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Trace Formulas for Deformed W-Algebras

Fabrizio Nieri

TL;DR

We address trace formulas in $W_{eps1,eps2}$ algebras and connect them to the modular double of $W_{q1,q2}$ via a continuous free-boson realization and Clavelli-Shapiro style traces. We show that torus correlators in the additive (Yangian) setting reproduce sphere correlators in the trigonometric setting, with a non-perturbative completion from the modular double, and interpret this as a 2d to 3d uplift in gauge theory. The work proposes a unified framework wherein trace formulas relate massive and massless form-factor approaches in integrable QFT and 2d CFT, and situates these results within the rational/trigonometric/elliptic hierarchy of deformed W-algebras. Together these results bridge algebraic, geometric, and physical perspectives on deformed W-algebras and suggest directions toward non-perturbative definitions, holographic questions, and extensions to holomorphic blocks and refined topological strings.

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.

Trace Formulas for Deformed W-Algebras

TL;DR

We address trace formulas in algebras and connect them to the modular double of via a continuous free-boson realization and Clavelli-Shapiro style traces. We show that torus correlators in the additive (Yangian) setting reproduce sphere correlators in the trigonometric setting, with a non-perturbative completion from the modular double, and interpret this as a 2d to 3d uplift in gauge theory. The work proposes a unified framework wherein trace formulas relate massive and massless form-factor approaches in integrable QFT and 2d CFT, and situates these results within the rational/trigonometric/elliptic hierarchy of deformed W-algebras. Together these results bridge algebraic, geometric, and physical perspectives on deformed W-algebras and suggest directions toward non-perturbative definitions, holographic questions, and extensions to holomorphic blocks and refined topological strings.

Abstract

We investigate trace formulas in -deformed W-algebras, highlighting a novel connection to the modular double of -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 2d3d 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.

Paper Structure

This paper contains 6 sections, 32 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Brief review and conventions
  3. $\boldsymbol{{\rm W}_{{\mathfrak{q}}_1,{\mathfrak{q}}_2}}$ sphere correlators as $\boldsymbol{{\rm W}_{\varepsilon_1,\varepsilon_2}}$ torus correlators
  4. Discussion and outlook
  5. Multiple Gamma and sine functions
  6. Clavelli--Shapiro trace technique

Figures (1)

  • Figure 1: Sketch of the rational/trigonometric/elliptic classification of the deformed W-algebras and the related limits and uplifts one can implement on gauge theoretic observables (for simplicity, at co-dimension 2 in 4d/5d/6d) and the involved special functions or observables.