Trace Formulas for Deformed W-Algebras

TL;DR

This paper explores trace formulas in deformed W-algebras, revealing connections to modular doubles, gauge theory uplifts, and integrable quantum field theories, offering new insights into algebraic and geometric structures.

Contribution

It introduces a novel connection between trace formulas in deformed W-algebras and the modular double, linking different algebraic and physical frameworks.

Findings

Torus correlators in Yangian reproduce sphere correlators in trigonometric setup.

A gauge theoretic 2d to 3d uplift mechanism is identified.

Unified picture of deformed W-algebras related to integrable QFT and 2d CFT is proposed.

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.