Counting meromorphic differentials on \(\mathbb{CP}^1\) (Q6606277)

While enumerative geometry is classicaly interested in counting maps, this paper is concerned with counting the number of curves with a differential form with prescribed orders of the singularities (zeros and poles) and some linear constraints on the residues at the poles. For this counting problem to be well-defined, we need a finite 0-dimensional moduli space, which only occurs in genus 0 and with two situations:\N\N-- Type I: all residues vanish and there are exactly two zeros.\N\N-- Type II: all but one residue vanish, and there is exactly one zero.\N\NThe first type is still a count of maps (Hurwitz number) because in genus 0, a meromophic differential without residue is the differential of meromophic function. These numbers were described earlier by [\textit{D. Chen} et al., Invent. Math. 222, No. 1, 283--373 (2020; Zbl 1446.14015)], but the authors provide a new interpretation of these results from the point of view of the representation theory of \(SL(2,\mathbb{C})\) by interpreting them as coefficients of the dispersonless KP hierarhcy. This is a follow-up on the paper of the authors with Zvonkine proving that the coefficients of KP can be recovered from integrals over moduli spaces of residueless differentials [\textit{A. Buryak} et al., Geom. Topol. 28, No. 6, 2793--2824 (2024; Zbl 07951662)].\N\NFor the Type II, the integral takes a very simple form that was already proved in [\textit{M. Costantini} et al., ``Integrals of $\psi$-classes on twisted double ramification cycles and spaces of differentials'', Preprint, \url{arXiv:2112.04238}] but the authors show that this formula follows directly from a WDVV type of argument. Again, this approach shows the revelance of these numbers/integrals from the perspective of mathematical physics.