Counting meromorphic differentials on $\mathbb{CP}^1$

Abstract: We give explicit formulas for the number of meromorphic differentials on $\mathbb{CP}^1$ with two zeros and any number of residueless poles and for the number of meromorphic differentials on $\mathbb{CP}^1$ with one zero, two poles with unconstrained residue and any number of residueless poles, in terms of the orders of their zeros and poles. These are the only two finite families of differentials on $\mathbb{CP}^1$ with vanishing residue conditions at a subset of poles, up to the action of $\mathrm{PGL}(2,\mathbb{C})$. The first family of numbers is related to triple Hurwitz numbers by simple integration and we show its connection with the representation theory of $\mathrm{SL}_2(\mathbb{C})$ and the equations of the dispersionless KP hierarchy. The second family has a very simple generating series, and we recover it through surprisingly involved computations using intersection theory of moduli spaces of curves and differentials.