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 ${\mathcal{A}}_{q,p}(\widehat{\mathfrak{gl}}(N{)}_c)$. 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 brackets structures are then computed for which some universal features are described.