Conformal invariants of twisted Dirac operators and positive scalar curvature

Abstract

For a closed, spin, odd dimensional Riemannian manifold (Y,g), we define the rho invariant ρspin(Y,E,H,[g]) for the twisted Dirac operator ⁄∂HE on Y, acting on sections of a flat Hermitian vector bundle E over Y, where H=∑ij+1H2j+1 is an odd-degree closed differential form on Y and H2j+1 is a real-valued differential form of degree 2j+1. We prove that it only depends on the conformal class [g] of the metric g. In the special case when H is a closed 3-form, we use a Lichnerowicz–Weitzenböck formula for the square of the twisted Dirac operator, which in this case has no first order terms, to show that ρspin(Y,E,H,[g])=ρspin(Y,E,[g]) for all |H| small enough, whenever g is conformally equivalent to a Riemannian metric of positive scalar curvature. When H is a top-degree form on an oriented three dimensional manifold, we also compute ρspin(Y,E,H).