Many-faced Painlev\'e I: irregular conformal blocks, topological recursion, and holomorphic anomaly approaches

In recent years, the Fourier series (Zak transform) structure of the Painlev\'e I tau function has emerged in multiple contexts. Its main building block admits several conjectural interpretations, such as the partition function of an Argyres-Douglas gauge theory, the topological recursion partition function for the Weierstrass elliptic curve, and a 1-point conformal block on the Riemann sphere with an irregular insertion of rank $\frac52$. We review and further develop a mathematical framework for these constructions, and formulate conjectures on their equivalence. In particular, we give a simple explanation of the Fourier series representation of the tau function based on the Jimbo-Miwa-Ueno differential extended to the space of Stokes data. We provide an algebraic construction of the rank $\frac52$ Whittaker state for the Virasoro algebra embedded into a rank $2$ Whittaker module, prove its existence and uniqueness, and fix its descendant structure. We also prove the conifold gap property of the relevant topological recursion partition function, which, on one hand, enables its efficient computation within the holomorphic anomaly approach and, on the other, establishes the existence of solution for the latter.