Conformal invariants of twisted Dirac operators and positive scalar curvature

TL;DR

This paper introduces a conformally invariant rho invariant for twisted Dirac operators on spin manifolds, explores its dependence on geometric data, and examines its behavior under positive scalar curvature and specific forms.

Contribution

It defines a new conformal invariant for twisted Dirac operators and analyzes its properties, including invariance and behavior in positive scalar curvature contexts.

Findings

The rho invariant depends only on the conformal class of the pair [H,g].

For small H, the rho invariant equals the untwisted case when scalar curvature is positive.

Explicit computation of the rho invariant for top-degree forms on 3D manifolds.

Abstract

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 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 $\rho_{spin}(Y,E,H, g)= \rho_{spin}(Y,E, g)$ 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 $\rho_{spin}(Y,E,H, g)$.