this back along $\operatorname{id}\times q$, using the following diagram:

---
$\textstyle{PG\times P\Omega G\ltimes
U(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\operatorname{Ad}}}$$\scriptstyle{\operatorname{pr}_{12}}$$\scriptstyle{\operatorname{id}\times
q}$$\textstyle{P\Omega G\ltimes
U(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{pr}_{1}}$$\scriptstyle{q}$$\textstyle{PG\times\widehat{\Omega
G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{\operatorname{Ad}}}$$\scriptstyle{\operatorname{id}\times\pi}$$\textstyle{\widehat{\Omega
G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{PG\times
P\Omega
G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Ad}}$$\scriptstyle{\operatorname{id}\times\operatorname{ev}_{2\pi}}$$\textstyle{P\Omega
G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{ev}_{2\pi}}$$\textstyle{PG\times\Omega
G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Ad}}$$\textstyle{\Omega
G}$

We have that

$\displaystyle(\operatorname{id}\times
q)^{*}\left(\widehat{\operatorname{Ad}}^{*}\mu-\operatorname{pr}_{2}^{*}\mu\right)$
$\displaystyle=\widetilde{\operatorname{Ad}}^{*}q^{*}\mu-\operatorname{pr}_{23}^{*}q^{*}\mu$
$\displaystyle=\widetilde{\operatorname{Ad}}^{*}\tilde{\mu}-\operatorname{pr}_{23}^{*}\tilde{\mu}$
$\displaystyle=\widetilde{\operatorname{Ad}}^{*}\left(\operatorname{pr}_{2}^{*}\theta^{-1}d\theta+\operatorname{pr}_{1}^{*}\int_{[0,2\pi]}\operatorname{ev}^{*}R\right)-\left(\operatorname{pr}_{3}^{*}\theta^{-1}d\theta+\operatorname{pr}_{2}^{*}\int_{[0,2\pi]}\operatorname{ev}^{*}R\right)$
$\displaystyle=\operatorname{pr}_{12}^{*}d\int_{[0,2\pi]}(\operatorname{id}\times\operatorname{ev})^{*}\rho+\operatorname{pr}_{12}^{*}\operatorname{Ad}^{*}\int_{[0,2\pi]}\operatorname{ev}^{*}R-\operatorname{pr}_{12}^{*}\int_{[0,2\pi]}\operatorname{ev}^{*}R$
$\displaystyle=\operatorname{pr}_{12}^{*}\left(d\int_{[0,2\pi]}(\operatorname{id}\times\operatorname{ev})^{*}\rho+\int_{[0,2\pi]}(\operatorname{id}\times\operatorname{ev})^{*}(\operatorname{Ad}^{*}R-R)\right)$
$\displaystyle=\operatorname{pr}_{12}^{*}\left(d\int_{[0,2\pi]}(\operatorname{id}\times\operatorname{ev})^{*}\rho+\int_{[0,2\pi]}d(\operatorname{id}\times\operatorname{ev})^{*}\rho\right)$
$\displaystyle=\operatorname{pr}_{12}^{*}((\operatorname{id}\times\operatorname{ev}_{2\pi})^{*}\rho-(\operatorname{id}\times\operatorname{ev}_{0})^{*}\rho)$
$\displaystyle=(\operatorname{id}\times
q)^{*}(\operatorname{id}\times\pi)^{*}\rho,$

where we have applied the fibrewise Stokes’ theorem (Proposition 3.9) in the
second-to-last line, and then the fact
$(\operatorname{id}\times\operatorname{ev}_{0})^{*}\rho=0$. This latter point
follows from the fact that the function $\operatorname{ev}_{0}$ is constant at
the constant loop $1\in\Omega G$. As $\operatorname{id}\times q$ is a
submersion, pullback along it is injective, so we have, finally,
$\widehat{\operatorname{Ad}}^{*}\mu-\operatorname{pr}_{2}^{*}\mu=\rho$. ∎

###  3.3. The Chern–Simons connective structure

Suppose that the principal $G$-bundle $Q\to X$ has connection 1-form $A$. In
the following Definition we refer to the items (1)–(6) in Definition 2.16.

######  Definition 3.13.

The data for the rigid connective structure on $CS(Q)$ is given by

  1. (i)

The bundle gerbe connection in (1) is given by the 1-form $\nabla$ in Lemma
1.6.

  2. (ii)

The curving 2-form $\beta_{A}$ on $Q\times PG$ in (2) is given by

$\beta_{A}=\operatorname{pr}_{2}^{*}B-\pi^{*}\langle\operatorname{pr}_{1}^{*}A,\operatorname{pr}_{2}^{*}\widehat{\Theta}^{G}\rangle,$

where $B$ is defined in Lemma 1.7, and $\widehat{\Theta}^{G}$ is the right
Maurer–Cartan form on $G$.

  3. (iii)

The 2-curving in (3) is the (negative of the) Chern–Simons 3-form

$-CS(A)=-\langle A,F_{A}\rangle+\frac{1}{6}\langle A,[A,A]\rangle,$

where $F_{A}$ is the curvature of $A$.

  4. (iv)

The 1-form $\alpha$ on $Z_{2}=Q\times PG^{2}$ for (4)–(6) is given at
$(x,p,q)\in Q\times PG^{2}$ by

$\alpha_{(x,p,q)}=2\int_{0}^{2\pi}\mathrm{d}\theta\,\langle\Theta_{p},\widehat{\phi}_{q}\rangle.$

(Recall that the 1-form $\alpha$ is denoted $\alpha_{2,1}$ in the course of
the proof of Proposition 2.17.)

That this defines a rigid connective structure is proved below.

Note that $d\alpha=\pi^{*}\kappa$, where $\kappa\in\Omega^{2}(G\times G)$ is
given by
$\kappa=\langle\operatorname{pr}_{1}^{*}\Theta,\operatorname{pr}_{2}^{*}\widehat{\Theta}\rangle$.
This also satisfies $d\kappa=\delta_{h}(\omega)$, where $\omega$ is the
curvature of the basic gerbe, from Lemma 1.8, and appears in [Wal13].

######  Proposition 3.14.

The forms $\nabla$, $\beta_{A}$ and $CS(A)$ give a connective structure on
$CS(Q)$.

###### Proof.

The connection $\nabla$ is a bundle gerbe connection, and there is no further
condition on it. The 2-form $\beta_{A}$ is a curving for $\nabla$, which
verifies conditions (1) and (2) of Definition 2.16; it remains to prove
condition (3): $\pi^{*}\delta_{h}(CS(A))=d\beta_{A}$.

We first calculate $\delta_{h}(-CS(A))$, use the identity
$(\operatorname{act}^{*}A)_{(y,g)}=\operatorname{Ad}_{g^{-1}}(A_{y}+dg\,g^{-1})$,
in stages. First,

$\displaystyle\left(\operatorname{act}^{*}\langle
A,F_{A}\rangle-\operatorname{pr}_{1}^{*}\langle A,F_{A}\rangle\right)_{(y,g)}$
$\displaystyle=\langle\operatorname{Ad}_{g^{-1}}(A_{y}+dg\,g^{-1}),\operatorname{Ad}_{g^{-1}}(F_{A})_{y}\rangle-\langle
A_{y},(F_{A})_{y}\rangle$ $\displaystyle=\langle
dg\,g^{-1},(F_{A})_{y}\rangle,$

and also then

$\displaystyle\tfrac{1}{6}\left(\operatorname{act}^{*}\langle
A,[A,A]\rangle-\operatorname{pr}^{*}\langle A,[A,A]\rangle\right)_{(y,g)}$
$\displaystyle=\tfrac{1}{6}\langle\operatorname{Ad}_{g^{-1}}(A_{y}+dg\,g^{-1}),[\operatorname{Ad}_{g^{-1}}(A_{y}+dg\,g^{-1}),\operatorname{Ad}_{g^{-1}}(A_{y}+dg\,g^{-1})]\rangle-\tfrac{1}{6}\langle
A_{y},[A_{y},A_{y}]\rangle$ $\displaystyle=\tfrac{1}{6}\langle
A_{y}+dg\,g^{-1},[A_{y}+dg\,g^{-1},A_{y}+dg\,g^{-1}]\rangle-\tfrac{1}{6}\langle
A_{y},[A_{y},A_{y}]\rangle$
$\displaystyle=\begin{multlined}\tfrac{1}{6}\left(\langle
dg\,g^{-1},[A_{y},A_{y}]\rangle+\langle
dg\,g^{-1},[dg\,g^{-1},A_{y}]\rangle+\langle
dg\,g^{-1},[A_{y},dg\,g^{-1}]\rangle\right.\\\ +\langle
dg\,g^{-1},[dg\,g^{-1},dg\,g^{-1}]\rangle+\langle
A_{y},[dg\,g^{-1},A_{y}]\rangle+\langle A_{y},[A_{y},dg\,g^{-1}]\rangle\\\
\left.+\langle
A_{y},[dg\,g^{-1},dg\,g^{-1}]\rangle\right)\end{multlined}\tfrac{1}{6}\left(\langle
dg\,g^{-1},[A_{y},A_{y}]\rangle+\langle
dg\,g^{-1},[dg\,g^{-1},A_{y}]\rangle+\langle
dg\,g^{-1},[A_{y},dg\,g^{-1}]\rangle\right.\\\ +\langle
dg\,g^{-1},[dg\,g^{-1},dg\,g^{-1}]\rangle+\langle
A_{y},[dg\,g^{-1},A_{y}]\rangle+\langle A_{y},[A_{y},dg\,g^{-1}]\rangle\\\
\left.+\langle A_{y},[dg\,g^{-1},dg\,g^{-1}]\rangle\right)$
$\displaystyle=\tfrac{1}{2}\left(\langle
dg\,g^{-1},[A_{y},A_{y}]\rangle+\langle
A_{y},[dg\,g^{-1},dg\,g^{-1}]\rangle\right)+\omega_{g},$

where $\omega\in\Omega^{3}(G)$ is defined in Lemma 1.8. So we have

$\displaystyle\delta_{h}(-CS(A))_{y,g}$ $\displaystyle=-\langle
dg\,g^{-1},(F_{A})_{y}\rangle+\tfrac{1}{2}\langle
dg\,g^{-1},[A_{y},A_{y}]\rangle+\tfrac{1}{2}\langle
A_{y},[dg\,g^{-1},dg\,g^{-1}]\rangle+\omega_{g}$
$\displaystyle=\omega_{g}-\langle
dA_{y},\widehat{\Theta}^{G}_{g}\rangle+\langle
A_{y},d\widehat{\Theta}^{G}_{g}\rangle$
$\displaystyle=\omega_{G}-d\langle\operatorname{pr}_{1}^{*}A,\operatorname{pr}_{2}^{*}\widehat{\Theta}^{G}\rangle_{(y,g)}.$

If we then pull this up to $Q\times PG$, we get

$\displaystyle\pi^{*}\omega-\pi^{*}d\langle\operatorname{pr}_{1}^{*}A,\operatorname{pr}_{2}^{*}\widehat{\Theta}^{G}\rangle$
$\displaystyle=dB-d\pi^{*}\langle\operatorname{pr}_{1}^{*}A,\operatorname{pr}_{2}^{*}\widehat{\Theta}^{G}\rangle$
$\displaystyle=d\beta_{A},$

as we needed to show. ∎

However, more is true: the data in Definition 3.13 will form a _rigid_
connective structure. We will next check condititions (4)–(6) in Definition
2.16.

######  Lemma 3.15.

Recall from Definition 3.4 (viii) the section $M\colon(PG\ltimes\Omega
G)^{2}\to\delta_{h}(PG\times\widehat{\Omega G})$. Then

$\delta_{v}\alpha=M^{*}(\operatorname{pr}_{1}^{*}\nabla-\mathrm{mult}^{*}\nabla+\operatorname{pr}_{2}^{*}\nabla)=M^{*}\delta_{h}\nabla,$

which is Definition 2.16 (4) (here $\mathrm{mult}$ is the multiplication map
on the semidirect product $PG\ltimes\Omega G$).

###### Proof.

Recall that
$\nu_{(\gamma,\eta)}=2\int_{0}^{2\pi}\mathrm{d}\theta\,\langle\Theta_{\gamma},\widehat{\phi}_{\eta}\rangle$
satisfies $\pi^{*}\nu=\delta(\nabla)$, where here $\delta$ is with respect to
the three maps $\widehat{\Omega G}^{2}\to\widehat{\Omega G}$, namely
multiplication and the two projections. Using the factorisation

(3.2)
$\begin{multlined}(p,\gamma,q,\eta)\mapsto(pq,q;\gamma,\eta)\xmapsto{\operatorname{id}\times\operatorname{inv}\times\operatorname{id}^{2}}(pq,q^{-1},\gamma,\eta)\xmapsto{\operatorname{id}\times\operatorname{Ad}\times\operatorname{id}}(pq,\operatorname{Ad}_{q^{-1}}(\gamma),\eta)\\\
\xmapsto{\operatorname{id}\times
m}(pq,\operatorname{Ad}_{q^{-1}}(\gamma)\eta)\end{multlined}(p,\gamma,q,\eta)\mapsto(pq,q;\gamma,\eta)\xmapsto{\operatorname{id}\times\operatorname{inv}\times\operatorname{id}^{2}}(pq,q^{-1},\gamma,\eta)\xmapsto{\operatorname{id}\times\operatorname{Ad}\times\operatorname{id}}(pq,\operatorname{Ad}_{q^{-1}}(\gamma),\eta)\\\
\xmapsto{\operatorname{id}\times
m}(pq,\operatorname{Ad}_{q^{-1}}(\gamma)\eta)$

of $\mathrm{mult}$, we will now calculate $M^{*}\delta_{h}\nabla$ at a point
$(p,\gamma,q,\eta)\in(PG\times\Omega G)^{2}$.

