L_{1}$ singular, and
it means, once again, that $d^{2}L_{1}\neq 0$ globally on $X$, hence

$d^{2}_{A}L_{1}-F\wedge^{\rhd}L_{1}=d^{2}L_{1}\neq 0.$

From the computations in (2.18), this leads to the anomalous 2-gauge
transformation

$K\rightarrow K+\mathcal{F}\wedge^{\rhd}(L_{0}+L_{1})+d^{2}L_{1}\sim
K+d^{2}L_{1},$

where $\sim$ means going on-shell of the fake-flatness condition
$\mathcal{F}=F-t\Sigma=0$.

As $K$ is smooth and regular within $X^{\prime}$, it cannot yield a non-
trivial 2-monopole charge. The sole contributor of $\tilde{q}_{m}$ must then
come from the anomalous component $L_{1}$:

$\tilde{q}_{m}=\frac{1}{8\pi}\int_{S^{3}}K\xrightarrow{\sim}\frac{1}{8\pi}\int_{S^{3}}d^{2}L_{1}=\frac{1}{4\pi}\int\limits_{S^{2}=H_{+}\cap
H_{-}}dL_{1}\neq 0,$

where we have used Stokes’s theorem on the two patches $D^{3}\cong
H_{\pm}\subset S^{3}$ that cover the 3-sphere. The singular 2-gauge parameter
$L_{1}$ has the structure of a "hedghog defect" [5, 2], and its 2-dimensional
winding number is identified with $\tilde{q}_{m}$.

######  Remark 3.4.

The 2-monopole defect, being point-like in 5D, behaves similarly to the
1-monopole in 4d studied in section 2.5.3. In particular, two 2-monopole
defects may fuse such that their charges add, corresponding to the additivity

$\kappa(\mathcal{P}\otimes\mathcal{P}^{\prime})=\kappa(\mathcal{P})+\kappa(\mathcal{P}^{\prime})$

of the Postnikov classes associated to the 2-gauge bundles
$\mathcal{P},\mathcal{P}^{\prime}\rightarrow X$; see [18].

This clutching construction [58] for the 2-monopole allows us to match the
2-curvature anomaly with an anomalous 2-gauge transformation,
$\kappa(A)=d^{2}L_{1}$, via (3.22) on the boundary 3-sphere $S^{3}$. Writing
$dL_{1}=\mathcal{L}$, we once again see a "2-shift transformation"

$\Sigma\rightarrow\Sigma+\mathcal{L},$ (3.23)

manifesting before us, generating a "$U(1)_{2}$ 2-form symmetry". This
suggests the 3-gauge structure we should define in order to resolve the
anomaly $\kappa(A)$.

##### Resolving the 2-monopole anomaly; the 3-gauge structure.

We can now introduce a 3-form field $\Gamma$ to absorb the 2-curvature
anomaly. This is encoded by the 2-monopole condition

$\int_{V}\Gamma=\int_{V}K,\qquad\forall\leavevmode\nobreak\ \text{closed
3-surfaces }V\subset X^{\prime},$ (3.24)

stating that the quantized 2-monopole charge $\tilde{q}_{m}\in\mathbb{Z}$ is
matched by the flux of the 3-form $\Gamma$.

If we take $t^{\prime}=\operatorname{id}$ as the identity (strictly speaking
an inclusion) as suggested by (3.23) and (3.24), to achieve the exactness
$tt^{\prime}=0$ of the complex

$\mathfrak{i}\xrightarrow{t^{\prime}=\operatorname{id}}\widehat{\Omega_{k=1}\mathfrak{g}}\xrightarrow{t}P\mathfrak{g}$
(3.25)

we must have
$\mathfrak{i}=\operatorname{im}t^{\prime}=\operatorname{ker}t=\mathbb{R}$.
This means that $P\mathfrak{g}$ acts trivially $\rhd^{\prime}=0$ on
$\mathfrak{i}=\mathbb{R}$, and hence on the $\mathfrak{i}$-valued 3-connection
$\Gamma$. Moreover, $\operatorname{ker}t^{\prime}=0$ means that the degree-4
Postnikov class $[\tau]$ for (3.25) is trivial.

As
$t^{\prime}:\mathfrak{i}=\mathbb{R}\hookrightarrow\mathfrak{h}=\widehat{\Omega_{1}\mathfrak{g}}$
is an inclusion, we have
$t^{\prime}\\{\cdot,\cdot\\}_{\text{Pf}}=\\{\cdot,\cdot\\}_{\text{Pf}}$. The
lifting condition (3.3) then gives

$\\{(\ell,c),(\ell^{\prime},c^{\prime})\\}_{\text{Pf}}=[(\ell,c),(\ell^{\prime},c^{\prime})]|_{\operatorname{ker}t}=2\int_{S^{1}}\langle\ell,\dot{\ell}^{\prime}\rangle$

for
$(\ell,c),(\ell^{\prime},c^{\prime})\in\widehat{\Omega_{1}\mathfrak{g}}\cong\Omega\mathfrak{g}\oplus\mathbb{R}$.
Here, $\\{\cdot,\cdot\\}_{\text{Pf}}$ is skew-symmetric, as can be seen by an
integration by parts, but it is not trivial. Equipped with this Peiffer
lifting map, the complex (3.25) forms a 2-crossed-module which we call the
string 2-crossed-module.

The quadratic form $Q$, however, does vanish when evaluated on forms of even
degrees, such as the 2-form connection $\Sigma=(\sigma,c)$,

$Q_{\Sigma}=\int_{S^{1}}\langle\sigma\wedge\dot{\sigma}\rangle=-\int_{S^{1}}\langle\dot{\sigma}\wedge\sigma\rangle=-Q_{\Sigma},$

thus the modified 3-curvature $\mathcal{H}=d\Gamma$ coincides with the
unmodified 3-curvature. The anomalous EOM
$d(K|_{\operatorname{ker}t})=\ast\tilde{j}_{m}$ and the 2-monopole condition
(3.24) together identify the integral of $\frac{1}{8\pi}\mathcal{H}$ over the
boundary 4-surface $\partial X^{\prime}=W=S^{3}\times l$ with the 2-monopole
charge.

In the interior of the 5-manifold $X^{\prime}$, we interpret the 2-monopole
condition (3.24) as a 2-fake-flatness condition

$t^{\prime}\Gamma=K.$

As $P\mathfrak{g}$ acts trivially on
$\mathbb{R}=\operatorname{ker}t=\operatorname{im}t^{\prime}$, we can express
the 3-curvature by taking the covariant derivative,

$\mathcal{H}\sim
d_{A}K=d_{A}^{2}\Sigma=F\wedge^{\rhd}\Sigma=([F\wedge\sigma],2\int_{S^{1}}\langle
F\wedge\dot{\sigma}\rangle).$

If we now go on-shell of the fake-flatness condition $\mathcal{F}=0$, we then
have the 3-flatness condition

$\mathcal{H}\sim([F\wedge F],2\int_{S^{1}}\langle
F\wedge\dot{F}\rangle)=(0,\langle F(1)\wedge F(1)\rangle)=0,$ (3.26)

where we recall that fake-flatness $F=\sigma$ implies $F$ is valued in
$\Omega\mathfrak{g}$, which forces $F(1)=0$. The fake-flatness, 2-fake-
flatness and 3-flatness conditions are therefore mutually consistent, and the
fields $(A,\Sigma,\Gamma)$ define a flat 3-connection based on (3.25) in the
interior of $X^{\prime}$.

######  Remark 3.5.

What about near the boundary $W=\partial X^{\prime}$? By construction, $W$
contains a tubular neighborhood of the 2-monopole worldline $l$, at which the
field $\Sigma$ is singular. The 3-curvature $\mathcal{H}$ then receives a
contribution from $d^{2}\Sigma=dK\neq 0$, which on-shell of the 2-fake-
flatness and fake-flatness conditions is given precisely by the 2-monopole
current $\ast\tilde{j}_{m}$, as we have mentioned previously.

We now demonstrate the gauge invariance of the 2-monopole/2-fake-flatness
condition (3.24). First, as $\kappa$ and $K$ are both valued in
$\mathbb{R}=\operatorname{ker}t$, the 2-monopole charge is invariant under a
1-gauge transformation $\lambda$. As such $\Gamma$ is also invariant. Next,
under the 3-gauge transformation (3.23), the 2-monopole charge $\tilde{q}_{m}$
receives contributions from the possibly non-trivial periods

$\frac{1}{8\pi}\int_{V}d\mathcal{L}\neq
0,\qquad\mathcal{L}\in\Omega^{3}(X)\otimes\mathfrak{i}.$

As such, the 3-form connection must transform accordingly

$\Gamma\rightarrow\Gamma+d\mathcal{L},$

in order to preserve the 2-monopole condition (3.24).

