Gaugings of Groupoids, Strings in Shadows, and Emergent Poisson $\sigma$-Models

The gauge principle is proposed for rigid Lie-groupoidal symmetries $G=>M$ of the Polyakov-Alvarez-Gaw\k{e}dzki 2$d$ non-linear $\sigma$-model with metric target $(M,g_M)$ and the WZ term given by a CS differential character coming from an abelian gerbe $\mathcal{G}$. The principle bases on the notion of principaloid bundle with connection $(P,\Theta)$, introduced by Strobl and the Author. The descent of the model to the shadow of $P$ is demonstrated to require a twisted $G$-equivariant structure on $\mathcal{G}$, prequantising a multiplicative extension $(H_M,\rho,0)$ of the gerbe's curvature $H_M$ to a 3-cocycle in the BSS cohomology of the groupoid's nerve. The descent is accompanied by a combined $g_M$-isometric/$\rho$-holonomic reduction of the structure group of $P$, and uses an augmentation of the original gerbe by a trivial one depending quadratically on $\Theta$. The latter couples to the field of the $\sigma$-model -- in an extension of the scheme worked out for the action groupoid by Gaw\k{e}dzki, Waldorf and the Author -- through the comomentum component of the Spencer pair of $\rho$, as described by Crainic et al. A fully fledged cohomological analysis of the relevant gauge anomaly and of inequivalent gaugings is presented. In the symplectic setting, the augmentation procedure is shown to lead to the emergence of the standard Poisson $\sigma$-model. Conversely, a coaugmentation of a (local) Poisson $\sigma$-model by a flat equivariant gerbe yields a novel field theory with the shadow of the underlying symplectic principaloid bundle as the configuration bundle, and with manifest lagrangean gauge symmetry. A simple Cartan-type associating mechanism is proposed to account for the reduction of the structure group of the principaloid bundle. The mechanism allows for the coupling of an arbitrary number of distinct charged matter-field species to a given gauge field.