# Spectral sections, twisted rho invariants and positive scalar curvature

### Moulay-Tahar Benameur

Université Montpellier 2, France### Varghese Mathai

University of Adelaide, Australia

## Abstract

We had previously defined in [10], the rho invariant $ρ_{spin}(Y,E,H,g)$ for the twisted Dirac operator $/∂_{H}$ on a closed odd dimensional Riemannian spin manifold $(Y,g)$, acting on sections of a flat hermitian vector bundle $E$ over $Y$, where $H=∑i_{j+1}H_{2j+1}$ is an odd-degree differential form on $Y$ and $H_{2j+1}$ is a real-valued differential form of degree $2j+1$. Here we show that it is a conformal invariant of the pair $(H,g)$. In this paper we express the defect integer $ρ_{spin}(Y,E,H,g)−ρ_{spin}(Y,E,g)$ in terms of spectral flows and prove that $ρ_{spin}(Y,E,H,g)∈Q$, whenever $g$ is a Riemannian metric of positive scalar curvature. In addition, if the maximal Baum–Connes conjecture holds for $π_{1}(Y)$ (which is assumed to be torsion-free), then we show that $ρ_{spin}(Y,E,H,rg)=0$ for all $r≫0$, significantly generalizing results in [10]. These results are proved using the Bismut–Weitzenböck formula, a scaling trick, the technique of noncommutative spectral sections, and the Higson–Roe approach [22].

## Cite this article

Moulay-Tahar Benameur, Varghese Mathai, Spectral sections, twisted rho invariants and positive scalar curvature. J. Noncommut. Geom. 9 (2015), no. 3, pp. 821–850

DOI 10.4171/JNCG/209