Much work is done at present connecting generating functions for algebro-geometric data to the tau functions of integrable hierarchies, and this paper is in this vein of study, proving that a partition function associated to the space \(M:=H_{0; 1, N-1}\) of meromorphic functions on the Riemann sphere with exactly one simple pole and one pole of order \(N-1\) is related to the KP hierarchy.
The paper assumes the reader has a background knowledge of Dubrovin-Froebnius manifolds, and of a method of Givental-Milanov-Tseng for constructing integrable systems from partition function data.
§1 considers \(M\) as a Dubrovin-Frobenius manifold, and defines the associated geometric structures. A “special point” in \(M\) is given about which, for simplicity, most computations on the manifold are completed.
§2 introduces the partition function in question, the total descendant potential, giving two forms in Equations (2.2) and (2.3) adapted for different uses. The expansion parameters of the descendant potential are related to the flat coordinates on \(M\). In Theorem 3 this descendant potential is expressed as an expansion in the number of isomorphism classes of rooted hypermaps, thus giving it an “enumerative meaning”; this theorem is proven with an application of topological recursion. By citing a connection between the generating function for the number of isomorphism classes and the KP hierarchy, it is shown that the descendant potential restricts to a KP tau function.
§3 defines period vectors from integrals of multivalued complex functions associated to \(M\), and discusses many of their properties such as the monodromy and asymptotics. This lays the groundwork for §4 which constructs Hirota equations which the descendant potential solves.
§5 uses the Hirota equation to give a Lax formulation for the associated integrable systems, which is identified with a version of the KP hierarchy.