[Paper Review] Interpolating Matrix Models for WLZZ series

A. Mironov, V. Mishnyakov+4 more|arXiv (Cornell University)|Jan 10, 2023

Algebraic structures and combinatorial models4 citations

TL;DR

This paper introduces a two-matrix model with three infinite parameter sets that interpolates between all WLZZ series models via W-representations. It establishes that these models are matrix models through explicit integrals, generalizes them using Borel subalgebra generators of the $w_{\infty}$-algebra, and identifies their integrable structure via skew hypergeometric $\tau$-functions, resolving ambiguities in topological recursion and $1/N$ expansion relations for both positive and negative branches.

ABSTRACT

We suggest a two-matrix model depending on three (infinite) sets of parameters which interpolates between all the models proposed in arXiv:2206.13038, and defined there through $W$-representations. We also discuss further generalizations of these WLZZ models realized by $W$-representations associated with infinite commutative families of generators of $w_\infty$-algebra which are presumably related to more sophisticated multi-matrix models. Integrable properties of these generalizations are described by what we call the skew hypergeometric $τ$-functions.

Motivation & Objective

• To unify and generalize the WLZZ series of $\tau$-functions through a two-matrix model framework.
• To establish that WLZZ models are indeed matrix models by constructing explicit two-matrix integrals.
• To extend the class of models using $W$-operators from a Borel subalgebra of the $w_{\infty}$-algebra.
• To clarify the distinct $W$-representation structures for the positive and negative branches of the WLZZ models.
• To characterize the integrable properties of these models using skew hypergeometric $\tau$-functions.

Proposed method

• Proposes a general partition function of the form $ Z_f(\bar{p}, p, g) = \sum_{R,Q} \prod_{i,j \in R/Q} f(j-i) S_{R/Q}\{\bar{p}\} S_R\{g\} S_Q\{p\} $, which defines skew hypergeometric $\tau$-functions.
• Constructs a two-matrix model integral depending on $\bar{p}_k$, $g_k$, and an external matrix $\Lambda$, with $p_k = \mathrm{Tr}\, \Lambda^k$, to realize the WLZZ models.
• Derives $W$-representations via differential operators in $p_k$-variables, explicitly constructing $\hat{F}_1$, $\hat{F}_2$ for the positive branch and analogous operators for the negative branch.
• Identifies the integrable structure through $W$-operators from a Borel subalgebra of the $w_\infty$-algebra, ensuring compatibility with Toda and KP hierarchies.
• Uses Schur function basis and scalar product $\langle S_R | S_Q \rangle = \delta_{R,Q}$ to compute matrix elements and verify $W$-action on $\tau$-functions.
• Applies generating function techniques and exponential $W$-representations to derive the Schur function expansions of $Z_f$, recovering the original partition functions from $W$-actions.

Experimental results

Research questions

• RQ1 Can the WLZZ series of $\tau$-functions be unified under a single two-matrix model framework?
• RQ2 How do the $W$-representations differ between the positive and negative branches of the WLZZ models?
• RQ3 What is the role of skew hypergeometric $\tau$-functions in characterizing the integrable structure of these models?
• RQ4 Can the $W$-operators generating these models be systematically derived from the $w_\infty$-algebra's Borel subalgebra?
• RQ5 How do the $W$-representations resolve the tension between AMM/CEO topological recursion and $1/N$ expansion in the WLZZ models?

Key findings

• The proposed two-matrix model provides a unified realization of all WLZZ series models, reducing to single-matrix models for specific cases like $n=\pm2$.
• The $W$-representations for the positive and negative branches are structurally distinct, despite the negative branch being a limiting case of the positive one.
• The skew hypergeometric $\tau$-functions, defined via $Z_f$, are shown to be $\tau$-functions of the KP and Toda hierarchies, confirming their integrable nature.
• Explicit differential operators $\hat{F}_1$ and $\hat{F}_2$ are derived for the positive branch, with $\hat{F}_1 = -\sum_b (b+1)p_b \partial_{p_{b+1}} - N \partial_{p_1}$ and $\hat{F}_2$ involving second-order derivatives and structure constants.
• The negative branch model, with $p_k=0$, is realized via a different $W$-representation, confirming the non-triviality of the $W$-action in this limit.
• The action of $W$-operators on Schur functions is computed explicitly, with $p_1 S_R = \sum_{R+\Box} S_{R+\Box}$ and $\hat{W}_0(N) S_R = \left(\sum_{(i,j)\in R}(j-i+N)\right) S_R$, providing a complete algebraic framework.