Affine deformations of cotangent groupoids

We study affine deformations of the cotangent groupoid $T^{\ast }\mathcal{G}\rightrightarrows A^{\ast }$, governed by a one-form $\gamma \in {\mathrm{\Omega }}^1({\mathcal{G}}^{(2)})$, and characterize the conditions on $\gamma$ under which this construction is valid. We show that these deformations arise naturally from ${\mathbb{S}}^1$-central extensions of Lie groupoids via symplectic reduction, and identify the reduced symplectic form as a multiplicative magnetic form. In particular, for Kac-Moody extensions, this construction yields nontrivial deformations of quotient stacks and ${\mathbb{S}}^1$-gerbes.