On the Kähler-Hodge structure of superconformal manifolds

Abstract

We show that conformal manifolds in $d\ge 3$ 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 $\mathcal{L}$, which encodes the operator mixing of supercharges under marginal deformations. Using conformal perturbation theory and superconformal Ward identities, we compute the curvature of $\mathcal{L}$ 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 $\mathcal{L}$, 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.

arXiv comment33 pages, typos corrected

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Cited by in corpus (1)

• Covariantly Constant Anomalies on Conformal Manifolds