$\displaystyle(M^{*}\delta_{h}\nabla)_{(p,\gamma,q,\eta)}$
$\displaystyle=M^{*}\big{[}\mu_{\hat{\gamma}}-\mathrm{mult}^{*}\mu+\mu_{\hat{\eta}}\big{]}-\epsilon_{(p,\gamma)}+\epsilon_{(pq,\operatorname{Ad}_{q^{-1}}(\gamma)\eta)}-\epsilon_{(q,\eta)}$
$\displaystyle=\begin{multlined}M^{*}\big{[}\mu_{\hat{\gamma}}+\mu_{\hat{\eta}}-(\operatorname{id}\times\operatorname{inv}\times\operatorname{id}^{2})^{*}(\operatorname{id}\times\operatorname{Ad}\times\operatorname{id})^{*}(\operatorname{pr}_{1}^{*}\mu+\operatorname{pr}_{2}^{*}\mu-\pi^{*}\nu)\big{]}\\\
-\epsilon_{(p,\gamma)}+\epsilon_{(pq,\operatorname{Ad}_{q^{-1}}(\gamma)\eta)}-\epsilon_{(q,\eta)}\end{multlined}M^{*}\big{[}\mu_{\hat{\gamma}}+\mu_{\hat{\eta}}-(\operatorname{id}\times\operatorname{inv}\times\operatorname{id}^{2})^{*}(\operatorname{id}\times\operatorname{Ad}\times\operatorname{id})^{*}(\operatorname{pr}_{1}^{*}\mu+\operatorname{pr}_{2}^{*}\mu-\pi^{*}\nu)\big{]}\\\
-\epsilon_{(p,\gamma)}+\epsilon_{(pq,\operatorname{Ad}_{q^{-1}}(\gamma)\eta)}-\epsilon_{(q,\eta)}$
$\displaystyle=M^{*}\big{[}\mu_{\hat{\gamma}}+\mu_{\hat{\eta}}-(\operatorname{inv}\times\operatorname{id})^{*}\operatorname{Ad}^{*}\operatorname{pr}_{1}^{*}\mu-\mu_{\hat{\eta}}\big{]}+\nu_{(\operatorname{Ad}_{q^{-1}}(\gamma),\eta)}-\epsilon_{(p,\gamma)}+\epsilon_{(pq,\operatorname{Ad}_{q^{-1}}(\gamma)\eta)}-\epsilon_{(q,\eta)}$
$\displaystyle=M^{*}\big{[}\mu_{\hat{\gamma}}-\mu_{\widehat{\operatorname{Ad}}_{q^{-1}}\hat{\gamma}}\big{]}+\nu_{(\operatorname{Ad}_{q^{-1}}(\gamma),\eta)}+\epsilon_{(pq,\operatorname{Ad}_{q^{-1}}(\gamma)\eta)}-\epsilon_{(q,\eta)}-\epsilon_{(p,\gamma)}$
$\displaystyle=M^{*}\operatorname{inv}_{q}^{*}\big{[}\mu_{\hat{\gamma}}-\mu_{\widehat{\operatorname{Ad}}_{q}\hat{\gamma}}\big{]}+\nu_{(\operatorname{Ad}_{q^{-1}}(\gamma),\eta)}+\epsilon_{(pq,\operatorname{Ad}_{q^{-1}}(\gamma)\eta)}-\epsilon_{(q,\eta)}-\epsilon_{(p,\gamma)}$
$\displaystyle=-\rho_{q^{-1},\gamma}+\nu_{(\operatorname{Ad}_{q^{-1}}(\gamma),\eta)}+\epsilon_{(pq,\operatorname{Ad}_{q^{-1}}(\gamma)\eta)}-\epsilon_{(q,\eta)}-\epsilon_{(p,\gamma)}.$

Now we can calculate that

$\displaystyle\nu_{(\operatorname{Ad}_{q^{-1}}(\gamma),\eta)}$
$\displaystyle=2\int_{0}^{2\pi}\mathrm{d}\theta\,\langle\Theta_{\operatorname{Ad}_{q^{-1}}(\gamma)},\widehat{\phi}_{\eta}\rangle$
$\displaystyle=2\int_{0}^{2\pi}\mathrm{d}\theta\,\langle\operatorname{Ad}_{q^{-1}}(\Theta_{\gamma}+\widehat{\Theta}_{q}-\operatorname{Ad}_{\gamma^{-1}}\widehat{\Theta}_{q}),\widehat{\phi}_{\eta}\rangle,$

and also

$\displaystyle\epsilon_{(pq,\operatorname{Ad}_{q^{-1}}(\gamma)\eta)}$
$\displaystyle=2\int_{0}^{2\pi}\mathrm{d}\theta\,\langle\Theta_{pq},\widehat{\phi}_{\operatorname{Ad}_{q^{-1}}(\gamma)\eta}\rangle$
$\displaystyle=2\int_{0}^{2\pi}\mathrm{d}\theta\,\langle\operatorname{Ad}_{q^{-1}}(\Theta_{p}+\widehat{\Theta}_{q}),\widehat{\phi}_{\operatorname{Ad}_{q^{-1}}(\gamma)}+\operatorname{Ad}_{\operatorname{Ad}_{q^{-1}}(\gamma)}\widehat{\phi}_{\eta}\rangle$
$\displaystyle=2\int_{0}^{2\pi}\mathrm{d}\theta\,\langle\operatorname{Ad}_{q^{-1}}(\Theta_{p}+\widehat{\Theta}_{q}),\operatorname{Ad}_{q^{-1}}(\widehat{\phi}_{\gamma}+\operatorname{Ad}_{\gamma}\widehat{\phi}_{q}-\widehat{\phi}_{q})+\operatorname{Ad}_{\operatorname{Ad}_{q^{-1}}(\gamma)}\widehat{\phi}_{\eta}\rangle$
$\displaystyle=2\int_{0}^{2\pi}\mathrm{d}\theta\,\langle\Theta_{p}+\widehat{\Theta}_{q},\widehat{\phi}_{\gamma}+\operatorname{Ad}_{\gamma}\widehat{\phi}_{q}-\widehat{\phi}_{q}+\operatorname{Ad}_{\gamma
q}\widehat{\phi}_{\eta}\rangle$
$\displaystyle=\begin{multlined}\epsilon_{(p,\gamma)}+2\int_{0}^{2\pi}\mathrm{d}\theta\,\Big{(}\langle\Theta_{p},\operatorname{Ad}_{\gamma}\widehat{\phi}_{q}\rangle-\langle\Theta_{p},\widehat{\phi}_{q}\rangle+\langle\Theta_{p},\operatorname{Ad}_{\gamma
q}\widehat{\phi}_{\eta}\rangle\\\
+\langle\widehat{\Theta}_{q},\widehat{\phi}_{\gamma}\rangle+\langle\widehat{\Theta}_{q},\operatorname{Ad}_{\gamma}\widehat{\phi}_{q}\rangle-\langle\widehat{\Theta}_{q},\widehat{\phi}_{q}\rangle+\langle\widehat{\Theta}_{q},\operatorname{Ad}_{\gamma
q}\widehat{\phi}_{\eta}\rangle\Big{)}.\end{multlined}\epsilon_{(p,\gamma)}+2\int_{0}^{2\pi}\mathrm{d}\theta\,\Big{(}\langle\Theta_{p},\operatorname{Ad}_{\gamma}\widehat{\phi}_{q}\rangle-\langle\Theta_{p},\widehat{\phi}_{q}\rangle+\langle\Theta_{p},\operatorname{Ad}_{\gamma
q}\widehat{\phi}_{\eta}\rangle\\\
+\langle\widehat{\Theta}_{q},\widehat{\phi}_{\gamma}\rangle+\langle\widehat{\Theta}_{q},\operatorname{Ad}_{\gamma}\widehat{\phi}_{q}\rangle-\langle\widehat{\Theta}_{q},\widehat{\phi}_{q}\rangle+\langle\widehat{\Theta}_{q},\operatorname{Ad}_{\gamma
q}\widehat{\phi}_{\eta}\rangle\Big{)}.$

Therefore, we have:

$\displaystyle(M^{*}\delta_{h}\nabla)_{(p,\gamma,q,\eta)}$
$\displaystyle=\begin{multlined}2\int_{0}^{2\pi}\mathrm{d}\theta\,\Big{\\{}-\langle\widehat{\Theta}_{q},\operatorname{Ad}_{\gamma}\widehat{\phi}_{q}+\widehat{\phi}_{\gamma}\rangle+\langle\Theta_{\gamma}+\widehat{\Theta}_{q},\widehat{\phi}_{q}\rangle+\langle\Theta_{\gamma}+\widehat{\Theta}_{q}-\operatorname{Ad}_{\gamma^{-1}}\widehat{\Theta}_{q},\operatorname{Ad}_{q}\widehat{\phi}_{\eta}\rangle\\\
+\langle\Theta_{p},\widehat{\phi}_{\gamma}\rangle-\langle\Theta_{p},\widehat{\phi}_{q}\rangle+\langle\Theta_{p},\operatorname{Ad}_{\gamma}\widehat{\phi}_{q}\rangle+\langle\Theta_{p},\operatorname{Ad}_{\gamma}\operatorname{Ad}_{q}\widehat{\phi}_{\omega}\rangle-\langle\operatorname{Ad}_{q}\Theta_{q},\widehat{\phi}_{q}\rangle\\\
+\langle\operatorname{Ad}_{q}\Theta_{q},\widehat{\phi}_{\gamma}\rangle+\langle\operatorname{Ad}_{q}\Theta_{q},\operatorname{Ad}_{\gamma}\widehat{\phi}_{q}\rangle+\langle\operatorname{Ad}_{q}\Theta_{q},\operatorname{Ad}_{\gamma
q}\widehat{\phi}_{\eta}\rangle-\langle\Theta_{q},\widehat{\phi}_{\eta}\rangle-\langle\Theta_{p},\widehat{\phi}_{\gamma}\rangle\Big{\\}}\end{multlined}2\int_{0}^{2\pi}\mathrm{d}\theta\,\Big{\\{}-\langle\widehat{\Theta}_{q},\operatorname{Ad}_{\gamma}\widehat{\phi}_{q}+\widehat{\phi}_{\gamma}\rangle+\langle\Theta_{\gamma}+\widehat{\Theta}_{q},\widehat{\phi}_{q}\rangle+\langle\Theta_{\gamma}+\widehat{\Theta}_{q}-\operatorname{Ad}_{\gamma^{-1}}\widehat{\Theta}_{q},\operatorname{Ad}_{q}\widehat{\phi}_{\eta}\rangle\\\
+\langle\Theta_{p},\widehat{\phi}_{\gamma}\rangle-\langle\Theta_{p},\widehat{\phi}_{q}\rangle+\langle\Theta_{p},\operatorname{Ad}_{\gamma}\widehat{\phi}_{q}\rangle+\langle\Theta_{p},\operatorname{Ad}_{\gamma}\operatorname{Ad}_{q}\widehat{\phi}_{\omega}\rangle-\langle\operatorname{Ad}_{q}\Theta_{q},\widehat{\phi}_{q}\rangle\\\
+\langle\operatorname{Ad}_{q}\Theta_{q},\widehat{\phi}_{\gamma}\rangle+\langle\operatorname{Ad}_{q}\Theta_{q},\operatorname{Ad}_{\gamma}\widehat{\phi}_{q}\rangle+\langle\operatorname{Ad}_{q}\Theta_{q},\operatorname{Ad}_{\gamma
q}\widehat{\phi}_{\eta}\rangle-\langle\Theta_{q},\widehat{\phi}_{\eta}\rangle-\langle\Theta_{p},\widehat{\phi}_{\gamma}\rangle\Big{\\}}$
$\displaystyle=2\int_{0}^{2\pi}\mathrm{d}\theta\,\begin{multlined}\Big{\\{}\langle\widehat{\Theta}_{q},\widehat{\phi}_{q}\rangle-\langle\Theta_{p},\widehat{\phi}_{q}\rangle-\langle\widehat{\Theta}_{q},\widehat{\phi}_{q}\rangle-\langle\widehat{\Theta}_{q},\operatorname{Ad}_{\gamma}\widehat{\phi}_{q}+\widehat{\phi}_{\gamma}\rangle+\langle\Theta_{\gamma},\widehat{\phi}_{q}\rangle\\\
+\langle\widehat{\Theta}_{\gamma},\operatorname{Ad}_{\gamma
q}\widehat{\phi}_{\eta}\rangle+\langle\Theta_{q},\widehat{\phi}_{\eta}\rangle-\langle\widehat{\Theta}_{q},\operatorname{Ad}_{\gamma
q}\widehat{\phi}_{\eta}\rangle+\langle\Theta_{p},\operatorname{Ad}_{\gamma}\widehat{\phi}_{q}\rangle\\\
+\langle\Theta_{p},\operatorname{Ad}_{\gamma
q}\widehat{\phi}_{\eta}\rangle+\langle\widehat{\Theta}_{q},\widehat{\phi}_{\gamma}\rangle+\langle\widehat{\Theta}_{q},\operatorname{Ad}_{\gamma}\widehat{\phi}_{q}\rangle+\langle\widehat{\Theta}_{q},\operatorname{Ad}_{\gamma
q}\widehat{\phi}_{\eta}\rangle-\langle\Theta_{q},\widehat{\phi}_{\eta}\rangle\Big{\\}}\end{multlined}\Big{\\{}\langle\widehat{\Theta}_{q},\widehat{\phi}_{q}\rangle-\langle\Theta_{p},\widehat{\phi}_{q}\rangle-\langle\widehat{\Theta}_{q},\widehat{\phi}_{q}\rangle-\langle\widehat{\Theta}_{q},\operatorname{Ad}_{\gamma}\widehat{\phi}_{q}+\widehat{\phi}_{\gamma}\rangle+\langle\Theta_{\gamma},\widehat{\phi}_{q}\rangle\\\
+\langle\widehat{\Theta}_{\gamma},\operatorname{Ad}_{\gamma
q}\widehat{\phi}_{\eta}\rangle+\langle\Theta_{q},\widehat{\phi}_{\eta}\rangle-\langle\widehat{\Theta}_{q},\operatorname{Ad}_{\gamma
q}\widehat{\phi}_{\eta}\rangle+\langle\Theta_{p},\operatorname{Ad}_{\gamma}\widehat{\phi}_{q}\rangle\\\
+\langle\Theta_{p},\operatorname{Ad}_{\gamma
q}\widehat{\phi}_{\eta}\rangle+\langle\widehat{\Theta}_{q},\widehat{\phi}_{\gamma}\rangle+\langle\widehat{\Theta}_{q},\operatorname{Ad}_{\gamma}\widehat{\phi}_{q}\rangle+\langle\widehat{\Theta}_{q},\operatorname{Ad}_{\gamma
q}\widehat{\phi}_{\eta}\rangle-\langle\Theta_{q},\widehat{\phi}_{\eta}\rangle\Big{\\}}$
$\displaystyle=2\int_{0}^{2\pi}\mathrm{d}\theta\,\langle\widehat{\Theta}_{\gamma}+\Theta_{p},\operatorname{Ad}_{\gamma}\widehat{\phi}_{q}\rangle+\langle\widehat{\Theta}_{\gamma}+\Theta_{p},\operatorname{Ad}_{\gamma
q}\widehat{\phi}_{\eta}\rangle-\langle\Theta_{p},\widehat{\phi}_{q}\rangle$
$\displaystyle=2\int_{0}^{2\pi}\mathrm{d}\theta\,\langle\Theta_{p\gamma},\widehat{\phi}_{q\eta}\rangle-\langle\Theta_{p},\widehat{\phi}_{q}\rangle$
$\displaystyle=\alpha_{(p\gamma,q\eta)}-\alpha_{(p,q)},$