Recall that the quadratic form $Q$ necessarily vanishes on 2-forms valued in
the loop algebra $\Omega\mathfrak{g}$, so under a regular 2-gauge
transformation $L_{0}=L$, the 3-curvature $\mathcal{H}$ must be invariant,
according to (3.10). This means that the 3-connection $\Gamma$ then transforms
by an exact 3-form $d\phi$, $\Gamma\rightarrow\Gamma+d\phi$. On-shell of the
2-fake-flatness condition, we can infer this 2-form $\phi$ from how the
2-curvature $K$ transforms. Indeed, recalling that the 2-curvature $K$ is
valued in $\operatorname{ker}t=\mathbb{R}$ on-shell of the fake-flatness
condition, we consider the sector $L|_{\operatorname{ker}t}$ and compute

$K\rightarrow
K^{L|_{\operatorname{ker}t}}=K+d^{2}L|_{\operatorname{ker}t}+\frac{1}{2}d[L\wedge
L]|_{\operatorname{ker}t}=K+\frac{1}{2}d[L\wedge L]|_{\operatorname{ker}t}.$

As such, in order to preserve (3.24), the exact 3-form $d\phi$ must be given
by $\phi=\frac{1}{2}[L\wedge L]|_{\operatorname{ker}t}$, up to an exact
1-form. To show that this is consistent with (3.10), we compute the 2-gauge
transformation

$\displaystyle\\{\Sigma^{L}\wedge
L\\}_{\text{Pf}}+\\{L\wedge\Sigma\\}_{\text{Pf}}$ $\displaystyle=$
$\displaystyle\\{\Sigma\wedge
L\\}_{\text{Pf}}+\\{L\wedge\Sigma\\}_{\text{Pf}}$
$\displaystyle+\\{d_{A}L\wedge L\\}_{\text{Pf}}+\frac{1}{2}\\{[L\wedge
L]\wedge L\\}_{\text{Pf}}\qquad\text{Skew-symmetry of
$\\{\cdot,\cdot\\}_{\text{Pf}}$}$ $\displaystyle=$
$\displaystyle\\{d_{A}L\wedge L\\}_{\text{Pf}}+\frac{1}{2}\\{[L\wedge L]\wedge
L\\}_{\text{Pf}}\qquad\text{Equivariance of $\\{\cdot,\cdot\\}_{\text{Pf}}$}$
$\displaystyle=$ $\displaystyle\frac{1}{2}d_{A}\\{L\wedge
L\\}_{\text{Pf}}+\frac{1}{2}\\{[L\wedge L]\wedge
L\\}_{\text{Pf}}\qquad\text{Jacobi identity}$ $\displaystyle=$
$\displaystyle\frac{1}{2}d_{A}\\{L\wedge L\\}_{\text{Pf}}\qquad\text{Action
$\rhd^{\prime}=0$ is trivial on $\mathfrak{i}=\operatorname{ker}t^{\prime}$}$
$\displaystyle=$ $\displaystyle\frac{1}{2}d\\{L\wedge L\\}_{\text{Pf}},$

which is indeed nothing else than $\phi=\frac{1}{2}\\{L\wedge
L\\}_{\text{Pf}}=\frac{1}{2}[L\wedge L]|_{\operatorname{ker}t}$ by the lifting
condition (3.3). This means that we have a fully-fledged 3-gauge theory based
on the 2-crossed-module (3.25) in the interior of $X^{\prime}$, as introduced
in section 3 and [33].

In summary, we have demonstrated a 3-group analogue of the observation made in
[23, 24] and section 2.5.3: an anomalous 2-group (mixed 0- and 1-form)
symmetry — in the sense that the 2-Bianchi identity $dK=\ast\tilde{j}_{m}$ is
violated by the presence of a 2-monopole — gauges into a 3-group symmetry,
which involves 0-, 1- and 2-form symmetries.

##### 3-Yang-Mills theory.

With the 3-gauge structure based on the string 2-crossed-module in hand, we
now construct a non-anomalous 3-Yang-Mills theory $S_{\text{3YM}}$ that
captures the 2-monopole. (We still consider $k=1$ in the following.)

Analogous to what was done in section 2.5.3, we begin by considering a trivial
2-gauge theory $\mathcal{P}\rightarrow X$ (ie. vanishing Postnikov class
$[\kappa]=0$) and $X$ a spin manifold of dimension 5, and insert a non-trivial
2-monopole charge through a 2-form $\mathcal{L}$ that has a non-trivial
quantized period,

$\tilde{q}_{m}=\frac{1}{8\pi}\int_{V}\Gamma\in\mathbb{Z},\qquad
d\mathcal{L}=\Gamma.$ (3.27)

The 2-monopole condition (3.24) allows us to treat $\Gamma$ as a pure-gauge
3-connection, which induces a 3-gauge shift symmetry

$\Gamma\rightarrow\Gamma+d\mathcal{L}^{\prime}$

parameterized by regular 2-forms $\mathcal{L}^{\prime}\in\Omega^{2}(X)$. Our
fields $(A,\Sigma,\Gamma)$ then acquire the 3-gauge symmetry structure based
on the Lie 3-algebra given by the string 2-crossed-module (3.25). The goal is
now to construct an action which equations of motion have such shape (3.27)
for solution.

We define the 3-Yang-Mills action [80]

$S_{\text{3YM}}=\int_{X}\ast\mathcal{F}\wedge\mathcal{F}+\ast\mathcal{G}\wedge\mathcal{G},\quad\textrm{with
}\mathcal{F}=F-t\Sigma,\quad\mathcal{G}=K-\Gamma.$ (3.28)

As pairing, we use the non-degenerate pairing on the path and loop algebras

$\langle p,p^{\prime}\rangle_{P\mathfrak{g}}=\int_{[0,1]}\langle
p,p^{\prime}\rangle,\qquad\langle(\ell,c),(\ell^{\prime},c^{\prime})\rangle_{\widehat{\Omega_{1}\mathfrak{g}}}=\int_{S^{1}}\langle\ell,\ell^{\prime}\rangle+cc^{\prime}$

composed with the Hodge pairing on $\Omega^{\bullet}(X)$. The action (3.28) is
the 3-gauge generalization of the higher-Yang-Mills theory developed in [63].

By construction, $S_{\text{3YM}}$ is invariant under 3-gauge transformations
(3.10). By varying the 3-gauge $\Gamma$, we obtain the 2-fake-flatness EOM
$\mathcal{G}=K-\Gamma=0$. Once again, we see that no non-trivial 2-monopole
charge can be introduced on-shell of this EOM, as the original 2-gauge bundle
$\mathcal{P}$ was by hypothesis trivial.

To amend this, we source the 1-, 2- and 3-form connections $A,\Sigma,\Gamma$,
respectively, with the 1-, 2- and 3-form currents $j,J,\mathcal{J}$. This
introduces the following term

$S_{\text{3cur}}=\int_{X}A\wedge\ast j+\Sigma\wedge\ast
J+\Gamma\wedge\ast\mathcal{J}$

to $S_{\text{3YM}}$ (3.28). The properties of the currents are listed below

Currents | Form degree | Valued in
---|---|---
$j$ | 1 | $P\mathfrak{g}$
$J$ | 2 | $\widehat{\Omega_{1}\mathfrak{g}}\cong\Omega\mathfrak{g}\oplus\mathbb{R}$
$\mathcal{J}$ | 3 | $\mathbb{R}=\operatorname{ker}t$

Upon introducing these currents, a variation of the 3-gauge $\Gamma$ then
yields the modified EOM

$\ast\mathcal{H}=\ast\mathcal{J}\implies d\Gamma=-d\mathcal{J},$

where we have used the 2-Bianchi identity $d_{A}K=0$ and the fact that
$P\mathfrak{g}$ acts trivially on $\operatorname{ker}t$. This makes it
possible to identify the flux $d\mathcal{J}$ of the 3-current $\mathcal{J}$
with the 2-monopole and obtain what we sought: as the quantized period of
$\mathcal{L}$, we have

$\tilde{q}_{m}=\frac{1}{8\pi}\int_{V}d\mathcal{L}=\frac{1}{8\pi}\int_{V\times
l}d\Gamma=-\frac{1}{8\pi}\int_{V\times l}d\mathcal{J},$

where $l$ is the worldline. This allows to relate the 2-monopole current
$\tilde{j}_{m}$ (3.21) with ${\cal J}$,

$d\mathcal{J}=-\ast\tilde{j}_{m}.$

##### 3-conservation laws and higher mobility constraints.

We now derive higher-conservation laws of $S_{\text{3YM}}+S_{\text{3cur}}$ by
making 3-gauge transformations.

  1. 1.

3-gauge transformation $\mathcal{L}^{\prime}$: the 1-connection $A$ is
unaffected, and the sourcing terms introduce

$S_{\text{3cur}}\rightarrow
S_{\text{3cur}}+\int_{X}\mathcal{L}^{\prime}\wedge\ast
J+d\mathcal{L}^{\prime}\wedge\ast\mathcal{J}.$

Note that $\mathcal{L}^{\prime}$ (the prime is to distinguish
$\mathcal{L}^{\prime}$ from the $\mathcal{L}$ chosen in (3.27)) is valued in
$\mathbb{R}=\operatorname{ker}t$; by writing $J=(J_{\ell},J_{c})$ in the
$\widehat{\Omega_{1}\mathfrak{g}}\cong\Omega\mathfrak{g}\oplus\mathbb{R}$-components,
we achieve

$\int_{X}\mathcal{L}^{\prime}\wedge\ast
J_{c}+d\mathcal{L}^{\prime}\wedge\ast\mathcal{J}=\int_{X}\mathcal{L}^{\prime}\wedge(\ast
J_{c}-d\ast\mathcal{J}),$

and hence the first-level conservation law

$d\ast\mathcal{J}=\ast J_{c},\qquad
J=(J_{\ell},J_{c})\in\Omega\mathfrak{g}\oplus\mathbb{R},\quad{\cal
J}\in\mathbb{R}.$ (3.29)

  2. 2.

2-gauge transformation $L$: the sourcing terms introduce

$S_{\text{3cur}}\rightarrow S_{\text{3cur}}+\int_{X}tL\wedge\ast
j+(d_{A}L+\frac{1}{2}[L\wedge L])\wedge\ast J+d\phi\wedge\ast\mathcal{J},$

where $\phi=\frac{1}{2}\\{L\wedge
L\\}_{\text{Pf}}=-\frac{1}{2}L^{2}|_{\operatorname{ker}t}$. To extract
conservation laws, we must introduce new operations.

Using the dual map
$t^{*}:P\mathfrak{g}\rightarrow\widehat{\Omega_{1}\mathfrak{g}}$ defined in
section 2.5.1, we can rewrite the first term as

$-\int_{X}L\wedge t^{*}\ast j.$

For the second term, we use an integration by parts and split
$L=(L_{\ell},L_{c})$ into $\Omega\mathfrak{g}\oplus\mathbb{R}$-components,
such that we have

$-\int_{X}L_{\ell}\wedge d_{A}\ast J_{\ell}+\frac{1}{2}\int_{X}[L_{\ell}\wedge
L_{\ell}]\wedge\ast J_{\ell}+\frac{1}{2}\int_{X}[L_{c}\wedge L_{c}]\wedge(\ast
J_{c}-d\ast\mathcal{J}),$

where $[L_{c}\wedge L_{c}]=[L\wedge L]|_{\operatorname{ker}t}$ as in our
previous computation. Note the final term vanishes on-shell of (3.29), and
hence does not give us anything new.

