[Paper Review] $(q,t)$-deformed (skew) Hurwitz $τ$-functions

Fan Liu, A. Mironov+5 more|arXiv (Cornell University)|Mar 1, 2023

Algebraic structures and combinatorial models4 citations

TL;DR

This paper introduces a $(q,t)$-deformation of skew Hurwitz $\tau$-functions using a commutative family of $W$-operators constructed via the quantum toroidal algebra $U_{q,t^{-1}}(\hat{\hat{\mathfrak{gl}}}_1)$ in the Fock representation. It generalizes the cut-and-join operator to the Macdonald polynomial basis, yielding a $(q,t)$-deformed version of the Hurwitz $\tau$-function that satisfies integrable hierarchy structures and provides a foundation for future matrix model and topological string applications.

ABSTRACT

We follow the general recipe for constructing commutative families of $W$-operators, which provides Hurwitz-like expansions in symmetric functions (Macdonald polynomials), in order to obtain a difference operator example that gives rise to a $(q,t)$-deformation of the earlier studied models. As before, a key role is played by an appropriate deformation of the cut-and-join rotation operator. We outline its expression both in terms of generators of the quantum toroidal algebra and in terms of the Macdonald difference operators.

Motivation & Objective

• To construct a $(q,t)$-deformation of the skew Hurwitz $\tau$-functions using a commutative family of $W$-operators.
• To generalize the cut-and-join rotation operator to the Macdonald polynomial basis, enabling a $(q,t)$-deformation of earlier $W_{1+\infty}$-based models.
• To realize the deformed operators in terms of generators of the quantum toroidal algebra $U_{q,t^{-1}}(\hat{\hat{\mathfrak{gl}}}_1)$ in the Fock representation $\mathcal{F}_{q,t^{-1}}^{(1,0)}$.
• To lay the groundwork for future exploration of matrix model representations and $(q,t)$-deformed integrable structures.

Proposed method

• Construct the deformed $W$-operators $\mathbf{W}_{-n}^{(m)}(q,t|\vec{N})$ as a commutative family via the $W_{1+\infty}$-algebra framework, extended to $(q,t)$-deformation.
• Define the $(q,t)$-deformed cut-and-join operator $\hat{\mathbf{O}}(q,t|N)$ using Macdonald difference operators $D_N^{(k)}$ and their generating function $D_N(Y)$.
• Express $\hat{\mathbf{O}}$ as a product of $D_N$ operators with $Y = -q^{-k}t^{-N}z$, leading to eigenvalue expressions in terms of $q,t$-deformed products over partitions.
• Realize $\hat{\mathbf{O}}$ in terms of the quantum toroidal algebra $U_{q,t^{-1}}(\hat{\hat{\mathfrak{gl}}}_1)$ in the Fock representation $\mathcal{F}_{q,t^{-1}}^{(1,0)}$, linking to $W$-algebra structures.
• Use the generating function identity $\prod_{i=1}^N \frac{1 - z q^{\lambda_i} t^{-i}}{1 - z t^{-i}} = \prod_{(i,j) \in \lambda} \frac{1 - z q^j t^{-i}}{1 - z q^{j-1} t^{-i}}$ to derive eigenvalue expressions for $\hat{\mathbf{O}}_{\text{aux}}(z)$.
• Derive the final form $\hat{\mathbf{O}} = (1-q)^{-\hat{E}} \hat{\mathbf{O}}_{\text{aux}}(q^{-1}t^{N+1}) / \prod_{i,j} (1 - t^{N+1-i} q^{-j})$, ensuring normalization and convergence.

Experimental results

Research questions

• RQ1 How can the $W_{1+\infty}$-based Hurwitz $\tau$-functions be generalized to a $(q,t)$-deformed framework using Macdonald polynomials?
• RQ2 What is the $(q,t)$-deformation of the cut-and-join operator, and how does it act on Macdonald polynomials?
• RQ3 How can the deformed $W$-operators be realized in terms of the quantum toroidal algebra $U_{q,t^{-1}}(\hat{\hat{\mathfrak{gl}}}_1)$?
• RQ4 What is the algebraic structure of the commutative family $\mathbf{W}_{-n}^{(m)}(q,t|\vec{N})$, and how does it relate to integrable systems?
• RQ5 Can the $\tau$-functions constructed here be linked to matrix models or topological string theories through $1/N$ expansions or spectral curves?

Key findings

• The paper constructs a $(q,t)$-deformed version of the skew Hurwitz $\tau$-function using Macdonald polynomials, generalizing the Schur function-based models of earlier works.
• The deformed $W$-operators $\mathbf{W}_{-n}^{(m)}(q,t|\vec{N})$ form a commutative family, implying integrability and suggesting a new class of integrable systems.
• The cut-and-join operator $\hat{\mathbf{O}}(q,t|N)$ is explicitly realized in terms of Macdonald difference operators, with eigenvalues expressed as infinite products over partition boxes.
• The operator $\hat{\mathbf{O}}$ is constructed as $\hat{\mathbf{O}} = (1-q)^{-\hat{E}} \hat{\mathbf{O}}_{\text{aux}}(q^{-1}t^{N+1}) / \prod_{i=1}^N \prod_{j=1}^\infty (1 - t^{N+1-i} q^{-j})$, ensuring proper normalization and convergence.
• The construction is realized in the Fock representation $\mathcal{F}_{q,t^{-1}}^{(1,0)}$ of the quantum toroidal algebra $U_{q,t^{-1}}(\hat{\hat{\mathfrak{gl}}}_1)$, providing a new algebraic framework for these $\tau$-functions.
• The results lay a foundation for future work on matrix model representations, AMM/EO topological recursion, and $(q,t)$-deformed integrable structures in $5d/6d$ theories.