The precise definition is given in sections \[sec:four-integr-oper\] and \[sec:unitary-maps\]. The main properties of these operators are the following.

-   $U(\Phi_L,\Psi)$ is not empty. For any sequences $(T_k)$ and $(T_k')$ in $U(\Phi_L,\Psi)$ we have that $$T'_k T_k^{-1} = {\operatorname{id}}+ O(k^{-1})$$ in uniform norm.

-   We have an asymptotic representation in the sense that $$U(\Phi_L,\Psi) U(\Phi_L',\Psi') = U(\Phi_L \circ \Phi_L',\Psi \circ_{{{\mathcal{D}}}}
    \Psi').$$

-   When the graph of $\Phi$ intersects transversally the diagonal, we can estimate the trace of any $(T_k)$ in $U(\Phi_L, \Psi)$: $$\begin{gathered}
     \label{eq:trace}
     \operatorname{ Tr} ( T_k) = \sum_{x = \Phi (x) } \frac{ i^ {m_x}  u_x^k } { | \det ( {\operatorname{id}}- T_x \Phi ) |^{1/2}} + O(k^{-1}) \end{gathered}$$ where for any fixed point $x$ of $\Phi$, $u_x \in {{\mathbb{C}}}$ is the trace of the endomorphism $\Phi_{L}(x)$ and $m_x \in {{\mathbb{Z}}}/ 4{{\mathbb{Z}}}$ depends only on $T_x \Phi$ and $\Psi (x)$.

Precise statements are given in theorems \[sec:unitary-maps-comp\] and \[sec:trace-estimates\]. These results are consequences of more general statements where the half-form line bundle is replaced by any auxiliary line bundle, cf. theorem \[the:P1\] and \[the:trace-estimate\]. However the quantization with metaplectic correction presented here has the following particular features. First, we have an asymptotic representation of a finite cover of 