Combining these terms, we obtain

$-\int_{X}L_{\ell}\wedge(t^{*}\ast j+d_{A}\ast
J_{\ell})+\frac{1}{2}\int_{X}[L_{\ell}\wedge L_{\ell}]\wedge\ast J_{\ell}.$

If we suppose, for simplicity, that both of these terms individually vanish,
then we obtain the set of second-level conservation laws

$d_{A}\ast J_{\ell}=-t^{*}\ast j,\qquad\ast J_{\ell}=0.$ (3.30)

Of course, the second equation is not imposed if we have kept only the terms
linear in $L$ in the 2-gauge transformations.

  3. 3.

1-gauge transform $\lambda$: the 3-connection $\Gamma$ is unaffected, and the
sourcing terms introduce

$\displaystyle S_{\text{3cur}}$ $\displaystyle\rightarrow$ $\displaystyle
S_{\text{3cur}}+\int_{X}d_{A}\lambda\wedge\ast j-\lambda\rhd\Sigma\wedge\ast
J$ $\displaystyle=$ $\displaystyle S_{\text{3cur}}-\int_{X}\lambda(d_{A}\ast
j)-\int_{X}[\lambda,\sigma]\wedge\ast
J_{\ell}-2\int_{X}\left[\int_{S^{1}}\langle\lambda,\dot{\sigma}\rangle\right]\wedge\ast
J_{c},$

where we have used the definition of the action $\rhd$, and written
$\Sigma=(\sigma,c)$ in components. Recall we have kept the pairing form
$\langle\cdot,\cdot\rangle_{P\mathfrak{g}}$ implicit, such that we have

$\int_{X}\lambda(d_{A}\ast j)=\int_{X}\langle\lambda,d_{A}\ast
j\rangle_{P\mathfrak{g}}=\int_{X}\int_{[0,1]}\langle\lambda,d_{A}\ast
j\rangle$

by definition, for instance.

Due to the cyclic symmetry of the function
$\langle\cdot,[\cdot,\cdot]\rangle$, we can write

$\langle[\lambda,\sigma]\wedge\ast
J_{\ell}\rangle_{\Omega\mathfrak{g}}=\langle[\sigma\wedge\ast
J_{\ell}],\lambda\rangle_{\Omega\mathfrak{g}},$

which brings the sourcing terms to the form

$-\int_{X}\lambda(d_{A}\ast j+[\sigma\wedge\ast
J_{\ell}])-2\int_{X}\int_{S^{1}}\langle\dot{\lambda},\sigma\wedge\ast
J_{c}\rangle.$ (3.31)

Now as $\sigma\in\Omega\mathfrak{g}$, we can perform an integration by parts
on the second term of (3.31) without a boundary term,

$\int_{S^{1}}\langle\dot{\lambda},\sigma\wedge\ast
J_{c}\rangle=-\int_{S^{1}}\langle\lambda,\dot{\sigma}\wedge\ast J_{c}\rangle.$

Therefore, (3.31) yields the third-level conservation law

$d_{A}\ast j+[\sigma\wedge\ast J_{\ell}]=\dot{\sigma}\wedge\ast J_{c}.$ (3.32)

It is clear that the 1-current $j$ is conserved provided the 2-current
$J=(J_{\ell},J_{c})=0$ vanishes. This is guaranteed by the condition
$J_{\ell}=0$ in (3.30), as well as the requirement that the 3-current
$\mathcal{J}$ be conserved, which implies $J_{c}=0$ from (3.29). Therefore the
conservation of the 1-current does not impose any new conditions.

A direct consequence of these 3-conservation laws (3.29)-(3.32) is the very
interesting mobility constraint. The immobility of the 1-form Kac-Moody
charges, $J_{\ell},J_{c}=0$, guarantees that both the 2-form $U(1)_{2}$ and
the $PG$ 0-form symmetries are non-anomalous, in the sense that the
corresponding 3-,1-currents $\mathcal{J},j$ are conserved,

$d\ast\mathcal{J}=0,\qquad d_{A}\ast j=0,\qquad d_{A}\ast J_{\ell}=t^{*}(\ast
j)=0.$

The last conservation law states that the 1-current $j$ must be valued in
$\operatorname{ker}t^{*}\subset P\mathfrak{g}$. By the rank-nullity theorem,
we have

$\operatorname{ker}t^{*}\cong\operatorname{coker}t=P\mathfrak{g}/\Omega\mathfrak{g}\cong\mathfrak{g},$

so it appears that the $PG$-charges that descend to those of $G$ can still
remain mobile. Therefore, the only charges that are allowed to be mobile are
the ones labeled by the 0-form $G$ and the 2-form $U(1)_{2}$ symmetries.
Interestingly, despite us starting with the loop model of string 2-algebra,
this "mobility data" constitutes precisely the skeletal model [35, 34]; see
Remark 2.4.

##  4 Conclusion

In this paper, we have motivated higher-gauge structures from the gauge
theoretic perspective. This was done by the procedure of "gauging the gauge",
or equivalently by shifting the $n$-connection in an $n$-gauge theory. We then
introduce a $(n+1)$-form connection which is related to the $n$-curvature, so
that $n$-curvature has really become gauge data. This allows us to absorb the
$n$-curvature anomaly — which may include contributions from topological
characteristic classes of degree $n$ — into a $(n+1)$-connection through
anomaly resolution. In other words, an anomaly in the $n$-curvature,
regardless if it is topological (eg. a monopole) or geometric (eg. a massive
particle), can be considered as a choice of the $(n+1)$-gauge (possibly gauge
fixed).

We have shown that this anomaly resolution procedure [23, 24] categorifies the
$n$-gauge structure, and yields a higher $(n+1)$-gauge theory. Furthermore,
the structure $(n+1)$-group in general has the identity as its "top-level"
morphism, therefore it is always possible to pick a flat $(n+1)$-gauge theory
(ie. hosts no curvature anomalies), at least up to elementary equivalence
[36]. We have demonstrated explicitly these concepts in the cases of $n=1$ and
$2$.

As such, one may be left with the impression that all $n$-gauge theories for
$n<\infty$ can be embedded in an $\infty$-gauge theory [81, 82], which by
construction does not have any curvature anomalies. This is true, but in
general this categorification procedure cannot continue forever if the
spacetime dimension $d=\operatorname{dim}X$ is fixed and finite. It must
terminate at the $(d-1)$-form connection, as one does not have the notion of
curvature on $X$ for connections of degrees higher than $d-1$.

Moreover, focusing specifically on the $n$-BF theory, one may notice from
sections 2.5.1 and 3.3.1 that $n$-BF theory has a dual given by a $n$-gauge
theory only when $d=\operatorname{dim}X=n+2$, in which case the symmetries of
the $(d-2)$-BF theory is naturally described by an appropriate notion of a
"$(d-2)$-Manin triple". Though a $(d-1)$-gauge theory may be well-defined on
$X$, there is just no sufficient room for the $n$-Manin triple to fit on a
$d$-manifold $X$ if $n>d-2$. This can be rephrased as saying essentially that
the dynamics of $k$-dimensional excitations for $k\leq n$ is most naturally
embedded in $(n+2)$-dimensions; indeed, one needs two additional ambient
dimensions in order to braid excitations, and braiding is one of the central
operations that is afforded by a Drinfel’d double [83].

As we have covered a very wide range of topics in physics and mathematics, our
treatment may not be completely rigorous. However, we believe to have given
sufficient references such that our readers can find further details in them.
Nevertheless, our treatment exposes many open problems that may be tackled by
future research. In the following, we organize a few of them.

##### Drinfel’d double and tube algebra.

It is known that the Drinfel’d double $D(SU(2))$ serves as the symmetries of
Euclidean 3d $BF$ theory [84]. It is in fact equivalent to the Ocnenau’s tube
algebra over $G=SU(2)$ in (2+1)d [61].

On the other hand, we have explained in section 2.5.2 how the structure of the
trivial 2-group $\mathcal{I}_{SU(2)}$ arises from Euclidean 3d $BF$ theory
when one sources a non-trivial curvature anomaly. The trivial 2-group
$\mathcal{I}_{U(1)}$ also arises from resolving the monopole anomaly in
section 2.5.3, and we have made the observation in Remark 2.11 that the fusion
algebra of these monopoles gives rise to Ocneanu’s tube algebra over $U(1)$
(or over its group characters $\widehat{U(1)}=\mathbb{Z}$). As such, there
seems to be an interesting interplay between the trivial 2-group
$\mathcal{I}_{G}$ and the Drinfel’d double $D(G)$.

It is known that $D(G)$ in certain circumstances admits a description as an
algebra of a certain groupoid [43], and hence one may model the tube algebra
as a groupoid algebra. Upon categorification, the (3+1)d tube algebra has been
constructed as a 2-groupoid algebra in [62]. It would then be interesting to
construct a notion of a "2-Drinfel’d double" (see for example in [85, 86])
that fits with the (3+1)d tube algebra structure.

##### The classification of (3+1)d SPT phases and the Kitaev model.

The Drinfel’d double $D(G)$ also makes an appearance in condensed matter
theory. Specifically for $G=\mathbb{Z}_{2}$, the 2d toric code is defined by
the representation category $\operatorname{Rep}(\mathbb{Z}_{2})$ [87]. The
effective (2+1)d bulk topological quantum field theory (TQFT) is described by
the Kitaev model, whose spectrum of excitations is given by the Drinfel’d
centre ${Z}(\operatorname{Rep}(\mathbb{Z}_{2}))$ [20, 88]. The Drinfel’d
double $D(\mathbb{Z}_{2})$ is by construction such that

$\operatorname{Rep}_{D(\mathbb{Z}_{2})}\simeq
Z(\operatorname{Rep}_{\mathbb{Z}_{2}}),$

which allows one to construct a 1-$BF$ $\mathbb{Z}_{2}$-gauge theory with the
symmetry $D(\mathbb{Z}_{2})$ that describes the Kitaev model [89].

