$M$. Such a bundle exists if and only if the second Stiefel-Whitney class of $M$ vanishes. We will define the notion of half-form bundle automorphisms. For the introduction, it is sufficient to know that any symplectomorphism of $M$ isotopic to the identity lifts to a half-form bundle automorphism. Furthermore on any component of $M$, this lifts is unique up to a sign.

Consider two automorphisms $\Phi_L$ and $\Psi$ of the prequantum bundle and the half-form bundle respectively which lifts the same symplectomorphism $\Phi$ of $M$. Then we will define a class $U(\Phi_L,\Psi)$ which consists of sequences $$T_k : H^0(M,L^k \otimes \delta ) \rightarrow H^0 (M, L^k \otimes
\delta), \qquad k=1, 2,\ldots$$ of unitary maps whose Schwartz kernel has a precise asymptotic. Without going into the complete definition, let us describe the main characteristics.

-   The Schwartz kernel concentrate on the graph of $\Phi^{-1}$ in the sense that $$T_k (y, x) = O ( k^{-N}), \quad \forall N$$ for any $y$ and $x \in M$ such that $y\neq \Phi (x)$.

-   The asymptotic on the graph is given in terms of $\Phi_L$ and $\Psi$ by $$T_k (\Phi(x), x) = \Bigl( \frac{k}{2 \pi} \Bigr)^n \Phi_L(x)^k
      \otimes ( \Psi (x)  + O(k^{-1}) ).$$

