Multitransgression and regulators

In a parallel way to the work of Wang, we define higher order characteristic classes associated with the Chern character, generalizing the work of Bott-Chern and Gillet-Soul'e on secondary characteristic classes. Our formalism is simplicial and the computations are easier. As a consequence, we obtain the comparison of Borel and Beilinson regulators and an explicit formula for the real single-valued function associated with the Grasmannian polylogarithm.

💡 Research Summary

This paper builds on Wang’s multitransgression framework to construct higher‑order characteristic classes associated with the Chern character, extending the classical secondary classes of Bott‑Chern and Gillet‑Soulé. The authors introduce a simplicial formalism that replaces the traditional Čech‑Deligne complex with a simplicial complex equipped with a family of higher transgression operators (d_r) (for (r\ge 1)). These operators act simultaneously on several degrees of the complex, allowing a Chern class of degree (k) to “transgress” to a form of degree (k-r). The resulting multitransgressed Chern forms (ch^{(r)}(E)) are closed real differential forms and provide a systematic way to define secondary characteristic classes in arbitrary cohomological degree.

A major application is the explicit comparison between the Borel regulator (\beta) and the Beilinson regulator (b) on higher K‑theory. By constructing a correction term (\tau^{(r)}) from the multitransgression forms, the authors prove the identity
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