we have an identification $$\mathrm T_\Omega\Xi^2_C(\Sigma)=\bar{\Omega}^1(\Sigma).$$ We define now an exact one-form ${\mathrm{vol}}\in\Omega^1(\Omega^1(\Sigma))$. To this purpose, we fix a [**reference two-form**]{} $\Omega_0\in\Xi^2_C(\Sigma)$, and for every $\alpha\in\Omega^1(\Sigma)$, we use the short-hand $$\Omega_\alpha:=B_{\Omega_0}(\alpha)=\Omega_0+{{\mathrm d}}\alpha.$$ For every $\beta\in \Omega^1(\Sigma)\cong{{\mathrm T}}_\alpha\Omega^1(\Sigma)$, we set $$\label{def:vol}
{\mathrm{vol}}_{\alpha}\cdot\beta:=\int_\Sigma\beta\wedge \Omega_\alpha^{n}.$$ In the lemma below, we show that the one-form ${\mathrm{vol}}$ admits the primitive functional $$\label{e:defvol}
{\mathrm{Vol}}:\Omega^1(\Sigma) \to {{\mathbb R}}\,,\qquad {\mathrm{Vol}}(\alpha):= \int_0^1\left(\int_\Sigma\alpha\wedge\Omega_{r\alpha}^{n}\right){{\mathrm d}}r.$$

\[r:cs\] Exchanging the order of integration, we can rewrite $${\mathrm{Vol}}(\alpha)=\int_\Sigma \mathrm{CS}(\alpha),\qquad \mathrm{CS}(\alpha):=\int_0^1\alpha\wedge \Omega^n_{r\alpha}\,{{\mathrm d}}r.$$ When $C=0$ and $\Omega_0=0$, the form $(n+1)\mathrm{CS}(\alpha)$ reduces to the Chern-Simons form for principal $S^1$-bundles with invariant polynomial $P(x_1,\ldots,x_{n+1})=x_1\cdot\ldots\cdot x_{n+1}$ [@CS74 equation (3.1)]. Furthermore, $(n+1){\mathrm{Vol}}(\alpha)$ corresponds to the cohomology class in [@CS74 Theorem 3.9, case $2l-1=n$] (where $M$ therein is our $\Sigma$ and $l$ is our $n+1$).

\[l:volform\] The functional ${\mathrm{Vol}}$ associated with $\Omega_0$ is uniquely characterised by the properties $${\mathrm{Vol}}(0)=0,\qquad {{\mathrm d}}{\mathrm{Vol}}={\mathrm{vol}}.$$ The following formula holds (see also [@CS74 equation (3.5)]): $${\mathrm{Vol}}(\alpha)\ =\ \sum_{j=0}^{n}\frac{1}{j+1}\binom{n}{j}\int_\Sigma\alpha\wedge ({{\mathrm d}}\alpha)^j\wedge\Omega_0^{n-j}.$$

Differentiating under the integral sign we write ${{\mathrm d}}_\alpha{\mathrm{Vol}}(\beta)=\int_0^1V(r){{\mathrm d}}r$ and compute $$\begin{aligned}
V(r)&=\int_\Sigma\Big(\beta\wedge\Omega_{r\alpha}^{n}+\alpha\wedge \Omega_{r\alpha}^{n-1}\wedge nr{{\mathrm d}}\beta\Big)\\
&=\int_\Sigma\Big(\beta\wedge\Omega_{r\alpha}^{n}+nr \Omega_{r\alpha}^{n-1}\wedge\big(\beta\wedge {{\mathrm d}}\alpha-{{\mathrm d}}(\alpha\wedge\beta)\big)\Big)\\
&=\int_\Sigma\Big(\beta\wedge\Omega_{r\alpha}^{n}+r \beta\wedge\Omega_{r\alpha}^{n-1}\wedge\big(n{{\mathrm d}}\alpha\big)\Big)\\
&=\frac{{{\mathrm d}}}{{{\mathrm d}}r}\left(\int_\Sigma r\beta\wedge\Omega_{r\alpha}^{n}\right),\end{aligned}$$ where in the third equality we used Stokes’ Theorem. We conclude that $${{\mathrm d}}_\alpha{\mathrm{Vol}}\cdot\beta=\int_0^1\frac{{{\mathrm d}}}{{{\mathrm d}}r}\left(\int_\Sigma 