Equivariant Poisson 2-Algebra Bundles over Configuration Spaces

We study equivariant vector bundles over configuration spaces with diagonals included, viewed as orbifold quotients $M^n\mathrm{/}{\mathfrak{S}}_n$ by permutation groups. Working in the equivalent language of equivariant vector bundles, we construct an induced-equivariance functor and prove its adjunction with restriction. We then define Hadamard and Cauchy tensor products and show that they form a symmetric $2$-monoidal structure. We construct the corresponding tensor and symmetric algebra bundles and prove that, for a local vector bundle $V\to M$, the bundle ${\mathbf{S}}^{⊠}({\mathbf{S}}^{\otimes }(V))$ is the free commutative $2$-algebra generated by $V$. Finally, we show that any skew-symmetric bundle map $k:V⊠V\to {\mathbf{I}}_{\otimes }$ induces a compatible Poisson bracket on this $2$-algebra bundle.