Enumerativity of Gromov-Witten invariants of orbifolds

Question

For smooth Deligne-Mumford stacks, there is a well-defined Gromov-Witten theory, see http://arxiv.org/pdf/math/0103156.pdf and http://arxiv.org/pdf/math/0603151.pdf.

Is there some sense, or some class of examples (that include true orbifolds, namely orbifolds that are not schemes), in which these numbers have some enumerative meaning?

One class of examples is Gorenstein orbifolds, namely those orbifolds whose all degree shifting numbers are integers. In this case one can use the crepant resolution conjecture to translate the question (at least in genus $0$) into a question of enumerativity of Gromov-Witten invariants for the crepant resolution.

But what about other cases? In particular, can one make some enumerative sense of the Gromov-Witten invariants that are obtained when at least one of the cohomology classes is chosen to be the cohomology class of a twisted sector?

(apologies if the question is too vague, and if it is a repetition of other similar questions. I am not sure if this should be community wiki.)

Answer 1

One simple example (although it is a genus 0 example) is the following.

Consider a global quotient $\mathscr{X} = [X/(\mathbb{Z}/2)]$. Then if we look at the genus 0 GW theory of $\mathscr{X}$ where the source curve has $2g + 2$ stacky $\mathbb{Z}/2$ points whose evaluations lie in the twisted sector of $\mathscr{X}$, then this will (at least in principle) be providing a count of hyperelliptic curves in $X$ with certain incidence conditions (which would lie in the fixed locus of the $\mathbb{Z}/2$-action) based on the twisted classes that you pull back.

Somewhat more generally, one could look at the stack symmetric square $[Sym^2 X]$, and use the same idea to count hyperelliptic curves in $X$ itself with more arbitrary incidence conditions.

This has been studied in a few places: Jonathan Wise's thesis (for $\mathbb{P}^2$), Danny Gillam's thesis (for more general $X$ with a $\mathbb{Z}/2$-action, and with constant maps), as well as mine (for the case of Abelian surfaces).

There is in principle no issue with continuing this to higher genera to study the number of curves in a target $X$ which are ramified covers of some higher genus curve, but the details I suspect will be fraught with the same issues that you usually find in higher genus GW theory.