On the K\"ahler-Hodge structure of superconformal manifolds

We show that conformal manifolds in $d\geq 3$ conformal field theories with at least 4 supercharges are K\"ahler-Hodge, thus extending to 3d ${\cal N}=2$ and 4d ${\cal N}=1$ similar results previously derived for 4d ${\cal N}=2$ and ${\cal N}=4$ and various types of 2d SCFTs. Conformal manifolds in SCFTs are equipped with a holomorphic line bundle ${\cal L}$, which encodes the operator mixing of supercharges under marginal deformations. Using conformal perturbation theory and superconformal Ward identities, we compute the curvature of ${\cal L}$ at a generic point on the conformal manifold. We show that the K\"ahler form of the Zamolodchikov metric is proportional to the first Chern class of ${\cal L}$, with a constant of proportionality given by the two-point function coefficient of the stress tensor, $C_T$. In cases where certain additional conditions about the nature of singular points on the conformal manifold hold, this implies a quantization condition for the total volume of the conformal manifold.