On the Kähler-Hodge structure of superconformal manifolds

We show that conformal manifolds in d ≥ 3 conformal field theories with at least 4 supercharges are Kähler-Hodge, thus extending to 3d $ \mathcal{N} $ = 2 and 4d $ \mathcal{N} $ = 1 similar results previously derived for 4d $ \mathcal{N} $ = 2 and $ \mathcal{N} $ = 4 and various types of 2d SCFTs. Conformal manifolds in SCFTs are equipped with a holomorphic line bundle ℒ, which encodes the operator mixing of supercharges under marginal deformations. Using conformal perturbation theory and superconformal Ward identities, we compute the curvature of ℒ at a generic point on the conformal manifold. We show that the Kähler form of the Zamolodchikov metric is proportional to the first Chern class of 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.