as we needed to show. ∎

######  Lemma 3.16.

The condition (5) in Definition 2.16 holds for $\beta_{A}$ and $\alpha$,
namely $d\alpha=\delta_{h}\beta_{A}$.

###### Proof.

Let us first calculate $d\alpha$, taking into account the fact that $\alpha$
is independent of the $Q$-coordinate, that $PG^{2}$ is a Lie group, and that
$\alpha$ is left invariant in the first coordinate. Take a pair of tangent
vectors $(pX_{1},qY_{1}),(pX_{2},qY_{2})\in T_{(p,q)}PG^{2}$, and evaluate,
where by an abuse of notation we are suppressing any mention of coordinates on
$Q$:

$\displaystyle d\alpha_{(p,q)}\left((pX_{1},qY_{1}),(pX_{2},qY_{2})\right)$
$\displaystyle=\tfrac{1}{2}\Big{\\{}(pX_{1},qY_{1})\cdot\alpha(X_{2},qY_{2})-(pX_{2},qY_{2})\cdot\alpha(X_{1},qY_{1})-\alpha(p[X_{1},X_{2}],q[Y_{1},Y_{2}])\Big{\\}}.$

To continue, we need the Lie derivative
$(pX,qY)\cdot\alpha(pX^{\prime},qY^{\prime})$, which is

$\displaystyle(pX,qY)\cdot\alpha(pX^{\prime},qY^{\prime})$
$\displaystyle=\left.\frac{d}{dt}\left\\{2\int_{0}^{2\pi}\mathrm{d}\theta\,\langle
X^{\prime},\partial_{\theta}[q(1+tY+O(t^{2}))](1-tY+O(t^{2}))q^{-1}\rangle\right\\}\right|_{t=0}$
$\displaystyle=\begin{multlined}2\int_{0}^{2\pi}\mathrm{d}\theta\,\frac{d}{dt}\langle
X^{\prime},(\partial_{\theta}q)(1+tY+O(t^{2}))(1-tY+O(t^{2}))q^{-1}\rangle\big{|}_{t=0}\\\
+\frac{d}{dt}\langle
X^{\prime},qt\partial_{\theta}(Y)(1-tY)q^{-1}+O(t^{2})\rangle\big{|}_{t=0}\end{multlined}2\int_{0}^{2\pi}\mathrm{d}\theta\,\frac{d}{dt}\langle
X^{\prime},(\partial_{\theta}q)(1+tY+O(t^{2}))(1-tY+O(t^{2}))q^{-1}\rangle\big{|}_{t=0}\\\
+\frac{d}{dt}\langle
X^{\prime},qt\partial_{\theta}(Y)(1-tY)q^{-1}+O(t^{2})\rangle\big{|}_{t=0}$
$\displaystyle=2\int_{0}^{2\pi}\mathrm{d}\theta\,\frac{d}{dt}\langle
X^{\prime},(\partial_{\theta}q)(1+O(t^{2}))q^{-1}\rangle\big{|}_{t=0}+\langle
X^{\prime},q\partial_{\theta}(Y)q^{-1}\rangle$
$\displaystyle=2\int_{0}^{2\pi}\mathrm{d}\theta\,\langle
X^{\prime},q\partial_{\theta}(Y)q^{-1}\rangle.$

Using this, we find

$d\alpha_{(p,q)}\left((pX_{1},qY_{1}),(pX_{2},qY_{2})\right)=2\int_{0}^{2\pi}\mathrm{d}\theta\,\langle
X_{2},\operatorname{Ad}_{q}\partial Y_{1}\rangle-\langle
X_{1},\operatorname{Ad}_{q}\partial
Y_{2}\rangle-\langle[X_{1},X_{2}],\widehat{\phi}_{q}\rangle.$

Let us check some particular evaluations of $B$, noting that it is left-
invariant:

$\displaystyle B(Y_{1},\operatorname{Ad}_{q}^{-1}X_{2})$
$\displaystyle=\int_{0}^{2\pi}\mathrm{d}\theta\,\langle
Y_{1},\operatorname{Ad}_{q^{-1}}[X_{2},\widehat{\phi}_{q}]\rangle+\langle
Y_{1},\operatorname{Ad}_{q^{-1}}\partial X_{2}\rangle$

and

$\displaystyle
B(\operatorname{Ad}_{q^{-1}}X_{1},\operatorname{Ad}_{q^{-1}}X_{2})$
$\displaystyle=\int_{0}^{2\pi}\mathrm{d}\theta\,\langle
q^{-1}X_{1}q,\partial(q^{-1}X_{2}q)\rangle$
$\displaystyle=\int_{0}^{2\pi}\mathrm{d}\theta\,\langle
X_{1},[X_{2},\widehat{\phi}_{q}]\rangle+\langle X_{1},\partial X_{2}\rangle.$

We also need to calculate
$-\delta_{h}\pi^{*}\langle\operatorname{pr}_{1}^{*}A,\operatorname{pr}_{2}^{*}\widehat{\Theta}^{G}\rangle=-\pi^{*}\delta_{h}\langle\operatorname{pr}_{1}^{*}A,\operatorname{pr}_{2}^{*}\widehat{\Theta}^{G}\rangle$,
and so we need

$\displaystyle\delta_{h}\langle\operatorname{pr}_{1}^{*}A,\operatorname{pr}_{2}^{*}\widehat{\Theta}^{G}\rangle_{(y,g,h)}$
$\displaystyle=\langle A_{yg},\widehat{\Theta}_{h}\rangle-\langle
A_{y},\widehat{\Theta}_{gh}\rangle+\langle A_{y},\widehat{\Theta}_{g}\rangle$
$\displaystyle=\langle\operatorname{Ad}_{g^{-1}}A_{y}+\Theta_{g},\widehat{\Theta}_{h}\rangle-\langle
A_{y},\widehat{\Theta}_{g}+\operatorname{Ad}_{g}\widehat{\Theta}_{h}\rangle+\langle
A_{y},\widehat{\Theta}_{g}\rangle$ $\displaystyle=\langle
A_{y},\operatorname{Ad}_{g}\widehat{\Theta}_{h}\rangle+\langle\Theta_{g},\widehat{\Theta}_{h}\rangle-\langle
A_{y},\widehat{\Theta}_{g}+\operatorname{Ad}_{g}\widehat{\Theta}_{h}\rangle+\langle
A_{y},\widehat{\Theta}_{g}\rangle$
$\displaystyle=\langle\Theta_{g},\widehat{\Theta}_{h}\rangle=\kappa_{(g,h)}.$

And now we can calculate $\delta_{h}(B)_{(p,q)}$:

$\displaystyle\delta_{h}(B)_{(p,q)}\left((pX_{1},qY_{1}),(pX_{2},qY_{2})\right)$
$\displaystyle=B(X_{1},X_{2})-B(\operatorname{Ad}_{q^{-1}}X_{1}+Y_{1},\operatorname{Ad}_{q^{-1}}X_{2}+Y_{2})+B(Y_{1},Y_{2})$
$\displaystyle=B(X_{1},X_{2})-B(\operatorname{Ad}_{q^{-1}}X_{1},\operatorname{Ad}_{q^{-1}}X_{2})-B(Y_{1},\operatorname{Ad}_{q^{-1}}X_{2})-B(\operatorname{Ad}_{q^{-1}}X_{1},Y_{2})$
$\displaystyle=\begin{multlined}\frac{1}{2}\int_{0}^{2\pi}\mathrm{d}\theta\,\langle
X_{1},\partial X_{2}\rangle-\langle X_{2},\partial
X_{1}\rangle-\langle\operatorname{Ad}_{q^{-1}}X_{1},\partial(q^{-1}X_{2}q)\rangle+\langle\operatorname{Ad}_{q^{-1}}X_{2},\partial(q^{-1}X_{1}q)\rangle\\\
-\langle
Y_{1},\partial(q^{-1}X_{2}q)\rangle+\langle\operatorname{Ad}_{q^{-1}}X_{2},\partial
Y_{1}\rangle-\langle\operatorname{Ad}_{q^{-1}}X_{1},\partial
Y_{2}\rangle+\langle
Y_{1},\partial(q^{-1}X_{2}q)\rangle\end{multlined}\frac{1}{2}\int_{0}^{2\pi}\mathrm{d}\theta\,\langle
X_{1},\partial X_{2}\rangle-\langle X_{2},\partial
X_{1}\rangle-\langle\operatorname{Ad}_{q^{-1}}X_{1},\partial(q^{-1}X_{2}q)\rangle+\langle\operatorname{Ad}_{q^{-1}}X_{2},\partial(q^{-1}X_{1}q)\rangle\\\
-\langle
Y_{1},\partial(q^{-1}X_{2}q)\rangle+\langle\operatorname{Ad}_{q^{-1}}X_{2},\partial
Y_{1}\rangle-\langle\operatorname{Ad}_{q^{-1}}X_{1},\partial
Y_{2}\rangle+\langle Y_{1},\partial(q^{-1}X_{2}q)\rangle$
$\displaystyle=\begin{multlined}\frac{1}{2}\int_{0}^{2\pi}\mathrm{d}\theta\,\langle
X_{1},\partial X_{2}\rangle-\langle X_{2},\partial X_{1}\rangle-\langle
X_{1},\partial X_{2}\rangle-\langle X_{1},[X_{2},\widehat{\phi}_{q}]\rangle\\\
-\langle
Y_{1},\partial(\operatorname{Ad}_{q^{-1}}X_{2})\rangle+\langle\operatorname{Ad}_{q^{-1}},\partial
Y_{1}\rangle+\langle X_{2},\partial X_{1}\rangle+\langle
X_{2},[X_{1},\widehat{\phi}_{q}]\rangle\\\
-\langle\operatorname{Ad}_{q^{-1}}X_{1},\partial Y_{2}\rangle+\langle
Y_{2},\partial(\operatorname{Ad}_{q^{-1}}X_{1})\rangle\end{multlined}\frac{1}{2}\int_{0}^{2\pi}\mathrm{d}\theta\,\langle
X_{1},\partial X_{2}\rangle-\langle X_{2},\partial X_{1}\rangle-\langle
X_{1},\partial X_{2}\rangle-\langle X_{1},[X_{2},\widehat{\phi}_{q}]\rangle\\\
-\langle
Y_{1},\partial(\operatorname{Ad}_{q^{-1}}X_{2})\rangle+\langle\operatorname{Ad}_{q^{-1}},\partial
Y_{1}\rangle+\langle X_{2},\partial X_{1}\rangle+\langle
X_{2},[X_{1},\widehat{\phi}_{q}]\rangle\\\
-\langle\operatorname{Ad}_{q^{-1}}X_{1},\partial Y_{2}\rangle+\langle
Y_{2},\partial(\operatorname{Ad}_{q^{-1}}X_{1})\rangle$
$\displaystyle=\begin{multlined}\frac{1}{2}\int_{0}^{2\pi}\mathrm{d}\theta\,\langle
X_{2},\operatorname{Ad}_{q}\partial Y_{1}\rangle-\langle
X_{1},\operatorname{Ad}_{q}Y_{2}\rangle-2\langle[X_{1},X_{2},\widehat{\phi}_{q}\rangle\\\
-\langle Y_{1},\partial(\operatorname{Ad}_{q^{-1}}X_{2})\rangle+\langle
Y_{2},\partial(\operatorname{Ad}_{q^{-1}}X_{1})\rangle\end{multlined}\frac{1}{2}\int_{0}^{2\pi}\mathrm{d}\theta\,\langle
X_{2},\operatorname{Ad}_{q}\partial Y_{1}\rangle-\langle
X_{1},\operatorname{Ad}_{q}Y_{2}\rangle-2\langle[X_{1},X_{2},\widehat{\phi}_{q}\rangle\\\
-\langle Y_{1},\partial(\operatorname{Ad}_{q^{-1}}X_{2})\rangle+\langle
Y_{2},\partial(\operatorname{Ad}_{q^{-1}}X_{1})\rangle$
$\displaystyle=\begin{multlined}\frac{1}{2}\int_{0}^{2\pi}\mathrm{d}\theta\,\Big{\\{}\langle
X_{2},\operatorname{Ad}_{q}\partial Y_{1}\rangle+\langle\partial
Y_{1},\operatorname{Ad}_{q^{-1}}X_{2}\rangle-\langle
X_{1},\operatorname{Ad}_{q}\partial Y_{2}\rangle-\langle\partial
Y_{2},\operatorname{Ad}_{q^{-1}}X_{1}\rangle\\\
-2\langle[X_{1},X_{2},\widehat{\phi}_{q}\rangle\Big{\\}}+\tfrac{1}{2}\left(-\langle
X_{2}(2\pi),\operatorname{Ad}_{q(2\pi)}Y_{1}(2\pi)\rangle+\langle
X_{1}(2\pi),\operatorname{Ad}_{q(2\pi)}Y_{2}(2\pi)\rangle\right)\end{multlined}\frac{1}{2}\int_{0}^{2\pi}\mathrm{d}\theta\,\Big{\\{}\langle
X_{2},\operatorname{Ad}_{q}\partial Y_{1}\rangle+\langle\partial
Y_{1},\operatorname{Ad}_{q^{-1}}X_{2}\rangle-\langle
X_{1},\operatorname{Ad}_{q}\partial Y_{2}\rangle-\langle\partial
Y_{2},\operatorname{Ad}_{q^{-1}}X_{1}\rangle\\\
-2\langle[X_{1},X_{2},\widehat{\phi}_{q}\rangle\Big{\\}}+\tfrac{1}{2}\left(-\langle
X_{2}(2\pi),\operatorname{Ad}_{q(2\pi)}Y_{1}(2\pi)\rangle+\langle
X_{1}(2\pi),\operatorname{Ad}_{q(2\pi)}Y_{2}(2\pi)\rangle\right)$
$\displaystyle=\begin{multlined}\int_{0}^{2\pi}\mathrm{d}\theta\,\Big{\\{}\langle
X_{2},\operatorname{Ad}_{q}\partial Y_{1}\rangle-\langle
X_{1},\operatorname{Ad}_{q}\partial
Y_{2}\rangle-\langle[X_{1},X_{2},\widehat{\phi}_{q}\rangle\Big{\\}}+\pi^{*}\kappa_{(p,q)}\left((pX_{1},qY_{1}),(pX_{2},qY_{2})\right)\end{multlined}\int_{0}^{2\pi}\mathrm{d}\theta\,\Big{\\{}\langle
X_{2},\operatorname{Ad}_{q}\partial Y_{1}\rangle-\langle
X_{1},\operatorname{Ad}_{q}\partial
Y_{2}\rangle-\langle[X_{1},X_{2},\widehat{\phi}_{q}\rangle\Big{\\}}+\pi^{*}\kappa_{(p,q)}\left((pX_{1},qY_{1}),(pX_{2},qY_{2})\right)$
$\displaystyle=d\alpha_{(p,q)}\left((pX_{1},qY_{1}),(pX_{2},qY_{2})\right)+\pi^{*}\kappa_{(p,q)}\left((pX_{1},qY_{1}),(pX_{2},qY_{2})\right).$

