CFT correlators, ${\cal W}$-algebras and Generalized Catalan Numbers

TL;DR

This paper explores the connection between ${ m W}_N$ algebra vacuum blocks in two-dimensional conformal field theory and generalized Catalan numbers, revealing new combinatorial structures and their relation to higher-dimensional correlators.

Contribution

It generalizes the relation between Catalan numbers and Virasoro blocks to ${ m W}_N$ blocks, linking them to linear extensions of posets and higher-dimensional conformal correlators.

Findings

${ m W}_N$ vacuum blocks relate to generalized Catalan numbers.

The generating function satisfies a differential equation similar to the Virasoro case.

Connections established between combinatorics and higher-dimensional conformal correlators.

Abstract

In two spacetime dimensions the Virasoro heavy-heavy-light-light (HHLL) vacuum block in a certain limit is governed by the Catalan numbers. The equation for their generating function can be generalized to a differential equation which the logarithm of the block satisfies. We show that a similar story holds for the HHLL ${\cal W}_N$ vacuum blocks, where a suitable generalization of the Catalan numbers plays the main role. Moreover, the ${\cal W}_N$ blocks have the same form as the stress tensor sector of HHLL near lightcone conformal correlators in $2(N-1)$ spacetime dimensions. In the latter case the Catalan numbers are generalized to the numbers of linear extensions of certain partially ordered sets.