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 ${\rho }_{spin}(Y,E,H,g)$ for the twisted Dirac operator $D_H^E$ 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 ${\rho }_{spin}(Y,E,H,g)$.

13+2 pages, Latex 2e. Statement of conformal invariance corrected in erratum

References in corpus (6)

• Index, eta and rho-invariants on foliated bundles
• Analytic Torsion of Z_2-graded Elliptic Complexes
• Twisted Analytic Torsion
• Spectral sections, twisted rho invariants and positive scalar curvature
• Conformal invariants of twisted Dirac operators and positive scalar curvature
• Index type invariants for twisted signature complexes and homotopy invariance

Cited by in corpus (6)

• The global anomaly of the self-dual field in general backgrounds
• Spectral sections, twisted rho invariants and positive scalar curvature
• Conformal invariants of twisted Dirac operators and positive scalar curvature
• Index type invariants for twisted signature complexes and homotopy invariance
• The Higson-Roe exact sequence and ${\ell }^2$ eta invariants
• Positive scalar curvature via end-periodic manifolds