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We construct models of the differential $KO$-theory and the twisteddifferential $KO$-theory, by refining Karoubi's $KO$-theory [Kar78] in terms ofgradations on Clifford modules. In order for this, we set up the generalizedClifford superconnection formalism which generalizes the Quillen'ssuperconnection formalism [Qui85]. One of our models can be regarded asclassifying "fermionic mass terms" in physics.

Differential $KO$-theory via gradations and mass terms