In joint work with Buryak, Pandharipande and Tessler (in preparation), we define equivariant stationary descendent integrals on the moduli of stable maps from surfaces with boundary to $(\mathbb{CP}^1,\mathbb{RP}^1)$. For stable maps of the disk, the definition is geometric and we prove a fixed-point formula involving contributions from all the corner strata. We use this fixed-point formula to give a closed formula for the integrals in this case.

We conjecture that the higher genus theory exists and give an explicit fixed-point formula for it. We show that this formula behaves as expected in the closed and disk-map sector, and satisfies a domain decomposition property. We consider open branched covers and prove they satisfy analogous properties. This implies a Gromov-Witten Hurwitz correspondence extending part of the work of Okounkov and Pandharipande from the closed to the open setting.

In the talk I'll discuss some of these ideas. I'll try to make the talk as self-contained as possible, and give examples.