A Gromov-Witten approach to G-equivariant birational invariants

In arXiv: 2404. 19088, we initiated a program linking birational invariants with smooth ones and offering new interpretations of classical invariants, such as the Kervaire-Milnor invariants. Here, we rely on the profound geometric reasoning provided by Lupercio and Uribe in the early 2000s to establish a connection between Chen-Ruan cohomology and the G-birational invariants introduced by Kontsevich, Pestun, and Tschinkel in recent pioneering work, along with presenting applications. The final section of this paper explores conjectures and preliminary results regarding gerbes, connections over orbifolds, discrete torsion, and potential approaches via motivic integration and twisted K-theory for G-birationality. Combined with the theory of atoms by Katzarkov, Kontsevich, Pantev, and Yu, the proposal in this paper program will lead to a theory of equivariant atoms.