Mapping partition functions

Abstract

We introduce an infinite group action on partition functions of WK type, meaning of the type of the partition function $Z^{\rm WK}$ in the famous result of Witten and Kontsevich expressing the partition function of $\psi$-class integrals on the compactified moduli space $\overline{\mathcal{M}}_{g,n}$ as a $\tau$-function for the Korteweg--de Vries hierarchy. Specifically, the group which acts is the group $\mathcal{G}$ of formal power series of one variable $\varphi(V)=V+O(V^2)$, with group law given by composition, acting in a suitable way on the infinite tuple of variables of the partition functions. In particular, any $\varphi \in \mathcal{G}$ sends the Witten--Kontsevich (WK) partition function $Z^{\rm WK}$ to a new partition function $Z^\varphi$, which we call the WK mapping partition function associated to $\varphi$. We show that the genus zero part of $\log Z^\varphi$ is independent of $\varphi$ and give an explicit recursive description for its higher genus parts (loop equation), and as applications of this obtain relationships of the $\psi$-class integrals to Gaussian Unitary Ensemble and generalized Br\'ezin--Gross--Witten correlators. In a different direction, we use $Z^\varphi$ to construct a new integrable hierarchy, which we call the WK mapping hierarchy associated to $\varphi$. We show that this hierarchy is a bihamiltonian perturbation of the Riemann--Hopf hierarchy possessing a $\tau$-structure, and prove that it is a universal object for all such perturbations. Similarly, for any $\varphi\in\mathcal{G}$, we define the Hodge mapping partition function associated to $\varphi$, prove that it is integrable, and study its role in hamiltonian perturbations of the Riemann--Hopf hierarchy possessing a $\tau$-structure. Finally, we establish a generalized Hodge--WK correspondence relating different Hodge mapping partition functions.