A generalization of the Witten conjecture through spectral curve

We propose a generalization of the Witten conjecture, which connects a descendent enumerative theory with a specific reduction of KP integrable hierarchy. Our conjecture is realized by two parts: Part I (Geometry) establishes a correspondence between the geometric descendent potential (apart from ancestors) and the topological recursion of specific spectral curve data $(\Sigma, x,y)$; Part II (Integrability) claims that the TR descendent potential, defined at the boundary points of the spectral curve (where $dx$ has poles), is a tau-function of a certain reduction of the multi-component KP hierarchy. In this paper, we show the geometric part of the conjecture for any formal descendent theory by using a generalized Laplace transform. Subsequently, we prove the integrability conjecture for the one-boundary cases. As applications, we generalize and prove the $r$KdV integrability of negative $r$-spin theory conjectured by Chidambaram, Garcia-Failde and Giacchetto. We also show the KdV integrability of the total descendent potential associated with the Hurwitz space $M_{1,1}$, whose Frobenius manifold was initially introduced by Dubrovin.