Similarly, we have seen in section 2.5.4 how to construct a (3+1)d topological
order using a discrete 2-group $\mathcal{D}(\mathbb{Z}_{2})$. It hosts a
magnetic string and an electric charge defect, both labeled by
$\mathbb{Z}_{2}$ [30]. When the emergent electric charge is bosonic, it has
been postulated [19] (and rigorously shown [90]) that the effective bulk TQFT
is the (3+1)d toric code, described by the Drinfel’d centre
$Z(\operatorname{2Rep}_{\mathbb{Z}_{2}})$ of the 2-representation 2-category
$\operatorname{2Rep}_{\mathbb{Z}_{2}}$ [20]. Thus, it is reasonable to expect
a 2-Drinfel’d double $D$ related to $\mathbb{Z}_{2}$ to manifest such that

$\operatorname{2Rep}_{D}\simeq Z(\operatorname{2Rep}_{\mathbb{Z}_{2}}).$ (4.1)

It would be valuable to explicitly construct this categorical equivalence, and
study the resulting 2-BF $d$-gauge theory that describes the (3+1)d toric
code. This is currently being undertaken by the authors.

In case when the electric charge is fermionic, however, there are two distinct
classes of (3+1)d TQFTs [90]. One is the spin $\mathbb{Z}_{2}$-gauge theory
described by $Z_{1}(\Sigma{\bf sVect})$, where $\Sigma{\bf sVect}$ is the
condensation completion [91] 2-category of supervector spaces. The other is
its anomalous version, which hosts the gravitational anomaly (2.52) briefly
studied in section 2.5.4. To achieve an effective field theory in these cases
would require to develop the notion of quantum 2-groups and their
2-representation theory.

##### String structures and the $S^{1}$-transgression map in holography.

In section 3.3.2, we gave an example of a 2-crossed-module (3.25) that had
arisen through the anomaly resolution procedure for the string 2-algebra in
the loop model [35]. We have shown that a consistent, flat 3-gauge theory can
be identified in the interior of the excised 5-manifold $X^{\prime}$, thereby
categorifying the statement made in [23, 24]: that an anomalous $(n-1)$-form
symmetry can be gauged by introducing an $n$-form symmetry.

Near the boundary 4-manifold $W=\partial X^{\prime}$ (which we recall contains
the 2-monopole worldline $l$), the 3-curvature $\mathcal{H}$ receives a
contribution from the singularity in the 2-form connection $\Sigma$ (see
Remark 3.5). On the other hand, if we instead relax the fake-flatness
condition to a weaker, differentiated version
$\dot{\mathcal{F}}=\dot{F}-t\dot{\Sigma}=0$, then we could obtain another
contribution

$\mathcal{H}\sim(0,\langle F(1)\wedge
F(1)\rangle)=\operatorname{tr}(F(1)^{2})$ (4.2)

aside from the 2-monopole current $\tilde{j}_{m}\neq 0$, as $F(1)$ is no
longer required to vanish. This contribution $-4\pi p_{1}\in
H^{4}(W,\mathbb{Z})$ is the first Pontrjagyn class171717As we have normalized
the length of $S^{1}$, there is a missing factor of $1/2\pi$ in the usual
normalization $p_{1}=\frac{1}{8\pi^{2}}\operatorname{tr}F^{2}$. of the
boundary 4-manifold $\partial X^{\prime}=W$.

In light of (4.2), if we now insist upon the 3-flatness $\mathcal{H}\sim 0$
across the boundary of $X^{\prime}$, then we have

$\mathcal{H}\sim
d^{2}\Sigma+\operatorname{tr}(F(1)^{2})=0\implies\tilde{j}_{m}=4\pi p_{1},$

which matches the 2-monopole current with the first Pontrjagyn class.

  1. 1.

Due to the factor of $1/8\pi$ in the definition (3.21) of the 2-monopole
charge, the above matching condition implies

$\tilde{q}_{m}=\frac{1}{8\pi}\int_{W}\ast\tilde{j}=\frac{1}{2}\int_{W}p_{1}.$

It is known that the fractional first Pontrjagyn class
$\frac{1}{2}p_{1}\mod\mathbb{Z}$ is an obstruction class for string structures
[22]. Therefore, the quantization of the 2-monopole charge
$\tilde{q}_{m}\in\mathbb{Z}$ in fact implies the existence of a string
structure on the boundary 4-manifold $\partial X^{\prime}=W$.

  2. 2.

Given the first Chern class $\operatorname{tr}F(1)=0$ vanishes, $p_{1}$
coincides with (minus) the second Chern class $c_{2}$ [67, 76]. This means
that the second Chern class controls the 2-monopole defect, analogous to how
the first Chern class controls the monopole defect; see section 2.5.3. The
procedure of anomaly resolution then allows us to associate the 2-curvature
anomaly given by the fundamental 3-cocycle $\omega$ to the second Chern class
$c_{2}$ — namely it constructs an inverse to the $S^{1}$-transgression map
$H^{4}(G,\mathbb{Z})\rightarrow H^{3}(G,\mathbb{Z})$ [43] seen in the Chern-
Simons/Wess-Zumino-Witten holography [21].

As such, it would be important to understand the tower of "higher order fake-
flatness conditions" $\frac{d^{n}}{d\tau^{n}}\mathcal{F}=0$ in a more rigorous
and complete manner.

##### Mobility constraints in 3-Yang-Mills theories.

In section 2.5.3, we constructed a 2-Yang-Mills theory based on the trivial
2-group $\mathcal{I}_{U(1)}$ that arises from the anomaly resolution of the
magnetic monopole in classical electromagnetism. We derived the higher
conservation law (2.48), which imposes a certain mobility constraint for the
electric charges provided the dipole charge is conserved. This is consistent
with what others have found in the literature [25].

In section 3.3.2, we have also derived a set of new mobility constraints from
the higher-conservation laws (3.29)-(3.32) associated to the 3-Yang-Mills
theory based on the string 2-crossed-module (3.25). In particular, the central
2-form current $J_{c}$ is rendered immobile, which seems to be consistent with
the non-dynamical nature of the central charge in Wess-Zumino-Witten conformal
field theory [92, 21]. It would be worthwhile to make this observation into a
more concrete statement, and to see what these mobility constraints mean
physically for the 3-group charges as a whole. As far as we know, these
mobility constraints have never been previously derived, so it seems to be a
novel direction to explore.

##  Appendix A Classification of Lie algebra crossed-modules

In this section we examine the classification of Lie algebra crossed-modules
by Lie algebra cohomology, following [36]. Recall that a given two Lie
algebras $\mathfrak{h},\mathfrak{g}$ over a fixed field $k$ of characteristic
zero, a Lie algebra crossed-module is a map
$t:\mathfrak{h}\rightarrow\mathfrak{g}$ and an action $\rhd$ of $\mathfrak{g}$
on $\mathfrak{h}$ such that the following Peiffer conditions

$t(X\rhd Y)=[X,tY]_{\mathfrak{g}},\qquad tY\rhd
Y^{\prime}=[Y,Y^{\prime}]_{\mathfrak{h}}$ (A.1)

are satisfied for each $Y,Y^{\prime}\in\mathfrak{h},X\in\mathfrak{g}$.
Mathematically, it is equivalent to a strict Lie 2-algebra181818Namely a two-
term differential graded $L_{\infty}$-algebra., where the homotopy map $\mu=0$
introduced in the main text vanishes.

Consider the following four-term algebra complex built from the Lie algebra
crossed module,

$0\rightarrow
V\hookrightarrow\mathfrak{h}\xrightarrow{t}\mathfrak{g}\rightarrow\mathfrak{n}\rightarrow
0,$ (A.2)

where $V=\operatorname{ker}t$ and $\mathfrak{n}=\operatorname{coker}t$. Due to
the Peiffer identity in (A.1), the Lie algebra $V\subset Z(\mathfrak{h})$ must
lie in the centre of $\mathfrak{h}$, and hence is Abelian. It admits an action
by $\mathfrak{n}$ induced by the crossed-module action $\rhd$.

######  Definition A.1.

We say that two crossed modules
$t:\mathfrak{h}\rightarrow\mathfrak{g},t^{\prime}:\mathfrak{h}^{\prime}\rightarrow\mathfrak{g}^{\prime}$
with the respective actions $\rhd,\rhd^{\prime}$ are elementary equivalent if

  1. 1.

$\operatorname{ker}t=\operatorname{ker}t^{\prime}=V$ and
$\operatorname{coker}t=\operatorname{coker}t^{\prime}=\mathfrak{n}$,

  2. 2.

there exists Lie algebra homomorphisms
$\phi:\mathfrak{h}\rightarrow\mathfrak{h}^{\prime},\psi:\mathfrak{g}\rightarrow\mathfrak{g}^{\prime}$
compatible with the actions $\rhd,\rhd^{\prime}$ such that

$\phi(X\rhd Y)=\psi(X)\rhd^{\prime}\phi(Y)$

for all $X\in\mathfrak{g}$ and $Y\in\mathfrak{h}$. Moreover, the diagram

${\mathfrak{h}}$${\mathfrak{g}}$${0}$${V}$${\mathfrak{n}}$${0}$${\mathfrak{h}^{\prime}}$${\mathfrak{g}^{\prime}}$$\scriptstyle{t}$$\scriptstyle{\phi}$$\scriptstyle{\psi}$$\scriptstyle{t^{\prime}}$

commutes.

Let us denote the set of elementary equivalence classes of Lie algebra
crossed-modules by ${\bf XMod}(\mathfrak{n},V)$.

###  A.1 Lie algebra cohomology

We first review some basic facts about Lie algebra cohomology, which is a very
powerful and important tool for classification of $L_{\infty}$-algebras. We
once again follow the treatment of [36].

Let $\mathfrak{n}$ be a Lie algebra over the field $k$ and let $V$ be an
Abelian $\mathfrak{n}$-module. Define its differential graded Chevalley-
Eilenberg complex

$(C^{\bullet}(\mathfrak{n},V),d),\qquad
C^{p}(\mathfrak{n},V)=\begin{cases}\Lambda(\mathfrak{n}^{p},V)&;p>0\\\
V&;p=0\end{cases},$

where $\Lambda(\mathfrak{n}^{p},V)$ denotes the exterior algebra of
alternating forms on $p$-copies of $\mathfrak{n}$ over $V$. The differential
$d:C^{p}(\mathfrak{n},V)\rightarrow C^{p+1}(\mathfrak{n},V)$ is given
explicitly by

$\displaystyle dc(x_{0},\dots,x_{p})$ $\displaystyle=$
$\displaystyle\sum_{i<j}(-1)^{i+j}c([x_{i},x_{j}],x_{0},\dots,\hat{x}_{i},\dots,\hat{x}_{j},\dots,x_{p})$
$\displaystyle-\sum_{i=1}^{p}(-1)^{i}x_{i}\rhd
c(x_{0},\dots,\hat{x}_{i},\dots,x_{p})$

for each cochain $c\in C^{p}(\mathfrak{n},V)$, where $\hat{\cdot}$ denotes an
omitted element.

