From the text: The purpose of this paper is to provide a detailed treatment of Gromov-Witten invariants of orbifolds for the simplest case, namely, when \(V\) is the classifying stack \({\mathcal B}G\) of a finite group \(G\). Many of the features and subtleties are present even here. The state space of this theory contains twisted sectors and the correlators in this theory are intersection numbers on \(\overline{\mathcal M}_{g,n}({\mathcal B}G)\), the moduli space of genus-\(g\), \(n\)-pointed orbifold stable maps into \({\mathcal B}G\). The correlators in this theory can be described in purely group theoretic terms, and we recover the result that the algebraic structure on the state space \({\mathcal H}\) is isomorphic to the center of the group algebra, \(Z\mathbb{C}[G]\), together with an invariant metric [Y. Ruan, Discrete torsion and twisted orbifold cohomology, preprint, http://arxiv.org/math.AG/0005299]. Furthermore, we show that on the large phase space of this theory, there are \(r\) commuting copies of “half” the Virasoro algebra in this theory, where \(r\) is the dimension of \({\mathcal H}\), all of which annihilate, and completely determine, the exponential of the large phase space potential function. We obtain a proof of the usual Virasoro conjecture [T. Eguchi, K. Hori and Ch. Sh. Xion, Phys. Lett. B402, 71–80 (1997; Zbl 0933.81050)] as the special case where a diagonal action (after a variable rescaling) of these Virasoro algebras is considered. Similarly, the relevant integrable hierarchy consists of \(r\) commuting copies of the KdV hierarchy.
For the entire collection see [Zbl 1003.00015].