# Question about the vacuum bundle on A- and B-model

Let us consider the topological string A- and B-model (twisted SUSY non-linear sigma model on CY 3-manifold $X$). They are realization of $N=2$ SCFT and there are ground-states vector bundle $\mathcal{H}$ and vacuum line bundle $\mathcal{L}$ over the moduli space of the theory. In the A-model, $\mathcal{H}=H^{even}(X,\mathbb{C})$ and $\mathcal{L}=H^0(X,\mathbb{C})$. In the B-model, $\mathcal{H}=H^{3}(X,\mathbb{C})$ and $\mathcal{L}=H^{3,0}(X,\mathbb{C})$. The genus $g$ string amplitude is given as a section of $\mathcal{L}$ in either theory. Mirror symmetry is an identification of the geometry of A- and B-model ground-state geometry on distinct CY 3manifolds.

My questions is following. In the A-model, it seems the splitting of the bundle $$\mathcal{H}_A=H^{even}(X,\mathbb{C})=\oplus_{i=0}^3 H^{2i}(X,\mathbb{C})$$ does not vary over the moduli space of the theory (Kahler moduli space). On the other hand, in the B-model, the splitting $$\mathcal{H}=H^{3}(X,\mathbb{C})=\oplus_{p+q=3}H^{p,q}(X,\mathbb{C})$$ varies over the moduli space (variation of Hodge structure). Moreover, $\mathcal{L}$ is the trivial line bundle in the A-model, while it is not in the B-model. Isn't this contradiction?

About the vacuum line bundle. On the A-model sigma model side, 1 in H^{0} gives a natural trivialization. But the sigma model description is generally only valid in some limit of the moduli space, some cusp which is topologically a punctured polydisk. In particular, any complex line bundle is trivial in restriction to this domain and this is also the case for the vacuum line bundle of the B-model. Deep inside the moduli space, the topology can be complicated and the vacuum bundle of the B-model can be non-trivial but it is also the case for the A-model which has no longer a sigma model description and so no longer a "1" to trivialize $\mathcal{L}$.
(remark: the genus g string amplitude is a section of $\mathcal{L}^{2-2g}$ and not $\mathcal{L}$.)