Thus

$\delta_{h}\beta_{A}=\delta_{h}B-\delta_{h}\pi^{*}\langle\operatorname{pr}_{1}^{*}A,\operatorname{pr}_{2}^{*}\widehat{\Theta}^{G}\rangle=d\alpha+\pi^{*}\kappa-\pi^{*}\kappa=d\alpha,$

as we needed to show. ∎

######  Lemma 3.17.

We have $\delta_{h}\alpha=0$, which is Definition 2.16 (6).

###### Proof.

Since integration is linear, it is sufficient to show that

$\delta_{h}\Big{(}\langle\Theta_{p},\widehat{\phi}_{q}\rangle\Big{)}=0.$

We shall first evaluate this form at $(p_{1},p_{2},p_{3})\in PG^{3}$.

$\displaystyle\delta_{h}\langle\Theta_{p},\widehat{\phi}_{q}\rangle_{(p_{1},p_{2},p_{3})}$
$\displaystyle=\langle\Theta_{p_{2}}-\Theta_{p_{1}p_{2}},\widehat{\phi}_{p_{3}}\rangle+\langle\Theta_{p_{1}},\widehat{\phi}_{p_{2}p_{3}}-\widehat{\phi}_{p_{2}}\rangle$
$\displaystyle=\langle\Theta_{p_{2}}-\operatorname{Ad}_{p_{2}^{-1}}(\Theta_{p_{1}}+\widehat{\Theta}_{p_{2}}),\widehat{\phi}_{p_{3}}\rangle+\langle\Theta_{p_{1}},\operatorname{Ad}_{p_{2}}\widehat{\phi}_{p_{3}}\rangle$
$\displaystyle=\langle\operatorname{Ad}_{p_{2}^{-1}}({\widehat{\Theta}_{p_{2}}}-{\Theta_{p_{1}}}-{\widehat{\Theta}_{p_{2}}}),\widehat{\phi}_{p_{3}}\rangle+{\langle\operatorname{Ad}_{p_{2}^{-1}}\Theta_{p_{1}},\widehat{\phi}_{p_{3}}\rangle}$
$\displaystyle=0.$

Hence $\delta_{h}\alpha=0$, as we needed to show. ∎

We now arrive at last at the main result of this paper, the raison d’etre of
the theory developed so far. Denote by $p_{1}(Q,A)$ the standard Chern–Weil
representative for the first Pontryagin class associated to the $G$-bundle $Q$
equipped with connection $A$. The following theorem is our version of [Wal13,
Theorem 2.7], modulo choices of signs of generators of cohomology.

######  Theorem 3.18.

The data in Definition 3.13 gives a rigid connective structure on $CS(Q)$,
with curvature 4-form

$\operatorname{curv}(\nabla,\beta_{A},-CS(A))=-\langle F_{A},F_{A}\rangle,$

which in the case of $G=\operatorname{Spin}(n)$ ($n\geq 5$), satisfies
$\operatorname{curv}(\nabla,\beta_{A},-CS(A))/2\pi=\frac{1}{2}p_{1}(Q,A)$.

###### Proof.

The first claim follows from adding Lemmata 3.16, 3.17 and 3.15 to Proposition
3.14. That the 4-form curvature, the descent along $Q\to X$ of $-d\,CS(A)$, is
$-\langle F_{A},F_{A}\rangle$ is a standard result (see e.g. [Fre95,
Proposition 1.27]). When $G=\operatorname{Spin}(n)$, the first Pontryagin form
is defined to be (a certain multiple of) the _trace_ of $F_{A}^{2}$, hence a
_negative_ multiple of the inner product—we just have to check the coefficient
is correct. From Example 1.4 we have that $\langle
X,Y\rangle=-\operatorname{tr}(XY)/8\pi$ for
$\mathfrak{g}=\mathfrak{spin(n)}\simeq\mathfrak{so}(n)$, $n\geq 5$, so:

$\displaystyle\operatorname{curv}(\nabla,\beta_{A},-CS(A))/2\pi$
$\displaystyle=-\langle F_{A},F_{A}\rangle/2\pi$
$\displaystyle=\big{(}\operatorname{tr}(F_{A}^{2})/8\pi\big{)}/2\pi$
$\displaystyle=\tfrac{1}{2}\operatorname{tr}(F_{A}^{2})/8\pi^{2}$
$\displaystyle=\tfrac{1}{2}p_{1}(Q,A).\qed$

######  Remark 3.19.

If one takes the connection and curving on the basic gerbe on $G$ from [MS03],
then using the 1-form $\varepsilon_{MS}$ from Remark 1.10, the only other
necessary change to the rigid connective structure in Theorem 3.18 is that one
replaces $\alpha$ by $\alpha+\delta_{h}(\varepsilon_{MS})$. The 2-curving (the
negative of the Chern–Simons form) and the 4-form curvature (the first
fractional Pontryagin form) are unchanged.

######  Corollary 3.20.

Modulo the previous Remark, the geometric content of the rigid bundle 2-gerbe
given here—the _actual differential forms_ $\nabla$, $\beta_{A}$ and $-CS(A)$
making up the connective structure—is the same as that in [CJM+05]. Thus we
have described a rigidification of the Chern–Simons 2-gerbe of _loc. cit._ ,
including with connection, curving and 2-curving. Moreover, as we have a
_rigid_ connective structure, this gives bundle 2-gerbe connection in the
stronger sense of [Wal13, Definition 4.1] (using the background definitions in
§6.2 of the same).

###### Proof.

This follows after one has the construction in Appendix A of the associated
bundle 2-gerbe with connective structure (Propositions A.8 and A.10) and that
the underlying rigid 2-gerbe here rigidifies the Chern–Simons bundle 2-gerbe.
∎

The point of Theorem 3.18, then, is that it allows easier calculations with
the geometry of the Chern–Simons gerbe, in particular in the presence of
trivialisations, which we will treat in a forthcoming paper.

##  Appendix A Associated bundle 2-gerbes and trivialisations

Here we give the details of what is only outlined above in Subsection 2.3. The
economy of our definitions will be seen in comparison to the amount of work
that needs to be done to verify the weaker general bundle 2-gerbe definitions
from our rigid data.

###  A.1. Additional technical requirements

Before we start, we need to build up some technical material related to
constructions with bundle gerbes, as well as the bigroupoid of bundle gerbes.

Recall the definition of a _morphism_ of bundle gerbes
$(\varphi,f)\colon(E,Y)\to(F,X)$ over a manifold $M$: it is a map $f\colon
Y\to X$ over the manifold $M$, together with an isomorphism $\varphi\colon
E\simeq(f^{[2]})^{*}F$ respecting the bundle gerbe product. Viewing bundle
gerbes as certain sort of Lie groupoids over $M$, a morphism is equivalent
data to a functor over $M$. However, recall the definition of a _stable
isomorphism_ of bundle gerbes from $(E,Y)$ to $(F,X)$, both on $M$ (of which
there are several equivalent versions), which is a trivialisation of the
bundle gerbe
$(E^{*},Y)\otimes(F,X):=\left(\operatorname{pr}_{12}^{*}E^{*}\otimes\operatorname{pr}_{34}^{*}F,Y\times_{M}X\right)$.
Such a trivialisation consists of a principal $U(1)$-bundle $L\to
Y\times_{M}X$ plus a bundle isomorphism
$\tau\colon\delta(L)\xrightarrow{\simeq}\operatorname{pr}_{12}^{*}E^{*}\otimes\operatorname{pr}_{34}^{*}F$
on $(Y\times_{M}X)^{[2]}\simeq Y^{[2]}\times_{M}X^{[2]}$ preserving bundle
gerbe multiplication. We shall denote a stable isomorphism with this data by
$(L,\tau)\colon(E,Y)\rightsquigarrow(F,X)$. If there is a morphism between a
pair of bundle gerbes, then the bundle gerbes are also stably isomorphic, and
we can construct a canonical such.

