CFT correlators, $$ \mathcal{W} $$-algebras and generalized Catalan numbers

Abstract

A bstract 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 $$ \mathcal{W} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>W</mml:mi> </mml:math> N vacuum blocks, where a suitable generalization of the Catalan numbers plays the main role. Moreover, the $$ \mathcal{W} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>W</mml:mi> </mml:math> 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.