Exploring $\mathcal{W}_{\infty}$ in the quadratic basis

TL;DR

This paper investigates the operator product expansions of the $ ext{W}_ ext{infinity}$ algebra using primary fields and a free field quadratic basis, proposing a closed-form OPE and exploring its algebraic properties and symmetries.

Contribution

It introduces a quadratic basis for $ ext{W}_ ext{infinity}$, derives a conjectured closed-form OPE, and demonstrates how to compute commutation relations and correlation functions efficiently.

Findings

Derived a compact quadratic form of the OPE in the free field basis

Proposed a conjecture for the complete OPE formula in this basis

Verified consistency with minimal models and $ ext{W}_N$ reductions

Abstract

We study the operator product expansions in the chiral algebra $\mathcal{W}_{\infty}$, first using the associativity conditions in the basis of primary generating fields and second using a different basis coming from the free field representation in which the OPE takes a simpler quadratic form. The results in the quadratic basis can be compactly written using certain bilocal combinations of the generating fields and we conjecture a closed-form formula for the complete OPE in this basis. Next we show that the commutation relations as well as correlation functions can be easily computed using properties of these bilocal fields. In the last part of this paper we verify the consistency with results derived previously by studying minimal models of $\mathcal{W}_{\infty}$ and comparing them to known reductions of $\mathcal{W}_{\infty}$ to $\mathcal{W}_N$. The results we obtain illustrate nicely the role of triality symmetry in the representation theory of $\mathcal{W}_{\infty}$.