CFT correlators, W-algebras and generalized Catalan numbers

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 W-N vacuum blocks, where a suitable generalization of the Catalan numbers plays the main role. Moreover, the 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.

Automated summary

This research investigates the role of Catalan numbers in the Virasoro heavy-heavy-light-light (HHLL) vacuum blocks and HHLL W-N vacuum blocks in two spacetime dimensions. It reveals that these numbers play a crucial role in the generating function equation and the correlation of stress tensor in the latter case.