In the paper by Freed et al. "Topological Quantum Field Theories From Compact Lie Groups" they say


...the stack of G-bundles with connections is $\star // G = BG$...


My question is what's the notation $\star // G$? Is it the same as the symplectic quotient; 
i.e., take a contractible space $\star$ and quotient out by the action of $G$? Or, is this notation used with stacks (which I am not familiar).



question about notation
I can't say much about being related to a symplectic quotient but it is a very important example in the theory of stacks. The short answer, without going into too many details, is that $[S/G]$ is category whose objects are principal homogeneous $G$-bundles with a $G$-equivariant morphism to S. The morphisms are pullbacks which are compatible with the morphism to $S$. If $S=pt$ then the $G$ action is the trivial action and it then follows that $[pt/G]=BG$. This example also shows the dimension of stack can be negative i.e. $\dim [pt/G]=-\dim G$. For more details and great introductions to stacks see the article *[Algebraic stacks](https://arxiv.org/abs/math/9911199)* by T. Gomez, and B. Fantechi's *[Stacks for Everybody](https://doi.org/10.1007/978-3-0348-8268-2_20)* ([author pdf](https://www.math.uni-bielefeld.de/%7Erehmann/ECM/cdrom/3ecm/pdfs/pant3/fantechi.pdf)).


