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Given an exact Lagrangian submanifold $L$ in $T^*N$, we want to construct acomplex of sheaves in the derived category of sheaves on $N\times {\mathbb R}$, such that its singular support, $SS({\mathcal F}^\bullet_L)$, is equal to$\widehat L$, the cone constructed over $L$. Its existence was stated in\cite{Viterbo-ISTST} in 2011, with a sketch of proof, which however contained agap (fixed here by the rectification). A complete proof was shortly afterprovided by Guillermou (\cite{Guillermou}) by a completely different method, inparticular Guillermou's method does not use Floer theory. The proof providedhere is, as originally planned, based on Floer homology. Besides theconstruction of the complex of sheaves, we prove that the filtered versions ofsheaf cohomology of the quantization and of Floer cohomology coincide, that is$FH^*(N\times ]-\infty, \lambda [, {\mathcal F}^\bullet_L)\simeqFH^*(L;0_N;\lambda)$, and so do their product structures.

Sheaf Quantization of Lagrangians and Floer cohomology