Abstract
In an attempt to combine non-commutative geometry and quantum gravity, Aastrup-Grimstrup-Nest construct a semi-finite spectral triple, modeling the space of G-connections for G = U(1) or SU (2). AGN show that the interaction between the algebra of holonomy loops \({\mathcal{B}}\) and the Dirac type operator \({\mathcal{D}}\) quantizes the Poisson structure of General Relativity in Ashtekar’s loop variables. This article generalizes AGN’s construction to any connected compact Lie group G. A construction of AGN’s semi-finite spectral triple in terms of an inductive limit of spectral triples is formulated. The refined construction permits the semi-finite spectral triple to be even when G is even dimensional. The Dirac-type operator \({\mathcal{D}}\) in AGN’s semi-finite spectral triple is a weighted sum of a basic Dirac operator on G. The weight assignment is a diverging sequence that governs the “volume” associated to each copy of G. The JLO cocycle of AGN’s triple is examined in terms of the weight assignment. An explicit condition on the weight assignment perturbations is given, so that the associated JLO class remains invariant. Such a condition leads to a functoriality property of AGN’s construction.
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Communicated by Y. Kawahigashi
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Lai, A. The JLO Character for the Noncommutative Space of Connections of Aastrup-Grimstrup-Nest. Commun. Math. Phys. 318, 1–34 (2013). https://doi.org/10.1007/s00220-013-1665-6
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DOI: https://doi.org/10.1007/s00220-013-1665-6