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 $ρ_{spin}(Y,E,H, g)$ for the twisted Dirac operator $D^E_H$ on $Y$, acting on sections of a flat hermitian vector bundle $E$ over $Y$, where $H = \sum i^{j+1} H_{2j+1}$ is an odd-degree closed differential form on $Y$ and $H_{2j+1}$ is a real-valued differential form of degree ${2j+1}$. We prove that it only depends on the conformal class of the pair $[H,g]$. In the special case when $H$ is a closed 3-form, we use a Lichnerowicz-Weitzenbock formula for the square of the twisted Dirac operator, to show that whenever $Y$ is a closed spin manifold, then $ρ_{spin}(Y,E,H, g)= ρ_{spin}(Y,E, g)$ for all $|H|$ small enough, whenever g is a Riemannian metric of positive scalar curvature. When $H$ is a top-degree form on an oriented three dimensional manifold, we also compute $ρ_{spin}(Y,E,H, g)$.