######  Lemma A.1.

$d^{2}=0$.

###### Proof.

Recall the Cartan formula

$L_{x}=d\iota_{x}+\iota_{x}d,\qquad x\in\mathfrak{n}$

where $\iota_{x}:C^{p+1}(\mathfrak{n},V)\rightarrow C^{p}(\mathfrak{n},V)$ is
the interior evaluation

$\iota_{x}:c\mapsto((x_{1},\dots,x_{p})\mapsto c(x,x_{1},\dots,x_{p}))$

and $L_{x}:C^{p}(\mathfrak{n},V)\rightarrow C^{p}(\mathfrak{n},V)$ is the Lie
evaluation

$L_{x}:c\mapsto((x_{1},\dots,x_{p})\mapsto x\rhd
c(x_{1},\dots,x_{p})-\sum_{i}c(x_{1},\dots,[x,x_{i}],\dots,x_{p})),$

which by construction commutes with $d$. Now let $v\in
V=C^{0}(\mathfrak{n},V)$ be a 0-form, then

$\displaystyle d^{2}v(x_{1},x_{2})$ $\displaystyle=$ $\displaystyle-
dv([x_{1},x_{2}])+x_{1}\rhd dv(x_{2})-x_{2}\rhd dv(x_{1})$ $\displaystyle=$
$\displaystyle[x_{2},x_{1}]\rhd v+x_{1}\rhd(x_{2}\rhd v)-x_{2}\rhd(x_{1}\rhd
v)=0,$

which vanishes by the $\mathfrak{n}$-module structure on $V$.

Now let $p>0$ and assume the induction hypothesis: $d^{2}=0$ on
$C^{p-1}(\mathfrak{n},V)$. Consider $c\in C^{p}(\mathfrak{n},V)$, then by the
Cartan formula

$\displaystyle d^{2}c(x_{-1},x_{0},x_{1},\dots,x_{p})$ $\displaystyle=$
$\displaystyle\iota_{x_{-1}}(d^{2}c)(x_{0},x_{1},\dots,x_{p})$
$\displaystyle=$
$\displaystyle(L_{x_{-1}}-d\iota_{x_{-1}})dc(x_{0},x_{1},\dots,x_{p})$
$\displaystyle=$
$\displaystyle(L_{x_{-1}}d-d(L_{x_{-1}}-d\iota_{x_{-1}})c(x_{0},x_{1},\dots,x_{p})$
$\displaystyle=$
$\displaystyle(L_{x_{-1}}d-dL_{x_{-1}}+d^{2}\iota_{x_{-1}})c(x_{0},x_{1},\dots,x_{p})=0,$

where the first two terms cancel by the property $L_{x}d=dL_{x}$, and the last
term vanishes due to the induction hypothesis (recall $\iota_{x_{-1}}c\in
C^{p-1}(\mathfrak{n},V)$). ∎

This nilpotency allows us to define the Lie algebra cohomology

$H^{\bullet}(\mathfrak{n},V)=\operatorname{ker}d/\operatorname{im}d.$

These groups are extremely useful, as they are isomorphic to the de Rham
cohomology of the topological group $G$ [76]. Moreover, they classify various
algebraic structures; for instance,

  1. 1.

Degree $p=0$: the group $H^{0}(\mathfrak{n},V)=V^{\mathfrak{n}}\subset V$
classifies the $\mathfrak{n}$-invariants: namely elements $v\in V$ annihilated
by $\mathfrak{n}$ via the action $\rhd$. Indeed, the 0-cocycle condition
merely states

$dv(x)=x\rhd v=0,\qquad v\in V=C^{0}(\mathfrak{n},V),$

which means that $v\in Z^{0}(\mathfrak{n},V)$ is $\mathfrak{n}$-invariant.

  2. 2.

Degree $p=1$: the group $H^{1}(\mathfrak{n},V)$ classifies algebra
representations of $\mathfrak{n}$ on $V$ (i.e. derivations
$\operatorname{Der}_{\mathfrak{n}}(V)$) modulo inner representations. Indeed,
the 1-cocycle condition reads

$dc(x_{1},x_{2})=c([x_{1},x_{2}])-x_{1}\rhd c(x_{2})+x_{2}\rhd c(x_{1})=0,$

which implies that $c\in Z^{1}(\mathfrak{n},V)$ is a linear representation of
$\mathfrak{n}$ on $V$. The 1-coboundaries are inner derivations
$c(x)=dv(x)=x\rhd v$ for some $v\in V=C^{0}(\mathfrak{n},V)$. If
$\mathfrak{n}$ acts trivially on $V$, then $H^{1}(\mathfrak{n},V)$ is in fact
isomorphic to the (dual of the) Abelianization
$\mathfrak{n}/[\mathfrak{n},\mathfrak{n}]$.

  3. 3.

Degree $p=2$: the group $H^{2}(\mathfrak{n},V)$ classifies central extensions
$\widehat{\mathfrak{n}}$ of $\mathfrak{n}$ by $V$, which fits in the three-
term exact sequence

$0\rightarrow
V\rightarrow\widehat{\mathfrak{n}}\rightarrow\mathfrak{n}\rightarrow 0.$

To see this at a glance, a set-theoretic section
$s:\mathfrak{n}\rightarrow\widehat{\mathfrak{n}}$ sees an obstruction to being
a Lie algebra-theoretic section given by

$c(x_{1},x_{2})=s([x_{1},x_{2}])-[s(x_{1}),s(x_{2})].$

It can be shown, with the $\mathfrak{n}$-module structure of $V$ and the
Jacobi identity, that $c\in Z^{2}(\mathfrak{n},V)$ is a 2-cocycle, and any two
choices of such sections $s$ yields 2-cocycles $c,c^{\prime}$ that differ by a
2-coboundary $c-c^{\prime}=da$.

In general, the set $H^{p}(\mathfrak{n},V)$ classifies ($p+1$)-term extensions
of $\mathfrak{n}$ by $V$. Moreover, equivalence classes of such extensions can
be equipped with an Abelian group structure such that $H^{p}(\mathfrak{n},V)$
coincides with it not just as a set, but also as a group.

######  Remark A.1.

Recall a 2-crossed-module $\mathscr{G}$ as defined in section 3. The exactness
$t^{\prime}t=0$ of the complex (3.2) states that $\mathscr{G}$ gives rise to a
5-term exact sequence

$0\rightarrow
V=\operatorname{ker}t^{\prime}\rightarrow\underbrace{\mathfrak{i}\xrightarrow{t^{\prime}}\mathfrak{h}\xrightarrow{t}\mathfrak{g}}_{=\mathscr{G}}\rightarrow\mathfrak{n}=\operatorname{coker}t\rightarrow
0$

of Lie algebras, which by the above statement is classified by a degree-4 Lie
algebra cohomology class $H^{4}(\mathfrak{n},V)$. This class has also appeared
as part of the data that classifies crossed-squares [9]; indeed, each crossed-
square has an associated 2-crossed-module [10].

We shall show in detail next that, at degree 3, $H^{3}(\mathfrak{n},V)$
classifies precisely the four-term complex (A.2) of a Lie algebra crossed-
module.

###  A.2 Theorem of Gerstenhaber

Before constructing the 3-cocycle $c\in Z^{3}(\mathfrak{n},V)$, we introduce
the notion of addition in the set of crossed-modules. Given two crossed-
modules
$t:\mathfrak{h}\rightarrow\mathfrak{g},t^{\prime}:\mathfrak{h}^{\prime}\rightarrow\mathfrak{g}^{\prime}$
with the same kernel $V$ and cokernel $\mathfrak{n}$, it can be shown that

$(t\oplus
t^{\prime}):\mathfrak{h}\oplus\mathfrak{h}^{\prime}/\,\overline{\Delta}\rightarrow\mathfrak{g}\oplus_{\mathfrak{n}}\mathfrak{g}^{\prime}$

is another crossed-module, called the crossed-module sum of $t$ and
$t^{\prime}$. Here, $\overline{\Delta}$ is the kernel of the addition map
$+:V\oplus V\rightarrow V$, while
$\mathfrak{g}\oplus_{\mathfrak{n}}\mathfrak{g}^{\prime}$ is the fibre
pullback; explicitly,

$\overline{\Delta}=\\{(v,-v)\mid v\in
V\\},\qquad\mathfrak{g}\oplus_{\mathfrak{n}}\mathfrak{g}^{\prime}=\\{(X,X^{\prime})\in\mathfrak{g}\oplus\mathfrak{g}^{\prime}\mid
pX=p^{\prime}X^{\prime}\\}.$

Note that as direct sums are commutative, we have $(t\oplus
t^{\prime})\cong(t^{\prime}\oplus t)$.

This notion descends to elementary equivalence classes of crossed-modules, and
endows the set ${\bf XMod}(\mathfrak{n},V)$ the structure of an Abelian group.
We shall show that this Abelian group is isomorphic precisely to
$H^{3}(\mathfrak{n},V)$. To begin, we construct a bilinear skew-symmetric map

$f(x_{1},x_{2})=s_{1}([x_{1},x_{2}])-[s_{1}(x_{1}),s_{1}(x_{2})],\qquad
x_{1},x_{2}\in\mathfrak{n}$

from a section $s_{1}:\mathfrak{n}\rightarrow\mathfrak{g}$ of the map
$p:\mathfrak{g}\rightarrow\operatorname{coker}t=\mathfrak{n}$ in (A.2). Though
$s_{1}$ may not be a Lie algebra map, the projection $p$ is, so $pf=0$ and $f$
is valued in $\operatorname{ker}p$. By the exactness
$\operatorname{ker}p=\operatorname{im}t$ of (A.2), there exists a bilinear
skew-symmetric map $e:\mathfrak{n}^{\wedge 2}\rightarrow\mathfrak{h}$ such
that $f=te$.

