Affine deformations of cotangent groupoids

We study affine deformations of the cotangent groupoid $T^*\mathcal{G} \rightrightarrows A^*$, governed by a one-form $\gamma\in\Omega^1(\mathcal G^{(2)})$, 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.