Chern character for twisted complexes

We construct a Chern character of a perfect complex of twisted modules over an algebroid stack.

💡 Research Summary

The paper addresses the long‑standing problem of defining a Chern character for perfect complexes of twisted modules over an algebroid stack, a setting that naturally arises in non‑commutative geometry, gerbe theory, and string‑theoretic B‑field backgrounds. The authors begin by recalling that the classical Chern character is a natural transformation from algebraic K‑theory to (periodic) cyclic homology, built from the trace of the exponential of a curvature form. This construction relies on the existence of a globally defined connection on vector bundles, an assumption that fails when the underlying category is twisted by a gerbe or an Azumaya algebra.

To overcome this obstacle, the authors first formalize the notion of a twisted module over an algebroid stack. An algebroid stack can be thought of as a sheaf of categories equipped with a “twisting datum” consisting of a 1‑cocycle of line bundles and a 2‑cocycle in (H^{2}(X,\mathcal{O}^{\times})). Locally, twisted modules look like ordinary differential graded (DG) modules, but globally their gluing is governed by the twisting datum, which introduces non‑trivial associators. Using homotopy limits and descent theory, the authors construct a derived category (\mathsf{D}_{\mathrm{tw}}(X)) of perfect twisted complexes and prove that it is well‑behaved (triangulated, compactly generated, etc.).

The central contribution is a definition of a twisted Chern character \