######  Construction A.1.

Given a morphism of bundle gerbes $(\varphi,f)\colon(E,Y)\to(F,X)$ on $M$,
consider the pullback $Y\times_{M}X$, and recall that there is a map $Y\to
Y\times_{M}X$, $y\mapsto(y,f(y))$. Then the projection $Y\times_{M}X\to X$ is
equal to the composite $Y\times_{M}X\to Y\to X$, so that the bundle on
$Y^{[2]}\times_{M}X^{[2]}$ gotten by pulling back $F$ along
$\operatorname{pr}_{34}$ is isomorphic to the pullback of $E$ along
$\operatorname{pr}_{12}$, and this isomorphism is the pullback along
$\operatorname{pr}_{12}$ of $\varphi$. Thus, taking the bundle
$(f\times\operatorname{id})^{*}F$ on $Y\times_{M}X$, we get a stable
isomorphism
$\Sigma(\varphi,f)=((f\times\operatorname{id})^{*}F,\tau_{f})\colon(E,Y)\rightsquigarrow(F,X)$
by forming the multiplication-preserving isomorphism
$\tau_{f}\colon\delta((f\times\operatorname{id})^{*}F)\xrightarrow{\simeq}\operatorname{pr}_{12}^{*}E^{*}\otimes\operatorname{pr}_{34}^{*}F$,
given fibrewise by

$\displaystyle F^{*}_{(f(y_{1}),x_{1})}\otimes F_{(f(y_{2}),x_{2})}$
$\displaystyle\simeq F^{*}_{(f(y_{1}),x_{1})}\otimes
F^{*}_{(x_{1},f(y_{2}))}\otimes F_{(x_{1},f(y_{2}))}\otimes
F_{(f(y_{2}),x_{2})}$ $\displaystyle\simeq F^{*}_{(f(y_{1}),f(y_{2}))}\otimes
F_{(x_{1},x_{2})}$ $\displaystyle\simeq E^{*}_{(y_{1},y_{2})}\otimes
F_{(x_{1},x_{2})}.$

Further, this construction is appropriately pseudofunctorial for the
composition of bundle gerbe morphisms. In particular given morphisms
$(\varphi,f)\colon(E,Y)\to(F,X)$ and $(\psi,g)\colon(F,X)\to(G,W)$ over $M$,
there is an isomorphism

$\Sigma^{2}_{(\psi,g),(\varphi,f)}\colon\Sigma(\psi,g)\circ\Sigma(\varphi,f)\Rightarrow\Sigma((\psi,g)\circ(\varphi,f)$

between the associated stable isomorphisms.161616The other coherence map for
$\Sigma$, comparing the identity morphism of bundle gerbes and the identity
stable isomorphism, is an equality, essentially by definition

If we compute the composite $\Sigma(\varphi,f)\circ\Sigma(\psi,g)$ according
to [Ste00, pp 31–32], we can use the fact there is a section of
$Y\times_{M}X\times_{M}W\to Y\times_{M}W$, so that instead of constructing a
bundle on $Y\times_{M}W$ via descent, we can just pull back the constructed
bundle on $Y\times_{M}X\times_{M}W$ by the section. The result is that the
trivialising $U(1)$-bundle on $Y\times_{M}W$ for the composite stable
isomorphism is $\mathbf{1}\otimes\mathbf{1}\otimes((g\circ
f)\times\operatorname{id})^{*}G$ (where, recall, $\mathbf{1}$ denotes the
trivial bundle). The canonical isomorphism to $((g\circ
f)\times\operatorname{id})^{*}G$ is then the data of a 2-arrow between stable
isomorphisms, where the latter bundle is the trivialising bundle for the
stable isomorphism $\Sigma((\psi,g)\circ(\varphi,f))$. In particular, the data
of this 2-arrow is independent of $(\psi,g)$ and $(\varphi,f)$.

Recall that bundle gerbes on a given manifold $M$ form a bicategory with
1-arrows stable isomorphisms [Ste00, Proposition 3.9]. In a bicategory, we can
talk about what it means for a 1-arrow to be an _adjoint_ inverse to another
[JY21, Proposition 6.2.1]. This is a stronger and better-behaved notion than
just being an inverse up to isomorphism, and it is the case that any pseudo-
inverse can be strengthened to be an adjoint inverse (e.g. [JY21, Proposition
6.2.4]).

######  Lemma A.2.

Let $(E,Y)$ and $(F,X)$ be bundle gerbes on $M$. The adjoint inverse of a
stable isomorphism $(L,\tau)\colon(E,Y)\rightsquigarrow(F,X)$ is
$(\sigma^{*}L^{*},\sigma^{*}(\tau)^{*})$, where $\sigma\colon X\times_{M}Y\to
Y\times_{M}X$ is the swap map and $(\tau)^{*}$ denotes the isomorphism between
the dual bundles arising from $\tau$. Thus is we start with a _morphism_
$(\varphi,f)\colon(E,Y)\to(F,X)$ of bundle gerbes, the adjoint inverse of
$\Sigma(\varphi,f)$ is $((\operatorname{id}\times
f)^{*}F^{*},\sigma^{*}(\tau_{f})^{*})\colon(F,X)\rightsquigarrow(E,Y)$. If the
morphism $(\varphi,f)$ is an _isomorphism_ of bundle gerbes, so that $f$ is
invertible, the adjoint inverse of $\Sigma(\varphi,f)$ is
$\Sigma((f^{[2]})^{*}\varphi^{-1},f^{-1})$ of $(\varphi,f)$ (here
$((f^{[2]})^{*}\varphi^{-1},f^{-1})=(\varphi,f)^{-1}$).

###### Proof.

The proof is a routine calculation using the definition. ∎

We need to use the concept of a _mate_ of a 2-commutative square in a
bicategory, where two parallel edges are equipped with adjoints (on the same
side), see for example [JY21, Definition 6.1.12]. We also need the fact this
is functorial with respect to pasting of squares, cf [KS74, Proposition 2.2]
(this follows in more general bicategories by strictification/coherence
results, and the result in 2-categories from _loc. cit._).

Since every stable isomorphism has an adjoint inverse, we can take the mate in
the bicategory of bundle gerbes of any (2-)commuting square of stable
isomorphisms in either or both directions. In particular given a commuting
square of _morphisms_ , there is a commuting square of stable isomorphisms,
and hence we can take mates. The following lemma deals with a special case
that we need to consider later.

######  Lemma A.3.

Given a commutative square of bundle gerbe morphisms

$\textstyle{(E_{1},Y_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\alpha,g)}$$\scriptstyle{(\phi_{1},f_{1})}$$\scriptstyle{\simeq}$$\textstyle{(E_{2},Y_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\phi_{2},f_{2})}$$\scriptstyle{\simeq}$$\textstyle{(F_{1},X_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\beta,h)}$$\textstyle{(F_{2},X_{2})}$

over $M$, where the vertical morphisms are bundle gerbe _isomorphisms_ ,
consider its mate

$\textstyle{(E_{1},Y_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(E_{2},Y_{2})}$$\textstyle{(F_{1},X_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\textstyle{(F_{2},X_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

in the bicategory of bundle gerbes and stable isomorphisms. Here the
horizontal stable isomorphisms are those associated to the given horizontal
morphisms, and the vertical stable isomorphisms are those adjoint inverses
from Lemma A.2. Then the 2-arrow filling this square is an _identity_ 2-arrow
and the square of stable isomorphisms is that associated to the commutative
square

$\textstyle{(E_{1},Y_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\alpha,g)}$$\textstyle{(E_{2},Y_{2})}$$\textstyle{(F_{1},X_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\beta,h)}$$\scriptstyle{\simeq}$$\scriptstyle{((f_{1}^{[2]})^{*}\phi_{1}^{-1},f_{1}^{-1})}$$\textstyle{(F_{2},X_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{((f_{1}^{[2]})^{*}\phi_{2}^{-1},f_{2}^{-1})}$

of morphisms, using the inverses of the original vertical morphisms.

###### Proof.

We use the last point in Lemma A.2, so that the upward-pointing vertical
stable isomorphisms in the mate are those associated to the upward-pointing
isomorphisms in the last diagram. Thus we only need to know that the 2-arrow
in the mate diagram is the identity 2-arrow. This is a routine calculation
with pasting diagrams for bicategories, together with the following fact: the
data of the coherence isomorphism $\Sigma^{2}_{(\varphi,f),(\psi,g)}$ only
depends on the composite $(\varphi,f)\circ(\psi,g)$, and not on the individual
factors, and hence
$\Sigma^{2}_{(\phi_{2},f_{2}),(\alpha,g)}(\Sigma^{2}_{(\beta,h),(\phi_{1},f_{1})})^{-1}$
is an identity 2-arrow. ∎

We also need to consider an analogue of Construction A.1 when one has bundle
gerbes with connection and curving. Following [Wal13, Definition 6.6], a
stable isomorphism $(L,\tau)$ between bundle gerbes $(E,Y;\nabla_{E},b_{E})$
and $(F,X;\nabla_{F},b_{F})$ (on $M$) with connections and curvings has a
_compatible connection_ if there is a connection $\nabla_{L}$ on $L\to
Y\times_{M}X$, the isomorphism $\tau$ is connection-preserving for the induced
connection on
$\operatorname{pr}_{12}^{*}E^{*}\otimes\operatorname{pr}_{34}^{*}F$, and the
curvature of $\nabla_{L}$ is _equal_ to
$\operatorname{pr}_{2}^{*}b_{F}-\operatorname{pr}_{1}^{*}b_{E}\in\Omega^{2}(Y\times_{M}X)$.
Here we will just say that the stable isomorphism _has a connection_ , since
we have no need to discuss non-compatible connections.

######  Construction A.4.

We start with the basic details as in Construction A.1, in particular the
stable isomorphism $\Sigma(\varphi,f)$. Suppose given bundle gerbe connections
$\nabla_{E}$ and $\nabla_{F}$ on $(E,Y)$ and $(F,X)$ respectively, and a
1-form $a$ on $Y$ such that
$\pi^{*}\delta(a)=\nabla_{E}-\varphi^{*}(f^{[2]})^{*}\nabla_{F}$. We need to
give a connection on $(f\times\operatorname{id})^{*}F$ and show that the
isomorphism $\tau_{f}$ is connection-preserving.

However, defining the connection on $(f\times\operatorname{id})^{*}F$ to be
$(f\times\operatorname{id})^{*}\nabla_{F}-\pi^{*}\operatorname{pr}_{1}^{*}a$,
the composite isomorphism in Construction A.1 is connection-preserving, by the
assumption on $a$ and $\varphi$, and because bundle gerbe multiplication
preserves connections.

Now assume that there are also curvings $b_{E}$ and $b_{F}$ on $(E,Y)$ and
$(F,X)$ respectively, satisfying $b_{E}-f^{*}b_{F}=da$. Then the curvature of
$(f\times\operatorname{id})^{*}\nabla_{F}-\pi^{*}\operatorname{pr}_{1}^{*}a$
can be easily checked to be equal to
$\operatorname{pr}_{2}^{*}b_{F}-\operatorname{pr}_{1}^{*}b_{E}$, so we have a
compatible connection on $\Sigma(L,\tau)$.

Further, we will need the analogue of Lemma A.2 with connections and curvings.
Following [Wal07, Proposition 1.13], the adjoint inverse of what amounts to a
stable isomorphism with connection is constructed—for Waldorf’s definition of
bundle gerbe morphism. That this should work for the definition of stable
morphism here won’t be an issue, we just need to record the construction of a
special case. Attentive readers might wonder what the definition of the
bicategory of bundle gerbes with connections and curvings, stable isomorphisms
with (compatible) connections and 2-isomorphisms is. We take it to be the
analogue of Stevenson’s bicategory of bundle gerbes, only now equipped with
connections and curvings as outlined here. That this should form a bicategory
(with a forgetful 2-functor to Stevenson’s bicategory) will follow by
considering the descent problems in [Ste00, Chapter 3] that arise when forming
composites as consisting of $U(1)$-bundles equipped with connections.

We can now give the version of Lemma A.2 with connections and curvings, where
a morphism of bundle gerbes with connections and curvings means a morphism of
the underlying bundle gerbes such that that the connection and curving on the
domain are given by pullback from the codomain. We will only use the part of
Lemma A.2 for stable morphisms arising from plain morphisms.

######  Lemma A.5.

Let $(E,Y)$ and $(F,X)$ be bundle gerbes on $M$, with connections $\nabla_{E}$
and $\nabla_{F}$ and curvings $b_{E}$ and $b_{F}$. Suppose we have a morphism
of bundle gerbes $(\varphi,f)\colon(E,Y)\to(F,X)$, such that
$\nabla_{E}=\varphi^{*}(f^{[2]})^{*}\nabla_{F}$ (this means that the
connection on the stable isomorphism is just
$(f\times\operatorname{id})^{*}\nabla_{F}$), and that $f^{*}b_{F}=b_{E}$. Then
the adjoint inverse of the stable isomorphism $\Sigma(\varphi,f)$ with its
connection is given by the adjoint inverse of the underlying stable
isomorphism (from Lemma A.2, together with the connection171717recall that the
underlying space of the dual bundle $F^{*}$ is the same as that of $F$, only
equipped with the action via inversion of $U(1)$. The ‘dual connection’ on
$F^{*}$ is therefore just $-\nabla_{F}$. $-(\operatorname{id}\times
f)^{*}\nabla_{F}$.

###### Proof.

Note that the compatibility condition of 2-arrows does not involve the
curvings at all. This proof only needs to check that the unit (and hence
counit) of the adjunction consists of a connection-preserving isomorphism of
bundles. But then the triangle identities hold automatically as this is just a
property, not structure. However, the isomorphism of $U(1)$-bundles underlying
the 2-arrow, given on [Ste00, page 33], is built using bundle gerbe
multiplication on $(F,X)$ and so preserves connections. ∎

As a corollary, the analogue of Lemma A.3 will hold using isomorphisms of
bundle gerbes _with connection and curving_. In particular, it will hold in
the case where the square of bundle gerbes is induced from a square of maps
between surjective submersions $Y_{i}\to M$,

$\textstyle{Y_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\,,}$

and a bundle gerbe ${\mathcal{E}}=(E,Y_{0})$, and the connections and curvings
on them all are given by pullback from a connection and curving on
${\mathcal{E}}$. This means that although we start from a strictly commuting
square, the $U(1)$-bundle isomorphism underlying the nontrivial 2-arrow that
is the mate of the identity 2-arrow will be a connection-preserving
isomorphism.

###  A.2. Associated bundle 2-gerbes

The following definition is adapted from Definitions 7.1 and 7.3 in [Ste00].
Note that here we say ‘bundle 2-gerbe’ for what _loc. cit._ calls a ‘stable’
bundle 2-gerbe.

