of dimension $2n+1$. We endow $\Sigma$ with an auxiliary Riemannian metric in order to measure distances and norms of various objects defined on $\Sigma$. We denote by $r$ a real number in $[0,1]$. We will often consider paths $r\mapsto\mathbf o_r$ with values in some target space $\mathbf O$. We use the short-hand $\{\mathbf o_r\}\subset\mathbf O$ for such a path and $[\mathbf o_r]$ for a homotopy class of paths with conveniently chosen boundary conditions. If $\{\mathbf o_r\}$ and $\{\mathbf o'_r\}$ are two paths with $\mathbf o_1=\mathbf o'_0$, we write $\{\mathbf o_r\}\#\{\mathbf o_r'\}$ for the concatenated path. Finally, if a mathematical object is expressed by a certain symbol, we add a *prime* to it in order to denote another object of the same kind.

The volume of a closed two-form {#c:vol}
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Definition of the volume
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Let $\Xi^k(\Sigma)$ be the set of closed $k$-forms on $\Sigma$ and $\Xi_C^2(\Sigma)$ the set of elements in $\Xi^2(\Sigma)$ representing the de Rham cohomology class $C\in H_{{\mathrm{dR}}}^2(\Sigma)$. The set $\Xi^2_C(\Sigma)$ is affine with underlying vector space $$\label{def:baromega}
\bar{\Omega}^1(\Sigma):=\frac{\Omega^1(\Sigma)}{\Xi^1(\Sigma)}.$$ Indeed, for every $\Omega\in\Xi^2_C(\Sigma)$, we have the surjective map $$\label{eq:B_Omega}
B_{\Omega}:\Omega^1(\Sigma)\longrightarrow\Xi^2_C(\Sigma),\qquad \alpha\mapsto\Omega+{{\mathrm d}}\alpha.$$ This map passes to the quotient under $\Xi^1(\Sigma)$ and yields a bijection $$\bar{B}_{\Omega}:\bar{\Omega}^1(\Sigma)\longrightarrow\Xi^2_C(\Sigma)$$ so that 