Differential Models for Anderson Dual to Twisted $\mathrm{Spin}^c$-Bordism and Twisted Anomaly Map

Abstract: We construct differential models for twisted $\mathrm{Spin}^c$-bordism and for its Anderson dual, and employ the latter to define a twisted anomaly map whose source is the differential twisted $K$-theory. Our differential model for the twisted Anderson dual follows the formalism developed in [YY23]. To connect these constructions with the geometric framework of the Atiyah-Singer index theory, we further present a gerbe-theoretic formulation of our models in terms of bundle gerbes and gerbe modules [Mur96] [BCMMS02]. Within this geometric setting, we define the twisted anomaly map [ \widehat{\Phi}{\widehat{\mathcal{G}}}\colon \widehat{K}^{0}(X,\widehat{\mathcal{G}}^{-1}) \longrightarrow \bigl(\widehat{I\Omega^{\mathrm{Spin}^c}{\mathrm{dR}}}\bigr)^{n}(X,\widehat{\mathcal{G}}), ] whose construction naturally involves the reduced eta-invariant of Dirac operators acting on Clifford modules determined by the twisted data. Conceptually, this map is expected to encode the anomalies of twisted $1|1$-dimensional supersymmetric field theories, in accordance with the perspectives developed in [ST11] and [FH21].