We now pick another section
$s_{2}:\operatorname{im}t\subset\mathfrak{g}\rightarrow\mathfrak{h}$ of the
crossed-module map $t:\mathfrak{h}\rightarrow\mathfrak{g}$, whence $e=s_{2}f$.
Let $\circlearrowright$ denote a summation over cyclic permutations of
$x_{1},x_{2},x_{3}$, then by construction,

$\displaystyle tde(x_{1},x_{2},x_{3})$ $\displaystyle=$ $\displaystyle
t\left[\circlearrowright e([x_{1},x_{2}],x_{3})-\circlearrowright
s_{1}(x_{1})\rhd e(x_{2},x_{3})\right]$ $\displaystyle=$
$\displaystyle\circlearrowright f([x_{1},x_{2}],x_{3})-\circlearrowright
t(s(x_{1})\rhd e(x_{2},x_{3}))\qquad\text{Peiffer conditions
(\ref{eq:pfeif})}$ $\displaystyle=$ $\displaystyle\circlearrowright
f([x_{1},x_{2}],x_{3})-\circlearrowright[s_{1}(x_{1}),\underbrace{te(x_{2},x_{3})}_{=f(x_{2},x_{3})}]\qquad\text{Definition
of $f$}$ $\displaystyle=$
$\displaystyle\circlearrowright\left([s_{1}([x_{1},x_{2}]),s_{1}(x_{3})]-s_{1}([[x_{1},x_{2}],x_{3}])\right)$
$\displaystyle-\circlearrowright\left([s_{1}(x_{1}),[s_{1}(x_{2}),s_{1}(x_{3})]]-[s_{1}(x_{1}),s_{1}([x_{2},x_{3}])]\right)\qquad\text{Jacobi
identity}$ $\displaystyle=$
$\displaystyle\circlearrowright\left([s_{1}([x_{1},x_{2}]),s_{1}(x_{3})]-[s_{1}([x_{2},x_{3}]),s(x_{1})]\right)\qquad\text{Cyclicity
of summation}$ $\displaystyle=$ $\displaystyle 0,$

as such $de$ is in fact valued in $\operatorname{ker}t$. Again by the
exactness of the sequence (A.2) we may find a skewsymmetric trilinear map
$c:\mathfrak{n}^{\wedge 3}\rightarrow V$ such that $ic=de$, where
$i:V\hookrightarrow\mathfrak{h}$ is the inclusion. Picking yet another section
$s_{3}:\mathfrak{h}\rightarrow V$ yields $c=s_{3}De$.

Now we must show that $dc=0$. It may be tempting to say that, since $ic=de$,
we have $idc=dic=d^{2}e=0$ by the nilpotency $d^{2}=0$. However, this does not
immediately follow, as $s_{1}$ is not necessarily a section and hence
$s_{1}(\cdot)\rhd$ is not necessarily a well-defined action. By explicit
computation, terms involving the problematic operation $s_{1}(\cdot)\rhd$ in
$idc$ read

$\displaystyle\sum_{i<j}(-1)^{i+j}s_{1}([x_{i},x_{j}])\rhd
e(x_{1},\dots,\hat{x}_{i},\dots,\hat{x}_{j},x_{4})$
$\displaystyle-\sum_{i=1}^{4}(-1)^{i}s_{1}(x_{i})\rhd\left[\sum_{j\neq
i}(-1)^{j}s_{1}(x_{j})\rhd
e(x_{1},\dots,\hat{x}_{j},\dots,x_{3})\right]\qquad\text{Rearrange terms}$
$\displaystyle=$
$\displaystyle\sum_{i<j}(-1)^{i+j}\left(s_{1}([x_{i},x_{j}])-[s_{1}(x_{i}),s_{1}(x_{j})]\right)\rhd
e(x_{1},\dots,\hat{x}_{i},\dots,\hat{x}_{j},x_{4})\qquad\text{Definition of
$f$}$ $\displaystyle=$
$\displaystyle\sum_{i<j}(-1)^{i+j}\underbrace{f(x_{i},x_{j})}_{=te(x_{i},x_{j})}\rhd
e(x_{1},\dots,\hat{x}_{i},\dots,\hat{x}_{j},x_{4})\qquad\text{Peiffer
conditions}$ $\displaystyle=$
$\displaystyle\sum_{i<j}(-1)^{i+j}[e(x_{i},x_{j}),e(x_{1},\dots,\hat{x}_{i},\dots,\hat{x}_{j},x_{4})]\qquad\text{Cyclicity
of summation}$ $\displaystyle=$ $\displaystyle 0,$

hence we nevertheless have $dc=0$. This allows us to conclude that $c\in
Z^{3}(\mathfrak{n},V)$.

We now wish to show that changing the choices of the sections $s_{1,2,3}$ adds
to $c$ a 3-coboundary. By linearity, we can write
$s_{1}^{\prime}=s_{1}+\delta$ for some map
$\delta:\mathfrak{n}\rightarrow\mathfrak{g}$. Defining a bilinear skew-
symmetric map $f^{\prime}$ analogously, we see that

$f^{\prime}(x_{1},x_{2})=f(x_{1},x_{2})+[s_{1}(x_{1}),\delta(x_{2})]+[\delta(x_{1}),s_{1}(x_{2})]+[\delta(x_{1}),\delta(x_{2})]-\delta([x_{1},x_{2}]).$

Notice the terms
$[s_{1}(x_{1}),\delta(x_{2})]+[\delta(x_{1}),s_{1}(x_{2})]-\delta([x_{1},x_{2}])$
constitute precisely the coboudnary $d\delta(x_{1},x_{2})$ of a cochain
$\delta:\mathfrak{n}\rightarrow\mathfrak{g}$, with
$x_{1},x_{2}\in\mathfrak{n}$ lifted up to $\mathfrak{g}$ by the map $s_{1}$.

Now as $f^{\prime},f$ are vlaued in $\operatorname{ker}p=\operatorname{im}t$,
we can find $\mathfrak{h}$-valued bilinear maps $\epsilon,\varepsilon$ such
that $t\epsilon(x_{1},x_{2})=d\delta(x_{1},x_{2})$ and
$t\varepsilon(x_{1},x_{2})=[\delta(x_{1}),\delta(x_{2})]$. Further, we can
also find a $\operatorname{ker}t=\operatorname{im}i$-valued bilinear map
$\varphi$ such that

$e^{\prime}(x_{1},x_{2})=e(x_{1},x_{2})+\epsilon(x_{1},x_{2})+\varepsilon(x_{1},x_{2})+i\varphi(x_{1},x_{2})$

when lifted by $s_{2}$. Our goal now is to apply the differential $d$;
however, the trouble here is that $d$ and $s_{2}$ need not commute, as $s_{2}$
is not in general a section. Now by computation

$\displaystyle tds_{2}\delta(x_{1},x_{2})$ $\displaystyle=$ $\displaystyle
t(s_{1}(x_{1})\rhd s_{2}\delta(x_{2})+s_{1}(x_{2})\rhd
s_{2}\delta(x_{1})-s_{2}\delta([x_{1},x_{2}])),\qquad\text{Peiffer
conditions}$ $\displaystyle=$ $\displaystyle
ts_{2}([s_{1}(x_{1}),\delta(x_{2})]-[s_{1}(x_{2}),\delta(x_{1})]-\delta([x_{1},x_{2}]))$
$\displaystyle=$ $\displaystyle ts_{2}d\delta(x_{1},x_{2}),$

so $\Delta_{1}=ds_{2}\epsilon-s_{2}d\epsilon$ is valued in
$\operatorname{ker}t$. Similarly, the difference
$\Delta_{2}=ds_{2}\varepsilon-s_{2}d\varepsilon$ also lies in
$\operatorname{ker}t$, which allows us to finally write

$c^{\prime}(x_{1},x_{2},x_{3})=c(x_{1},x_{2},x_{3})+d\epsilon(x_{1},x_{2},x_{3})+d\varepsilon(x_{1},x_{2},x_{3})+i\left(\Delta_{1}+\Delta_{2}\right)(x_{1},x_{2},x_{3})+di\varphi(x_{1},x_{2},x_{3}).$

Using the injectivity of $i$, we have $di\varphi=i(d|_{V}\varphi)$, hence
defining $\sigma=\epsilon+\varepsilon$ and
$\gamma=\Delta_{1}+\Delta_{2}+d|_{V}\varphi$ yields

$c^{\prime}=c+d\sigma+i\gamma=c+d\sigma\mod\operatorname{ker}t,$

whence lifting by $s_{3}$ up to $V$ yields $c^{\prime}=c+d\sigma$. This shows
that the cohomology class of $c$ does not depend on the choice of the section
$s_{1}$.

Now suppose we have distinct sections $s_{2},s_{2}^{\prime}$, defining
$e=s_{2}f$ and $e^{\prime}=s_{2}^{\prime}f$. It is clear that
$t(e-e^{\prime})=ts_{2}f-ts_{2}^{\prime}f=f-f=0$, hence $e-e^{\prime}$ is
valued in $\operatorname{ker}t=\operatorname{im}i$. This means that $s_{3}$
lifts $d(e-e^{\prime})$ to a coboundary $d\omega$ such that
$c^{\prime}=c+d\omega$, demonstrating that the cohomnology class of $c$ does
not depend on the choice of the section $s_{2}$ as well. Lastly, any two
sections $s_{3},s_{3}^{\prime}$ must coincide, at least on the image
$\operatorname{im}i=\operatorname{ker}t$, hence the cocycle itself $c$ does
not depend on the choice of $s_{3}$.

######  Lemma A.2.

Let $t,t^{\prime}$ denote two elementary equivalent crossed-modules, then the
3-cocycles $c,c^{\prime}$ they define coincide $[c]=[c^{\prime}]\in
H^{3}(\mathfrak{n},V)$ in cohomology.

