Geometry of Conformal Manifolds and the Inversion Formula

TL;DR

This paper develops a method to compute the curvature of conformal manifolds using inversion formulae, linking geometric properties to correlation functions and operator spectra, with applications to 2D superconformal theories.

Contribution

It introduces inversion formulae-based techniques to calculate conformal manifold curvatures from correlation functions and operator data, providing new sum rules and divergence criteria.

Findings

Curvature can be computed from four-point functions and operator spectra.

Divergences in curvature relate to conserved currents or continuum limits.

Explicit verification in 2D superconformal theories confirms the theoretical results.

Abstract

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