arXiv:2611.14784v1 Announce Type: new Abstract: The Poisson $\sigma$-model (PSM) over a closed worldsheet $\Sigma$ is reinterpreted as a theory of connections on a trivial principaloid bundle $P_0 = \Sigma \times G \to \Sigma$ with the symplectic structure groupoid $(G,\omega)$ integrating the Poisson target $(M,\Pi)$, and that of sections of the shadow bundle $F_0 = \Sigma \times M \to \Sigma$ - as defined in 2503.09886 [math.DG]. The flat directions of the PSM are explained as linearisations of automorphisms of $P_0$, modelled on maps from $\Sigma$ into the group of lagrangean bisections of $G$, which induce Poisson automorphisms of $(M,\Pi)$. The gauge anomaly of the ensuing field theory is derived and demonstrated to vanish for prequantisable $\omega$. A general anomaly-cancellation mechanism is proposed, which couples the bare model - rewritten as the holonomy of a connection-dependent trivial gerbe on $F_0$ - to a background gerbe over $M$ endowed with a $G$-equivariant structure. The coupling takes the form of the tensor product of the two gerbes and the extended PSM is defined as the holonomy of the product gerbe. An essential generalisation of the theory is then proposed - to shadows $F$ of principaloid bundles $P$ of arbitrary topology, in keeping with the interpretation of the gauging as descent to the orbistack $M//G$. The generalisation uses local product gerbes depending on local connections over trivialisations of $F$. The $G$-equivariant structure is proven to ensure a 2-stacky descent of the local gerbes to a global one over $F$. Its holonomy is taken as the definition of the generalised PSM. A cohomological discussion of obstructions against the descent is provided. Finally, a reformulation of the generalised PSM as a field theory with a topological gauge-symmetry defect is given, in which the $G$-equivariant structure is employed - alongside the transition cocycle of $P$ - as the datum of the defect.
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