A generalization of Witten's conjecture for the Pixton class and the noncommutative KdV hierarchy

TL;DR

This paper conjectures that the partition function of the Pixton class on moduli spaces corresponds to the tau function of a noncommutative KdV hierarchy, supported by computational evidence and linking to the double ramification cycle.

Contribution

It formulates a new conjecture connecting Pixton classes with the noncommutative KdV hierarchy and provides evidence and implications related to the double ramification cycle.

Findings

Conjecture that Pixton class partition function equals noncommutative KdV tau function.

Evidence supporting the conjecture through computational checks.

Relation established between the conjecture and the Double Ramification/Dubrovin--Zhang equivalence.

Abstract

In this paper, we formulate and present ample evidence towards the conjecture that the partition function (i.e. the exponential of the generating series of intersection numbers with monomials in psi classes) of the Pixton class on the moduli space of stable curves is the topological tau function of the noncommutative KdV hierarchy, which we introduced in a previous work. The specialization of this conjecture to the top degree part of Pixton's class states that the partition function of the double ramification cycle is the tau function of the dispersionless limit of this hierarchy. In fact, we prove that this conjecture follows from the Double Ramification/Dubrovin--Zhang equivalence conjecture. We also provide several independent computational checks in support of it.