the distribution ${{\mathcal Y}}$ on $e_m$.

Finally, we define $\Delta_{\rm Neu}$ (resp. $\Delta_{\rm Dir}$) as the Neumann (resp. Dirichlet) Laplacian acting on $\varphi\in{\ensuremath{{{\mathcal S}}_{{\ensuremath{\operatorname{Neu}}}}}}$ and ${{\mathcal Y}}\in{\ensuremath{{{\mathcal S}}_{{\ensuremath{\operatorname{Neu}}}}}}'$ (resp. $\varphi \in {\ensuremath{{{\mathcal S}}_{{\ensuremath{\operatorname{Dir}}}}}}$ and ${{\mathcal Y}} \in {\ensuremath{{{\mathcal S}}_{{\ensuremath{\operatorname{Dir}}}}}}'$) as follows: $$\Delta_{\rm Neu} {{\mathcal Y}}(\varphi) = {{\mathcal Y}}(\Delta \varphi), \qquad \text{resp. } \quad \Delta_{\rm Dir} {{\mathcal Y}}(\varphi) = {{\mathcal Y}}(\Delta \varphi).$$ Let ${\mathcal{H}_{{\ensuremath{\operatorname{Neu}}}}^{-k}}$ be the topological dual space of ${{\mathcal H}}_{\rm Neu}^k$, both defined similarly to ${{\mathcal H}}_{\rm Dir}^k$ and ${\mathcal{H}_{{\ensuremath{\operatorname{Dir}}}}^{-k}}$ but replacing the basis $e_m(u)=\sqrt{2}\sin(m\pi u)$ by ${{\widetilde}{e}_m(u)}=\sqrt{2}\cos(m\pi u)$. In the next sections we will also need to consider one last operator, denoted by $\nabla_{\rm Dir}$ and defined as follows: given $k > 0$, ${{\mathcal Y}} \in{\ensuremath{{{\mathcal S}}_{{\ensuremath{\operatorname{Neu}}}}}}'$ and $\varphi \in{\ensuremath{{{\mathcal S}}_{{\ensuremath{\operatorname{Dir}}}}}}$ we set $$\label{eq:nabladir}
   \nabla_{\rm Dir} {{\mathcal Y}}(\varphi) = - {{\mathcal Y}}(\nabla \varphi).$$

Fluctuation fields {#sec:fields}
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Now we introduce all the microscopic fluctuation fields for which we will prove convergence to some infinite dimensional stochastic partial differential equations (SPDEs). In the following, for any space ${{\mathcal S}}$ of distributions, we denote by $\mathcal{D}([0,T], {{\mathcal S}})$ (resp. $\mathcal C([0,T], {{\mathcal S}}$) the set of càdlàg (resp. continuous) trajectories taking values in ${{\mathcal S}}$ and endowed with the Skorohod 