We construct a version of Beilinson’s regulator as a map of sheaves of commutative ring spectra and use it to define a multiplicative variant of differential algebraic -theory. We use this theory to give an interpretation of Bloch’s construction of -classes and the relation with dilogarithms. Furthermore, we provide a relation to Arakelov theory via the arithmetic degree of metrized line bundles, and we give a proof of the formality of the algebraic -theory of number rings.
"Multiplicative differential algebraic $K$-theory and applications." Ann. K-Theory 1 (3) 227 - 258, 2016. https://doi.org/10.2140/akt.2016.1.227