######  Definition A.6.

A _simplicial bundle gerbe_ on a semisimplicial manifold $B_{\bullet}$
consists of the following:

  1. (1)

A bundle gerbe $\mathcal{E}$ on $B_{1}$;

  2. (2)

A bundle gerbe stable isomorphism $\overline{m}\colon
d_{0}^{*}{\mathcal{E}}\otimes d_{2}^{*}{\mathcal{E}}\to
d_{1}^{*}{\mathcal{E}}$ over $B_{2}$;

  3. (3)

A 2-arrow $\tilde{\alpha}$, the _associator_ , between stable isomorphisms
over $B_{3}$ as in the following diagram:

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where ${\mathcal{E}}_{ab}$ denotes the pullback of ${\mathcal{E}}$ by the
composite $B_{3}\to B_{1}$ of face maps induced by the map $(a,b)\to(0,1,2,3)$
in the simplex category (thinking of a simplicial manifold as a presheaf
$\Delta^{op}\to\operatorname{Mfld}$).

  4. (4)

Such that the following pair of 2-arrows over $B_{4}$ are equal, where by
${\mathcal{E}}_{ab}$ we mean the pullback of ${\mathcal{E}}$ by the composite
$B_{4}\to B_{1}$ of face maps induced by the inclusion $(a,b)\to(0,1,2,3,4)$
in the simplex category $\Delta$:

A _bundle 2-gerbe_ is the special case where the simplicial manifold is that
associated to a surjective submersion $Y\to X$, namely $Y^{[\bullet+1]}$. A
_strict_ bundle 2-gerbe is a bundle 2-gerbe where the stable isomorphism
$\overline{m}$ arises from a plain morphism of bundle gerbes, and
$\tilde{\alpha}$ is the identity 2-arrow.

Our framework is defined specifically to avoid having to deal with the
complexity of this definition, which, even as written, is hiding a lot of
detail.

We note that given a map of simplicial manifolds $B^{\prime}_{\bullet}\to
B_{\bullet}$, and a simplicial bundle gerbe on $B_{\bullet}$, one can pull
back to get a simplicial bundle gerbe on $B^{\prime}_{\bullet}$. It is by this
process that the Chern–Simons bundle 2-gerbe is constructed ([CJM+05, Lemma
5.4]), whereby one considers the (strictly) multiplicative bundle gerbe
arising from the _basic gerbe_ on $G$, and then pulls it back along the
simplicial map $Q^{[\bullet+1]}\to NG_{\bullet}$.

The following construction is given at a reasonably high level, since it is
intended for those who wish to see how a rigid bundle 2-gerbe gives rise to a
bundle 2-gerbe, which we presume might be a minority of readers, namely those
already familiar with bundle 2-gerbes.

######  Construction A.7.

Consider a rigid bundle 2-gerbe on the manifold $X$, with surjective
submersion $Y\to X$, simplicial submersion $Z_{\bullet}\to Y^{[\bullet+1]}$,
bundle gerbe $(E,Z_{1})$ on $Y^{[2]}$, and 2-gerbe multiplication $M$. The
_associated bundle 2-gerbe_ is given by the following data

  1. (1)

The bundle gerbe ${\mathcal{E}}$ on $Y^{[2]}$ is just $(E,Z_{1})$ as given;

  2. (2)

The strong trivialisation of $(\delta_{h}(E),Z_{2})$ gives an isomorphism of
bundle gerbes

$(d_{0}^{*}E\otimes d_{2}^{*}E,Z_{2})\simeq(d_{1}^{*}E,Z_{2})$

which, if we denote the left bundle gerbe by ${\mathcal{E}}_{012}$, gives a
span of morphisms of bundle gerbes

${\mathcal{E}}_{12}\otimes{\mathcal{E}}_{01}\leftarrow{\mathcal{E}}_{012}\to{\mathcal{E}}_{02}$

where ${\mathcal{E}}_{12}=d_{0}^{*}{\mathcal{E}}$, and so on, a notation which
is consistent with the notation in Definition A.6. The left-pointing morphism
${\mathcal{E}}_{012}\to{\mathcal{E}}_{12}\otimes{\mathcal{E}}_{01}$ has
adjoint inverse stable isomorphism
${\mathcal{E}}_{12}\otimes{\mathcal{E}}_{01}\rightsquigarrow{\mathcal{E}}_{012}$
as in Lemma A.2. The bundle 2-gerbe multiplication
$\overline{m}\colon{\mathcal{E}}_{12}\otimes{\mathcal{E}}_{01}\rightsquigarrow{\mathcal{E}}_{02}$
is the composition of this adjoint inverse with the stable isomorphism
associated to ${\mathcal{E}}_{012}\to{\mathcal{E}}_{02}$.

  3. (3)

We need to define the associator, and here is where we will use mates. First,
we consider the induced maps filling the diagram181818You should think of this
diagram as being indexed by the face lattice of a square. of (domains of)
submersions to $Y^{[4]}$:

$\textstyle{d_{0}^{*}d_{0}^{*}Z_{1}\times_{Y^{[4]}}d_{3}^{*}d_{0}^{*}Z_{1}\times_{Y^{[4]}}d_{3}^{*}d_{2}^{*}Z_{1}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
d_{0}^{*}Z_{2}\times_{Y^{[4]}}d_{3}^{*}d_{2}^{*}Z_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{d_{0}^{*}d_{1}^{*}Z_{1}\times_{Y^{[4]}}d_{3}^{*}d_{2}^{*}Z_{1}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
d_{0}^{*}d_{0}^{*}Z_{1}\times_{Y^{[4]}}d_{3}^{*}Z_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
Z_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
d_{2}^{*}Z_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{d_{0}^{*}d_{0}^{*}Z_{1}\times_{Y^{[4]}}d_{3}^{*}d_{1}^{*}Z_{1}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
d_{1}^{*}Z_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{d_{1}^{*}d_{1}^{*}Z_{1}}$

where we have used some simplicial identities to make sure the domains and
codomains match, and some squares commute. The associativity condition
$\delta_{h}(M)=1$ then allows us to define four commuting squares of morphisms
of bundle gerbes on $Y^{[4]}$:

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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
74.08064pt\raise-67.4422pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
127.46681pt\raise-38.63776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{013}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawsquiggles@}}\ignorespaces{\hbox{\kern
138.0696pt\raise-4.80444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawsquiggles@}}\ignorespaces{\hbox{\lx@xy@drawsquiggles@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
138.0696pt\raise-67.4422pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathcal{E}}_{23}\otimes{\mathcal{E}}_{02}}$}}}}}}}{\hbox{\kern
63.47786pt\raise-77.27551pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{023}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawsquiggles@}}\ignorespaces{\hbox{\kern
19.29448pt\raise-77.27551pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawsquiggles@}}\ignorespaces{\hbox{\lx@xy@drawsquiggles@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
128.8668pt\raise-77.27551pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
128.8668pt\raise-77.27551pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathcal{E}}_{03}}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}$

where the bundle gerbe ${\mathcal{E}}_{0123}$ has $U(1)$-bundle induced by
pulling back the $U(1)$-bundle of
${\mathcal{E}}_{23}\otimes{\mathcal{E}}_{12}\otimes{\mathcal{E}}_{01}$ along
the composite $Z_{3}\to
d_{0}^{*}d_{0}^{*}Z_{1}\times_{Y^{[4]}}d_{3}^{*}d_{0}^{*}Z_{1}\times_{Y^{[4]}}d_{3}^{*}d_{2}^{*}Z_{1}$
from the previous diagram. In fact we can define everything in this diagram
without using associativity of $M^{\prime}$, except the commutativity of the
lower right square, which follows from condition (5)(b’) in Proposition 2.5 on
$M^{\prime}$ and the universal property of pullback of $U(1)$-bundles. Now to
get the associator, we consider the mates of the top right, bottom left, and
top left squares using the adjoint inverses for the arrows denoted
$\rightsquigarrow$, and then paste the resulting diagram of 2-arrows between
stable isomorphisms.

  4. (4)

Now we need to prove that this associator is coherent. The strategy is to
build up a commuting diagram of bundle gerbes and morphisms on $Y^{[5]}$, of
shape the face lattice of a cube, as displayed in Figure 3. The arrows denoted
$\rightsquigarrow$ are again considered as being equipped with their adjoint
inverses as stable isomorphisms.

---
$\textstyle{\mathcal{X}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
o\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{o}$$\textstyle{o\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
o\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{o\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{o}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
o\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{o}$$\textstyle{o\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
o\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{o\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{o\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{01234}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{o\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{o\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
o\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{o\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{o}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
o\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{o}$$\textstyle{o\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
o\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{o\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{o}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
o\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{04}}$
Figure 3. Diagram of bundle gerbes and morphisms over $Y^{[5]}$. The objects
with placeholders marked $o$ are specified in Figure 4.

Then taking mates involving all such adjoint inverses, pasting, and using the
result that pasting preserves mates, the resulting boundary cube will
2-commute as needed. Two specific objects have been named, to help orient the
reader, and the object labelled $\mathcal{X}$ is in fact
${\mathcal{E}}_{34}\otimes{\mathcal{E}}_{23}\otimes{\mathcal{E}}_{12}\otimes{\mathcal{E}}_{01}$.
The cube should be considered as encoding associator coherence for the various
ways of performing iterated bundle 2-gerbe multiplication to get a stable
isomorphism
$X={\mathcal{E}}_{34}\otimes{\mathcal{E}}_{23}\otimes{\mathcal{E}}_{12}\otimes{\mathcal{E}}_{01}\rightsquigarrow{\mathcal{E}}_{04}$.
The bundle gerbe ${\mathcal{E}}_{1234}$ uses the submersion $Z_{4}\to
Y^{[5]}$, and its $U(1)$-bundle is defined to make the top-left-back cube
commute.

In Figure 4 we display the outer faces in two pieces, as in the definition of
coherence for the associator in Definition A.6, and describe the inner
structure seperately.

Since the cubical diagram in the 1-category of bundle gerbes and morphisms
commutes on the nose, we have that the image of this diagram, up to insertion
of associativity isomorphisms, commutes in the bigroupoid of bundle gerbes and
stable isomorphisms. Coherence for bicategories ensures that the squares
2-commute when associator isomorphisms are added back in. Then by the
compatibility of pasting and mates, the eight 2-commuting small cubes still
2-commute when passing to the corresponding diagram with the mates. Then the
outer cube 2-commutes, which means that the associator satisfies the required
coherence condition.

Back-right-bottom faces:

---
$\textstyle{{\mathcal{E}}_{34}\otimes{\mathcal{E}}_{23}\otimes{\mathcal{E}}_{12}\otimes{\mathcal{E}}_{01}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{234}\otimes{\mathcal{E}}_{12}\otimes{\mathcal{E}}_{01}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{24}\otimes{\mathcal{E}}_{12}\otimes{\mathcal{E}}_{01}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{34}\otimes{\mathcal{E}}_{23}\otimes{\mathcal{E}}_{012}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{234}\otimes{\mathcal{E}}_{012}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{24}\otimes{\mathcal{E}}_{012}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{124}\otimes{\mathcal{E}}_{01}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{34}\otimes{\mathcal{E}}_{23}\otimes{\mathcal{E}}_{02}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{234}\otimes{\mathcal{E}}_{02}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{24}\otimes{\mathcal{E}}_{02}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{0124}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{14}\otimes{\mathcal{E}}_{01}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{34}\otimes{\mathcal{E}}_{023}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{0234}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{024}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{014}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{34}\otimes{\mathcal{E}}_{03}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{034}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{04}}$

Front-left-top faces:

---
$\textstyle{{\mathcal{E}}_{34}\otimes{\mathcal{E}}_{23}\otimes{\mathcal{E}}_{12}\otimes{\mathcal{E}}_{01}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{234}\otimes{\mathcal{E}}_{12}\otimes{\mathcal{E}}_{01}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{24}\otimes{\mathcal{E}}_{12}\otimes{\mathcal{E}}_{01}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{34}\otimes{\mathcal{E}}_{23}\otimes{\mathcal{E}}_{012}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{34}\otimes{\mathcal{E}}_{123}\otimes{\mathcal{E}}_{01}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{1234}\otimes{\mathcal{E}}_{01}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{124}\otimes{\mathcal{E}}_{01}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{34}\otimes{\mathcal{E}}_{23}\otimes{\mathcal{E}}_{02}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{34}\otimes{\mathcal{E}}_{0123}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{34}\otimes{\mathcal{E}}_{13}\otimes{\mathcal{E}}_{01}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{134}\otimes{\mathcal{E}}_{01}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{14}\otimes{\mathcal{E}}_{01}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{34}\otimes{\mathcal{E}}_{023}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{34}\otimes{\mathcal{E}}_{013}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{0134}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{014}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{34}\otimes{\mathcal{E}}_{03}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{034}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{04}}$

Figure 4. Encoding of associator coherence from rigid 2-gerbe data, giving the
outer faces of Figure 3

This completes the construction of the associated bundle 2-gerbe.

Now on top of this, we can also consider what happens to the connective
structure on a rigid bundle 2-gerbe, once we form the associated bundle
2-gerbe. Here, in fact, things are relatively much easier, if we only work
with [Ste04, Definition 8.1]. This is because the connection data (connection,
curving, 2-curving) on a bundle 2-gerbe is actually insensitive to the
specifics of the bundle 2-gerbe multiplication and its coherence; all that is
needed is that the 3-form curvature $H$ of the bundle gerbe on $Y^{[2]}$
satisfies191919The existence of such 3-form curvature such that $\delta(H)$ is
_exact_ is immediate after we know that _any_ bundle 2-gerbe multiplication
exists, coherent or not. That one can improve the curving on the bundle gerbe
to get $\delta(H)=0$ on the nose is the hard part of [Ste00, Chapter 10].
$\delta(H)=0$. The proof of the following proposition is immediate.

