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

Thermal stress tensor correlators, OPE and holography

“…The result has many similarities to the Virasoro HHLL vacuum block (see e.g. [22][23][24][25]) but at the moment the full resummed correlator in d > 2 is only known in the ∆L→∞ limit [26]. (see [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44] for related work).…”

“…Holography provides a powerful tool to study hydrodynamics of strongly coupled quantum field theories and transport coefficients can be read off from the stress tensor two-point function at finite temperature. 25 The conformal bootstrap provides another window into strongly coupled phenomena when perturbation theory is not applicable. While the bootstrap program for vacuum correlators has led to significant developments in the past decade, the corresponding tools for thermal correlators are still developing, see e.g.…”

Approximate symmetries in d = 4 CFTs with an Einstein gravity dual

“…As observed in a recent paper [18], the d = 4 structure (3.17) is identical to the central term of the W 3 algebra in d = 2 CFT [44], and one can exactly reproduce the d = 4 single stress-tensor exchange contribution via a mode summation using such a central term. See also [36] for related observations. Let us here make a transformation:…”

“…The stress-tensor three-point function is important because it provides information about the graviton mixing. In d = 2, the mixing has the form L m L m L −m−n , which contributes to the two stress-tensor exchanges; see [36,39] for the d = 2 discussion. To compute the d = 4 scalar correlator at subleading ∆, a similar graviton-mixing contribution should be included.…”

“…central charge [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]. Using AdS effective field theory, the operator product expansion (OPE) coefficients corresponding to multi-stress-tensor exchanges were computed for d = 4 CFTs dual to semiclassical Einstein gravity with a minimally coupled scalar [20].…”