Open $\mathbb{CP}^1$ descendent theory I: The stationary sector

We define stationary descendent integrals on the moduli space of stable maps from disks to $(\mathbb{CP}^1,\mathbb{RP}^1)$. We prove a localization formula for the stationary theory involving contributions from the fixed points and from all the corner-strata. We use the localization formula to prove a recursion relation and a closed formula for all genus $0$ disk cover invariants in the stationary case. For all higher genus invariants, we propose a conjectural formula.

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