Abstract
In this paper we introduce the curvature of densely defined universal connections on Hilbert C⁎-modules relative to a spectral triple (or unbounded Kasparov module), obtaining a well-defined curvature operator. Fixing the spectral triple, we find that modulo junk forms, the curvature only depends on the represented form of the universal connection. We refine our definition of curvature to factorizations of unbounded Kasparov modules. Our definition recovers all the curvature data of a Riemannian submersion of compact manifolds, viewed as a KK-factorization.
Highlights
This paper offers a new approach to defining and effectively computing curvature of Hilbert modules and unbounded Kasparov modules
Our approach does not rely on the heat kernel coefficient analogy, and so our results differ from the recent work of [14, 11, 12, 20, 22, 34, 35]
Examples where curvature appears in the context of unbounded Kasparov theory is in the factorisation of Dirac operators on Riemannian submersions and G-spectral triples [7, 9, 29]
Summary
This paper offers a new approach to defining and effectively computing curvature of Hilbert modules and unbounded Kasparov modules. We may consider an unbounded representative of the internal Kasparov product given by the essentially self-adjoint and regular operator S ⊗ 1 + 1 ⊗∇ T , defined on the appropriate domain in X ⊗B Y (see [7, 28, 37, 38, 39] for more details). The left hand side of Equation (0.4) is a well-defined, direct and constructive way of representing the curvature of a module XB: all we require is the differential structure provided by a spectral triple or unbounded Kasparov module. Examples where curvature appears in the context of unbounded Kasparov theory is in the factorisation of Dirac operators on Riemannian submersions and G-spectral triples [7, 9, 29]. BM thanks Matthias Lesch for numerous conversations related to some of the technical aspects of this work
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