Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 094, 18 pages      arXiv:2005.03579      https://doi.org/10.3842/SIGMA.2020.094
Contribution to the Special Issue on Elliptic Integrable Systems, Special Functions and Quantum Field Theory

On Abelianity Lines in Elliptic $W$-Algebras

Jean Avan ^a, Luc Frappat ^b and Eric Ragoucy ^b
a) Laboratoire de Physique Théorique et Modélisation, CY Cergy Paris Université, CNRS, F-95302 Cergy-Pontoise, France
^b) Laboratoire d'Annecy-le-Vieux de Physique Théorique LAPTh, Université Grenoble Alpes, USMB, CNRS, F-74000 Annecy, France

Received May 08, 2020, in final form September 22, 2020; Published online September 30, 2020

Abstract
We present a systematic derivation of the abelianity conditions for the $q$-deformed $W$-algebras constructed from the elliptic quantum algebra $\mathcal{A}_{q,p}\big(\widehat{\mathfrak{gl}}(N)_{c}\big)$. 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.

Key words: elliptic quantum algebras; $W$-algebras.

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