Classifying space BG as a 2-shifted symplectic stack

Shifted symplectic structure should be the correct concept when one tries to put symplectic structures on stacks. These guys will pop out as reduced phase space of (AKSZ, or BV) sigma model when dimension goes higher. When we do this for the Chern-Simons sigma model, we end up with $BG$. In this talk, we explore various differential-geometric (1-group, 2-group, double-group if time allows) models to realise this (2-shift) symplectic structure in concrete formulas and show the equivalences between them.

In the infinite dimensional models (2-group, double-group), Segal's symplectic form on based loop groups turns out to be additionally multiplicative or almost so. These models are equivalent to a finite dimensional model with Cartan 3-form and Karshon-Weinstein 2-form via Morita Equivalence. All these forms give rise to the first Pontryagin class on $BG$. Moreover, they are related to the original invariant pairing on the Lie algebra through an explicit integration and Van Est procedure.

It's a joint work with Miquel Cueca Ten.