###### Proof.

First, pick sections $s_{1,2,3},s_{1,2,3}^{\prime}$ in the respective crossed-
modules $t,t^{\prime}$ and construct the 3-cocycles $c,c^{\prime}\in
C^{3}(\mathfrak{n},V)$. Suppose an elementary equivalence $(\phi,\psi)$
between the two crossed-modules exists, then $\psi s_{1}$ is a section of
$p^{\prime}$. The above shows that the 3-cocycle $\tilde{c}^{\prime}$
constructed from the sections $(\psi s_{1},s_{2}^{\prime},s_{3}^{\prime})$
differ from that $c^{\prime}$ constructed from
$(s_{1}^{\prime},s_{2}^{\prime},s_{3}^{\prime})$ only by a coboundary. Our
task is thus to show that $\tilde{c}^{\prime}$ also coincides with $c$ up to
coboundary.

Toward this, we define $s_{2}^{\prime}\psi f\equiv\tilde{e}^{\prime}$ and
compare this to $\phi e=\phi s_{2}f$. First, we know that
$t^{\prime}s_{2}^{\prime}=1$, hence $\tilde{e}^{\prime}-\phi e$ is valued in
$\operatorname{ker}t^{\prime}=\operatorname{im}i^{\prime}$, so we can find a
map $v:\mathfrak{n}^{\wedge 2}\rightarrow V$ such that
$\tilde{e}^{\prime}-\phi e=i^{\prime}v$.

We now take the differential $d$ of this equation. By definition of the
elementary equivalence, we can rewrite contributions
$\psi(x_{i})\rhd\phi(e)=\phi(x_{i}\rhd e)$ in the differential, as such
$d(\phi e)=\phi de$. Now $s_{3}\phi$ is a section of $i^{\prime}$, hence

$\tilde{c}^{\prime}-c=s_{3}D\tilde{e}^{\prime}-(s_{3}\phi)de=dv$

is a coboundary. This proves the lemma. ∎

The lemma allows us to put a well-defined map $b:{\bf
XMod}(\mathfrak{n},V)\rightarrow H^{3}(\mathfrak{n},V)$.

######  Theorem A.1.

(Gerstenhaber, attr. by MacLane). $b$ is an isomorphism of Abelian groups.

The classifying data of a Lie algebra crossed-module
$t:\mathfrak{h}\rightarrow\mathfrak{g}$ is exactly $(\mathfrak{n},V,c)$ with
$c\in H^{3}(\mathfrak{n},V)$.

###  A.3 The Postnikov class

Let us now turn to the reason why we called an element in
$H^{3}(\mathfrak{n},V)$ a "Postnikov class" in the main text. Formally, a Lie
2-algebra integrates to a Lie 2-group $t:H\rightarrow G$ [37, 35], for which a
"Gerstenhaber theorem" also holds: $t:H\rightarrow G$ is classified by its
Hoang data $(N,V,\kappa)$ [10, 9], where
$N=\operatorname{coker}t,V=\operatorname{ker}t$ and $\kappa\in H^{3}(N,V)$ is
a group cohomology class (as opposed to a Lie algebra cohomology class).

The name "Postnikov class" comes from topology. Given any "nice" space $X$ (a
finite CW complex), its fundamental group $\pi_{1}(X)$ in general acts on
higher homotopy groups $\pi_{\geq 2}(X)$ via monodromy. The homotopy 2-type
$\Pi_{2}(X)=(\pi_{1}(X),\pi_{2}(X),\operatorname{Ptn}(X))$ is modeled by the
group crossed-module [9]

$1\rightarrow\operatorname{ker}\partial=\pi_{2}(X)\rightarrow\pi_{2}(X,Y)\xrightarrow{\partial}\pi_{1}(Y)\rightarrow\pi_{1}(X)=\operatorname{coker}\partial\rightarrow
1,$

where $Y\subset X$ is a closed subspace and $\partial$ is the natural boundary
map. Up to homotopy, it is classified by the Postnikov class
$\operatorname{Ptn}(X)\in H^{3}(\pi_{1}(X),\pi_{2}(X))$, which determines how
2-cells are glued upon the 1-cells.

It is possible to construct the classifying space $B(N,V)$ satisfying the
condition $\Pi_{2}B(N,V)=(N,V,\kappa)$ [18, 39]. Such a space sits in the
Postnikov tower fibration sequence

$B^{2}V\rightarrow B(N,V)\rightarrow BN,$

where $BN=K(N,1)$ is the classifying Eilenberg-MacLane space of $N$ and
$B^{2}V=K(V,2)$ is the second delooping of $V$, satisfying $\pi_{2}(B^{2}V)=V$
with other homotopy groups vanishing.

In other words, the Postnikov class determines how $B(N,V)$ is constructed
from the base $BN$ by gluing the second delooping space $B^{2}V$. The homotopy
classification theorem states that gauge-equivalent discrete flat
2-connections $H^{1}(X,(N,V))$ are isomorphic to homotopy classes of
classifying maps $X\rightarrow B(N,V)$ [10, 39]; this is how 2-gauge
topological field theories are constructed [18, 19].

##  Appendix B 2-bundle homomorphisms

In this appendix, we show that an elementary equivalence gives rise to a
homomorphism between 2-gauge bundles. We also generalize this perspective to
the weak case.

Let $\mathcal{P},\mathcal{P}^{\prime}\rightarrow X$ denote two 2-gauge bundles
on $X$, equipped with connections $(A,\Sigma)$ and
$(A^{\prime},\Sigma^{\prime})$, respectively. Intuitively, from the gauge
theory perspective, a 2-bundle homomorphism
$g:\mathcal{P}\rightarrow\mathcal{P}^{\prime}$ should satisfy two properties:
(1) it is a bundle map over $X$; namely the triangle

${\mathcal{P}}$${\mathcal{P}^{\prime}}$${X}$$\scriptstyle{g}$

commutes, and (2) preserves all gauge-invariant data.

From our computations in the main text, the gauge-invariant data consist
precisely of the fake-flatness $\mathcal{F}$ (2.6) and the 2-curvature
$\mathcal{G}=K$. As such homomorphisms $\psi$ must satisfy

$\mathcal{F}=g^{*}\mathcal{F}^{\prime},\qquad\mathcal{G}=g^{*}\mathcal{G}^{\prime}.$

Let us write, locally, $g^{*}=f^{*}\otimes\Psi$ in terms of components, where
$f^{*}$ is the pullback of $f:X\rightarrow X$ on forms and $\Psi=(\phi,\psi)$
is a map on the Lie algebras

$\phi:\mathfrak{h}^{\prime}\rightarrow\mathfrak{h},\qquad\psi:\mathfrak{g}^{\prime}\rightarrow\mathfrak{g}.$

The fake-flatness condition $\mathcal{F}=\psi^{*}\mathcal{F}^{\prime}$ implies

$F=(f^{*}\otimes\psi)F^{\prime},\qquad
t\Sigma=(f^{*}\otimes\psi)t^{\prime}\Sigma^{\prime}=t(f^{*}\otimes\phi)\Sigma^{\prime};$
(B.1)

by linearity and $F=d_{A}A,F^{\prime}=d_{A^{\prime}}A^{\prime}$, the first
condition in (B.1) means that $f^{*}$ commutes with the de Rham differential
$d$, and that $\psi$ is a Lie algebra homomorphism191919This means that
$A=\psi A^{\prime}$ and $[A\wedge A]=\psi[A^{\prime}\wedge A^{\prime}]=[\psi
A^{\prime}\wedge\psi A^{\prime}]$.. The second condition means $t\phi=\psi
t^{\prime}$ commutes with the crossed-module maps $t,t^{\prime}$.

##### Equivalence of 2-gauge bundles.

The 2-curvature condition reads

$\mathcal{G}=d_{A}\Sigma=(f^{*}\otimes\phi)d_{A^{\prime}}\Sigma^{\prime}=(f^{*}\otimes\phi)(d\Sigma^{\prime}+A\wedge^{\rhd^{\prime}}\Sigma^{\prime}),$

where $\rhd^{\prime}$ is the crossed-module action in $\mathcal{P}^{\prime}$.
Using the second condition from (B.1), the first term reads

$(f^{*}\otimes\phi)d\Sigma^{\prime}=d\Sigma=d(f^{*}\otimes\phi)\Sigma^{\prime},$

while the second term reads

$A\wedge^{\rhd}\Sigma=(f^{*}\otimes\phi)A^{\prime}\wedge^{\rhd^{\prime}}\Sigma^{\prime}.$

However, the condition $A=(f^{*}\otimes\psi)A^{\prime}$ means that we must
have

$(f^{*}\otimes\phi)A^{\prime}\wedge^{\rhd^{\prime}}\Sigma^{\prime}=((f^{*}\otimes\psi)A^{\prime})\wedge^{\rhd}(f^{*}\otimes\phi)\Sigma^{\prime}.$

This tells us that, not only does $g_{-1}$ also has to be a Lie algebra
homomorphism, but also the condition

$\phi(X\rhd^{\prime}Y)=(\psi X)\rhd(\phi Y),\qquad\forall\leavevmode\nobreak\
X\in\mathfrak{g}^{\prime},Y\in\mathfrak{h}^{\prime}.$ (B.2)

This is precisely the definition of an elementary equivalence of Lie algebra
crossed modules [36, 35].

As such, we may interpret elementary equivalence as an equivalence of the
gauge-invariant data on the 2-gauge bundles
$\mathcal{P},\mathcal{P}^{\prime}$. The Gerstenhaber Theorem A.1 then implies

######  Corollary B.1.

If the 2-gauge bundles $\mathcal{P},\mathcal{P}^{\prime}$ exhibit distinct
Postnikov classes $\kappa\neq\kappa^{\prime}\in H^{3}(\mathfrak{n},V)$ as
2-curvature anomalies, then there does not exist a 2-bundle homomorphism
between them.

##### Extension to weak Lie 2-algebras.

The above notions of elementary equivalence was formulated in the context of
strict Lie 2-algebras. We can extend the notion of an elementary equivalence
to the weak case, by appending an additional component $\varphi$ in the map
$(\phi,\psi)$ between the 2-algebras.