######  Proposition A.8.

Consider a rigid bundle 2-gerbe on the manifold $X$, with surjective
submersion $Y\to X$, simplicial submersion $Z_{\bullet}\to Y^{[\bullet+1]}$,
and bundle gerbe $(E,Z_{1})$ on $Y^{[2]}$. Now assume also that the rigid
bundle 2-gerbe has a connective structure $(\nabla_{E},\beta,\lambda)$. The
associated bundle 2-gerbe then has a connection $\nabla_{E}$, curving $\beta$,
and 2-curving $\lambda$ in the sense of [Ste04, Definition 8.1], all of which
are exactly the same pieces of data as for the rigid bundle 2-gerbe.

Note that here we are not saying what happens to the data
$\alpha\in\Omega^{1}(Z_{2})$ in a _rigid_ connective structure. One could in
fact include this information, if presenting a bundle 2-gerbe with connective
structure as descent data valued in the bigroupoid of bundle gerbes equipped
with connections and curvings (cf [NS11]), as is done in [Wal13, Definition
4.1]. In fact, it seems202020Konrad Waldorf, personal communication that the
definition of connection and curving on a bundle 2-gerbe in [Ste04] is not
rigid enough, and that the proof that bundle 2-gerbes with connection, curving
and 2-curving are classified by an appropriate Deligne cohomology group may
fail. So we shall here also prove that the connective structure on the
associated bundle 2-gerbe satisfies Waldorf’s stronger definition, which we
now give.

######  Definition A.9 ([Wal13]).

Assume given a bundle 2-gerbe, with connection $\nabla$, curving $\beta$ and
2-curving $\lambda$ as in [Ste04, Definition 8.1]. Then we require the
following conditions on $\nabla$ and $\beta$

  1. (1)

There is a connection $\nabla_{\overline{m}}$ on the $U(1)$-bundle encoding
the stable isomorphism $\overline{m}$ on $Y^{[3]}$

  2. (2)

The associator isomorphism $\alpha$ in Definition A.6, which is an isomorphism
between certain $U(1)$-bundles, preserves the induced connections.

######  Proposition A.10.

If we assume the connection $\nabla_{E}$ and curving $\beta$ in A.8 are part
of a _rigid_ connective structure, with 1-form $\alpha$, then they satisfy the
additional conditions in A.9.

###### Proof.

Lemma A.5 guarantees that the stable isomorphism $\overline{m}$ has a
compatible connection, built using $\alpha$. The identity
$\delta_{h}(\alpha)=0$ ensures that the diagram (A.1) becomes a diagram of
bundle gerbes with connection and curving, and morphisms compatible with
these. Further, since the associator is built from mates, and these mates are
2-arrows with data connection-preserving bundle isomorphisms, as per the
discussion after Lemma A.5, then the associator preserves connections. The
coherence equation holds for free, since it is an equality of 2-arrows, which
neither places or required extra constraints on the differential form data. ∎

###  A.3. The proof that the Chern–Simons bundle 2-gerbe is rigidified

The following Proposition captures a feature that holds for the rigid
Chern–Simons 2-gerbe, in a more abstract way.

######  Proposition A.11.

Suppose we are given a rigid bundle 2-gerbe on $X$ with submersion $Y\to X$
and simplicial submersion $Z_{\bullet}\to Y^{[\bullet+1]}$.

  1. (1)

If the map $Z_{2}\to d_{0}^{*}Z_{1}\times_{Y^{[3]}}d_{2}^{*}Z_{1}$ over
$Y^{[3]}$ induced by the universal properties of the pullbacks is an
isomorphism, then in the span of morphisms of bundle gerbes giving rise to the
associated bundle 2-gerbe multiplication, the left-pointing morphism is an
_isomorphism_ of bundle gerbes.

  2. (2)

Suppose additionally that either the indued map $Z_{3}\to
d_{1}^{*}d_{0}^{*}Z_{1}\times_{Y^{[4]}}d_{3}^{*}Z_{2}$, or the induced map
$Z_{3}\to d_{0}^{*}Z_{2}\times_{Y^{[4]}}d_{2}^{*}d_{2}^{*}Z_{1}$, is an
isomorphisms. Then the associator 2-arrow is an identity.

###### Proof.

  1. (1)

This follows from Lemma A.2, so that the inverse of the bundle gerbe
isomorphism involving $Z_{2}\to d_{0}^{*}Z_{1}\times_{Y^{[3]}}d_{2}^{*}Z_{1}$
gives the adjoint inverse as required.

  2. (2)

Here we apply Lemma A.3 twice, since the relevant square of bundle gerbe
morphisms involves the commutative square

$\textstyle{d_{1}^{*}d_{0}^{*}Z_{1}\times_{Y^{[4]}}d_{3}^{*}(d_{0}^{*}Z_{1}\times_{Y^{[3]}}d_{2}^{*}Z_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{d_{0}^{*}(d_{0}^{*}Z_{1}\times_{Y^{[3]}}d_{2}^{*}Z_{1})\times_{Y^{[4]}}d_{2}^{*}d_{2}^{*}Z_{1}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
d_{0}^{*}Z_{2}\times_{Y^{[4]}}d_{2}^{*}d_{2}^{*}Z_{1}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
d_{1}^{*}d_{0}^{*}Z_{1}\times_{Y^{[4]}}d_{3}^{*}Z_{2}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
Z_{3}}$

The hypothesis from part (1) implies that the left and top arrows are
invertible (hence giving rise to isomorphisms of bundle gerbes), and the extra
hypothesis from part (2) implies that either the bottom or the right arrow is
invertible, implying that the whole square consists of isomorphisms, and hence
the top left square in (A.1) consists of isomorphisms of bundle gerbes. Thus
the mate of this square (using both adjoint inverse pairs) is the on-the-nose
commutative square corresponding to the inverses of the bundle gerbe
isomorphisms. A similar argument works for the bottom left and top right
squares of (A.1), except in this case we only need to check that the morphisms
of bundle gerbes induced by

$d_{1}^{*}Z_{2}\to
d_{1}^{*}(d_{0}^{*}Z_{1}\times_{Y^{[3]}}d_{2}^{*}Z_{1})=d_{0}^{*}d_{0}^{*}Z_{1}\times_{Y^{[4]}}d_{3}^{*}d_{1}^{*}Z_{1}$

and

$d_{2}^{*}Z_{2}\to
d_{2}^{*}(d_{0}^{*}Z_{1}\times_{Y^{[3]}}d_{2}^{*}Z_{1})=d_{0}^{*}d_{1}^{*}Z_{1}\times_{Y^{[4]}}d_{3}^{*}d_{2}^{*}Z_{1}$

are isomorphisms, but this holds by assumption from (1). Thus the associator,
which is the pasting of the four squares, all of which commute on the nose, is
an identity 2-arrow. ∎

######  Remark A.12.

  1. (1)

In Proposition A.11 (1), the bundle 2-gerbe multiplication is a stable
isomorphism arising from a _morphism_ of bundle gerbes, namely the composite
of the inverse isomorphism of bundle gerbes with the forward-pointing morphism
of bundle gerbes.

Thus in this instance the associated bundle 2-gerbe is a bundle 2-gerbe (in
the sense of Definition A.6) arising from the slightly stricter notion of
bundle 2-gerbe given as [Ste00, Definition 7.2], via [Ste00, Note 7.1.1]
(recall that the objects given by Definition A.6 are called ‘stable bundle
2-gerbes’ in [Ste00]).

  2. (2)

In the situation of Proposition A.11 (2) the associated bundle 2-gerbe is the
bundle 2-gerbe arising from a _strict_ bundle 2-gerbe.

Thus our notion of rigid bundle 2-gerbe is a different way of encoding a
2-gerbe-like object in a strict fashion, only sometimes corresponding to the
more obvious strict notion coming from general higher categorical
considerations (e.g. honest morphisms, trivial associators etc).

Here we give the postponed proof of Theorem 3.7, that our rigid Chern–Simons
2-gerbe really is a rigidification of the Chern–Simons bundle 2-gerbe
appearing in the literature. It is now just a corollary of the previous
Proposition.

###### Proof.

(Of Theorem 3.7) Recall that the definition in the literature of the
Chern–Simons bundle 2-gerbe associated to the $G$-bundle $Q\to X$ [Ste00,
§9.3], [CJM+05, Definition 6.5] is as the pullback along the simplicial map
$Q^{[\bullet]}\to NG_{\bullet}$ of the multiplicative bundle gerbe on $G$,
namely the basic gerbe. The multiplicative structure on the latter, defined
using the crossed module structure on $\widehat{\Omega G}\to PG$, is a strict
2-group, and hence the Chern–Simons bundle 2-gerbe is a strict bundle 2-gerbe.
The only caveat is that in loc. cit. a different, but isomorphic, model for
the basic gerbe is given. One can in fact easily check that the conditions of
Proposition A.11 hold, and that the resulting bundle 2-gerbe multplication
_morphism_ (not stable isomorphism) is exactly analogous to that from the
literature, modulo the different model of the basic gerbe. Thus the associated
bundle 2-gerbe of our rigid Chern–Simons 2-gerbe is isomorphic to the existing
bundle 2-gerbe model, and hence we have a rigidification. ∎

That the data of the connective structure works correctly for the Chern–Simons
2-gerbe has been mentioned in Corollary 3.20 discussed above. Thus the only
other piece of data to look at is the 1-form $\alpha$ in the _rigid_
connective structure, which is not even considered in [CJM+05]. But then
Proposition A.10 takes care of this last piece of data, which has never before
been given explicitly.

###  A.4. Trivialisations

Now, of course the other construction we need to wrap up is that of a bundle
2-gerbe trivialisation from a rigid trivialisation, and also the version with
connections. We note that the definition we are using here is adapted from
[Ste00, Definition 12.2] (and [CJM+05, Definition 4.6]), which is different to
[Ste00, Definition 12.1], [Ste04, Definition 11.1], and [Wal13, Definition
3.5] in that the latter essentially use the dual bundle gerbe
${\mathcal{F}}^{*}$ in place of ${\mathcal{F}}$ in the below definition (which
leads to what can be considered a _left_ action of ${\mathcal{E}}$ on
${\mathcal{F}}$, rather than a right action, as in [Ste00, Definition 12.1]).
The choice we are making as to what gets dualised is made so as to be
compatible with the notion of coboundary, and to be consistent with how duals
are taken when pulling back by face maps in a simplicial space. However, the
definition we adopt is _in the style of_ [Wal13, Definition 3.5], and does not
lead to any substantial difference in any results here (or elsewhere).

######  Definition A.13.

A _trivialisation_ of a bundle 2-gerbe on $M$, with surjective submersion
$Y\to M$ and data $({\mathcal{E}},\overline{m},\alpha)$ is given by the
following data:

  1. (1)

A bundle gerbe ${\mathcal{F}}$ on $Y$;

  2. (2)

A stable isomorphism $\eta\colon{\mathcal{E}}\otimes d_{1}^{*}{\mathcal{F}}\to
d_{0}^{*}{\mathcal{F}}$ of bundle gerbes on $Y^{[2]}$;

  3. (3)

A 2-arrow $\theta$ (the ‘actionator’) between stable isomorphisms over
$Y^{[3]}$ as in the following diagram:

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where here ${\mathcal{E}}_{ab}$ denotes the pullback along $Y^{[3]}\to
Y^{[2]}$ induced by $(a,b)\hookrightarrow(0,1,2)$ in the simplex category
$\Delta$, and ${\mathcal{F}}_{c}$ is the pullback along the map $Y^{[3]}\to Y$
induced by $(c)\hookrightarrow(0,1,2)$.

  4. (4)

Such that the following pair of 2-arrows over $Y^{[4]}$ are equal:

where again we have used the notation ${\mathcal{E}}_{ab}$ from Definition
A.6, and now also use the notation ${\mathcal{F}}_{a}$ to denote the pullback
along the map $Y^{[4]}\to Y$ induced by $(a)\hookrightarrow(0,1,2,3)$ in the
simplex category $\Delta$.

We will use the equivalent definition of a rigid trivialisation from
Proposition 2.11 in the following construction, since it matches the
definition of bundle 2-gerbe trivialisation more closely.

######  Construction A.14.

Assume we are given a rigid bundle 2-gerbe with data $Y\to X$, $Z_{\bullet}\to
Y^{[\bullet+1]}$, $(E,Z_{1})$ and $M$, and a rigid trivialisation with data
$(F,Z_{0})$ and $t^{\prime}$ (as in Proposition 2.11), let us construct a
trivialisation in the sense of Definition A.13 of the associated bundle
2-gerbe.

  1. (1)

We already have the bundle gerbe on $Y$ as needed.

  2. (2)

We construct the stable isomorphism $\eta$ via the span of morphisms of bundle
gerbes involving the span of covers

$Z_{1}\times_{Y^{[2]}}d_{1}^{*}Z_{0}\xleftarrow{(\operatorname{id},d_{1})}Z_{1}\xrightarrow{d_{0}}d_{0}^{*}Z_{0}$

of $Y^{[2]}$, and the isomorphism
$(t^{\prime},\operatorname{id})\colon(E\otimes
d_{1}^{*}F,Z_{1})\to(d_{0}^{*}F,Z_{1})$ of bundle gerbes. As in Construction
A.7, we get a span

${\mathcal{E}}\otimes
d_{1}^{*}{\mathcal{F}}\mathrel{\mathchoice{\reflectbox{$\displaystyle\rightsquigarrow$}}{\reflectbox{$\textstyle\rightsquigarrow$}}{\reflectbox{$\scriptstyle\rightsquigarrow$}}{\reflectbox{$\scriptscriptstyle\rightsquigarrow$}}}({\mathcal{E}}{\mathcal{F}}_{1})\to
d_{0}^{*}{\mathcal{F}}$

where $({\mathcal{E}}{\mathcal{F}}_{1}):=(E\otimes d_{1}^{*}F,Z_{1})$. Passing
to the associated stable isomorphisms, then taking the adjoint inverse to the
arrow denoted $\rightsquigarrow$ and composing, this will then be the stable
isomorphism $\eta$.

