General gerbe decomposition equivalence

Establish in full generality the equivalence between the Gromov–Witten theory of a smooth proper one-dimensional Deligne–Mumford stack with generic stabilizer and the corresponding Gromov–Witten theory of its gerbe-decomposition data, including the associated twisted theory.

Background

The paper uses gerbe decomposition to relate the Gromov–Witten theory of a one-dimensional Deligne–Mumford stack to a disconnected theory involving a twisted -gerbe. This reduction underlies the proposed extension of Virasoro constraints beyond effective orbifold curves.

The authors explicitly note that the required equivalence of Gromov–Witten theories has not been established in full generality, leaving the general validity of this reduction unresolved.