Abstract. For a closed, spin, odd dimensional Riemannian manifold pY, gq, we define the rho invariant ρ spin pY, E, H, rgsq for the twisted Dirac operator C E H on Y , acting on sections of a flat hermitian vector bundle E over Y , where H " ř 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 rgs of the metric g. In the special case when H is a closed 3-form, we use a Lichnerowicz-Weitzenböck formula for the square of the twisted Dirac operator, which in this case has no first order terms, to show that ρ spin pY, E, H, rgsq " ρ spin pY, E, rgsq 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 ρ spin pY, E, Hq.
Given a constant magnetic field on Euclidean space ${\mathbb R}^p$ determined
by a skew-symmetric $(p\times p)$ matrix $\Theta$, and a ${\mathbb
Z}^p$-invariant probability measure $\mu$ on the disorder set $\Sigma$ which is
by hypothesis a Cantor set, where the action is assumed to be minimal, the
corresponding Integrated Density of States of any self-adjoint operator
affiliated to the twisted crossed product algebra $C(\Sigma) \rtimes_\sigma
{\mathbb Z}^p$, where $\sigma$ is the multiplier on ${\mathbb Z}^p$ associated
to $\Theta$, takes on values on spectral gaps in the magnetic gap-labelling
group. The magnetic frequency group is defined as an explicit countable
subgroup of $\mathbb R$ involving Pfaffians of $\Theta$ and its sub-matrices.
We conjecture that the magnetic gap labelling group is a subgroup of the
magnetic frequency group. We give evidence for the validity of our conjecture
in 2D, 3D, the Jordan block diagonal case and the periodic case in all
dimensions.Comment: 43 pages. Exposition improve
Abstract. For a closed, oriented, odd dimensional manifold X, we define the rho invariant ρ(X, E, H) for the twisted odd signature operator valued in a flat hermitian vector bundle E, where H = i j+1 H 2j+1 is an odd-degree closed differential form on X and H 2j+1 is a real-valued differential form of degree 2j + 1. We show that ρ(X, E, H) is independent of the choice of metrics on X and E and of the representative H in the cohomology class [H]. We establish some basic functorial properties of the twisted rho invariant. We express the twisted eta invariant in terms of spectral flow and the usual eta invariant. In particular, we get a simple expression for it on closed oriented 3-dimensional manifolds with a degree three flux form. A core technique used is our analogue of the Atiyah-Patodi-Singer theorem, which we establish for the twisted signature operator on a compact, oriented manifold with boundary. The homotopy invariance of the rho invariant ρ(X, E, H) is more delicate to establish, and is settled under further hypotheses on the fundamental group of X.
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