This component $\varphi$ is used to control the failures of $(\phi,\psi)$ from
being an elementary equivalence, as well as the Jacobiators
$\mu,\mu^{\prime}$. The full definition is [35]

######  Definition B.1.

A 2-homomorphism between weak Lie 2-algebras
$t:\mathfrak{h}\rightarrow\mathfrak{g},t^{\prime}:\mathfrak{h}^{\prime}\rightarrow\mathfrak{g}^{\prime}$
— with respective Jacobiators $\mu,\mu^{\prime}$ — are given by the set of
chain maps

$\varphi:\mathfrak{g}\times\mathfrak{g}\rightarrow\mathfrak{h}^{\prime},\qquad\phi:\mathfrak{h}\rightarrow\mathfrak{h}^{\prime},\qquad\psi:\mathfrak{g}\rightarrow\mathfrak{g}^{\prime},$

such that $t^{\prime}\phi=\psi t$ and the following conditions are satisfied:

$\displaystyle t^{\prime}\varphi(X,X^{\prime})$ $\displaystyle=$
$\displaystyle\psi([X,X^{\prime}])-[\psi X,\psi X^{\prime}]^{\prime},$
$\displaystyle\varphi(X,tY)$ $\displaystyle=$ $\displaystyle\phi(X\rhd
Y)-(\psi X)\rhd^{\prime}(\phi Y),$
$\displaystyle\mu^{\prime}(\psi(X),\psi(X^{\prime}),\psi(X^{\prime\prime}))-\phi(\mu(X,X^{\prime},X^{\prime\prime}))$
$\displaystyle=$
$\displaystyle\circlearrowright\varphi(X,[X^{\prime},X^{\prime\prime}])+\circlearrowright[(\psi
X)\rhd^{\prime}\varphi(X^{\prime},X^{\prime\prime})]$ (B.3)

for each $X,X^{\prime},X^{\prime\prime}\in\mathfrak{g},Y\in\mathfrak{h}$ and
$\circlearrowright$ denotes a summation over cyclic permutations of the
arguments.

In mathematically more sophisticated terms, $\Psi=(\varphi,\phi,\psi)$ is a
chain homotopy between two-term $L_{\infty}$-algebras [35, 42]. In this way,
it can be understood that $\varphi$ can only appear between weak 2-algebras,
and not strict 2-algebras.

Definition B.1 gives us a weaker notion of elementary equivalence: that two
weak Lie 2-algebras are weakly equivalent if there exist a 2-homomorphism
$\Psi=(\varphi,\phi,\psi)$ between them whose kernel and cokernel are
(strictly) elementary equivalent to the trivial Lie 2-algebras. This was the
notion of equivalence that was used in [35], which we shall examine in detail
next.

###  B.1 Loop model of the weak string 2-algebra

Let $\mathfrak{g}$ be a simple Lie algebra. Recall the loop model
$\mathfrak{l}_{k}$ for the string 2-algebra
$\mathfrak{string}_{k}(\mathfrak{g})$ of level $k\in\mathbb{Z}$ described in
the main text. It is a strict 2-algebra [34], but non-trivial with the
Postnikov class $[\kappa]\in H^{3}(\mathfrak{g},\mathbb{R})$ given by the
generating fundamental 3-cocycle $\omega=\langle\cdot,[\cdot,\cdot]\rangle$ on
$\mathfrak{g}$ [76]; see section 3.3.2.

The main result in [35] is that $\mathfrak{l}_{k}$ is equivalent to the
skeletal model [34] at level $k$:

######  Theorem B.1.

(Baez-Crans-Stevenson-Schreiber). Let $\mathfrak{l}_{k}$ denote the non-
trivial crossed-module constructed in section 3.3.2. Define the maps

$\displaystyle\psi:P\mathfrak{g}\rightarrow\mathfrak{g},$ $\displaystyle
p\mapsto p(1),$
$\displaystyle\phi:\widehat{\Omega_{k}\mathfrak{g}}\rightarrow\mathbb{R},$
$\displaystyle(\ell,c)\mapsto c,$
$\displaystyle\varphi:P\mathfrak{g}^{2\otimes}\rightarrow\mathbb{R},$
$\displaystyle(p,p^{\prime})\mapsto k\int_{[0,1]}(\langle
p,\dot{p}^{\prime}\rangle-\langle p^{\prime},\dot{p}\rangle),$

then
$\Psi=(\varphi,\phi,\psi):\mathfrak{l}_{k}\rightarrow\mathfrak{string}_{k}(\mathfrak{g})$
is a weak equivalence.

More concretely, it was shown that $\Psi$ is a surjective 2-homomorphism
inducing a split short exact sequence

$0\rightarrow\mathfrak{I}_{\Omega\mathfrak{g}}\rightarrow\mathfrak{l}_{k}\xrightarrow{\Psi}\mathfrak{string}_{k}(\mathfrak{g})\rightarrow
0.$ (B.4)

Combined with the fact that any 2-algebra $\mathfrak{I}_{\mathfrak{g}}\simeq
0$ with $t=\operatorname{id}$ are trivial for any $\mathfrak{g}$ up to
elementary equivalence, the theorem follows.

##### Gauge-theoretic interpretation.

So what is happening in the 2-gauge theory picture? Since $\Psi$ is a (weak)
elementary equivalence, it can be considered as part of a 2-bundle map
$\mathcal{P}\rightarrow\mathcal{P}^{\prime}$. Moreover, the sequence (B.4)
induces also a short exact sequence of bundles

$\mathcal{P}_{0}\rightarrow\mathcal{P}\rightarrow\mathcal{P}^{\prime},$

in which $\mathcal{P}_{0}$ is a trivial 2-gauge bundle based on
$\mathfrak{I}_{\Omega\mathfrak{g}}$. Our goal is to understand the effect of
the 2-homomorphism $\Psi$ on the 2-gauge kinematical data. We shall focus on
the level $k=1$ case.

The 2-gauge data on $\mathcal{P}$ has already been done in section 3.3.2. We
just copy down the gauge-covariant curvature quantities derived there,

$\displaystyle\mathcal{F}$ $\displaystyle=$ $\displaystyle
F-t\Sigma=F-\sigma,$ $\displaystyle\mathcal{G}$ $\displaystyle=$
$\displaystyle K-\kappa(A)=d_{A}\Sigma-\kappa(A),$

where the 2-curvature anomaly $\kappa(A)$ is given by (3.19). We now describe
the 2-gauge data on $\mathcal{P}^{\prime}$. The 1-connection $A^{\prime}$ is
valued in $\mathfrak{g}$, while the 2-connection $\Sigma^{\prime}$ is valued
in $\mathbb{R}$. The action is trivial

$A^{\prime}\wedge^{\rhd}\Sigma^{\prime}=0,$

whence the 1- and 2-curvatures read

$F^{\prime}=d_{A^{\prime}}A^{\prime}=dA^{\prime}+\frac{1}{2}[A^{\prime}\wedge
A^{\prime}],\qquad K^{\prime}=d_{A^{\prime}}\Sigma^{\prime}=d\Sigma^{\prime}.$

The Jacobiator is given by the fundamental 3-cocycle
$\mu(A,A,A)=\omega(A,A,A)=\frac{1}{3!}\langle A\wedge[A\wedge A]\rangle$,
whence we have the gauge-invariant data

$\mathcal{F}^{\prime}=F^{\prime},\qquad\mathcal{G}^{\prime}=K^{\prime}+\frac{1}{3!}\langle
A\wedge[A\wedge A]\rangle.$

The 2-homomorphism $\Psi=(\varphi,\phi,\psi)$ given in Theorem B.1 should then
induce a 2-bundle homomorphim that sends the 2-gauge data
$(\mathcal{F},\mathcal{G})$ to $(\mathcal{F}^{\prime},\mathcal{G}^{\prime})$.
To see this, we first note that $2\int_{S^{1}}\langle
A\wedge\dot{\sigma}\rangle=\phi(A\wedge^{\rhd}\Sigma)=\varphi(A,t\Sigma)$ by
Eq, (B.3). Then, recalling from section 3.3.2 that $dc^{\prime}$ can be gauged
away, this yields

$F^{\prime}=\psi F=F(1),\qquad K^{\prime}=\phi K=-2\int_{S^{1}}\langle
A\wedge\dot{\sigma}\rangle,$ (B.5)

where we have made the pullback map
$f^{*}:\Omega^{\bullet}(X)\rightarrow\Omega^{\bullet}(X)$ on forms that comes
with a 2-bundle homomorphism implicit. With $A^{\prime}=\psi A=A(1)$
understood, the Jacobiator $\mu$ can be reconstructed from (B.3) as

$\mu(A^{\prime},A^{\prime},A^{\prime})=\leavevmode\nobreak\
\circlearrowright\varphi(A,[A\wedge A])=3!\varphi(A,[A\wedge A]),$

where we have used the total skew-symmetry of $\varphi$.

This implies that the right-hand side should factor through the $\kappa$. More
precisely, we should have

$\phi(\kappa(A))=\varphi(A,[A\wedge A]),$

such that

$\frac{1}{3!}\langle A^{\prime},[A^{\prime}\wedge
A^{\prime}]\rangle=\frac{1}{3!}\mu(A^{\prime},A^{\prime},A^{\prime})=\varphi(A,[A\wedge
A]).$

This is nothing but the statement that the 2-curvature anomaly $\kappa(A)$
coincides with $\frac{1}{3!}\omega(A^{\prime},A^{\prime},A^{\prime})$, which
is precisely (3.19). Indeed, one can show

$\int_{0}^{1}\langle
p_{1},\frac{d}{d\tau}[p_{2},p_{3}]\rangle=\frac{1}{3!}\langle
p_{1}(1),[p_{2}(1),p_{3}(1)]\rangle$

for any $p_{1},p_{2},p_{3}\in P\mathfrak{g}$ by performing an integration by
parts then using the total skew-symmetry of the form
$\langle\cdot,[\cdot,\cdot]\rangle$ [35]. The 2-homormophism $\Psi$ then
implements

$K^{\prime}-\frac{1}{3!}\mu(A^{\prime},A^{\prime},A^{\prime})=\mathcal{G}^{\prime}=\phi\mathcal{G}=\phi(K-k\omega(A,A,A)),$

as desired.

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