Do we need the $W^{(n>3)}$ constraints to solve the $(1,q)$ models coupled to 2D gravity?

TL;DR

This paper demonstrates that correlation functions in (1,q) models coupled to 2D gravity can be computed solely using Virasoro and W^{(3)} constraints, simplifying the analysis by focusing on lower contact degenerations.

Contribution

The authors prove that higher W^{(n>3)} constraints are unnecessary for calculating correlators in (1,q) models, providing an efficient algorithm for any q and genus.

Findings

Correlation functions are determined by Virasoro and W^{(3)} constraints.

An explicit algorithm for correlator computation is developed.

New polynomial identities related to Abel's identity are discovered.

Abstract

We prove that all the correlation functions in the $(1,q)$ models are calculable using only the Virasoro and the $W^{(3)}$ constraints. This result is based on the invariance of correlators with respect to an interchange of the order of the operators they contain. In terms of the topological recursion relations, it means that only two and three contacts and the corresponding degenerations of the underlying surfaces are relevant. An algorithm to compute correlators for any $q$ and at any genus is presented and demonstrated through some examples. On route to these results, some interesting polynomial identities, which are generalizations of Abel's identity, were discovered. }