Very twisted stable maps (Q641529)

The conjectural Donaldson-Thomas/Gromov-Witten correspondence for Calabi-Yau 3-folds has been established by \textit{D. Maulik, N. Nekrasov, A. Okounkov} and \textit{R. Pandharipande} [Compos. Math. 142, No. 5, 1263--1285 (2006; Zbl 1108.14046)]. A natural question is to set up the foundations of the GW/DT equivalence when the Calabi-Yau 3-fold is an orbifold. J. Bryan and R. Pandharipande noticed a difficulty that in Donaldson-Thomas theory, there could be curves with generic orbifold structure, in contrast to orbifold Gromov-Witten theory. So one needs to study the so-called ``very twisted stable maps'' to allow generic stabilizers on the source curve, which is the main goal of the paper under review. Let \(X\) be a 3-dimensional Calabi-Yau orbifold. To construct the moduli of very twisted stable maps \(\widetilde{\mathcal K}_{g,n}(X,\beta)\), the authors first constructed the stack \(\mathcal G_X\) of étale gerbes in \(X\) as a rigidification of the stack \(\mathcal S_X\) of subgroups of the inertia stack \(\mathcal I(X)\). Then they prove that the two universal objects on \(\widetilde{\mathcal K}_{g,n}(X,\beta)\) in fact give the same invariants following the arguments of \textit{D. Abramovich, T. Graber} and \textit{A. Vistoli} [Am. J. Math. 130, No. 5, 1337--1398 (2008; Zbl 1193.14070)] As pointed out in the final remark of the paper, the theory they developed has problems of nonzero virtual dimension and so needs further improvements.