Nonlocal gravity with worldline inversion symmetry

We construct a quadratic curvature theory of gravity whose graviton propagator around the Minkowski background respects wordline inversion symmetry, the particle approximation to modular invariance in string theory. This symmetry automatically yields a corresponding gravitational theory that is nonlocal, with the action containing infinite order differential operators. As a consequence, despite being a higher order derivative theory, it is ghost-free and has no degrees of freedom besides the massless spin-$2$ graviton of Einsteins general relativity. By working in the linearised regime we show that the point-like singularities that afflict the (local) Einsteins theory are smeared out.