By the work of H.-L. Chang and J. Li, the (arbitrary genus) Gromov-Witten invariants of a quintic threefold can be computed using the relatively simple moduli space of stable maps to P^4 with a p-field.
This is an example of a GLSM moduli space. Another example is the moduli of r-spin curves with a field, which by the work of Chang-Li-Li computes Witten's r-spin class. One subtlety in using these moduli spaces for computations lies in the cosection localized virtual class that is necessary to produce invariants from these non-compact moduli spaces.
In my talk, I will discuss joint work in progress with Q. Chen, Y.
Ruan on how to derive the same invariants from an ordinary virtual class on a compactified moduli space constructed using logarithmic geometry. This virtual class is amenable to torus localization methods.
In the case of a quintic threefold, in joint work in progress with S.
Guo and Y. Ruan, we apply this method to prove the holomorphic anomaly equations. There are further applications in joint work with A.
Sauvaget and D. Zvonkine, and with X. Wang.