1210.0301 (Moulay-Tahar Benameur et al.)

Conformal invariants of twisted Dirac operators and positive scalar
curvature    [PDF]

Moulay-Tahar Benameur, Varghese Mathai

For a closed, spin, odd dimensional Riemannian manifold (Y,g), we define the rho invariant $\rho_{spin}(Y,\E,H, [g])$ 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 [g] of the metric 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 $X$ is a closed spin manifold, then rho_{spin}(Y,E,H, [g])= rho_{spin}(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 rho_{spin}(Y,E,H).

View original: http://arxiv.org/abs/1210.0301