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

Question

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?

Answer 1

Saying that a splitting varies over the moduli space is not completely well defined: you have to say how to identify the total spaces at different points of the moduli i.e. to specify a flat connection on the bundle of total spaces.

In the B-model, if you take the Gauss-Manin connection as the flat connection then the Hodge splitting varies over the moduli space (because the Gauss-Manin connection does not preserve the splitting in general).

In the A-model, if you take the trivial connection as the flat connection then the splitting does not vary over the moduli space (the trivial connection preserves the degree decomposition). But it is not the trivial connection which appears in mirror symmetry on the A-model side but a flat connexion which is the trivial one corrected by contributions of holomorphic world-sheet instantons (i.e. Gromov-Witten invariants) and this connection does not preserve the degree decomposition in general.

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 $\mathcal{L}$.)