On Abelianity Lines in Elliptic 𝑊-Algebras

Abstract

We present a systematic derivation of the abelianity conditions for the q-deformed 𝑊-algebras constructed from the elliptic quantum algebra Aq,ₚ(𝖌𝔩ˆ(𝘕)c). We identify two sets of conditions on a given critical surface yielding abelianity lines in the moduli space (𝑝, 𝑞, 𝘤). Each line is identified as an intersection of a countable number of critical surfaces obeying diophantine consistency conditions. The corresponding Poisson bracket structures are then computed, for which some universal features are described.