Homological contractibility of a prestack

Question

This question is in reference to Gaitsgory's preprint Contractibility of the space of rational maps. On p. 5 of the preprint, Gaitsgory defines a prestack $\mathscr{Y}$ (say over affine $\mathbb{C}$-schemes) to be homologically contractible if the functor $\text{Vect}\longrightarrow\mathfrak{D}(\mathscr{Y})$ taking $V\mapsto V\otimes\omega_{\mathscr{Y}}$ is fully faithful. Here $\mathfrak{D}(\mathscr{Y})$ denotes the DG category of D-modules on $\mathscr{Y}$. Further down the same page, Gaitsgory mentions that homological contractibility of $\mathscr{Y}$ is equivalent to the condition that $H_{\bullet}(\mathscr{Y}(\mathbb{C})^{\text{top}},\mathbb{Q})\cong\mathbb{Q}$. What is the argument for the equivalence of these two formulations of homological contractibility? I would be happy to see the argument even in a simple case, like the case where $\mathscr{Y}$ is a smooth complex variety.

Answer 1

This is proven in some detail in section 3 Gaitsgory's writeup of his the Atiyah-Bott formula. He starts with the fully faithfulness definition, then proves the equivalance with homological statement at the very end of the section.