[Paper Review] Deformations of $\mathcal W$ algebras via quantum toroidal algebras

Boris Feigin, M. Jimbo+2 more|arXiv (Cornell University)|Mar 9, 2020

Algebraic structures and combinatorial models19 references4 citations

TL;DR

This paper provides a uniform construction of deformed $\mathcal{W}$ algebras in all classical types (A, B, C, D), including supersymmetric and twisted cases, using the quantum toroidal $\mathfrak{gl}_1$ algebra $\mathcal{E}$. It introduces a comodule algebra $\mathcal{K}$ over $\mathcal{E}$ that generates deformed $\mathcal{W}$ currents and screening operators, and proves the existence of a commutative family of integrals of motion via Bethe ansatz, valid for all non-exceptional types except $\textsf{D}^{(2)}_{\ell+1}$.

ABSTRACT

The deformed $\mathcal W$ algebras of type $ extsf{A}$ have a uniform description in terms of the quantum toroidal $\mathfrak{gl}_1$ algebra $\mathcal E$. We introduce a comodule algebra $\mathcal K$ over $\mathcal E$ which gives a uniform construction of basic deformed $\mathcal W$ currents and screening operators in types $ extsf{B}, extsf{C}, extsf{D}$ including twisted and supersymmetric cases. We show that a completion of algebra $\mathcal K$ contains three commutative subalgebras. In particular, it allows us to obtain a commutative family of integrals of motion associated with affine Dynkin diagrams of all non-exceptional types except $ extsf{D}^{(2)}_{\ell+1}$. We also obtain in a uniform way deformed finite and affine Cartan matrices in all classical types together with a number of new examples, and discuss the corresponding screening operators.

Motivation & Objective

• To provide a uniform algebraic framework for deformed $\mathcal{W}$ algebras across all classical types, including supersymmetric and twisted cases.
• To extend the known $q$-deformation of $\mathcal{W}$ algebras of type A to other classical types using quantum toroidal algebras.
• To construct a comodule algebra $\mathcal{K}$ over the quantum toroidal $\mathfrak{gl}_1$ algebra $\mathcal{E}$ that generates deformed $\mathcal{W}$ currents and screening operators uniformly.
• To establish a commutative family of integrals of motion associated with affine Dynkin diagrams, using Bethe ansatz, for all non-exceptional types except $\textsf{D}^{(2)}_{\ell+1}$.

Proposed method

• The construction uses the quantum toroidal $\mathfrak{gl}_1$ algebra $\mathcal{E}$ as a universal algebraic framework for deformed $\mathcal{W}$ algebras.
• Deformed $\mathcal{W}$ currents $A_i(z)$ are defined as ratios of neighboring vertex operators in a sum of $\ell+1$ vertex operators arising from the action of the generating current $e(z)$ on tensor products of Fock modules with arbitrary colors.
• The comodule algebra $\mathcal{K}$ over $\mathcal{E}$ is introduced to systematically generate deformed $\mathcal{W}$ currents and screening operators in types B, C, D, including twisted and supersymmetric cases.
• The deformed Cartan matrices (finite and affine) are derived from the contraction structure of the $A_i(z)$ currents, with explicit matrix forms provided for various configurations.
• Dressed currents $\boldsymbol{e}(z) = e(z)\tilde{Z}_\mu(z)$ are used to construct integrals of motion via multiple integrals with the Feigin-Odesskii kernel.
• The spectrum of the integrals of motion is computed using the Bethe ansatz, and the construction is shown to be consistent across all classical types except $\textsf{D}^{(2)}_{\ell+1}$.

Experimental results

Research questions

• RQ1 How can deformed $\mathcal{W}$ algebras in all classical types (A, B, C, D) be constructed in a uniform way using quantum toroidal algebras?
• RQ2 What is the role of the comodule algebra $\mathcal{K}$ over the quantum toroidal $\mathfrak{gl}_1$ algebra $\mathcal{E}$ in generating deformed $\mathcal{W}$ currents and screening operators?
• RQ3 Can a commutative family of integrals of motion be constructed for all non-exceptional affine Dynkin types using this framework?
• RQ4 How do the deformed Cartan matrices in finite and affine types emerge from the algebraic structure of $\mathcal{E}$ and its Fock modules?
• RQ5 What is the significance of the $qq$-character interpretation of the dressed current $\boldsymbol{e}(z)$ in relation to quantum affine $\mathfrak{sl}_{\ell+1}$?

Key findings

• The comodule algebra $\mathcal{K}$ over the quantum toroidal $\mathfrak{gl}_1$ algebra $\mathcal{E}$ contains three commutative subalgebras, enabling a uniform construction of deformed $\mathcal{W}$ currents and screening operators in types B, C, D, including supersymmetric and twisted cases.
• A commutative family of integrals of motion is constructed for all non-exceptional affine Dynkin types except $\textsf{D}^{(2)}_{\ell+1}$, with the spectrum computed via Bethe ansatz.
• The deformed finite and affine Cartan matrices are explicitly derived for all classical types, with specific matrix forms provided for various configurations of colors and types.
• The dressed current $\boldsymbol{e}(z)$, obtained by multiplying $e(z)$ with a vertex operator $\tilde{Z}_\mu(z)$, yields a $qq$-character of the first fundamental representation of quantum affine $\mathfrak{sl}_{\ell+1}$, linking the construction to known integrable systems.
• The construction allows for the generation of new examples of deformed Cartan matrices by permuting parameters $s_1, s_2, s_3$ or reversing the order of data, with stable matrices obtained via superposition of maximal finite-type submatrices.
• The method successfully generalizes the $q$-deformation of $\mathcal{W}$ algebras from type A to all classical types, providing explicit formulas for currents, screening operators, and Cartan matrices.