Differential Models for the Anderson Dual to Twisted ${\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^c$-Bordism and a Twisted Anomaly Map

We construct differential models for degree-3 twisted ${\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^c$-bordism and for its Anderson dual. The model for the differential Anderson dual is based on the framework of Yamashita–Yonekura. Using these differential models, we define a twisted anomaly map from differential twisted $K$-theory with inverse twist to the differential Anderson dual of twisted ${\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^c$-bordism. The construction is described geometrically in terms of bundle gerbes, gerbe modules, and reduced eta-invariants of Dirac operators associated to the twisted data. Conceptually, this map is expected to be related to the anomalies of twisted $1\mathrm{\mid }1$-dimensional supersymmetric field theories, in line with the perspectives of Stolz–Teichner and Freed–Hopkins.