Geometry of conformal manifolds and the inversion formula

Families of conformal field theories are naturally endowed with a Riemannian geometry which is locally encoded by correlation functions exactly marginal operators. We show that the curvature such manifolds can be computed using Euclidean and Lorentzian inversion formulae, combine operator content theory into an analytic function. Analogously, operators fixed dimension define bundles over manifold whose curvatures also formulae. These results relate to integrated four-point sensitive only behavior at separated points. apply these formulae derive convergent sum rules expressing in terms spectrum local their three-point function coefficients. further smoothly diverge if conserved current appears spectrum, or develops continuum. verify our explicitly $2d$ examples. In particular, for (2,2) superconformal we lower bound on scalar curvature, saturated free when central charge multiple three.