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

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 ${\mathcal{W}}_N$ vacuum blocks, where a suitable generalization of the Catalan numbers plays the main role. Moreover, the ${\mathcal{W}}_N$ 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.

36 pages, 6 figures. v2: Minor correction of reference. v3: Minor clarifications and one added reference

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• Approximate Symmetries in $d=4$ CFTs with an Einstein Gravity Dual
• Toward Null State Equations in $d>2$
• Resummation of Multi-Stress Tensors in Higher Dimensions
• Differential equations for classical Virasoro blocks with heavy and light operators