On affine coordinates of the tau-function for open intersection numbers

TL;DR

This paper computes the affine coordinates of a tau-function related to open intersection numbers, translating Virasoro constraints into recursive formulas and deriving new expressions for the partition function and n-point functions.

Contribution

It provides the first explicit computation of affine coordinates for this tau-function and reformulates Virasoro constraints as recursion relations in the fermionic framework.

Findings

Affine coordinates are explicitly computed for the tau-function.

Virasoro constraints are rewritten as recursion formulas.

New formulas for the extended partition function and n-point functions are derived.

Abstract

In Alexandrov's work \cite{al2, al3} it has been shown that the extended partition function $\exp(F^{o,ext}+F^c)$ introduced by Buryak in \cite{bu, bu2} is a tau-function of the KP hierarchy. In this work, we compute the affine coordinates of this tau-function on the Sato Grassmannian, and rewrite the Virasoro constraints as recursions for the affine coordinates in the fermionic picture. As applications we derive some formulas for the extended partition function and the connected $n$-point functions using methods developed by Zhou in \cite{zhou1} based on the boson-fermion correspondence.