  3. (3)

We then need to build the 2-arrow $\theta$. Consider the diagram of (domains
of) submersions to $Y^{[3]}$

$\textstyle{d_{0}^{*}Z_{1}\times_{Y^{[3]}}d_{2}^{*}Z_{1}\times_{Y^{[3]}}d_{2}^{*}d_{1}^{*}Z_{0}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
Z_{2}\times_{Y^{[3]}}d_{2}^{*}d_{1}^{*}Z_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{d_{1}^{*}Z_{1}\times_{Y^{[3]}}d_{2}^{*}d_{1}^{*}Z_{0}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
d_{0}^{*}Z_{1}\times_{Y^{[3]}}d_{2}^{*}Z_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
Z_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
d_{1}^{*}Z_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{d_{0}^{*}Z_{1}\times_{Y^{[3]}}d_{2}^{*}d_{0}^{*}Z_{0}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
d_{0}^{*}Z_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{d_{0}^{*}d_{0}^{*}Z_{0}}$

where we have used simplicial identities implicitly to define maps using the
universal property of the pullback. This gives rise to a diagram of bundle
gerbe morphisms

(A.2)
$\textstyle{{\mathcal{E}}_{12}\otimes{\mathcal{E}}_{01}\otimes{\mathcal{F}}_{0}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{012}\otimes{\mathcal{F}}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{02}\otimes{\mathcal{F}}_{0}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{12}\otimes({\mathcal{E}}_{01}{\mathcal{F}}_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces({\mathcal{E}}_{012}{\mathcal{F}}_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces({\mathcal{E}}_{02}{\mathcal{F}}_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{12}\otimes{\mathcal{F}}_{1}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces({\mathcal{E}}_{12}{\mathcal{F}}_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{F}}_{2}}$

where the $U(1)$-bundle part of the bundle gerbe
$({\mathcal{E}}_{012}{\mathcal{F}}_{0})$ is given by pullback along
$Z_{2}^{[2]}\to(d_{0}^{*}Z_{1}\times_{Y^{[3]}}d_{2}^{*}Z_{1}\times_{Y^{[3]}}d_{2}^{*}d_{1}^{*}Z_{0})^{[2]}$
of the $U(1)$-bundle from
${\mathcal{E}}_{12}\otimes{\mathcal{E}}_{01}\otimes{\mathcal{F}}_{0}$. The
commutativity of the bottom right square follows from the condition (3’) on
$t^{\prime}$ in Proposition 2.11. The coherence morphism $\theta$ is defined
to be the pasting of the mates (in the bicategory of bundle gerbes and stable
isomorphisms) of the top left, top right and bottom left squares, taken with
respect to the adjoint equivalences denoted $\rightsquigarrow$.

  4. (4)

We now need to establish the coherence of $\theta$. We have a cube of the same
form as before, shown in Figure 5, which is a diagram of bundle gerbes and
morphisms on $Y^{[4]}$, and where the arrows of the form $\rightsquigarrow$
are intended to be considered as being equipped with their adjoint inverses as
stable isomorphisms. The object denoted here $\mathcal{X}^{\prime}$ is the
bundle gerbe
${\mathcal{E}}_{23}\otimes{\mathcal{E}}_{12}\otimes{\mathcal{E}}_{01}\otimes{\mathcal{F}}_{0}$.

---
$\textstyle{\mathcal{X}^{\prime}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
o\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{o}$$\textstyle{o\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
o\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{o\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{o}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
o\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{o}$$\textstyle{o\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
o\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{o\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{o\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces({\mathcal{E}}_{0123}{\mathcal{F}}_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{o\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{o\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
o\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{o\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{o}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
o\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{o}$$\textstyle{o\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
o\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{o\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{o}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
o\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{F}}_{3}}$
Figure 5. Diagram of bundle gerbes and morphisms over $Y^{[4]}$ to derive the
coherence for $\theta$. The objects with placeholders marked $o$ are specified
in Figure 6.

Cutting open this cube to reveal the faces and the specific bundle gerbes at
the nodes, we get Figure 6. Note that the bundle gerbe
$({\mathcal{E}}_{0123}{\mathcal{F}}_{0})$ in the cubical diagram has
submersion $Z_{3}\to Y^{[4]}$, and has $U(1)$-bundle defined to be the
pullback along $Z_{3}\to
Z_{3}\times_{Y^{[4]}}d_{1}^{*}d_{1}^{*}d_{1}^{*}Z_{0}$ of the $U(1)$-bundle of
the bundle gerbe ${\mathcal{E}}_{0123}\otimes
d_{1}^{*}d_{1}^{*}d_{1}^{*}{\mathcal{F}}$.

Back-right-bottom faces:

---
$\textstyle{{\mathcal{E}}_{23}\otimes{\mathcal{E}}_{12}\otimes{\mathcal{E}}_{01}\otimes{\mathcal{F}}_{0}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{123}\otimes{\mathcal{E}}_{01}\otimes{\mathcal{F}}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{13}\otimes{\mathcal{E}}_{01}\otimes{\mathcal{F}}_{0}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{23}\otimes{\mathcal{E}}_{12}\otimes({\mathcal{E}}_{01}{\mathcal{F}}_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{123}\otimes({\mathcal{E}}_{01}{\mathcal{F}}_{0})}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{13}\otimes({\mathcal{E}}_{01}{\mathcal{F}}_{0})}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{013}\otimes{\mathcal{F}}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{23}\otimes{\mathcal{E}}_{12}\otimes{\mathcal{F}}_{1}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{123}\otimes{\mathcal{F}}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{13}\otimes{\mathcal{F}}_{1}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces({\mathcal{E}}_{013}{\mathcal{F}}_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{03}\otimes{\mathcal{F}}_{0}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{23}\otimes({\mathcal{E}}_{12}{\mathcal{F}}_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces({\mathcal{E}}_{123}{\mathcal{F}}_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces({\mathcal{E}}_{13}{\mathcal{F}}_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces({\mathcal{E}}_{03}{\mathcal{F}}_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{23}\otimes{\mathcal{F}}_{2}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces({\mathcal{E}}_{23}{\mathcal{F}}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{F}}_{3}}$

Front-left-top faces:

---
$\textstyle{{\mathcal{E}}_{23}\otimes{\mathcal{E}}_{12}\otimes{\mathcal{E}}_{01}\otimes{\mathcal{F}}_{0}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{123}\otimes{\mathcal{E}}_{01}\otimes{\mathcal{F}}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{13}\otimes{\mathcal{E}}_{01}\otimes{\mathcal{F}}_{0}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{23}\otimes{\mathcal{E}}_{12}\otimes({\mathcal{E}}_{01}{\mathcal{F}}_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{23}\otimes{\mathcal{E}}_{012}\otimes{\mathcal{F}}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{0123}\otimes{\mathcal{F}}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{013}\otimes{\mathcal{F}}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{23}\otimes{\mathcal{E}}_{12}\otimes{\mathcal{F}}_{1}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{1234}\otimes{\mathcal{E}}_{01}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{23}\otimes{\mathcal{E}}_{02}\otimes{\mathcal{F}}_{0}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{023}\otimes{\mathcal{F}}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{03}\otimes{\mathcal{F}}_{0}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{23}\otimes({\mathcal{E}}_{12}{\mathcal{F}}_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{E}}_{23}\otimes({\mathcal{E}}_{02}{\mathcal{F}}_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces({\mathcal{E}}_{023}{\mathcal{F}}_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces({\mathcal{E}}_{03}{\mathcal{F}}_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{23}\otimes{\mathcal{F}}_{2}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces({\mathcal{E}}_{23}{\mathcal{F}}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{F}}_{3}}$

Figure 6. Encoding of action coherence from rigid trivialisation data, giving
the outer faces of Figure 5.

The same pasting argument with mates for the arrows indicated
$\rightsquigarrow$ as in Construction A.7 establishes the required coherence
axiom for $\theta$.

This completes the construction of the associated bundle 2-gerbe
trivialisation.

Now consider the case where the rigid 2-gerbe is equipped with a rigid
connective structure, and the rigid trivialisation has a trivialisation
connection. Then, as before in Proposition A.8, we get the data of a
connection, curving and 2-curving on the associated bundle 2-gerbe, and a
trivialisation in the sense of Definition A.13. We need to prove that the
trivialisation connection gives rise to a compatible connection in the sense
of [Wal13, Definition 4.3]. We outline Waldorf’s definition here in more
elementary terms.

######  Definition A.15 (Waldorf).

Consider a bundle 2-gerbe with submersion $Y\to X$, bundle gerbe
${\mathcal{E}}=(E,Z_{1})$ on $Y^{[2]}$, multiplication stable morphism
$\overline{m}$ and associator $\tilde{\alpha}$, and also connection, curving
and 2-curving. The data of a compatible connection on a trivialisation with
data ${\mathcal{F}}=(F,Z_{0})$, $\eta$ and $\theta$, is a connection and
curving on ${\mathcal{F}}$, together with a connection $\nabla_{L}$ on the
$U(1)$-bundle
$L\to(Z_{1}\times_{Y^{[2]}}d_{1}^{*}Z_{0})\times_{Y^{[2]}}d_{0}^{*}Z_{0}$
underlying the stable isomorphism $\eta$ such that the curvature
$F_{\nabla_{L}}$ is the difference between the (pullback of the) curving
$\beta$ of the bundle 2-gerbe and the (pullback of) the curving of
$\delta({\mathcal{F}})$. Further, the 2-arrow $\theta$ in Definition A.13, the
data of which is an isomorphism of trivialising $U(1)$-bundles (satisfying a
compatibility condition), needs to be an isomorphism of bundles _with
connection_.

The following construction is in some sense the keystone of this paper,
linking our notion of rigid trivialisation to the construction of geometric
string structures à la Waldorf [Wal13]. Here we only give it in outline, as it
repeats a lot of steps shown previously.

######  Construction A.16.

Start with a rigid bundle 2-gerbe with rigid trivialisation, rigid connective
structure, and trivialisation connection. Build the associated bundle 2-gerbe
from Construction A.7, the associated trivialisation from Construction A.14,
and the associated connection, curving and 2-curving from Proposition A.8. The
1-form $a$ in the trivialisation connection allows us to define a connection
on the stable isomorphism $\eta$, as in Lemma A.5. That this is compatible
follows from the assumptions on $a$. Then following the same argument as in
the proof of Proposition A.10, we get that the diagram (A.2) is a diagram of
bundle gerbes with connection and curving, and morphisms between these, and
that the 2-arrow $\theta$ preserves the required connection. Again, the
coherence equation for $\theta$ holds for free.

We will rely on this construction in a future paper to give a large family of
examples of geometric string structures.

##  Appendix B Table of specific differential forms

Form |  Description |  Page
---|---|---
$\Theta_{g}=g^{-1}dg\in\Omega^{1}(G,\mathfrak{g})$ |  left-invariant Maurer–Cartan form |
$\widehat{\Theta}_{g}=dg\,g^{-1}\in\Omega^{1}(G,\mathfrak{g})$ |  right-invariant Maurer–Cartan form |
$\mu\in\Omega^{1}(\widehat{\Omega G})$ |  connection on the central extension of $\Omega G$ |  1.2
$R\in\Omega^{2}(\Omega G)$ |  curvature of $\mu$, left-invariant 2-form |  1.4
$\nu\in\Omega^{1}(\Omega G\times\Omega G)$ |  measures the failure of the connection $\mu$ to be multiplicative on the nose |  1.2
$\phi\colon PG\to\Omega\mathfrak{g}$ |  the function $p\mapsto p^{-1}\partial p$ (a ‘Higgs field’) |
$\widehat{\phi}\colon PG\to\Omega\mathfrak{g}$ |  the function $p\mapsto\partial p\,p^{-1}=\operatorname{Ad}_{p}(p^{-1}\partial p)$ |
$B\in\Omega^{2}(PG)$ |  curving on the basic gerbe on $G$ |  1.2
$\epsilon\in\Omega^{1}(PG\times\Omega G)$ |  1-form used to fix the connection $\mu$ into a _bundle gerbe_ connection, |  1.5
$\nabla=\mu-\pi^{*}\epsilon$
xxxx $\in\Omega^{1}(PG\times\widehat{\Omega G})$ |  bundle gerbe connection on the basic gerbe |  1.5
$\kappa\in\Omega^{2}(G\times G)$ |  measures the failure of the curvature of the basic gerbe to be a multiplicative 3-form |  3.13
$\rho\in\Omega^{1}(PG\times\Omega G)$ |  measures the failure of the crossed module action to preserve the connection $\mu$ |  3.10
$A\in\Omega^{1}(Q,\mathfrak{g})$ |  connection on the principal $G$-bundle $Q$ |
$\beta_{A}\in\Omega^{2}(Q\times PG)$ |  curving on the rigid Chern–Simons bundle 2-gerbe of $Q$ |  3.3
$\alpha\in\Omega^{1}(Q\times PG\times PG)$ |  strong trivialisation connection witnessing Chern–Simons bundle 2-gerbe product |  3.3
$-CS(A)\in\Omega^{3}(Q)$ |  2-curving on the rigid Chern–Simons bundle 2-gerbe of $Q$ |  3.3

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