Conformal invariants of twisted Dirac operators and positive scalar curvature

Abstract

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.

Spectral sections, twisted rho invariants and positive scalar curvature

“…In this paper we express the defect integer spi n .Y; E; H; g/ spi n .Y; E; g/ in terms of spectral flows and prove that spi n .Y; E; H; g/ 2 Q, whenever g is a Riemannian metric of positive scalar curvature. In addition, if the maximal Baum-Connes conjecture holds for 1 .Y / (which is assumed to be torsion-free), then we show that spi n .Y; E; H; rg/ D 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].…”

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We had previously defined in [10], the rho invariant spi n .Y; E; H; g/ for the twisted Dirac operator = @ E H on a closed odd dimensional Riemannian spin manifold .Y; g/, acting on sections of a flat hermitian vector bundle E over Y , whereHere we show that it is a conformal invariant of the pair .H; g/. In this paper we express the defect integer spi n .Y; E; H; g/ spi n .Y; E; g/ in terms of spectral flows and prove that spi n .Y; E; H; g/ 2 Q, whenever g is a Riemannian metric of positive scalar curvature.

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Two-cocycle twists and Atiyah–Patodi–Singer index theory

“…Our goal is to define and study secondary invariants, in particular twisted η-and ρ-invariants. (Note that the sense of "twisted" here is different from the one in [BM1,BM2,BM3], also dealing with secondary invariants, and in general from papers concerned with twisted K-theory.) To do so, we sharpen some of the tools developed by Mathai, with particular attention to the dependence on the choices involved, and to the extension to manifolds with boundary.…”

Index type invariants for twisted signature complexes and homotopy invariance

“…[44,23]) and superstring theory ( [43]) via Riemannian connections ∇ with totally skew-symmetric, (de Rham) closed torsion tensor H, that is, ∇ X Y = ∇ representation theory [34]. In recent work, we study the eta and rho invariants for the twisted Dirac operator and the relation to positive scalar curvature, [8,9]. Acknowledgments M.B.…”

“…It also appears in the study of Dirac operators on loop groups and their representation theory [34]. In recent work, we study the eta and rho invariants for the twisted Dirac operator and the relation to positive scalar curvature [8,9].…”