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

SIGMA 22 (2026), 003, 11 pages      arXiv:2507.02831      https://doi.org/10.3842/SIGMA.2026.003

Trace Formulas for Deformed W-Algebras

Fabrizio Nieri ab
a) Dipartimento di Matematica ''Giuseppe Peano'', Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy
b) INFN, Sezione di Torino, Via Pietro Giuria 1, 10125 Torino, Italy

Received September 08, 2025, in final form January 04, 2026; Published online January 13, 2026

Abstract
We investigate trace formulas in $\varepsilon$-deformed W-algebras, highlighting a novel connection to the modular double of $\mathfrak{q}$-deformed W-algebras. In particular, we show that torus correlators in the additive (Yangian) setting reproduce sphere correlators in the trigonometric setup, possibly with the inclusion of a non-perturbative completion. From a dual perspective, this mechanism implements a gauge theoretic 2d$\to$3d uplift, where a circle direction in the world-sheet transmutes to a compact space-time direction in a non-trivial manner. We further discuss a unified picture of deformed W-algebras driven by trace formulas, suggesting a deeper algebraic layer related to the massive and massless form-factor approach to integrable QFT and 2d CFT.

Key words: deformed W-algebras; bosonization; trace formulas; supersymmetry; integrable form-factors.

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