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Noncommutative geometry and conformal geometry. III: Vafa-Witten inequality and Poincaré duality. (English) Zbl 1311.53037

Summary: This paper is the third part of a series of papers whose aim is to use the framework of twisted spectral triples to study conformal geometry from a noncommutative geometric viewpoint. In this paper we reformulate the inequality of C. Vafa and E. Witten [“Eigenvalue inequalities for fermions in gauge theories”, Commun. Math. Phys. 95, No. 3, 257–276 (1984; doi:10.1007/BF01212397)] in the setting of twisted spectral triples. This involves a notion of Poincaré duality for twisted spectral triples. Our main results have various consequences. In particular, we obtain a version in conformal geometry of the original inequality of Vafa-Witten, in the sense of an explicit control of the Vafa-Witten bound under conformal changes of metrics. This result has several noncommutative manifestations for conformal deformations of ordinary spectral triples, spectral triples associated with conformal weights on noncommutative tori, and spectral triples associated with duals of torsion-free discrete cocompact subgroups satisfying the Baum-Connes conjecture.

MSC:

53C20 Global Riemannian geometry, including pinching
14J80 Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants)
46L87 Noncommutative differential geometry
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