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

Understand

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 $\mathcal{W}_N$.