Affine deformations of cotangent groupoids

Abstract

We study affine deformations of the cotangent groupoid $T^*\mathcal{G} \rightrightarrows A^*$, governed by a one-form $γ\inΩ^1(\mathcal G^{(2)})$, and characterize the conditions on $γ$ 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.